Chapter 2 · સંબંધ અને વિધેય

#1 SUB 0M ✎ Edit ⚠ Log Error

Question

òu (x + 1, y – 2) = (3, 1) íkku, x yLku y þkuÄku.

Answer

ynª, (x + 1, y – 2) = (3, 1) nkuðkÚke,

x + 1 = 3, y – 2 = 1 ÚkkÞ.

\ x = 2, y = 3

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu (x + 1, y – 2) = (3, 1) íkku, x yLku y þkuÄku.

Answer

ynª, (x + 1, y – 2) = (3, 1) nkuðkÚke,

x + 1 = 3, y – 2 = 1 ÚkkÞ.

\ x = 2, y = 3

✓ Apply Cancel

#2 SUB 0M ✎ Edit ⚠ Log Error

Question

òu P = {a, b, c}, Q = {r}, íkku P × Q yLku Q × P þkuÄku. þwt yk fkíkuorÍÞ økwýkfkh Mk{kLk Au ?

Answer

P × Q = {a, b, c} × {r}

= {(a, r), (b, r), (c, r)}...(1)

Q × P = {r} × {a, b, c}

= {(r, a), (r, b), (r, c)}...(2)

Ãkrhýk{ (1) yLku (2) ÃkhÚke òuE þfkÞ Au fu, ¢{Þwõík òuz (a, r) yLku (r, a) Mk{kLk Lk nkuðkÚke yk fkíkuorÍÞ økwýkfkh Mk{kLk LkÚke.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu P = {a, b, c}, Q = {r}, íkku P × Q yLku Q × P þkuÄku. þwt yk fkíkuorÍÞ økwýkfkh Mk{kLk Au ?

Answer

P × Q = {a, b, c} × {r}

= {(a, r), (b, r), (c, r)}...(1)

Q × P = {r} × {a, b, c}

= {(r, a), (r, b), (r, c)}...(2)

Ãkrhýk{ (1) yLku (2) ÃkhÚke òuE þfkÞ Au fu, ¢{Þwõík òuz (a, r) yLku (r, a) Mk{kLk Lk nkuðkÚke yk fkíkuorÍÞ økwýkfkh Mk{kLk LkÚke.

✓ Apply Cancel

#3 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, íkku Lke[uLkk øký þkuÄku.

(i) A × (B ∩ C) (ii) (A × B) ∩ (A × C)

(iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)

Answer

(i) A × (B ∩ C) {kxu,

B ∩ C = {4}

\ A × (B ∩ C) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}

(ii) (A × B) ∩ (A × C)

A × B = {1, 2, 3} × {3, 4}

= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3),

(3, 4)}........(1)

A × C = {1, 2, 3} × {4, 5, 6}

= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5),

(2, 6), (3, 4), (3, 5), (3, 6)}........(2)

\ (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}

(iii) A × (B ∪ C)

B ∪ C = {3, 4, 5, 6}

\ A × (B ∪ C) = {1, 2, 3} × {3, 4, 5, 6}

= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),

(2, 4), (2, 5), (2, 6), (3, 3),

(3, 4), (3, 5), (3, 6)}

(iv) (A × B) ∪ (A × C)

Ãkrhýk{ (1) yLku (2)Lkku WÃkÞkuøk fhíkkt,

(A × B) ∪ (A × C)

= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),

(2, 4), (2, 5), (2, 6), (3, 3),

(3, 4), (3, 5), (3, 6)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, íkku Lke[uLkk øký þkuÄku.

(i) A × (B ∩ C) (ii) (A × B) ∩ (A × C)

(iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)

Answer

(i) A × (B ∩ C) {kxu,

B ∩ C = {4}

\ A × (B ∩ C) = {1, 2, 3} × {4}

= {(1, 4), (2, 4), (3, 4)}

(ii) (A × B) ∩ (A × C)

A × B = {1, 2, 3} × {3, 4}

= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3),

(3, 4)}........(1)

A × C = {1, 2, 3} × {4, 5, 6}

= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5),

(2, 6), (3, 4), (3, 5), (3, 6)}........(2)

\ (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}

(iii) A × (B ∪ C)

B ∪ C = {3, 4, 5, 6}

\ A × (B ∪ C) = {1, 2, 3} × {3, 4, 5, 6}

= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),

(2, 4), (2, 5), (2, 6), (3, 3),

(3, 4), (3, 5), (3, 6)}

(iv) (A × B) ∪ (A × C)

Ãkrhýk{ (1) yLku (2)Lkku WÃkÞkuøk fhíkkt,

(A × B) ∪ (A × C)

= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3),

(2, 4), (2, 5), (2, 6), (3, 3),

(3, 4), (3, 5), (3, 6)}

✓ Apply Cancel

#4 SUB 0M ✎ Edit ⚠ Log Error

Question

òu P = {1, 2}, íkku P × P × P þkuÄku.

Answer

P × P × P = {1, 2} × {1, 2} × {1, 2}

= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2),

(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu P = {1, 2}, íkku P × P × P þkuÄku.

Answer

P × P × P = {1, 2} × {1, 2} × {1, 2}

= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2),

(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}

✓ Apply Cancel

#5 SUB 0M ✎ Edit ⚠ Log Error

Question

òu R ðkMíkrðf MktÏÞkykuLkku øký nkuÞ, íkku R × R yLku R × R × R þwt Ëþkoðþu ?

Answer

fkíkuorÍÞ økwýkfkh R × R = {(x, y) : x, y ∈ R} yu rîÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.

fkíkuorÍÞ økwýkfkh R × R × R = {(x, y, z) : x, y, z ∈ R} yu rºkÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu R ðkMíkrðf MktÏÞkykuLkku øký nkuÞ, íkku R × R yLku R × R × R þwt Ëþkoðþu ?

Answer

fkíkuorÍÞ økwýkfkh R × R = {(x, y) : x, y ∈ R} yu rîÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.

fkíkuorÍÞ økwýkfkh R × R × R = {(x, y, z) : x, y, z ∈ R} yu rºkÃkrh{kýeÞ Þk{-Mk{ík÷Lkk «íÞuf ®çkËwLkwt rLkYÃký Ëþkoðu Au.

✓ Apply Cancel

#6 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

òu A × B = {(p, q), (p, r), (m, q), (m, r)}, íkku A yLku B þkuÄku.

Answer

øký A = A × BLke Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf

\ A = {p, m}

øký B = A × BLke Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf

\ B = {q, r}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A × B = {(p, q), (p, r), (m, q), (m, r)}, íkku A yLku B þkuÄku.

Answer

øký A = A × BLke Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf

\ A = {p, m}

øký B = A × BLke Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf

\ B = {q, r}

✓ Apply Cancel

#7 SUB 0M 🖼 12 ✎ Edit ⚠ Log Error

Question

òu = , íkku x yLku y þkuÄku.

Answer

ynª, =

\ + 1 = , y – =

\ = , =

\ x + 3 = 5 , 3y – 2 = 1

\ x = 2 , 3y = 3

\ y = 1

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu = , íkku x yLku y þkuÄku.

Answer

ynª, =

\ + 1 = , y – =

\ = , =

\ x + 3 = 5 , 3y – 2 = 1

\ x = 2 , 3y = 3

\ y = 1

✓ Apply Cancel

#8 SUB 0M ✎ Edit ⚠ Log Error

Question

òu øký A{kt 3 ½xfku nkuÞ yLku B = {3, 4, 5}, íkku A × BLkk ½xfkuLke MktÏÞk þkuÄku.

Answer

ynª, n(A) = 3, n(B) = 3

\ n(A × B) = n(A) ⋅ n(B) = 3 × 3 = 9 ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu øký A{kt 3 ½xfku nkuÞ yLku B = {3, 4, 5}, íkku A × BLkk ½xfkuLke MktÏÞk þkuÄku.

Answer

ynª, n(A) = 3, n(B) = 3

\ n(A × B) = n(A) ⋅ n(B) = 3 × 3 = 9 ÚkkÞ.

✓ Apply Cancel

#9 SUB 0M ✎ Edit ⚠ Log Error

Question

òu G = {7, 8}, H = {5, 4, 2}, íkku G × H yLku
H × G þkuÄku.

Answer

G × H = {7, 8} × {5, 4, 2}

= {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

H × G = {5, 4, 2} × {7, 8}

= {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu G = {7, 8}, H = {5, 4, 2}, íkku G × H yLku
H × G þkuÄku.

Answer

G × H = {7, 8} × {5, 4, 2}

= {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

H × G = {5, 4, 2} × {7, 8}

= {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}

✓ Apply Cancel

#10 SUB 0M ✎ Edit ⚠ Log Error

Question

Lke[u ykÃku÷kt rðÄkLkku{ktÚke fÞwt rðÄkLk MkíÞ Au yLku fÞwt rðÄkLk yMkíÞ Au, íku sýkðku íkÚkk yMkíÞ rðÄkLk MkíÞ çkLku íku VheÚke ÷¾ku.

(i) òu P = {m, n}, Q = {n, m}, íkku P × Q = {(m, n), (n, m)}.

Answer

ykÃku÷ rðÄkLk yMkíÞ Au.

MkíÞ rðÄkLk : P × Q = {(m, n), (m, m), (n, n), (n, m)}

(ii) òu A yLku B yrhõík øký nkuÞ, íkku ßÞkt, x ∈ A íkÚkk y ∈ B nkuÞ íkuðe ík{k{ ¢{Þwõík òuz (x, y)Úke çkLkíkku yrhõík øký A × B Au.

ykÃku÷ rðÄkLk MkíÞ Au.

(iii) òu A = {1, 2}, B = {3, 4}, íkku A × (B ∩ f) = f

ykÃku÷ rðÄkLk MkíÞ Au.

ynª, B ∩ f = {3, 4} ∩ f = f Au.

\ A × (B ∩ f) = {1, 2} × f = f ÚkkÞ.

fu{ fu, çktLku{ktÚke fkuE Ãký yuf øký òu ¾k÷e øký nkuÞ, íkku íku{Lkku fkíkuorÍÞ økwýkfkh ¾k÷e øký s ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[u ykÃku÷kt rðÄkLkku{ktÚke fÞwt rðÄkLk MkíÞ Au yLku fÞwt rðÄkLk yMkíÞ Au, íku sýkðku íkÚkk yMkíÞ rðÄkLk MkíÞ çkLku íku VheÚke ÷¾ku.

(i) òu P = {m, n}, Q = {n, m}, íkku P × Q = {(m, n), (n, m)}.

Answer

ykÃku÷ rðÄkLk yMkíÞ Au.

MkíÞ rðÄkLk : P × Q = {(m, n), (m, m), (n, n), (n, m)}

(ii) òu A yLku B yrhõík øký nkuÞ, íkku ßÞkt, x ∈ A íkÚkk y ∈ B nkuÞ íkuðe ík{k{ ¢{Þwõík òuz (x, y)Úke çkLkíkku yrhõík øký A × B Au.

ykÃku÷ rðÄkLk MkíÞ Au.

(iii) òu A = {1, 2}, B = {3, 4}, íkku A × (B ∩ f) = f

ykÃku÷ rðÄkLk MkíÞ Au.

ynª, B ∩ f = {3, 4} ∩ f = f Au.

\ A × (B ∩ f) = {1, 2} × f = f ÚkkÞ.

fu{ fu, çktLku{ktÚke fkuE Ãký yuf øký òu ¾k÷e øký nkuÞ, íkku íku{Lkku fkíkuorÍÞ økwýkfkh ¾k÷e øký s ÚkkÞ.

✓ Apply Cancel

#11 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A = {–1, 1}, íkku A × A × A {u¤ðku.

Answer

A × A × A = {–1, 1} × {–1, 1} × {–1, 1}

= {(–1, –1, –1), (–1, –1, 1), (–1, 1, –1), (–1, 1, 1), (1, –1, –1), (1, –1, 1),

(1, 1, 1), (1, 1, –1)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {–1, 1}, íkku A × A × A {u¤ðku.

Answer

A × A × A = {–1, 1} × {–1, 1} × {–1, 1}

= {(–1, –1, –1), (–1, –1, 1), (–1, 1, –1), (–1, 1, 1), (1, –1, –1), (1, –1, 1),

(1, 1, 1), (1, 1, –1)}

✓ Apply Cancel

#12 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A × B = {(a, x), (a, y), (b, x), (b, y)}, íkku A yLku B þkuÄku.

