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Page 1 CBSE XII  Mathematics Sample Paper – 10 Solution Mathematics Class XII Sample Paper – 10 Solution SECTION – A 1. By observation we find that 2 + x = 10 x = 8. 2. ? ? d cos x dx ? ? ? ? d sin x x dx dy sin x dx 2x ?? ? ? 3. DE: 32 32 d y d y dy y siny 0 dx dx dx ? ? ? ? It is linear, since y siny ? is product of two different functions, and their individual power is one. Page 2 CBSE XII  Mathematics Sample Paper – 10 Solution Mathematics Class XII Sample Paper – 10 Solution SECTION – A 1. By observation we find that 2 + x = 10 x = 8. 2. ? ? d cos x dx ? ? ? ? d sin x x dx dy sin x dx 2x ?? ? ? 3. DE: 32 32 d y d y dy y siny 0 dx dx dx ? ? ? ? It is linear, since y siny ? is product of two different functions, and their individual power is one. CBSE XII  Mathematics Sample Paper – 10 Solution 4. Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a2 a3 b1 b2 c3 c1 11 cos 14 ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? OR Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a a b b c c 2 1 7 2 3 4 0 cos 0 2 ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Page 3 CBSE XII  Mathematics Sample Paper – 10 Solution Mathematics Class XII Sample Paper – 10 Solution SECTION – A 1. By observation we find that 2 + x = 10 x = 8. 2. ? ? d cos x dx ? ? ? ? d sin x x dx dy sin x dx 2x ?? ? ? 3. DE: 32 32 d y d y dy y siny 0 dx dx dx ? ? ? ? It is linear, since y siny ? is product of two different functions, and their individual power is one. CBSE XII  Mathematics Sample Paper – 10 Solution 4. Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a2 a3 b1 b2 c3 c1 11 cos 14 ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? OR Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a a b b c c 2 1 7 2 3 4 0 cos 0 2 ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper – 10 Solution SECTION – B 5. Let X be the nonempty set for which P(X) is the power set. ARB ? A ? B i. ARA ? A ? A, every set is a subset of itself. R is reflexive ii. If A, B, C ? P(X) ARB ? A ? B, BRC ? B ? C A ? B and B ? C ?A ? C So ARC; Hence R is transitive. iii. ARB ? A ? B does not imply B ? A So B R A R is not symmetric R is reflexive, transitive but not symmetric ?R is not an equivalence relation 6. We have, 2A – 3B + 5C = O 2A = 3B – 5C 2 2 0 2 0 2 2A 3 5 3 1 4 7 1 6 6 6 0 10 0 10 2A 9 3 12 35 5 30 16 6 10 2A 26 2 18 ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? ? ?? ? ? ? ?? 16 6 10 1 A 2 26 2 18 ??? ? ?? ? ? ? ?? 8 3 5 A 13 1 9 ??? ? ?? ? ? ? ?? Page 4 CBSE XII  Mathematics Sample Paper – 10 Solution Mathematics Class XII Sample Paper – 10 Solution SECTION – A 1. By observation we find that 2 + x = 10 x = 8. 2. ? ? d cos x dx ? ? ? ? d sin x x dx dy sin x dx 2x ?? ? ? 3. DE: 32 32 d y d y dy y siny 0 dx dx dx ? ? ? ? It is linear, since y siny ? is product of two different functions, and their individual power is one. CBSE XII  Mathematics Sample Paper – 10 Solution 4. Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a2 a3 b1 b2 c3 c1 11 cos 14 ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? OR Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a a b b c c 2 1 7 2 3 4 0 cos 0 2 ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper – 10 Solution SECTION – B 5. Let X be the nonempty set for which P(X) is the power set. ARB ? A ? B i. ARA ? A ? A, every set is a subset of itself. R is reflexive ii. If A, B, C ? P(X) ARB ? A ? B, BRC ? B ? C A ? B and B ? C ?A ? C So ARC; Hence R is transitive. iii. ARB ? A ? B does not imply B ? A So B R A R is not symmetric R is reflexive, transitive but not symmetric ?R is not an equivalence relation 6. We have, 2A – 3B + 5C = O 2A = 3B – 5C 2 2 0 2 0 2 2A 3 5 3 1 4 7 1 6 6 6 0 10 0 10 2A 9 3 12 35 5 30 16 6 10 2A 26 2 18 ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? ? ?? ? ? ? ?? 