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Page 1 CBSE XII  Mathematics Sample Paper 3 – Solution Mathematics Class XII Sample Paper – 3 Solution Time: 3 hour Total Marks: 100 SECTION A 1. If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function. Graph should have no vertical lines 2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector. 2 2 2 2 .(i+j+k) 3 3 1 1 3 3. 3 2 2 x0 x0 y x x 1 dy 3x 1 dx At a point where the curve cuts yaxis, x = 0 dy 3x 1 1 dx ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? Page 2 CBSE XII  Mathematics Sample Paper 3 – Solution Mathematics Class XII Sample Paper – 3 Solution Time: 3 hour Total Marks: 100 SECTION A 1. If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function. Graph should have no vertical lines 2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector. 2 2 2 2 .(i+j+k) 3 3 1 1 3 3. 3 2 2 x0 x0 y x x 1 dy 3x 1 dx At a point where the curve cuts yaxis, x = 0 dy 3x 1 1 dx ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper 3 – Solution 4. 64 women lost their jobs in factory III in the last two months. OR ? A(adjA)  AI ? ? ? ? ? ? 8 0 1 0  AI 8 8I 0 8 0 1 det(A)  A 8 SECTION B 5. ? ?? ? ?? ?? ?? ?? ?? ?? 1 1 sin sin 32 ?? ?? ?? ? ? ? ?? ?? ?? ?? sin 36 ?? ?? ?? ?? ?? ? ?? sin 36 sin 1 2 6. ? ? ? ?? ? ?? ? ?? 1 1 2x I log dx ...(1) 2x ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ? ?? 1 1 2x Let f(x) log and 2x 2 x 2 x f( x) log log f(x). 2 x 2 x f(x) is an odd function. 2x Thus, log dx 0 2x Page 3 CBSE XII  Mathematics Sample Paper 3 – Solution Mathematics Class XII Sample Paper – 3 Solution Time: 3 hour Total Marks: 100 SECTION A 1. If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function. Graph should have no vertical lines 2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector. 2 2 2 2 .(i+j+k) 3 3 1 1 3 3. 3 2 2 x0 x0 y x x 1 dy 3x 1 dx At a point where the curve cuts yaxis, x = 0 dy 3x 1 1 dx ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper 3 – Solution 4. 64 women lost their jobs in factory III in the last two months. OR ? A(adjA)  AI ? ? ? ? ? ? 8 0 1 0  AI 8 8I 0 8 0 1 det(A)  A 8 SECTION B 5. ? ?? ? ?? ?? ?? ?? ?? ?? 1 1 sin sin 32 ?? ?? ?? ? ? ? ?? ?? ?? ?? sin 36 ?? ?? ?? ?? ?? ? ?? sin 36 sin 1 2 6. ? ? ? ?? ? ?? ? ?? 1 1 2x I log dx ...(1) 2x ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ? ?? 1 1 2x Let f(x) log and 2x 2 x 2 x f( x) log log f(x). 2 x 2 x f(x) is an odd function. 2x Thus, log dx 0 2x CBSE XII  Mathematics Sample Paper 3 – Solution 7. ? ? ? ? Let p =2i 3j 4k and q = ai 6j 8k ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Two vectors p and q will be collinear if, i j k p q 0 2 3 4 0 a 6 8 i(24 24) j ( 16 4a) k(12 3a) 0 0i (16 4a)j (12 3a)k 0i 0j 0k 16 4a 0 and 12+3a = 0 a4 8. ? ?? ?? ?? ? ? ? ? ?? ?? ? ?? ? ? ?? ?? ? ? ?? ?? ? ? ? ? ? ? ?? 1 25 Given A = , then 13 A 6 5 0 35 and adjA = , obtained from A by interchanging the main diagonal 12 elements and multiplying by ( 1) the non  diagonal elements. 35 35 12 adjA A 12 A1 ? Page 4 CBSE XII  Mathematics Sample Paper 3 – Solution Mathematics Class XII Sample Paper – 3 Solution Time: 3 hour Total Marks: 100 SECTION A 1. If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function. Graph should have no vertical lines 2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector. 2 2 2 2 .(i+j+k) 3 3 1 1 3 3. 3 2 2 x0 x0 y x x 1 dy 3x 1 dx At a point where the curve cuts yaxis, x = 0 dy 3x 1 1 dx ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper 3 – Solution 4. 64 women lost their jobs in factory III in the last two months. OR ? A(adjA)  AI ? ? ? ? ? ? 8 0 1 0  AI 8 8I 0 8 0 1 det(A)  A 8 SECTION B 5. ? ?? ? ?? ?? ?? ?? ?? ?? 1 1 sin sin 32 ?? ?? ?? ? ? ? ?? ?? ?? ?? sin 36 ?? ?? ?? ?? ?? ? ?? sin 36 sin 1 2 6. ? ? ? ?? ? ?? ? ?? 1 1 2x I log dx ...(1) 2x ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ? ?? 