Answer

ykÚke, (a, x) ∈ A × B

\ a ∈ A, x ∈ B

(a, y) ∈ A × B

\ a ∈ A, y ∈ B

(b, x) ∈ A × B

\ b ∈ A, x ∈ B

ykÚke, øký A = A × BLke «íÞuf ¢{Þwõík òuzLkku «Úk{ ½xf

= {a, b}

øký B = A × BLke «íÞuf ¢{Þwõík òuzLkku çkeòu ½xf

= {x, y}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A × B = {(a, x), (a, y), (b, x), (b, y)}, íkku A yLku B þkuÄku.

Answer

ykÚke, (a, x) ∈ A × B

\ a ∈ A, x ∈ B

(a, y) ∈ A × B

\ a ∈ A, y ∈ B

(b, x) ∈ A × B

\ b ∈ A, x ∈ B

ykÚke, øký A = A × BLke «íÞuf ¢{Þwõík òuzLkku «Úk{ ½xf

= {a, b}

øký B = A × BLke «íÞuf ¢{Þwõík òuzLkku çkeòu ½xf

= {x, y}

✓ Apply Cancel

#13 SUB 0M ✎ Edit ⚠ Log Error

Question

Äkhku fu, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6},
D = {5, 6, 7, 8}, íkku Lke[uLkkt Ãkrhýk{ku [fkMkku :

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

Answer

zk.çkk. A × (B ∩ C) {kxu,

B ∩ C = f

\ A × (B ∩ C) = {1, 2} × f = f...(1)

s.çkk. (A × B) ∩ (A × C)

A × B = {1, 2} × {1, 2, 3, 4}

= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1),

(2, 2), (2, 3), (2, 4)}

A × C = {1, 2} × {5, 6}

= {(1, 5), (1, 6), (2, 5), (2, 6)}

\ (A × B) ∩ (A × C) = f...(2)

\ Ãkrhýk{ (1) yLku (2) ÃkhÚke

zk.çkk. = s.çkk.

(ii) A × C yu B × DLkku WÃkøký Au.

ynª, A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}

B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6),
(3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}

ynª, òuE þfkÞ Au fu, A × CLke Ëhuf ¢{Þwõík òuz B × D{kt ykðu÷ nkuðkÚke A × C yu B × DLkku WÃkøký Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Äkhku fu, A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6},
D = {5, 6, 7, 8}, íkku Lke[uLkkt Ãkrhýk{ku [fkMkku :

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

Answer

zk.çkk. A × (B ∩ C) {kxu,

B ∩ C = f

\ A × (B ∩ C) = {1, 2} × f = f...(1)

s.çkk. (A × B) ∩ (A × C)

A × B = {1, 2} × {1, 2, 3, 4}

= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1),

(2, 2), (2, 3), (2, 4)}

A × C = {1, 2} × {5, 6}

= {(1, 5), (1, 6), (2, 5), (2, 6)}

\ (A × B) ∩ (A × C) = f...(2)

\ Ãkrhýk{ (1) yLku (2) ÃkhÚke

zk.çkk. = s.çkk.

(ii) A × C yu B × DLkku WÃkøký Au.

ynª, A × C = {(1, 5), (1, 6), (2, 5), (2, 6)}

B × D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6),
(3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}

ynª, òuE þfkÞ Au fu, A × CLke Ëhuf ¢{Þwõík òuz B × D{kt ykðu÷ nkuðkÚke A × C yu B × DLkku WÃkøký Au.

✓ Apply Cancel

#14 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A = {1, 2}, B = {3, 4}, íkku A × B þkuÄku. A × BLku fux÷k WÃkøkýku nþu ? íku ík{k{ WÃkøkýkuLke ÞkËe çkLkkðku.

Answer

A × B = {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}

ynª, n(A × B) = 4 nkuðkÚke íkuLkk WÃkøkýkuLke MktÏÞk 24 = 16 Úkþu.

• WÃkøkýku : f, A × B, {(1, 3)}, {(1, 4)}, {(2, 3)},
  {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}, {(1, 4),
  (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4),
  (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 4),
  (2, 3), (2, 4)}, {(2, 3), (2, 4), (1, 3)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {1, 2}, B = {3, 4}, íkku A × B þkuÄku. A × BLku fux÷k WÃkøkýku nþu ? íku ík{k{ WÃkøkýkuLke ÞkËe çkLkkðku.

Answer

A × B = {1, 2} × {3, 4} = {(1, 3), (1, 4), (2, 3), (2, 4)}

ynª, n(A × B) = 4 nkuðkÚke íkuLkk WÃkøkýkuLke MktÏÞk 24 = 16 Úkþu.

• WÃkøkýku : f, A × B, {(1, 3)}, {(1, 4)}, {(2, 3)},
  {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2, 4)}, {(1, 4), (2, 3)}, {(1, 4),
  (2, 4)}, {(2, 3), (2, 4)}, {(1, 3), (1, 4),
  (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1, 4),
  (2, 3), (2, 4)}, {(2, 3), (2, 4), (1, 3)}

✓ Apply Cancel

#15 SUB 0M ✎ Edit ⚠ Log Error

Question

òu n(A) = 3, n(B) = 2 nkuÞ íkuðk çku øký A yLku B nkuÞ yLku r¼LLk ½xfku x, y, z {kxu (x, 1), (y, 2),
(z, 1) yu A × BLkk ½xfku nkuÞ, íkku A yLku B þkuÄku.

Answer

ynª, (x, 1), (y, 2), (z, 1) ∈ A × B nkuðkÚke,

A = {x, y, z}, B = {1, 2} ÚkkÞ, fkhý fu n(A) = 3, n(B) = 2 ykÃku÷ Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu n(A) = 3, n(B) = 2 nkuÞ íkuðk çku øký A yLku B nkuÞ yLku r¼LLk ½xfku x, y, z {kxu (x, 1), (y, 2),
(z, 1) yu A × BLkk ½xfku nkuÞ, íkku A yLku B þkuÄku.

Answer

ynª, (x, 1), (y, 2), (z, 1) ∈ A × B nkuðkÚke,

A = {x, y, z}, B = {1, 2} ÚkkÞ, fkhý fu n(A) = 3, n(B) = 2 ykÃku÷ Au.

✓ Apply Cancel

#16 SUB 0M ✎ Edit ⚠ Log Error

Question

òu fkíkuorÍÞ økwýkfkh A × ALkk ½xfkuLke MktÏÞk 9 nkuÞ yLku íku{ktLkk çku ½xfku (–1, 0) yLku (0, 1) nkuÞ, íkku A þkuÄku íkÚkk A × ALkk çkkfeLkk ½xfku ÷¾ku.

Answer

ynª, n(A × A) = 9 \ n(A) = 3 ÚkkÞ íkÚkk (–1, 0), (0, 1) ∈ A × A nkuðkÚke A = {–1, 0, 1} ÚkkÞ.

{kxu, A × ALkk çkkfeLkk ½xfku (–1, 1), (–1, –1),
(0, –1), (0, 0), (1, –1), (1, 0) yLku (1, 1) Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu fkíkuorÍÞ økwýkfkh A × ALkk ½xfkuLke MktÏÞk 9 nkuÞ yLku íku{ktLkk çku ½xfku (–1, 0) yLku (0, 1) nkuÞ, íkku A þkuÄku íkÚkk A × ALkk çkkfeLkk ½xfku ÷¾ku.

Answer

ynª, n(A × A) = 9 \ n(A) = 3 ÚkkÞ íkÚkk (–1, 0), (0, 1) ∈ A × A nkuðkÚke A = {–1, 0, 1} ÚkkÞ.

{kxu, A × ALkk çkkfeLkk ½xfku (–1, 1), (–1, –1),
(0, –1), (0, 0), (1, –1), (1, 0) yLku (1, 1) Au.

✓ Apply Cancel

#17 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

òu A = {1, 2, 3, 4, 5, 6}, R = {(x, y) : y = x + 1} ÚkkÞ íku heíku MktçktÄ R Au, su AÚke A Ãkh ÔÞkÏÞkrÞík Au, íkku

(i) yk MktçktÄLku rfhý ykf]rík îkhk Ëþkoðku.

(ii) RLkku «Ëuþ, Mkn«Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x, y ∈ A = {1, 2, 3, 4, 5, 6} íkÚkk y = x + 1 Au.

x = 1, 2, 3, 4, 5 ÷uíkkt yLke ®f{íkku yLkw¢{u 2, 3, 4, 5, 6 {¤u. R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

RLkku «Ëuþ : {1, 2, 3, 4, 5},

rðMíkkh : {2, 3, 4, 5, 6}

Mkn«Ëuþ : {1, 2, 3, 4, 5, 6}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {1, 2, 3, 4, 5, 6}, R = {(x, y) : y = x + 1} ÚkkÞ íku heíku MktçktÄ R Au, su AÚke A Ãkh ÔÞkÏÞkrÞík Au, íkku

(i) yk MktçktÄLku rfhý ykf]rík îkhk Ëþkoðku.

(ii) RLkku «Ëuþ, Mkn«Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x, y ∈ A = {1, 2, 3, 4, 5, 6} íkÚkk y = x + 1 Au.

x = 1, 2, 3, 4, 5 ÷uíkkt yLke ®f{íkku yLkw¢{u 2, 3, 4, 5, 6 {¤u. R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

RLkku «Ëuþ : {1, 2, 3, 4, 5},

rðMíkkh : {2, 3, 4, 5, 6}

Mkn«Ëuþ : {1, 2, 3, 4, 5, 6}

✓ Apply Cancel

#18 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

økwýÄ{oLke heík : R : {(x, y) : x = y2, x ∈ P, y ∈ Q}

ÞkËeLke heík : R : {(9, 3), (9, –3), (4, 2), (4, –2),

(25, 5), (25, –5)}

«Ëuþ : {4, 9, 25}, rðMíkkh : {–2, 2, –3, 3, –5, 5}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

økwýÄ{oLke heík : R : {(x, y) : x = y2, x ∈ P, y ∈ Q}

ÞkËeLke heík : R : {(9, 3), (9, –3), (4, 2), (4, –2),

(25, 5), (25, –5)}

«Ëuþ : {4, 9, 25}, rðMíkkh : {–2, 2, –3, 3, –5, 5}

✓ Apply Cancel

#19 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

òu A = {1, 2}, B = {3, 4}, íkku AÚke BLkk MktçktÄkuLke MktÏÞk þkuÄku.

Answer

A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

ynª, n(A) = 2, n(B) = 2 íkÚkk

n(A×B) = n(A) ⋅ n(B) = 2 × 2 = 4 {kxu MktçktÄkuLke MktÏÞk = 24 = 16 ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {1, 2}, B = {3, 4}, íkku AÚke BLkk MktçktÄkuLke MktÏÞk þkuÄku.

Answer

A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

ynª, n(A) = 2, n(B) = 2 íkÚkk

n(A×B) = n(A) ⋅ n(B) = 2 × 2 = 4 {kxu MktçktÄkuLke MktÏÞk = 24 = 16 ÚkkÞ.

✓ Apply Cancel

#20 SUB 0M ✎ Edit ⚠ Log Error

Question

A = {1, 2, 3, 4, ..., 14}, R = {(x, y) : 3x – y = 0,
x, y ∈ A}. òu R yu AÚke ALkku MktçktÄ nkuÞ, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh {u¤ðku.

Answer

ynª, 3x – y = 0, x, y ∈ {1, 2, 3, ..., 14}

\ 3x = y

ynª, x = 1, 2, 3, 4 ÷uíkkt, yLke ®f{íkku yLkw¢{u
3, 6, 9, 12 {¤u.

\ R = {(1, 3), (2, 6), (3, 9), (4, 12)} {¤u.

\ RLkku «Ëuþ : {1, 2, 3, 4}

(Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf)

\ RLkku rðMíkkh : {3, 6, 9, 12}

(Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf)

\ RLkku Mkn«Ëuþ : øký A Ãkkuíku = {1, 2, 3, 4, ..., 14}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

A = {1, 2, 3, 4, ..., 14}, R = {(x, y) : 3x – y = 0,
x, y ∈ A}. òu R yu AÚke ALkku MktçktÄ nkuÞ, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh {u¤ðku.

