Science-Class-12 Maths DPP Talent Search Examinations

Mathematics-XII MATHEMATICS 1. The value of lim sin4 x  cos2 x 4 2 is x cos x  sin x (a) 0 (b) 1 (c) ½ (d) not defined 2. Let f(x) = sinx + x. Then f–1(x) is (a) many-one and even (b) one-one and even (c) one one and odd (d) many-one and odd 3. The locus of point r satisfying | r – a | = | r – b | = | r – c |, a, b, c are collinear points, is a (a) straight line (b) circle (c) plane (d) null set dy d 2 y 4. For the growth of a function, which of the factors y, , dx dx2 is most important? (a) y (b) dy dx (c) d 2 y dx2 (d) none of these 5. a, b, c and d are complex numbers, then the value of 2 a  b  c  d ac  bd a  b  c  d 2(a  c)(b  d ) ac(b  d )  bd (a  c) is ac  bd ac(b  d )  bd (a  c) 2abcd (a) 0 (b) a + b + c + d (c) abcd 6. If A, B, C, P, Q, R  R, then the value of cos( A  P) cos( A  Q) cos( A  R) (d) ac + bd cos(B  P) cos(B  Q) cos(B  R) is cos(C  P) cos(C  Q) cos(C  R) (a) cos A cos P + cos B cos Q + cos C cos R (b) cos (A – P) cos (B – Q) cos(C– R) (c) cos (A + B + C) – cos (P – Q – R) (d) none of the above - : Rough Space : - [2] Mathematics-XII 7. lim n (n!)1/ n n equals (a) e (b) e–1 (c) 1 (d) none of these 8. lim sin x  (sin x)sin x is equal to x / 2 1  sin x  ln sin x (a) 4 (b) 2 (c) 1 (d) none of these 9. A man of height 2m walks directly away from a lamp of height 5m, on a level road at 3m/s. The rate at which the length of his shadow is increasing is (a) 1 m/s (b) 2 m/s (c) 3 m/s (d) 4 m/s 10. By LMVT, which of the following is true for x > 1 (a) 1 + x ln x < x < 1 + ln x (b) 1 + ln x < x < 1 + x ln x (c) x < 1 + x ln x < 1 + ln x (d) 1 + ln x < 1 + x ln x < x 11. (x2 1)  x3 (2x4  2x2  1) dx is equal to (a) (2x4  2x2  1) 2  c (b) (2x4  2x2  1) c x(x 1) x3 (c) (2x4  2x2  1) x2 c (d) (2x4  2x2  1) 2x2 c 12. (2x12  5x9 )  (x5  x3  1)3 dx is equal to (a) x2  2x (x5  x3 1)2 c (b) x10 2(x5  x3  1)2 c (c) ln | x5 + x3 + 1| + (2x7  5x4 )  c (d) none of the above - : Rough Space : - [3] Mathematics-XII 13. Solution of the differential equation dy  y(x  y ln y) is dx x(x ln x  y) (a) x ln x  y ln y  c xy (b) x ln x  y ln y  c xy (c) l n x  ln y  c x y (d) l n x  ln y  c x y 14. If y = f(x) passing through (1, 2) satisfies the differential equation y(1 + xy) dx – x dy = 0, then (a) f (x)  2x 2  x2 (b) f (x)  x  1 x2  1 (c) f (x)  x 1 4  x2 (d) f (x)  4x 1 2x2 15. If the non-zero vectors a and b are perpendicular to each other, then the solution of the equation r  a  b → → 1 → → → → 1 → → → → (a) r  xa  → → (a  b ) (b) a.a r  xb  → → (a  b ) (c) b.b r  x (a  b ) (d) none of the above 16. Let → a, b c be mutually perpendicular vectors of the same magnitude. If a vector x satisfies the equation →  →  b )  →  b  →  →  b ]  →  →  →  →  0 , then x is given by a [(x 1 → → a] [(x c ) → 1 → c [(x a) c] → → 1 → → → 1 → → → (a) (a  b  c ) 2 (b) (2a  b  c ) 3 (c) (a  b  c ) 3 (d) (a  b  2c) 2 17. The locus of point r satisfying | r – a | = | r – b | = | r – c | = | r – d |, no three of a, b, c, d are collinear points, a, b, c, d are coplanar but non-concyclic points, is a (a) straight line (b) circle (c) plane (d) null set 18. The locus of point r satisfying | r – a | = | r – b | = | r – c | = | r – d |, no three of a, b, c, d are collinear points, a, b, c, d are non-coplanar points, is a (a) straight line (b) circle (c) plane (d) a point 19. For any vector A , the value of iˆ ( A  iˆ)  ˆj  ( A  ˆj)  kˆ  ( A  kˆ) is equal to (a) 0 (b) 2 A (c) 2 A (d) none of these 20. Top 8 players P1, P2, P3, ...., P8 play tennis tournament. Let pi denote the probability of Pi reaching 8 final. Then  pi is i 1 (a) 1 (b) 1½ (c) 2 (d) 4 🙤🙦🙥🙧 [4]