1 1 2x Let f(x) log and 2x 2 x 2 x f( x) log log f(x). 2 x 2 x f(x) is an odd function. 2x Thus, log dx 0 2x CBSE XII  Mathematics Sample Paper 3 – Solution 7. ? ? ? ? Let p =2i 3j 4k and q = ai 6j 8k ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Two vectors p and q will be collinear if, i j k p q 0 2 3 4 0 a 6 8 i(24 24) j ( 16 4a) k(12 3a) 0 0i (16 4a)j (12 3a)k 0i 0j 0k 16 4a 0 and 12+3a = 0 a4 8. ? ?? ?? ?? ? ? ? ? ?? ?? ? ?? ? ? ?? ?? ? ? ?? ?? ? ? ? ? ? ? ?? 1 25 Given A = , then 13 A 6 5 0 35 and adjA = , obtained from A by interchanging the main diagonal 12 elements and multiplying by ( 1) the non  diagonal elements. 35 35 12 adjA A 12 A1 ? CBSE XII  Mathematics Sample Paper 3 – Solution OR ? Since A is skew symmetric matrix. ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? T T T3 T T Therefore,A A AA A ( 1) A AA A A ......(Since A A ) 2 A 0 A0 9. ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 1 1 1 Let sec x . Then,x sec and for x < 1, 2 Given expression = cot ( cot ), 2 cot (cot( )) sec as 0 < 2 10. ? ?? ? ?? ? ? ? ?? 1 2 11 22 5x y = tan 1 6x y = tan 3x tan 2x dy 3 2 dx 1 9x 1 4x Page 5 CBSE XII  Mathematics Sample Paper 3 – Solution Mathematics Class XII Sample Paper – 3 Solution Time: 3 hour Total Marks: 100 SECTION A 1. If a vertical line intersects the graph of a relation in two or more points, then the relation is not a function. Graph should have no vertical lines 2. Magnitude of ?(î + j + k )= 1, for it to be a unit vector. 2 2 2 2 .(i+j+k) 3 3 1 1 3 3. 3 2 2 x0 x0 y x x 1 dy 3x 1 dx At a point where the curve cuts yaxis, x = 0 dy 3x 1 1 dx ? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? CBSE XII  Mathematics Sample Paper 3 – Solution 4. 64 women lost their jobs in factory III in the last two months. OR ? A(adjA)  AI ? ? ? ? ? ? 8 0 1 0  AI 8 8I 0 8 0 1 det(A)  A 8 SECTION B 5. ? ?? ? ?? ?? ?? ?? ?? ?? 1 1 sin sin 32 ?? ?? ?? ? ? ? ?? ?? ?? ?? sin 36 ?? ?? ?? ?? ?? ? ?? sin 36 sin 1 2 6. ? ? ? ?? ? ?? ? ?? 1 1 2x I log dx ...(1) 2x ? ? ? ?? ? ?? ? ?? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ?? ? ?? 1 1 2x Let f(x) log and 2x 2 x 2 x f( x) log log f(x). 2 x 2 x f(x) is an odd function. 2x Thus, log dx 0 2x CBSE XII  Mathematics Sample Paper 3 – Solution 7. ? ? ? ? Let p =2i 3j 4k and q = ai 6j 8k ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Two vectors p and q will be collinear if, i j k p q 0 2 3 4 0 a 6 8 i(24 24) j ( 16 4a) k(12 3a) 0 0i (16 4a)j (12 3a)k 0i 0j 0k 16 4a 0 and 12+3a = 0 a4 8. ? ?? ?? ?? ? ? ? ? ?? ?? ? ?? ? ? ?? ?? ? ? ?? ?? ? ? ? ? ? ? ?? 1 25 Given A = , then 13 A 6 5 0 35 and adjA = , obtained from A by interchanging the main diagonal 12 elements and multiplying by ( 1) the non  diagonal elements. 35 35 12 adjA A 12 A1 ? CBSE XII  Mathematics Sample Paper 3 – Solution OR ? Since A is skew symmetric matrix. ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? T T T3 T T Therefore,A A AA A ( 1) A AA A A ......(Since A A ) 2 A 0 A0 9. ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 1 1 1 Let sec x . Then,x sec and for x < 1, 2 Given expression = cot ( cot ), 2 cot (cot( )) sec as 0 < 2 10. ? ?? ? ?? ? ? ? ?? 1 2 11 22 5x y = tan 1 6x y = tan 3x tan 2x dy 3 2 dx 1 9x 1 4x CBSE XII  Mathematics Sample Paper 3 – Solution OR Let x be the length of an edge of the cube, V be the volume and S be the surface area at any time t. ? ?? ? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? ? ? ?? ?? 32 3 3 2 2 2 2 2 x 10 Then,V x and S 6x . It is given that, dV 9 cm / sec dt d (x ) 9 dt dx 3x 9 dt dx 3 dt x Now, S = 6x dS dx 12x dt dt dS 3 12x dt x dS 36 dt x dS 36 3.6 cm / sec dt 10 11. ? ? ? ? ? ? ? 2 2 2 2 2 2 2 x (y b) a b or x y 2by a .......(1) ? ? ? ? ? ? ? ? ? ? 2 2 2 dy 2x 2y dy dy dx 2x 2y 2b 0 2b .......(2) dy dx dx dx Substituting in (1), dy (x y a ) 2xy 0 dxRead More
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