Answer

ynª, 3x – y = 0, x, y ∈ {1, 2, 3, ..., 14}

\ 3x = y

ynª, x = 1, 2, 3, 4 ÷uíkkt, yLke ®f{íkku yLkw¢{u
3, 6, 9, 12 {¤u.

\ R = {(1, 3), (2, 6), (3, 9), (4, 12)} {¤u.

\ RLkku «Ëuþ : {1, 2, 3, 4}

(Ëhuf ¢{Þwõík òuzLkku «Úk{ ½xf)

\ RLkku rðMíkkh : {3, 6, 9, 12}

(Ëhuf ¢{Þwõík òuzLkku çkeòu ½xf)

\ RLkku Mkn«Ëuþ : øký A Ãkkuíku = {1, 2, 3, 4, ..., 14}

✓ Apply Cancel

#21 SUB 0M ✎ Edit ⚠ Log Error

Question

R = {(x, y) : y = x + 5, x yu 4Úke LkkLke «kf]ríkf MktÏÞk Au, x, y ∈ N} ÚkkÞ íku heíku yuf MktçktÄ N Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku. RLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x = 1, 2, 3 ÷uíkkt, yLke ®f{íkku

y = x + 5 ÃkhÚke yLkw¢{u 6, 7, 8 {¤u.

\ R = {(1, 6), (2, 7), (3, 8)}

\ RLkku «Ëuþ : {1, 2, 3} íkÚkk RLkku rðMíkkh = {6, 7, 8}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

R = {(x, y) : y = x + 5, x yu 4Úke LkkLke «kf]ríkf MktÏÞk Au, x, y ∈ N} ÚkkÞ íku heíku yuf MktçktÄ N Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku. RLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x = 1, 2, 3 ÷uíkkt, yLke ®f{íkku

y = x + 5 ÃkhÚke yLkw¢{u 6, 7, 8 {¤u.

\ R = {(1, 6), (2, 7), (3, 8)}

\ RLkku «Ëuþ : {1, 2, 3} íkÚkk RLkku rðMíkkh = {6, 7, 8}

✓ Apply Cancel

#22 SUB 0M ✎ Edit ⚠ Log Error

Question

A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : x yLku yLkku íkVkðík yÞwø{ MktÏÞk Au. x ∈ A, y ∈ B} ÚkkÞ íku heíku MktçktÄ AÚke B Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku.

Answer

ynª, x ∈ A = {1, 2, 3, 5}, y ∈ B = {4, 6, 9} íkÚkk R : x yLku y Lkku íkVkðík yÞwø{ MktÏÞk Au.

R : {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

A = {1, 2, 3, 5}, B = {4, 6, 9}, R = {(x, y) : x yLku yLkku íkVkðík yÞwø{ MktÏÞk Au. x ∈ A, y ∈ B} ÚkkÞ íku heíku MktçktÄ AÚke B Ãkh ÔÞkÏÞkrÞík Au. RLku ÞkËeLke heíku ÷¾ku.

Answer

ynª, x ∈ A = {1, 2, 3, 5}, y ∈ B = {4, 6, 9} íkÚkk R : x yLku y Lkku íkVkðík yÞwø{ MktÏÞk Au.

R : {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}

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#23 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

Answer

P = {5, 6, 7}, Q = {3, 4, 5}

(i) økwýÄ{oLke heík : R = {(x, y) : x ∈ P, y ∈ Q,

x – y = 2}

(ii) ÞkËeLke heík : R = {(5, 3), (6, 4), (7, 5)}

RLkku «Ëuþ : {5, 6, 7},

RLkku rðMíkkh : {3, 4, 5}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLke ykf]rík{kt PÚke QLkku MktçktÄ Ëþkoðu÷ Au. yk MktçktÄLku (i) økwýÄ{oLke heíku, (ii) ÞkËeLke heíku ÷¾ku. íkuLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

Answer

P = {5, 6, 7}, Q = {3, 4, 5}

(i) økwýÄ{oLke heík : R = {(x, y) : x ∈ P, y ∈ Q,

x – y = 2}

(ii) ÞkËeLke heík : R = {(5, 3), (6, 4), (7, 5)}

RLkku «Ëuþ : {5, 6, 7},

RLkku rðMíkkh : {3, 4, 5}

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#24 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b yu a ðzu rð¼kßÞ Au} íku heíku MktçktÄ R yu A Ãkh ÔÞkÏÞkrÞík Au. íkku RLku ÞkËeLke heíku ÷¾ku íkÚkk RLkku «Ëuþ yLku rðMíkkh {u¤ðku.

Answer

ynª, a, b ∈ A = {1, 2, 3, 4, 6} íkÚkk R : b yu a ðzu rð¼kßÞ Au.

R = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2),
(2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}

«Ëuþ : {1, 2, 3, 4, 6}

rðMíkkh : {1, 2, 3, 4, 6}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {1, 2, 3, 4, 6}, R = {(a, b) : a, b ∈ A, b yu a ðzu rð¼kßÞ Au} íku heíku MktçktÄ R yu A Ãkh ÔÞkÏÞkrÞík Au. íkku RLku ÞkËeLke heíku ÷¾ku íkÚkk RLkku «Ëuþ yLku rðMíkkh {u¤ðku.

Answer

ynª, a, b ∈ A = {1, 2, 3, 4, 6} íkÚkk R : b yu a ðzu rð¼kßÞ Au.

R = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2),
(2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (6, 6)}

«Ëuþ : {1, 2, 3, 4, 6}

rðMíkkh : {1, 2, 3, 4, 6}

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#25 SUB 0M ✎ Edit ⚠ Log Error

Question

R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x ∈ {0, 1, 2, 3, 4, 5} íkÚkk (x, x + 5) {kxuLke ¢{Þwõík òuz R : {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} Au.

• \ RLkku «Ëuþ : {0, 1, 2, 3, 4, 5} íkÚkk

RLkku rðMíkkh : {5, 6, 7, 8, 9, 10}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄLkku «Ëuþ íku{s rðMíkkh {u¤ðku.

Answer

ynª, x ∈ {0, 1, 2, 3, 4, 5} íkÚkk (x, x + 5) {kxuLke ¢{Þwõík òuz R : {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)} Au.

• \ RLkku «Ëuþ : {0, 1, 2, 3, 4, 5} íkÚkk

RLkku rðMíkkh : {5, 6, 7, 8, 9, 10}

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#26 SUB 0M ✎ Edit ⚠ Log Error

Question

MktçktÄ R = {(x, x3) : x yu 10 fhíkkt LkkLke yrð¼kßÞ MktÏÞk Au}Lku ÞkËeLkk MðYÃk{kt ÷¾ku.

Answer

ynª, x = {2, 3, 5, 7} Au, {kxu (x, x3) {kxuLke ¢{Þwõík òuzku R : {(2, 8), (3, 27), (5, 125), (7, 343)} ÚkkÞ.

R = {(2, 23), (3, 33), (5, 53), (7, 73)}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

MktçktÄ R = {(x, x3) : x yu 10 fhíkkt LkkLke yrð¼kßÞ MktÏÞk Au}Lku ÞkËeLkk MðYÃk{kt ÷¾ku.

Answer

ynª, x = {2, 3, 5, 7} Au, {kxu (x, x3) {kxuLke ¢{Þwõík òuzku R : {(2, 8), (3, 27), (5, 125), (7, 343)} ÚkkÞ.

R = {(2, 23), (3, 33), (5, 53), (7, 73)}

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#27 SUB 0M ✎ Edit ⚠ Log Error

Question

òu A = {x, y, z}; B = {1, 2}, íkku A Úke B Lkk MktçktÄkuLke MktÏÞk þkuÄku.

Answer

ynª, n(A) = 3, n(B) = 2, íkku n(A × B) = n(A) ⋅ n(B) ÃkhÚke n(A × B) = 6 ÚkkÞ. íkuÚke AÚke BLkk MktçktÄkuLke MktÏÞk = 26 = 64 ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu A = {x, y, z}; B = {1, 2}, íkku A Úke B Lkk MktçktÄkuLke MktÏÞk þkuÄku.

Answer

ynª, n(A) = 3, n(B) = 2, íkku n(A × B) = n(A) ⋅ n(B) ÃkhÚke n(A × B) = 6 ÚkkÞ. íkuÚke AÚke BLkk MktçktÄkuLke MktÏÞk = 26 = 64 ÚkkÞ.

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#28 SUB 0M 🖼 28 ✎ Edit ⚠ Log Error

Question

R yu Z Ãkh R = {(a, b) : a, b ∈ Z, a – b yu ÃkqýkOf Au} îkhk ÔÞkÏÞkrÞík Au. RLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

ynª, Z = {..., –2, –1, 0, 1, 2, ...} Au íkÚkk
(a, b) ∈ Z {kxu a – b yu ÃkqýkOf Au.

RLkku «Ëuþ yLku rðMíkkh çktLku Z Úkþu.

rðÄuÞ yu ÞtºkLke su{ Au, su yLkLÞ ykWxÃkwx Ëhuf RLkÃkwx {kxu ykÃku Au.

ÔÞkÏÞk : yrhõík øký A yLku B {kxu, MktçktÄ f îkhk øký ALkk «íÞuf ½xfLku Mktøkík øký B{kt yLkLÞ «rík®çkçk {¤u Au, íkku yk MktçktÄ fLku AÚke B ÃkhLkwt rðÄuÞ fnu Au. AÚke B ÃkhLkk rðÄuÞLku f : A → B ÷¾kÞ Au.

òu a ∈ A, f(a) = b, b ∈ B íkku bLku f îkhk {¤íkwt aLkwt «rík®çkçk fnuðkÞ Au. aLku f îkhk bLkwt Ãkqðo«rík®çkçk fnuðkÞ Au.

yk{, rðÄuÞ yu rðrþ»x «fkhLkku MktçktÄ Au.

fkuE MktçktÄ yu rðÄuÞ íÞkhu s çkLku ßÞkhu, «ËuþLkku Ëhuf ½xf Mkn«Ëuþ{ktLkk {kºk yLku {kºk yuf s ½xf MkkÚku òuzkÞu÷ nkuÞ yLÞÚkk ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.

WËknhý íkhefu,

ynª ykf]rík (i){kt Ëþkoðu÷ MktçktÄ R1 yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 3 yu Mkn«ËuþLkk fkuE Ãký ½xf MkkÚku òuzkÞu÷ LkÚke.

ynª ykf]rík (ii){kt Ëþkoðu÷ MktçktÄ R2 yu rðÄuÞ Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

ynª ykf]rík (iii){kt Ëþkoðu÷ MktçktÄ R3 yu rðÄuÞ LkÚke, fkhý fu «Úk{ ½xf 1 yu (Mkn«ËuþLkkt çku) ½xfku MkkÚku òuzkÞu÷ Au.

Äkhku fu, f : A → B nkuÞ, íkku øký ALku fLkku «Ëuþ fnu Au íkÚkk øký BLku fLkku Mkn«Ëuþ fnu Au íkÚkk ALkk çkÄk ½xfkuLkk «rík®çkçkLku fLkku rðMíkkh fnu Au.

ynª, MÃk»x Au fu rðMíkkh ⊆ Mkn«Ëuþ.

òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf fhíkkt ðÄkhu ®çkËwyku{kt AuËu íkku ð¢ yu Võík MktçktÄ Ëþkoðu Au, Ãkhtíkw rðÄuÞ Lk Ëþkoðu.

Ãkhtíkw òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf s ®çkËw{kt AuËu íkku ð¢ yu rðÄuÞ Ëþkoðu Au.

(1) íkËuð rðÄuÞ (Identity Function) :

f : R → R, ∀x ∈ R {kxu f(x) = x Lku íkËuð rðÄuÞ fnu Au

ykLkku yÚko yu ÚkkÞ fu, su RLkÃkwx nkuÞ íku s ykWxÃkwx {¤u.

ynª, íkËuð rðÄuÞLkku «Ëuþ yLku rðMíkkh R Au.

íkËuð rðÄuÞLkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh Úkíke hu¾k Ëþkoðu Au.

(2) y[¤ rðÄuÞ (Constant Function) :

f : R → R, ∀x ∈ R, f(x) = c, (ßÞkt, c yu fkuE y[¤ Au)Lku y[¤ rðÄuÞ fnu Au.

y[¤ rðÄuÞLkku «Ëuþ R yLku íkuLkku rðMíkkh {c} Au.

y[¤ rðÄuÞLkku yk÷u¾ X-yûkLku Mk{ktíkh hu¾k Au.