16 6 10 1 A 2 26 2 18 ??? ? ?? ? ? ? ?? 8 3 5 A 13 1 9 ??? ? ?? ? ? ? ?? CBSE XII  Mathematics Sample Paper – 10 Solution 7. ? ? 22 sin x cos x dx sin xcosx ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ? 22 22 sin x cos x dx sin xcosx (cos x sin x) 2 dx 2sin xcosx cos2x 2 dx sin2x 2cos2x dx sin2x f '(x) log sin2x C .......( dx log f(x) c) f(x) 8. ? ?? 2 dx 5 8x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? 2 2 2 22 dx (x 8x 5) dx (x 8x 16 16 5) dx [(x 4) 21] dx ( 21) (x 4) 1 21 (x 4) log C 2 21 (x 4) 21 1 x 4 21 log C 2 21 x 4 21 Page 5 CBSE XII  Mathematics Sample Paper – 10 Solution Mathematics Class XII Sample Paper – 10 Solution SECTION – A 1. By observation we find that 2 + x = 10 x = 8. 2. ? ? d cos x dx ? ? ? ? d sin x x dx dy sin x dx 2x ?? ? ? 3. DE: 32 32 d y d y dy y siny 0 dx dx dx ? ? ? ? It is linear, since y siny ? is product of two different functions, and their individual power is one. CBSE XII  Mathematics Sample Paper – 10 Solution 4. Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a2 a3 b1 b2 c3 c1 11 cos 14 ?? ?? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? OR Let ? be the angles between, the given two lines So, the angle between them given their direction cosines is given by ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 1 2 1 2 2 2 2 2 2 2 1 1 1 2 2 2 1 2 1 2 1 2 1 a a b b c c cos a b c a b c substituting we get a a b b c c 2 1 7 2 3 4 0 cos 0 2 ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper – 10 Solution SECTION – B 5. Let X be the nonempty set for which P(X) is the power set. ARB ? A ? B i. ARA ? A ? A, every set is a subset of itself. R is reflexive ii. If A, B, C ? P(X) ARB ? A ? B, BRC ? B ? C A ? B and B ? C ?A ? C So ARC; Hence R is transitive. iii. ARB ? A ? B does not imply B ? A So B R A R is not symmetric R is reflexive, transitive but not symmetric ?R is not an equivalence relation 6. We have, 2A – 3B + 5C = O 2A = 3B – 5C 2 2 0 2 0 2 2A 3 5 3 1 4 7 1 6 6 6 0 10 0 10 2A 9 3 12 35 5 30 16 6 10 2A 26 2 18 ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ??? ? ?? ? ? ? ?? 16 6 10 1 A 2 26 2 18 ??? ? ?? ? ? ? ?? 8 3 5 A 13 1 9 ??? ? ?? ? ? ? ?? CBSE XII  Mathematics Sample Paper – 10 Solution 7. ? ? 22 sin x cos x dx sin xcosx ? ? ? ? ? ? ?? ? ?? ?? ? ? ? ? ? 22 22 sin x cos x dx sin xcosx (cos x sin x) 2 dx 2sin xcosx cos2x 2 dx sin2x 2cos2x dx sin2x f '(x) log sin2x C .......( dx log f(x) c) f(x) 8. ? ?? 2 dx 5 8x x ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? ?? ?? ?? ?? ?? 2 2 2 22 dx (x 8x 5) dx (x 8x 16 16 5) dx [(x 4) 21] dx ( 21) (x 4) 1 21 (x 4) log C 2 21 (x 4) 21 1 x 4 21 log C 2 21 x 4 21 CBSE XII  Mathematics Sample Paper – 10 Solution OR Let I = 2 2 4 2 2 1 1 x1 x dx dx 1 x1 x x ?? ? ? ? ? ? (Dividing numerator and denominator by x 2 ) I 2 2 1 1 x dx 1 x2 x ? ? ? ?? ?? ?? ?? Substituting x  1 x = t, 2 1 1 dx dt x ?? ? ? ? ?? ?? we get, ? ? 22 2 1 dt dt I t2 t2 1t tan 22 ? ?? ?? ? ? ? 1 2 1 1 x 1 x tan 22 1 x 1 tan c 2 2x ? ? ?? ? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ??Read More
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