Solution - Career Point

10. The foci of a hyperbola coincide with the foci of thex 2ellipse25y 2+ 9= 1. If eccentricity of thehyperbola is 2, then its equation is(A) x 2 – 3y 2 – 12 = 0 (B) 3x 2 – y 2 – 12 = 0(C) x 2 – y 2 – 4 = 0 (D) None of these11.→α and→β are two mutually perpendicular unit vectora → α + a → β + c( → α × → β ), → α + ( → α × → β ) and c → α + c → β+ b ( → α × → β ) are coplaner then c is(A) A.M. of a & b (B) G.M. of a & b(C) H.M. of a & b (D) None of these12. The point of contact of the spheresx 2 + y 2 + z 2 + 2x – 4y – 4z – 7 = 0x 2 + y 2 + z 2 + 2x – 4y – 16z + 65 = 0(A) (1, 2, 6) (B) (1, 2, –6)(C) (1, –2, 6) (D) (–1, 2, 6)13. If f(x) = 3 – 4{x 2 – 4x + 8} –1 then range of f(x) is(A) (–∞, 1) ∪ (3, ∞) (B) (2, 3)(C) [2, 3](D) None of these14. If x > 0 and g is a bounded function thennxf (x)e + g(x)limisn →∞ nxe + 1(A) 0 (B) f(x) (C) g(x) (D) None15. If a 1 = 1 and a n = n(1 + a n–1 ) ∀ n ≥ 2 than⎛ 1 ⎞⎛ 1 ⎞ ⎛ 1 ⎞lim ⎜1+⎟⎜1+⎟...⎜1+⎟ =n→∞⎝a1⎠⎝a 2 ⎠ ⎝ a n ⎠(A) 1 (B) e (C) 1/e (D) None16. Let f(x) = |2 sgn 2x| + 2 then f(x) has(A) removable discontinuity(B) infinte discontinuity(C) No discontinuity(D) essential discontinuity17. If f(x) = cosπ 3⎧⎨ [x] − x⎩2⎫⎬ , 1 < x < 2 and [.] = G.I.F.⎭⎛ ⎞then f´ ⎜π3 ⎟ is⎝ 2 ⎠(A) 0 (B) 3(π/2) 2/3(C) –3(π/2) 2/3 (D) None of these18. If ye x = cos x then, y 4 / y =(A) –1 (B) 2 (C) –4 (D) None19. Let f & g be differentiable function satisfying g´(a) = 2,g(a) = b and fog = I (Identity function), then f´(b) isequal to(A) 1/2 (B) 2 (C) 2/3 (D) None20. Tangents are drawn from origin to the curve y = sin xpoints of contact lie on the curve(A) x 2 + y 2 = x 2 y 2 (B) x 2 – y 2 = xy(C) x 2 – y 2 = x 2 y 2 (D) None of these21. Two positive numbers whose sum is 16 and sum ofwhose cubes is maximum are given by(A) 8, 8(B) no such number exist(C) 0, 16(D) None of these1 122. Let f(x) =2 , g(x) = on [a, b], 0 < a < b. Letx xf (b) − f (a) f´(c)= for same a < c < b then c isg(b) − g(a) g´(c)(A) A.M. of a & b(C) H.M. of a & b2(B) G. M. of a & b(D) None of these23.∫1+ cos x . sin 2x cos 2x dx =(A) 52 (1 + cos 2 x) 3/2 (3 – 2cos 2 x) 2 + c(B) 52 (1 + cos 2 x) 3/2 (3 – 2 cos 2 x) + c(C) 52 (1 + cos 2 x) 3/2 (3 + 2 cos 2 x) + c(D) None of these⎛ 2 424.∫ ⎟ ⎟ ⎞⎜ x x1− + .... dx⎜ 2 4⎝⎠(A) sin x (B) – sin x (C) cos x (D) None25.26.πx→2∫∫xπ / 22x2π / 4−cos t(2 −1)dtlim =(A)(log e 2(B)πt − π / 2)dtln22π2nlim 1 ⎞2/⎜1 ⎟n→∞2 ⎠(C)2ln2π(D) None224 / nn⎛ ⎛22 ⎞ ⎛2+ ⎜1⎟3 ⎞6/+ ⎜12 + ⎟⎝ nn2⎝ ⎠ n⎝ ⎠22n / n⎛2n ⎞.... ⎜1+ ⎟2n⎝ ⎠=(A) 4/e (B) 3/e (C) 2/e (D) NoneXtraEdge for IIT-JEE 83APRIL 2010