òu c > 0, íkku hu¾k X-yûkLke WÃkh Au.

òu c = 0, íkku hu¾k X-yûk Ãkh MktÃkkíke nþu.

òu c < 0, íkku hu¾k X-yûkLke Lke[u Au.

(3) çknwÃkËe rðÄuÞ (Polynomial Function) :

f : R → R, ∀x ∈ R {kxu

f(x) = a0 + a1x + a2x2 + ... + anxn, Lku n ½kíkLkwt çknwÃkËe rðÄuÞ fnu Au. ynª, n yu yLk]ý ÃkqýkOf Au.

a0, a1, a2, ..., an ∈ R íkÚkk an ≠ 0.

WËk. f (x) = x3 – x2 + 2, g(x) = x4 + x Au.

h(x) = + 2x LkÚke.

(4) Mkt{uÞ rðÄuÞ (Rational Function) :

g(x) ≠ 0 nkuÞ íkuðk «Ëuþ{kt ÔÞkÏÞkrÞík çknwÃkËe rðÄuÞ f(x) yLku g(x) {kxu Lku Mkt{uÞ rðÄuÞ fnu Au.

(5) {kLkktf rðÄuÞ (Modulus Function) :

f(x) = |x| , f : R → RÚke ÔÞkÏÞkrÞík rðÄuÞLku {kLkktf rðÄuÞ fnu Au.

f(x) = |x| =

íkuðe s heíku,

|x – 1| = íkÚkk

|x + 2| = ÷¾e þfkÞ.

{kLkktf rðÄuÞLkku yk÷u¾ Y-yûk «íÞu Mktr{ík nkuÞ Au íkÚkk X-yûkLke WÃkh nkuÞ Au íkÚkk f(x) = |x|Lkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh ÚkkÞ Au.

{kLkktf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Au.

(6) r[nTLk rðÄuÞ (Signum Function) :

f : R → R {kxu, f(x) =

ðzu ÔÞkÏÞkrÞík rðÄuÞLku r[nTLk rðÄuÞ fnu Au.

«Ëuþ R yLku rðMíkkh {–1, 0, 1} Au.

r[nTLk rðÄuÞ f(x) = ðzu Ãký ËþkoðkÞ Au.

(7) {n¥k{ ÃkqýkOf rðÄuÞ (Greatest Integer Function) :

rðÄuÞ f : R → R, f(x) = [x] yÚkðk x,
∀x ∈ R Lku {n¥k{ ÃkqýkOf rðÄuÞ fnu Auu. ßÞkt, [x] yÚkðk
x yu xÚke LkkLkk yÚkðk xLku Mk{kLk nkuÞ íkuðk ík{k{ ÃkqýkOfku{kt MkkiÚke {kuxku ÃkqýkOf Ëþkoðu Au. íkuLku floor function íkhefu Ãký yku¤¾ðk{kt ykðu Au.

{n¥k{ ÃkqýkOf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh Z Au.

fkuE Ãký x ∈ R Lkku {n¥k{ ÃkqýkOf Lke[u {wsçk þkuÄe þfkÞ Au :

[2.3] = 2, [– 3.2] = – 4, [5.99] = 5

[3] = 3, [0] = 0

fkuE Ãký ÃkqýkOf MktÏÞkLke {n¥k{ ÃkqýkOf íku MktÏÞk Ãkkuíku s ÚkkÞ.

(8) rMk®÷øk rðÄuÞ (Ceiling Function) :

rðÄuÞ f : R → R, f(x) = x Lku rMk®÷øk rðÄuÞ fnu Au.

rMk®÷øk rðÄuÞLkku «Ëuþ R yLku rðMíkkh ‘Z’ Au.

fkuE Ãký x ∈ R Lkku rMk®÷øk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ Au :

2.3 = 3, –3.2 = – 4, 5.99 = 5

3 = 3, 0 = 0

fkuE Ãký ÃkqýkOf MktÏÞkLkku rMk®÷øk rðÄuÞ íku MktÏÞk Ãkkuíku s ÚkkÞ.

(9) ÃkqýkOf ¼køk rðÄuÞ (Fractional Part function) :

rðÄuÞ f : R → R, f(x) = {x}

= x – [x]

Lku ÃkqýkOf ¼køk rðÄuÞ fnu Au.

ÃkqýkOf ¼køk rðÄuÞLkku R yLku rðMíkkh [0, 1) Au.

fkuE Ãký x ∈ R Lkku ÃkqýkOf ¼køk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ :

{2.3} = 0.3, {– 3.2} = 0.8

{5.99} = 0.99, {3} = 0

fkuE Ãký ÃkqýkOf MktÏÞkLkku ÃkqýkOf ¼køk rðÄuÞLkwt {qÕÞ ‘þqLÞ’ {¤u.

Äkhku fu, f : D1 → R íkÚkk g : D2 → R çku ðkMíkrðf rðÄuÞku Au.

(i) (f + g) (x) = f(x) + g(x), ∀x ∈ D1 ∩ D2

(ii) (f – g) (x) = f(x) – g(x), ∀x ∈ D1 ∩ D2

(iii) (fg) (x) = f(x) ⋅ g(x), ∀x ∈ D1 ∩ D2

(iv) (x) = ,

∀x ∈ (D1 ∩ D2) – {x : g(x) = 0}

(v) ðkMíkrðf rðÄuÞLkku y[¤ MkkÚku økwýkfkh : òu c yu fkuE y[¤ nkuÞ, íkku (c ⋅ f) (x) = c ⋅ f(x) ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

R yu Z Ãkh R = {(a, b) : a, b ∈ Z, a – b yu ÃkqýkOf Au} îkhk ÔÞkÏÞkrÞík Au. RLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

ynª, Z = {..., –2, –1, 0, 1, 2, ...} Au íkÚkk
(a, b) ∈ Z {kxu a – b yu ÃkqýkOf Au.

RLkku «Ëuþ yLku rðMíkkh çktLku Z Úkþu.

rðÄuÞ yu ÞtºkLke su{ Au, su yLkLÞ ykWxÃkwx Ëhuf RLkÃkwx {kxu ykÃku Au.

ÔÞkÏÞk : yrhõík øký A yLku B {kxu, MktçktÄ f îkhk øký ALkk «íÞuf ½xfLku Mktøkík øký B{kt yLkLÞ «rík®çkçk {¤u Au, íkku yk MktçktÄ fLku AÚke B ÃkhLkwt rðÄuÞ fnu Au. AÚke B ÃkhLkk rðÄuÞLku f : A → B ÷¾kÞ Au.

òu a ∈ A, f(a) = b, b ∈ B íkku bLku f îkhk {¤íkwt aLkwt «rík®çkçk fnuðkÞ Au. aLku f îkhk bLkwt Ãkqðo«rík®çkçk fnuðkÞ Au.

yk{, rðÄuÞ yu rðrþ»x «fkhLkku MktçktÄ Au.

fkuE MktçktÄ yu rðÄuÞ íÞkhu s çkLku ßÞkhu, «ËuþLkku Ëhuf ½xf Mkn«Ëuþ{ktLkk {kºk yLku {kºk yuf s ½xf MkkÚku òuzkÞu÷ nkuÞ yLÞÚkk ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.

WËknhý íkhefu,

ynª ykf]rík (i){kt Ëþkoðu÷ MktçktÄ R1 yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 3 yu Mkn«ËuþLkk fkuE Ãký ½xf MkkÚku òuzkÞu÷ LkÚke.

ynª ykf]rík (ii){kt Ëþkoðu÷ MktçktÄ R2 yu rðÄuÞ Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

ynª ykf]rík (iii){kt Ëþkoðu÷ MktçktÄ R3 yu rðÄuÞ LkÚke, fkhý fu «Úk{ ½xf 1 yu (Mkn«ËuþLkkt çku) ½xfku MkkÚku òuzkÞu÷ Au.

Äkhku fu, f : A → B nkuÞ, íkku øký ALku fLkku «Ëuþ fnu Au íkÚkk øký BLku fLkku Mkn«Ëuþ fnu Au íkÚkk ALkk çkÄk ½xfkuLkk «rík®çkçkLku fLkku rðMíkkh fnu Au.

ynª, MÃk»x Au fu rðMíkkh ⊆ Mkn«Ëuþ.

òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf fhíkkt ðÄkhu ®çkËwyku{kt AuËu íkku ð¢ yu Võík MktçktÄ Ëþkoðu Au, Ãkhtíkw rðÄuÞ Lk Ëþkoðu.

Ãkhtíkw òu y-yûkLku Mk{ktíkh Ëkuhu÷ hu¾k ykÃku÷ ð¢Lku yuf s ®çkËw{kt AuËu íkku ð¢ yu rðÄuÞ Ëþkoðu Au.

(1) íkËuð rðÄuÞ (Identity Function) :

f : R → R, ∀x ∈ R {kxu f(x) = x Lku íkËuð rðÄuÞ fnu Au

ykLkku yÚko yu ÚkkÞ fu, su RLkÃkwx nkuÞ íku s ykWxÃkwx {¤u.

ynª, íkËuð rðÄuÞLkku «Ëuþ yLku rðMíkkh R Au.

íkËuð rðÄuÞLkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh Úkíke hu¾k Ëþkoðu Au.

(2) y[¤ rðÄuÞ (Constant Function) :

f : R → R, ∀x ∈ R, f(x) = c, (ßÞkt, c yu fkuE y[¤ Au)Lku y[¤ rðÄuÞ fnu Au.

y[¤ rðÄuÞLkku «Ëuþ R yLku íkuLkku rðMíkkh {c} Au.

y[¤ rðÄuÞLkku yk÷u¾ X-yûkLku Mk{ktíkh hu¾k Au.

òu c > 0, íkku hu¾k X-yûkLke WÃkh Au.

òu c = 0, íkku hu¾k X-yûk Ãkh MktÃkkíke nþu.

òu c < 0, íkku hu¾k X-yûkLke Lke[u Au.

(3) çknwÃkËe rðÄuÞ (Polynomial Function) :

f : R → R, ∀x ∈ R {kxu

f(x) = a0 + a1x + a2x2 + ... + anxn, Lku n ½kíkLkwt çknwÃkËe rðÄuÞ fnu Au. ynª, n yu yLk]ý ÃkqýkOf Au.

a0, a1, a2, ..., an ∈ R íkÚkk an ≠ 0.

WËk. f (x) = x3 – x2 + 2, g(x) = x4 + x Au.

h(x) = + 2x LkÚke.

(4) Mkt{uÞ rðÄuÞ (Rational Function) :

g(x) ≠ 0 nkuÞ íkuðk «Ëuþ{kt ÔÞkÏÞkrÞík çknwÃkËe rðÄuÞ f(x) yLku g(x) {kxu Lku Mkt{uÞ rðÄuÞ fnu Au.

(5) {kLkktf rðÄuÞ (Modulus Function) :

f(x) = |x| , f : R → RÚke ÔÞkÏÞkrÞík rðÄuÞLku {kLkktf rðÄuÞ fnu Au.

f(x) = |x| =

íkuðe s heíku,

|x – 1| = íkÚkk

|x + 2| = ÷¾e þfkÞ.

{kLkktf rðÄuÞLkku yk÷u¾ Y-yûk «íÞu Mktr{ík nkuÞ Au íkÚkk X-yûkLke WÃkh nkuÞ Au íkÚkk f(x) = |x|Lkku yk÷u¾ Wøk{®çkËw{ktÚke ÃkMkkh ÚkkÞ Au.

{kLkktf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Au.

(6) r[nTLk rðÄuÞ (Signum Function) :

f : R → R {kxu, f(x) =

ðzu ÔÞkÏÞkrÞík rðÄuÞLku r[nTLk rðÄuÞ fnu Au.

«Ëuþ R yLku rðMíkkh {–1, 0, 1} Au.

r[nTLk rðÄuÞ f(x) = ðzu Ãký ËþkoðkÞ Au.

(7) {n¥k{ ÃkqýkOf rðÄuÞ (Greatest Integer Function) :

rðÄuÞ f : R → R, f(x) = [x] yÚkðk x,
∀x ∈ R Lku {n¥k{ ÃkqýkOf rðÄuÞ fnu Auu. ßÞkt, [x] yÚkðk
x yu xÚke LkkLkk yÚkðk xLku Mk{kLk nkuÞ íkuðk ík{k{ ÃkqýkOfku{kt MkkiÚke {kuxku ÃkqýkOf Ëþkoðu Au. íkuLku floor function íkhefu Ãký yku¤¾ðk{kt ykðu Au.

{n¥k{ ÃkqýkOf rðÄuÞLkku «Ëuþ R íkÚkk rðMíkkh Z Au.

fkuE Ãký x ∈ R Lkku {n¥k{ ÃkqýkOf Lke[u {wsçk þkuÄe þfkÞ Au :

[2.3] = 2, [– 3.2] = – 4, [5.99] = 5

[3] = 3, [0] = 0

fkuE Ãký ÃkqýkOf MktÏÞkLke {n¥k{ ÃkqýkOf íku MktÏÞk Ãkkuíku s ÚkkÞ.

(8) rMk®÷øk rðÄuÞ (Ceiling Function) :

rðÄuÞ f : R → R, f(x) = x Lku rMk®÷øk rðÄuÞ fnu Au.

rMk®÷øk rðÄuÞLkku «Ëuþ R yLku rðMíkkh ‘Z’ Au.

fkuE Ãký x ∈ R Lkku rMk®÷øk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ Au :

2.3 = 3, –3.2 = – 4, 5.99 = 5

3 = 3, 0 = 0

fkuE Ãký ÃkqýkOf MktÏÞkLkku rMk®÷øk rðÄuÞ íku MktÏÞk Ãkkuíku s ÚkkÞ.

(9) ÃkqýkOf ¼køk rðÄuÞ (Fractional Part function) :

rðÄuÞ f : R → R, f(x) = {x}

= x – [x]

Lku ÃkqýkOf ¼køk rðÄuÞ fnu Au.

ÃkqýkOf ¼køk rðÄuÞLkku R yLku rðMíkkh [0, 1) Au.

fkuE Ãký x ∈ R Lkku ÃkqýkOf ¼køk rðÄuÞ Lke[u {wsçk þkuÄe þfkÞ :

{2.3} = 0.3, {– 3.2} = 0.8

{5.99} = 0.99, {3} = 0

fkuE Ãký ÃkqýkOf MktÏÞkLkku ÃkqýkOf ¼køk rðÄuÞLkwt {qÕÞ ‘þqLÞ’ {¤u.

Äkhku fu, f : D1 → R íkÚkk g : D2 → R çku ðkMíkrðf rðÄuÞku Au.

(i) (f + g) (x) = f(x) + g(x), ∀x ∈ D1 ∩ D2

(ii) (f – g) (x) = f(x) – g(x), ∀x ∈ D1 ∩ D2

(iii) (fg) (x) = f(x) ⋅ g(x), ∀x ∈ D1 ∩ D2

(iv) (x) = ,

∀x ∈ (D1 ∩ D2) – {x : g(x) = 0}

(v) ðkMíkrðf rðÄuÞLkku y[¤ MkkÚku økwýkfkh : òu c yu fkuE y[¤ nkuÞ, íkku (c ⋅ f) (x) = c ⋅ f(x) ÚkkÞ.

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#29 SUB 0M ✎ Edit ⚠ Log Error

Question

N yu «kf]ríkf MktÏÞkykuLkku øký Au yLku íkuLke Ãkh ÔÞkÏÞkrÞík fkuE MktçktÄ R yuðku Au fu, R = {(x, y) :
y = 2x, x, y ∈ N}, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh þkuÄku. þwt yk MktçktÄ rðÄuÞ Au ?

Answer

ynª, x, y ∈ N íkÚkk y = 2x Au. {kxu ynª, «Ëuþ N, Mkn«Ëuþ N íkÚkk rðMíkkh yu Þwø{ «kf]ríkf MktÏÞk Úkþu.

ynª, «íÞuf x ∈ N {kxu yLkLÞ y ∈ N {¤u Au, {kxu ykÃku÷ MktçktÄ yu rðÄuÞ çkLkþu.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

N yu «kf]ríkf MktÏÞkykuLkku øký Au yLku íkuLke Ãkh ÔÞkÏÞkrÞík fkuE MktçktÄ R yuðku Au fu, R = {(x, y) :
y = 2x, x, y ∈ N}, íkku RLkku «Ëuþ, Mkn«Ëuþ yLku rðMíkkh þkuÄku. þwt yk MktçktÄ rðÄuÞ Au ?

Answer

ynª, x, y ∈ N íkÚkk y = 2x Au. {kxu ynª, «Ëuþ N, Mkn«Ëuþ N íkÚkk rðMíkkh yu Þwø{ «kf]ríkf MktÏÞk Úkþu.

ynª, «íÞuf x ∈ N {kxu yLkLÞ y ∈ N {¤u Au, {kxu ykÃku÷ MktçktÄ yu rðÄuÞ çkLkþu.

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#30 SUB 0M 🖼 13 ✎ Edit ⚠ Log Error

Question

Lke[uLkkt WËknhýku{kt ykÃku÷ MktçktÄ [fkMkku yLku «íÞuf MktçktÄ rðÄuÞ Au fu Lknª íku fkhý ykÃke sýkðku.

(i) R = {(2, 1), (3, 1), (4, 2)}

(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

Answer

(i) R = {(2, 1), (3, 1), (4, 2)}Lku Lke[u {wsçk Ëþkoðe þfkÞ :

2

3

4

R

1

2

ynª, «Ëuþ{ktLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku.

(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

2

3

4

R

2

3

4

ynª, «Ëuþ{ktLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.

(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

ynª, «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLkkt WËknhýku{kt ykÃku÷ MktçktÄ [fkMkku yLku «íÞuf MktçktÄ rðÄuÞ Au fu Lknª íku fkhý ykÃke sýkðku.

(i) R = {(2, 1), (3, 1), (4, 2)}

(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

Answer

(i) R = {(2, 1), (3, 1), (4, 2)}Lku Lke[u {wsçk Ëþkoðe þfkÞ :

2

3

4

R

1

2

ynª, «Ëuþ{ktLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku.

(ii) R = {(2, 2), (2, 4), (3, 3), (4, 4)}

2

3

4

R

2

3

4

ynª, «Ëuþ{ktLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ çkLku Lknª.

(iii) R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)}

ynª, «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ nkuðkÚke ykÃku÷ MktçktÄ yu rðÄuÞ Au.

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#31 SUB 0M ▦ 1 ✎ Edit ⚠ Log Error

Question

N yu «kf]ríkf MktÏÞkykuLkku øký Au f : N → N, f(x) = 2x + 1 îkhk ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ Au. yk ÔÞkÏÞkLke {ËËÚke Lke[uLkwt fku»xf Ãkqýo fhku :

x	1	2	3	4	5	6	7
y	f(1)=...	f(2)=...	f(3)=...	f(4)=...	f(5)=...	f(6)=...	f(7)=...

Answer

ynª, f(x) = 2x + 1 Au.

f(1) = 2(1) + 1 = 3, f(2) = 2(2) + 1 = 5, f(3) = 2(3) + 1 = 7

f(4) = 2(4) + 1 = 9, f(5) = 2(5) + 1 = 11, f(6) = 2(6) + 1 = 13,

f(7) = 2(7) + 1 = 15

x	1	2	3	4	5	6	7
y	f(1)=3	f(2)=5	f(3)=7	f(4)=9	f(5)=11	f(6)=13	f(7)=15

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

N yu «kf]ríkf MktÏÞkykuLkku øký Au f : N → N, f(x) = 2x + 1 îkhk ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ Au. yk ÔÞkÏÞkLke {ËËÚke Lke[uLkwt fku»xf Ãkqýo fhku :

Answer

ynª, f(x) = 2x + 1 Au.

f(1) = 2(1) + 1 = 3, f(2) = 2(2) + 1 = 5, f(3) = 2(3) + 1 = 7

f(4) = 2(4) + 1 = 9, f(5) = 2(5) + 1 = 11, f(6) = 2(6) + 1 = 13,

f(7) = 2(7) + 1 = 15

x	1	2	3	4	5	6	7
y	f(1)=3	f(2)=5	f(3)=7	f(4)=9	f(5)=11	f(6)=13	f(7)=15

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#32 SUB 0M 🖼 1 ▦ 1 ✎ Edit ⚠ Log Error

Question

f : R → R, y = f(x) = x2, x ∈ R Úke ÔÞkÏÞkrÞík yuf rðÄuÞ Au. yk ÔÞkÏÞkLku ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ? fLkku yk÷u¾ Ëkuhku.

x	– 4	– 3	– 2	– 1	0	1	2	3	4
y=f(x)=x2

Answer

ynª, y = f(x) = x2 Au.

\ f(– 4) = (– 4)2 = 16, f(– 3) = (– 3)2 = 9,
f(– 2) = (– 2)2 = 4, f(– 1) = (– 1)2 = 1,

f(0) = (0)2 = 0, f(1) = (1)2 = 1, f(2) = (2)2 = 4,

f(3) = (3)2 = 9, f(4) = (4)2 = 16 ÚkkÞ.

x	– 4	– 3	– 2	– 1	0	1	2	3	4
y=f(x)=x2	16	9	4	1	0	1	4	9	16

ynª, fLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Úkþu.

y = f(x) = x2Lkku yk÷u¾ Lke[u {wsçk Au: su Ãkhð÷Þ Ëþkoðu Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f : R → R, y = f(x) = x2, x ∈ R Úke ÔÞkÏÞkrÞík yuf rðÄuÞ Au. yk ÔÞkÏÞkLku ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ? fLkku yk÷u¾ Ëkuhku.

Answer

ynª, y = f(x) = x2 Au.

\ f(– 4) = (– 4)2 = 16, f(– 3) = (– 3)2 = 9,
f(– 2) = (– 2)2 = 4, f(– 1) = (– 1)2 = 1,

f(0) = (0)2 = 0, f(1) = (1)2 = 1, f(2) = (2)2 = 4,

f(3) = (3)2 = 9, f(4) = (4)2 = 16 ÚkkÞ.

x	– 4	– 3	– 2	– 1	0	1	2	3	4
y=f(x)=x2	16	9	4	1	0	1	4	9	16

ynª, fLkku «Ëuþ R íkÚkk rðMíkkh [0, ∞) Úkþu.

y = f(x) = x2Lkku yk÷u¾ Lke[u {wsçk Au: su Ãkhð÷Þ Ëþkoðu Au.

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#33 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

f : R → R, f(x) = x3, x ∈ RÚke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾ Ëkuhku.

Answer

f(x) = x3 {kxu,

f(0) = (0)3 = 0, f(1) = (1)3 = 1,

f(– 1) = (– 1)3 = – 1, f(2) = (2)3 = 8,

f(– 2) = (– 2)3 = – 8, f(–3) = –27

f(3) = 27 ðøkuhu ...

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f : R → R, f(x) = x3, x ∈ RÚke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾ Ëkuhku.

Answer

f(x) = x3 {kxu,

f(0) = (0)3 = 0, f(1) = (1)3 = 1,

f(– 1) = (– 1)3 = – 1, f(2) = (2)3 = 8,

f(– 2) = (– 2)3 = – 8, f(–3) = –27

f(3) = 27 ðøkuhu ...

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#34 SUB 0M 🖼 4 ▦ 1 ✎ Edit ⚠ Log Error

Question

f : R – {0} → R, f(x) = , x ∈ R – {0}Úke ÔÞkÏÞkrÞík yuf rðÄuÞ ykÃku÷ Au. yk ÔÞkÏÞkLkk ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

x	– 2	– 1.5	– 1	– 0.5	0.25	0.5	1	1.5	2
y = f(x) =	–	–	–	–	–	–	–	–	–

Answer

y = f(x) = ykÃku÷ Au.

x	– 2	– 1.5	– 1	– 0.5	0.25	0.5	1	1.5	2
y =	– 0.5	–0.67	– 1	– 2	4	2	1	0.67	0.5

fk[ku yk÷u¾ :

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f : R – {0} → R, f(x) = , x ∈ R – {0}Úke ÔÞkÏÞkrÞík yuf rðÄuÞ ykÃku÷ Au. yk ÔÞkÏÞkLkk ykÄkhu Lke[uLkwt fku»xf Ãkqýo fhku. yk rðÄuÞLkku «Ëuþ yLku rðMíkkh þwt Úkþu ?

Answer

y = f(x) = ykÃku÷ Au.

x	– 2	– 1.5	– 1	– 0.5	0.25	0.5	1	1.5	2
y =	– 0.5	–0.67	– 1	– 2	4	2	1	0.67	0.5

fk[ku yk÷u¾ :

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#35 SUB 0M 🖼 5 ✎ Edit ⚠ Log Error

Question

f(x) = x2, g(x) = 2x + 1 yu çku ðkMíkrðf rðÄuÞku nkuÞ, íkku (f + g)(x), (f – g)(x), (fg)(x), (x) þkuÄku.

Answer

(f + g)(x) = f(x) + g(x) = x2 + 2x + 1

(f – g)(x) = f(x) – g(x) = x2 – (2x + 1) = x2 – 2x – 1

(f ⋅ g)(x) = f(x) ⋅ g(x) = x2 (2x + 1) = 2x3 + x2

(x) = , g(x) ≠ 0, = , x ≠

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = x2, g(x) = 2x + 1 yu çku ðkMíkrðf rðÄuÞku nkuÞ, íkku (f + g)(x), (f – g)(x), (fg)(x), (x) þkuÄku.

Answer

(f + g)(x) = f(x) + g(x) = x2 + 2x + 1

(f – g)(x) = f(x) – g(x) = x2 – (2x + 1) = x2 – 2x – 1

(f ⋅ g)(x) = f(x) ⋅ g(x) = x2 (2x + 1) = 2x3 + x2

(x) = , g(x) ≠ 0, = , x ≠

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#36 SUB 0M 🖼 14 ✎ Edit ⚠ Log Error

Question

f(x) = , g(x) = x yu çku yLk]ý ðkMíkrðf MktÏÞkLkk øký Ãkh ÔÞkÏÞkrÞík rðÄuÞ nkuÞ, íkku (f + g)(x), (f – g) (x), (f ⋅ g)(x), (x) þkuÄku.

Answer

(f + g)(x) = f(x) + g(x) = + x

(f – g)(x) = f(x) – g(x) = – x

(f ⋅ g)(x) = f(x) ⋅ g(x) = ⋅ x = x ⋅ x = x+1 = x

(x) = , g(x) ≠ 0, = , x ≠ 0,

= x–1 = x–, x ≠ 0

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = , g(x) = x yu çku yLk]ý ðkMíkrðf MktÏÞkLkk øký Ãkh ÔÞkÏÞkrÞík rðÄuÞ nkuÞ, íkku (f + g)(x), (f – g) (x), (f ⋅ g)(x), (x) þkuÄku.

Answer

(f + g)(x) = f(x) + g(x) = + x

(f – g)(x) = f(x) – g(x) = – x

(f ⋅ g)(x) = f(x) ⋅ g(x) = ⋅ x = x ⋅ x = x+1 = x

(x) = , g(x) ≠ 0, = , x ≠ 0,

= x–1 = x–, x ≠ 0

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#37 SUB 0M 🖼 3 ✎ Edit ⚠ Log Error

Question

Lke[uLkk Ãkife fÞku MktçktÄ rðÄuÞ Au ? fkhý ykÃkku. òu íku rðÄuÞ nkuÞ, íkku íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

Answer

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

«Ëuþ : {2, 5, 8, 11, 14, 17}

rðMíkkh : {1}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

«Ëuþ : {2, 4, 6, 8, 10, 12, 14}

rðMíkkh : {1, 2, 3, 4, 5, 6, 7}

(iii) {(1, 3), (1, 5), (2, 5)}

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 1 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLkk Ãkife fÞku MktçktÄ rðÄuÞ Au ? fkhý ykÃkku. òu íku rðÄuÞ nkuÞ, íkku íkuLkku «Ëuþ yLku rðMíkkh þkuÄku.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

Answer

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

«Ëuþ : {2, 5, 8, 11, 14, 17}

rðMíkkh : {1}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ Ëþkoðu Au, fkhý fu «Ëuþ{ktLkku Ëhuf ½xf yu Mkn«ËuþLkk yLkLÞ ½xf MkkÚku òuzkÞu÷ Au.

«Ëuþ : {2, 4, 6, 8, 10, 12, 14}

rðMíkkh : {1, 2, 3, 4, 5, 6, 7}

(iii) {(1, 3), (1, 5), (2, 5)}

ynª, ykÃku÷ MktçktÄ yu rðÄuÞ LkÚke, fkhý fu «Ëuþ{ktLkku ½xf 1 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷ Au.

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#38 SUB 0M 🖼 4 ✎ Edit ⚠ Log Error

Question

Lke[uLkkt ðkMíkrðf rðÄuÞkuLkk «Ëuþ yLku rðMíkkh þkuÄku.

(i) f(x) = – |x|

Answer

ynª, f(x) = – |x|{kt xLke fkuE Ãký ðkMíkrðf ®f{ík {qfe þfkíke nkuðkÚke «Ëuþ R Úkþu.

íkÚkk |x|Lkku rðMíkkh [0, ∞) nkuðkÚke – |x|Lkku rðMíkkh
(– ∞, 0] ÚkkÞ.

(ii) f(x) =

«Ëuþ {u¤ððk {kxu,

\ 9 – x2 ≥ 0

\ 9 > x2

\ x2 < 9

\ < 3

\ |x| < 3

\ – 3 < x < 3

\ «Ëuþ : x ∈ [– 3, 3]

rðMíkkh {u¤ððk {kxu,

x ∈ [– 3, 3] ⇒ – 3 < x < 3

⇒ 0 < x2 < 9

⇒ – 9 < – x2 < 0

⇒ 0 < 9 – x2 < 9

⇒ 0 < < 3

⇒ 0 < f(x) < 3

\ f(x) ∈ [0, 3]

\ f(x) = Lkku rðMíkkh [0, 3] Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLkkt ðkMíkrðf rðÄuÞkuLkk «Ëuþ yLku rðMíkkh þkuÄku.

(i) f(x) = – |x|

Answer

ynª, f(x) = – |x|{kt xLke fkuE Ãký ðkMíkrðf ®f{ík {qfe þfkíke nkuðkÚke «Ëuþ R Úkþu.

íkÚkk |x|Lkku rðMíkkh [0, ∞) nkuðkÚke – |x|Lkku rðMíkkh
(– ∞, 0] ÚkkÞ.

(ii) f(x) =

«Ëuþ {u¤ððk {kxu,

\ 9 – x2 ≥ 0

\ 9 > x2

\ x2 < 9

\ < 3

\ |x| < 3

\ – 3 < x < 3

\ «Ëuþ : x ∈ [– 3, 3]

rðMíkkh {u¤ððk {kxu,

x ∈ [– 3, 3] ⇒ – 3 < x < 3

⇒ 0 < x2 < 9

⇒ – 9 < – x2 < 0

⇒ 0 < 9 – x2 < 9

⇒ 0 < < 3

⇒ 0 < f(x) < 3

\ f(x) ∈ [0, 3]

\ f(x) = Lkku rðMíkkh [0, 3] Au.

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#39 SUB 0M ✎ Edit ⚠ Log Error

Question

f(x) = 2x – 5Úke ÔÞkÏÞkrÞík rðÄuÞ {kxu Lke[uLke ®f{íkku þkuÄku.

(i) f(0)

Answer

2(0) – 5 = 0 – 5 = – 5

(ii) f(7)

2(7) – 5 = 14 – 5 = 9

(iii) f(– 3)

2(– 3) – 5 = – 6 – 5 = – 11

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = 2x – 5Úke ÔÞkÏÞkrÞík rðÄuÞ {kxu Lke[uLke ®f{íkku þkuÄku.

(i) f(0)

Answer

2(0) – 5 = 0 – 5 = – 5

(ii) f(7)

2(7) – 5 = 14 – 5 = 9

(iii) f(– 3)

2(– 3) – 5 = – 6 – 5 = – 11

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#40 SUB 0M 🖼 11 ✎ Edit ⚠ Log Error

Question

rðÄuÞ ‘t’ yu MkuÂÕMkÞMk{kt W»ýíkk{kLk yLku VuhLknex{kt W»ýíkk{kLk ðå[u YÃkktíkh fhíkwt Mkqºk, t(c) = + 32 îkhk ÔÞkÏÞkrÞík Au, íkku Lke[uLkkt {qÕÞku þkuÄku.

Answer

t(c) = + 32

(i) t(0)

(0) + 32 = 0 + 32 = 0

(ii) t(28)

(28) + 32 = + 32 = =

(iii) t(– 10)

(– 10) + 32 = –18 + 32 = 14

(iv) òu t(c) = 212 nkuÞ, íkku c þkuÄku.

\ 212 = c + 32

\ 212 – 32 = c

\ = c

\ c = 100

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

rðÄuÞ ‘t’ yu MkuÂÕMkÞMk{kt W»ýíkk{kLk yLku VuhLknex{kt W»ýíkk{kLk ðå[u YÃkktíkh fhíkwt Mkqºk, t(c) = + 32 îkhk ÔÞkÏÞkrÞík Au, íkku Lke[uLkkt {qÕÞku þkuÄku.

Answer

t(c) = + 32

(i) t(0)

(0) + 32 = 0 + 32 = 0

(ii) t(28)

(28) + 32 = + 32 = =

(iii) t(– 10)

(– 10) + 32 = –18 + 32 = 14

(iv) òu t(c) = 212 nkuÞ, íkku c þkuÄku.

\ 212 = c + 32

\ 212 – 32 = c

\ = c

\ c = 100

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#41 SUB 0M ✎ Edit ⚠ Log Error

Question

Lke[uLkk rðÄuÞkuLkk rðMíkkh þkuÄku.

(i) f(x) = 2 – 3x, x ∈ R, x > 0

Answer

heík 1 :

ynª, f(x) = 2 – 3x{kt x > 0Lke y÷øk y÷øk ®f{íkku {qfíkkt,

x	0.01	0.1	0.9	1	2	2.5	4	5	...ðøkuhu
f(x) =	1.97	1.7	– 0.7	– 1	– 4	– 5.5	– 10	– 13	...ðøkuhu

ynª, fku»xf ÃkhÚke òuE þfkÞ Au fu, ∀x > 0 {kxu,

f(x) < 2 \ rðMíkkh : (– ∞, 2) Úkþu.

heík 2 :

ynª, x > 0

\ 3x > 0

\ – 3x < 0

\ 2 – 3x < 0 + 2

\ 2 – 3x < 2

\ 2 – 3x ∈ (– ∞, 2)

\ f(x) ∈ (– ∞, 2)

(ii) f(x) = x2 + 2, x ∈ R

ynª, x2 ≥ 0

\ x2 + 2 ≥ 0 + 2

\ x2 + 2 ≥ 2

\ rðMíkkh : [2, ∞) Úkþu.

(iii) f(x) = x, x ∈ R

ynª, f(x) = x yu íkËuð rðÄuÞ nkuðkÚke íkuLkku rðMíkkh R ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

Lke[uLkk rðÄuÞkuLkk rðMíkkh þkuÄku.

(i) f(x) = 2 – 3x, x ∈ R, x > 0

Answer

heík 1 :

ynª, f(x) = 2 – 3x{kt x > 0Lke y÷øk y÷øk ®f{íkku {qfíkkt,

x	0.01	0.1	0.9	1	2	2.5	4	5	...ðøkuhu
f(x) =	1.97	1.7	– 0.7	– 1	– 4	– 5.5	– 10	– 13	...ðøkuhu

ynª, fku»xf ÃkhÚke òuE þfkÞ Au fu, ∀x > 0 {kxu,

f(x) < 2 \ rðMíkkh : (– ∞, 2) Úkþu.

heík 2 :

ynª, x > 0

\ 3x > 0

\ – 3x < 0

\ 2 – 3x < 0 + 2

\ 2 – 3x < 2

\ 2 – 3x ∈ (– ∞, 2)

\ f(x) ∈ (– ∞, 2)

(ii) f(x) = x2 + 2, x ∈ R

ynª, x2 ≥ 0

\ x2 + 2 ≥ 0 + 2

\ x2 + 2 ≥ 2

\ rðMíkkh : [2, ∞) Úkþu.

(iii) f(x) = x, x ∈ R

ynª, f(x) = x yu íkËuð rðÄuÞ nkuðkÚke íkuLkku rðMíkkh R ÚkkÞ.

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#42 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

ðkMíkrðf MktÏÞk øký R Ãkh ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ f : R → R, f(x) = x + 10, íkku rðÄuÞ fLkku yk÷u¾ Ëkuhku.

Answer

f(x) = x + 10, x ∈ R ykÃku÷ Au. f(0) = 0 + 10 = 10,
f(1) = 1 + 10 = 11, f(–5) = –5 + 10 = 5, f(5) = 5 + 10 = 15 ðøkuhu ... ÷uíkkt yk÷u¾ Lke[u {wsçk {¤u :

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

ðkMíkrðf MktÏÞk øký R Ãkh ÔÞkÏÞkrÞík ðkMíkrðf rðÄuÞ f : R → R, f(x) = x + 10, íkku rðÄuÞ fLkku yk÷u¾ Ëkuhku.

Answer

f(x) = x + 10, x ∈ R ykÃku÷ Au. f(0) = 0 + 10 = 10,
f(1) = 1 + 10 = 11, f(–5) = –5 + 10 = 5, f(5) = 5 + 10 = 15 ðøkuhu ... ÷uíkkt yk÷u¾ Lke[u {wsçk {¤u :

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#43 SUB 0M ✎ Edit ⚠ Log Error

Question

òu R yu QÚke Q ÃkhLkku

R = {(a, b) : a, b ∈ Q yLku a – b ∈ Z} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄ Au, íkku çkíkkðku fu,

(i) «íÞuf a ∈ Q {kxu (a, a) ∈ R

(ii) òu (a, b) ∈ R, íkku (b, a) ∈ R

(iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku (a, c) ∈ R.

Answer

(i) Äkhku fu, a ∈ Q ÷uíkkt, {kxu a – a = 0 ∈ Z. íkuÚke (a, a) ∈ R ÚkkÞ.

(ii) òu (a, b) ∈ R, íkku a – b ∈ Z. íkuÚke b – a ∈ Z. íkuÚke (b, a) ∈ R.

(iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku a – b ∈ Z.
b – c ∈ Z. íkuÚke a – c = (a – b) + (b – c) ∈ Z. íkuÚke (a, c) ∈ R ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu R yu QÚke Q ÃkhLkku

R = {(a, b) : a, b ∈ Q yLku a – b ∈ Z} ÚkkÞ íku heíku ÔÞkÏÞkrÞík MktçktÄ Au, íkku çkíkkðku fu,

(i) «íÞuf a ∈ Q {kxu (a, a) ∈ R

(ii) òu (a, b) ∈ R, íkku (b, a) ∈ R

(iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku (a, c) ∈ R.

Answer

(i) Äkhku fu, a ∈ Q ÷uíkkt, {kxu a – a = 0 ∈ Z. íkuÚke (a, a) ∈ R ÚkkÞ.

(ii) òu (a, b) ∈ R, íkku a – b ∈ Z. íkuÚke b – a ∈ Z. íkuÚke (b, a) ∈ R.

(iii) òu (a, b) ∈ R yLku (b, c) ∈ R, íkku a – b ∈ Z.
b – c ∈ Z. íkuÚke a – c = (a – b) + (b – c) ∈ Z. íkuÚke (a, c) ∈ R ÚkkÞ.

✓ Apply Cancel

#44 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

f = {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} ÚkkÞ íku heíku
Z Ãkh ÔÞkÏÞkrÞík Mkwhu¾ rðÄuÞ Au, íkku f(x) þkuÄku.

Answer

Äkhku fu, Mkwhu¾ rðÄuÞ f(x) = mx + c ÷uíkkt,

f(x) = mx + c íkÚkk f(1) = 1 f(2) = 3

\ f(1) = m(1) + c \ 3 = m(2) + c

\ 1 = m + c ...(1) \ 3 = 2m + c ...(2)

Ãkrhýk{ (1) yLku (2) ÃkhÚke,

m = 2Lku (1){kt {qfíkkt,

\ m + c = 1

\ c = 1 – 2

\ c = – 1

\ f(x) = mx + c ÃkhÚke,

f(x) = 2x – 1 ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f = {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} ÚkkÞ íku heíku
Z Ãkh ÔÞkÏÞkrÞík Mkwhu¾ rðÄuÞ Au, íkku f(x) þkuÄku.

Answer

Äkhku fu, Mkwhu¾ rðÄuÞ f(x) = mx + c ÷uíkkt,

f(x) = mx + c íkÚkk f(1) = 1 f(2) = 3

\ f(1) = m(1) + c \ 3 = m(2) + c

\ 1 = m + c ...(1) \ 3 = 2m + c ...(2)

Ãkrhýk{ (1) yLku (2) ÃkhÚke,

m = 2Lku (1){kt {qfíkkt,

\ m + c = 1

\ c = 1 – 2

\ c = – 1

\ f(x) = mx + c ÃkhÚke,

f(x) = 2x – 1 ÚkkÞ.

✓ Apply Cancel

#45 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

f(x) = nkuÞ, íkku rðÄuÞLkku «Ëuþ þkuÄku.

Answer

ykÃku÷ rðÄuÞ f yu Mkt{uÞ rðÄuÞ Au. {kxu fkuE Ãký Mkt{uÞ rðÄuÞLkku «Ëuþ = R – {x | AuË = 0},

ynª, x2 – 5x + 4 = 0 ÷uíkkt,

\ (x – 4) (x – 1) = 0

\ x = 4, x = 1

\ «Ëuþ : R – {1, 4}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = nkuÞ, íkku rðÄuÞLkku «Ëuþ þkuÄku.

Answer

ykÃku÷ rðÄuÞ f yu Mkt{uÞ rðÄuÞ Au. {kxu fkuE Ãký Mkt{uÞ rðÄuÞLkku «Ëuþ = R – {x | AuË = 0},

ynª, x2 – 5x + 4 = 0 ÷uíkkt,

\ (x – 4) (x – 1) = 0

\ x = 4, x = 1

\ «Ëuþ : R – {1, 4}

✓ Apply Cancel

#46 SUB 0M 🖼 2 ✎ Edit ⚠ Log Error

Question

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾

Ëkuhku.

Answer

x = 0, íkku f(0) = 1 Au.

x = – 1,

íkku f(– 1) = 1 – (– 1) = 2, x = 2,

íkku f(2) = 2 + 1 = 3, x = 3,

íkku f(3) = 3 + 1 = 4, x = – 2,

íkku f(– 2) = 1 – (– 2) = 3 ÚkkÞ.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku yk÷u¾

Ëkuhku.

Answer

x = 0, íkku f(0) = 1 Au.

x = – 1,

íkku f(– 1) = 1 – (– 1) = 2, x = 2,

íkku f(2) = 2 + 1 = 3, x = 3,

íkku f(3) = 3 + 1 = 4, x = – 2,

íkku f(– 2) = 1 – (– 2) = 3 ÚkkÞ.

✓ Apply Cancel

#47 SUB 0M 🖼 6 ✎ Edit ⚠ Log Error

Question

MktçktÄ f yu f(x) = Úke ÔÞkÏÞkrÞík Au

yLku MktçktÄ g yu g(x) = Úke ÔÞkÏÞkrÞík

Au, íkku Mkkrçkík fhku fu, f yu rðÄuÞ Au yLku g yu rðÄuÞ LkÚke.

Answer

ynª, f(x) = ÃkhÚke,

ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ({kºk yuf s) ½xf MkkÚku òuzkÞu÷ Au.

{kxu f yu rðÄuÞ Au.

g(x) =

ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷k nkuðkÚke g yu rðÄuÞ LkÚke.

x2 = (2)2 = 4 = g(2)

3(2) = 6 = g(2)

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

MktçktÄ f yu f(x) = Úke ÔÞkÏÞkrÞík Au

yLku MktçktÄ g yu g(x) = Úke ÔÞkÏÞkrÞík

Au, íkku Mkkrçkík fhku fu, f yu rðÄuÞ Au yLku g yu rðÄuÞ LkÚke.

Answer

ynª, f(x) = ÃkhÚke,

ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku Ëhuf ½xf Mkn«ËuþLkk yLkLÞ ({kºk yuf s) ½xf MkkÚku òuzkÞu÷ Au.

{kxu f yu rðÄuÞ Au.

g(x) =

ynª, ykf]rík ÃkhÚke òuE þfkÞ Au fu, «ËuþLkku ½xf 2 yu Mkn«ËuþLkk yuf fhíkkt ðÄw ½xf MkkÚku òuzkÞu÷k nkuðkÚke g yu rðÄuÞ LkÚke.

x2 = (2)2 = 4 = g(2)

3(2) = 6 = g(2)

✓ Apply Cancel

#48 SUB 0M 🖼 4 ✎ Edit ⚠ Log Error

Question

f(x) = x2, íkku þkuÄku.

Answer

f(1.1) = (1.1)2 = 1.21, f(1) = (1)2 = 1

\ =

=

=

= = 2.1

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = x2, íkku þkuÄku.

Answer

f(1.1) = (1.1)2 = 1.21, f(1) = (1)2 = 1

\ =

=

=

= = 2.1

✓ Apply Cancel

#49 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

rðÄuÞ f(x) = Lkku «Ëuþ þkuÄku.

Answer

f(x) =

«Ëuþ {u¤ððk {kxu,

AuË ≠ 0

\ x2 – 8x + 12 ≠ 0

\ (x – 6) (x – 2) ≠ 0

\ x – 6 ≠ 0 \ x – 2 ≠ 0

\ x ≠ 6 \ x ≠ 2

\ x ∈ R – {2, 6}

\ «Ëuþ = R – {2, 6}

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

rðÄuÞ f(x) = Lkku «Ëuþ þkuÄku.

Answer

f(x) =

«Ëuþ {u¤ððk {kxu,

AuË ≠ 0

\ x2 – 8x + 12 ≠ 0

\ (x – 6) (x – 2) ≠ 0

\ x – 6 ≠ 0 \ x – 2 ≠ 0

\ x ≠ 6 \ x ≠ 2

\ x ∈ R – {2, 6}

\ «Ëuþ = R – {2, 6}

✓ Apply Cancel

#50 SUB 0M 🖼 3 ✎ Edit ⚠ Log Error

Question

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

f(x) =

«Ëuþ {u¤ððk {kxu,

x – 1 > 0

\ x > 1

\ x ∈ [1, ∞)

\ «Ëuþ : [1, ∞)

rðMíkkh {u¤ððk {kxu,

x ∈ [1, ∞) ⇒ x > 1

⇒ x – 1 > 0

⇒ > 0

⇒ f(x) > 0

⇒ f(x) ∈ [0, ∞)

\ f Lkku rðMíkkh = [0, ∞)

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

f(x) =

«Ëuþ {u¤ððk {kxu,

x – 1 > 0

\ x > 1

\ x ∈ [1, ∞)

\ «Ëuþ : [1, ∞)

rðMíkkh {u¤ððk {kxu,

x ∈ [1, ∞) ⇒ x > 1

⇒ x – 1 > 0

⇒ > 0

⇒ f(x) > 0

⇒ f(x) ∈ [0, ∞)

\ f Lkku rðMíkkh = [0, ∞)

✓ Apply Cancel

#51 SUB 0M ✎ Edit ⚠ Log Error

Question

f(x) = |x – 1|Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

x ∈ R ⇒ x – 1 ∈ R

\ f(x) Lkku «Ëuþ R Au.

x – 1 ∈ R ⇒ |x – 1| > 0

⇒ f(x) > 0

⇒ f(x) ∈ [0, ∞)

\ f(x)Lkku rðMíkkh = [0, ∞) Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f(x) = |x – 1|Úke ÔÞkÏÞkrÞík rðÄuÞLkku «Ëuþ yLku rðMíkkh þkuÄku.

Answer

x ∈ R ⇒ x – 1 ∈ R

\ f(x) Lkku «Ëuþ R Au.

x – 1 ∈ R ⇒ |x – 1| > 0

⇒ f(x) > 0

⇒ f(x) ∈ [0, ∞)

\ f(x)Lkku rðMíkkh = [0, ∞) Au.

✓ Apply Cancel

#52 SUB 0M 🖼 5 ✎ Edit ⚠ Log Error

Question

òu f = yu RÚke RLkwt rðÄuÞ

nkuÞ, íkku fLkku rðMíkkh þkuÄku.

Answer

f =

ynª, x ∈ R ⇒ x2 > 0

⇒ 1 + x2 > 1

⇒ > 0

nðu, x2 < x2 + 1

\ < 1

\ 0 < < 1

\ 0 < f(x) < 1

\ f(x) ∈ [0, 1)

\ f(x)Lkku rðMíkkh [0, 1) Au.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu f = yu RÚke RLkwt rðÄuÞ

nkuÞ, íkku fLkku rðMíkkh þkuÄku.

Answer

f =

ynª, x ∈ R ⇒ x2 > 0

⇒ 1 + x2 > 1

⇒ > 0

nðu, x2 < x2 + 1

\ < 1

\ 0 < < 1

\ 0 < f(x) < 1

\ f(x) ∈ [0, 1)

\ f(x)Lkku rðMíkkh [0, 1) Au.

✓ Apply Cancel

#53 SUB 0M 🖼 5 ✎ Edit ⚠ Log Error

Question

f, g : R → R, f(x) = x + 1, g(x) = 2x – 3Úke ÔÞkÏÞkrÞík rðÄuÞ Au, íkku f + g, f – g, þkuÄku.

Answer

(f + g) (x) = f(x) + g(x) = x + 1 + 2x – 3 = 3x – 2

(f – g) (x) = f(x) – g(x) = x + 1 – (2x – 3)

= x + 1 – 2x + 3

= 4 – x

(x) = , g(x) ≠ 0, = , x ≠

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f, g : R → R, f(x) = x + 1, g(x) = 2x – 3Úke ÔÞkÏÞkrÞík rðÄuÞ Au, íkku f + g, f – g, þkuÄku.

Answer

(f + g) (x) = f(x) + g(x) = x + 1 + 2x – 3 = 3x – 2

(f – g) (x) = f(x) – g(x) = x + 1 – (2x – 3)

= x + 1 – 2x + 3

= 4 – x

(x) = , g(x) ≠ 0, = , x ≠

✓ Apply Cancel

#54 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

òu f : {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} yu f(x) = ax + bÚke ÔÞkÏÞkrÞík MktçktÄ nkuÞ, íkku a yLku b þkuÄku.

Answer

(1, 1) : f(1) = 1 ÃkhÚke,

f(x) = ax + b

\ f(1) = a(1) + b

\ 1 = a + b ...(1)

(2, 3) : f(2) = 3 ÃkhÚke,

f(x) = ax + b

\ f(2) = a(2) + b

\ 3 = 2a + b ...(2)

(1) yLku (2) ÃkhÚke,

a = 2Lku Mk{e. (1){kt {qfíkkt,

\ a + b = 1

\ 2 + b = 1

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

òu f : {(1, 1), (2, 3), (0, – 1), (– 1, – 3)} yu f(x) = ax + bÚke ÔÞkÏÞkrÞík MktçktÄ nkuÞ, íkku a yLku b þkuÄku.

Answer

(1, 1) : f(1) = 1 ÃkhÚke,

f(x) = ax + b

\ f(1) = a(1) + b

\ 1 = a + b ...(1)

(2, 3) : f(2) = 3 ÃkhÚke,

f(x) = ax + b

\ f(2) = a(2) + b

\ 3 = 2a + b ...(2)

(1) yLku (2) ÃkhÚke,

a = 2Lku Mk{e. (1){kt {qfíkkt,

\ a + b = 1

\ 2 + b = 1

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#55 SUB 0M ✎ Edit ⚠ Log Error

Question

R yu NÚke NLkku MktçktÄ Au. R : {(a, b) : a, b ∈ N yLku a = b2} ÚkkÞ íku heíku ÔÞkÏÞkrÞík Au, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ?

(i) ∀a ∈ N {kxu (a, a) ∈ R

(ii) òu (a, b) ∈ R, íkku (b, a) ∈ R

(iii) òu (a, b) ∈ R, (b, c) ∈ R, íkku (a, c) ∈ R

«íÞuf rðÄkLk{kt ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

ynª, R = {(a, b) : a, b ∈ N, a = b2} ykÃku÷ Au.

(i) Äkhku fu, 3 ∈ N {kxu, 3 = (3)2 = 9 su þõÞ LkÚke, {kxu ∀a ∈ N {kxu (a, a) ∈ R MkíÞ LkÚke.

(ii) a = 9, b = 3 ÷uíkkt, (9, 3) ∈ R, fkhý fu 32 = 9 Ãkhtíkw (3, 9) ∉ R, fkhý fu 92 ≠ 3

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

(iii) a = 16, b = 4 ÷uíkkt, 42 = 16 ÚkkÞ.

\ (16, 4) ∈ R ÚkkÞ.

b = 4, c = 2 ÷uíkkt, 22 = 4 ÚkkÞ.

\ (4, 2) ∈ R Ãkhtíkw

(a, c) = (16, 2), ∉ R, fkhý fu 22 ≠ 16

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

R yu NÚke NLkku MktçktÄ Au. R : {(a, b) : a, b ∈ N yLku a = b2} ÚkkÞ íku heíku ÔÞkÏÞkrÞík Au, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ?

(i) ∀a ∈ N {kxu (a, a) ∈ R

(ii) òu (a, b) ∈ R, íkku (b, a) ∈ R

(iii) òu (a, b) ∈ R, (b, c) ∈ R, íkku (a, c) ∈ R

«íÞuf rðÄkLk{kt ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

ynª, R = {(a, b) : a, b ∈ N, a = b2} ykÃku÷ Au.

(i) Äkhku fu, 3 ∈ N {kxu, 3 = (3)2 = 9 su þõÞ LkÚke, {kxu ∀a ∈ N {kxu (a, a) ∈ R MkíÞ LkÚke.

(ii) a = 9, b = 3 ÷uíkkt, (9, 3) ∈ R, fkhý fu 32 = 9 Ãkhtíkw (3, 9) ∉ R, fkhý fu 92 ≠ 3

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

(iii) a = 16, b = 4 ÷uíkkt, 42 = 16 ÚkkÞ.

\ (16, 4) ∈ R ÚkkÞ.

b = 4, c = 2 ÷uíkkt, 22 = 4 ÚkkÞ.

\ (4, 2) ∈ R Ãkhtíkw

(a, c) = (16, 2), ∉ R, fkhý fu 22 ≠ 16

\ ykÃku÷ rðÄkLk MkíÞ LkÚke.

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#56 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} yLku
f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ?

(i) f yu AÚke B ÃkhLkku MktçktÄ Au.

(ii) f yu AÚke B ÃkhLkwt rðÄuÞ Au. ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

ynª, f yu A × BLkku WÃkøký nkuðkÚke f yu MktçktÄ Au, {kxu rðÄkLk (i) MkíÞ Au.

íkÚkk ykf]rík 5hÚke òuE þfkÞ Au fu, «ËuþLkku ½xf ‘2’ yu Mkn«ËuþLkk çku ½xfku MkkÚku òuzkÞu÷ Au, {kxu f yu rðÄuÞ LkÚke. {kxu rðÄkLk (ii) MkíÞ LkÚke.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} yLku
f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}, íkku þwt Lke[uLkkt rðÄkLkku MkíÞ Au ?

(i) f yu AÚke B ÃkhLkku MktçktÄ Au.

(ii) f yu AÚke B ÃkhLkwt rðÄuÞ Au. ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

ynª, f yu A × BLkku WÃkøký nkuðkÚke f yu MktçktÄ Au, {kxu rðÄkLk (i) MkíÞ Au.

íkÚkk ykf]rík 5hÚke òuE þfkÞ Au fu, «ËuþLkku ½xf ‘2’ yu Mkn«ËuþLkk çku ½xfku MkkÚku òuzkÞu÷ Au, {kxu f yu rðÄuÞ LkÚke. {kxu rðÄkLk (ii) MkíÞ LkÚke.

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#57 SUB 0M ✎ Edit ⚠ Log Error

Question

f yu Z × ZLkku WÃkøký Au.

òu f = {ab, a + b) : a, b ∈ Z} Úke ÔÞkÏÞkrÞík Au, íkku þwt f yu ZÚke ZLkwt rðÄuÞ Au ? ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

f = {ab, a + b}, a, b ∈ Z

\ f (ab) = a + b

a = 1, b = 2 ÷uíkkt, f(1 × 2) = 1 + 2 \ f(2) = 3 {¤u.

a = –1, b = –2 ÷uíkkt, f(–1 × –2) = – 1 – 2

\ f(2) = – 3 {¤u.

yk{, (2, 3) yLku (2, – 3) çku ¢{Þwõík òuz {¤u.

yk{, 2 yu çku ½xfku MkkÚku òuzkÞu÷ Au.

\ f yu Z Úke Z Lkwt rðÄuÞ LkÚke.

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

f yu Z × ZLkku WÃkøký Au.

òu f = {ab, a + b) : a, b ∈ Z} Úke ÔÞkÏÞkrÞík Au, íkku þwt f yu ZÚke ZLkwt rðÄuÞ Au ? ík{khk sðkçkLke MkíÞkÚkoíkk [fkMkku.

Answer

f = {ab, a + b}, a, b ∈ Z

\ f (ab) = a + b

a = 1, b = 2 ÷uíkkt, f(1 × 2) = 1 + 2 \ f(2) = 3 {¤u.

a = –1, b = –2 ÷uíkkt, f(–1 × –2) = – 1 – 2

\ f(2) = – 3 {¤u.

yk{, (2, 3) yLku (2, – 3) çku ¢{Þwõík òuz {¤u.

yk{, 2 yu çku ½xfku MkkÚku òuzkÞu÷ Au.

\ f yu Z Úke Z Lkwt rðÄuÞ LkÚke.

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#58 SUB 0M 🖼 1 ✎ Edit ⚠ Log Error

Question

A = {9, 10, 11, 12, 13}, f : A → N, f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð Au. fLkku rðMíkkh þkuÄku.

Answer

f = A → N íkÚkk f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð

f(9) = 9Lkku {n¥k{ yrð¼kßÞ yðÞð = 3 íkuðe s heíku, f(10) = 5, f (9) = 3, f(11) = 11, f(12) = 3, f(13) = 13 {¤u.

\ fLkku rðMíkkh = {3, 5, 11, 13} ÚkkÞ.

2

«fhý Mkqr[

yøkkWLkk ð»ko{kt ÃkwAkÞu÷k JEE-Main «&LkkuLkwt «fhýðkh rð&÷u»ký

Type SUB — Subjective MCQ — Multiple Choice TF — True / False YN — Yes / No IA — Interactive Activity FB — Fill in the Blanks OW — One Word MT — Match the Following GLO — Glossary RCEMT — Reading Comp. – Extracts – MT SN — Short Notes CBTFQ — Content-Based T/F RCMTP — Reading Comp. – MT – Poem RCSR — Reading Comp. – Supp. Reader RCNV — Reading Comp. – Nonverbal RCUDI — Reading Comp. – Unseen D/I IFMC — Identify Function – Match Cols CTD — Complete the Dialogue CTS — Complete the Sentences VCPB — Vocab: Complete Para – Blanks VFNM — Vocab: Nearest Meaning – SR INDS — Indirect Speech CONJ — Conjunction EDTNG — Editing DASD — Do as Directed STCQ — Select the Correct PARAW — Paragraph Writing EFRW — Email / Report Writing PICD — Picture Description PARA — Paragraph CS — Case Study Marks

Question

A = {9, 10, 11, 12, 13}, f : A → N, f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð Au. fLkku rðMíkkh þkuÄku.

Answer

f = A → N íkÚkk f(n) = nLkku {n¥k{ yrð¼kßÞ yðÞð

f(9) = 9Lkku {n¥k{ yrð¼kßÞ yðÞð = 3 íkuðe s heíku, f(10) = 5, f (9) = 3, f(11) = 11, f(12) = 3, f(13) = 13 {¤u.

\ fLkku rðMíkkh = {3, 5, 11, 13} ÚkkÞ.

2

«fhý Mkqr[

yøkkWLkk ð»ko{kt ÃkwAkÞu÷k JEE-Main «&LkkuLkwt «fhýðkh rð&÷u»ký

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