May 2011 - Career Point

SECTION – IIMultiple correct Choice TypeThis section contains 4 multiple choice questions. Eachquestion has four choices (A), (B), (C) and (D), out ofwhich ONE OR MORE may be correct.⎧ π π⎪−x − , x ≤ −2 2⎪π49. If f ( x)= ⎨ − cos x,− < x ≤ 0 , then⎪2⎪x −10 < x ≤1⎪⎩ln x,x > 1π(A) f(x) is continuous at x = −2(B) f(x) is not differentiable at x = 0(C) f(x) is differentiable at x = 13(D) f(x) is differentiable at x = −2Ans. [A, B, C, D]πSol. At x = −2⎛ π ⎞LHL = 0, RHL = 0, f ⎜− ⎟ = 0, So f(x) is⎝ 2 ⎠πcontinuous at x = −2At x = 0LHD = 0; RHD = 1So f(x) is not differentiable at x = 0At x = 1LHD = 1, RHD = 1So f(x) is differentiable at x = 1⎛ π ⎤in ⎜− , 0⎥ ; f(x) = – cos x⎝ 2 ⎦so f(x) is differentiable at x =3−250. Let L be a normal to the parabola y 2 = 4x. If Lpasses through the point (9, 6), then L is given by(A) y – x + 3 = 0 (B) y + 3x – 33 = 0(C) y + x – 15 = 0 (D) y – 2x + 12 = 0Ans. [A, B, D]Sol. y = mx – 2m – m 3It passes through (9, 6)6 = 9m –2m – m 3m 3 – 7m + 6 = 0(m –1) (m –2) (m + 3) = 0∴ m = –3, 1, 2Hence equations will bey = x – 3, y = 2x –12 and y = –3x + 3351. Let E and F be two independent events. Theprobability that exactly one of them occurs is11/25 and the probability of none of themoccurring is 2/25. If P(T) denotes the probabilityof occurrence of the event T, then -4 3(A) P ( E)= , P(F)=5 51 2(B) P ( E)= , P(F)=5 52 1(C) P ( E)= , P(F)=5 53 4(D) P ( E)= , P(F)=5 5Ans. [A, D]11Sol. P(E) (1 – P(F)) + (1 – P(E)) P(F) = 2511P(E) + P(F) –2P (E) P(F) = 252(1 – P(E)) (1 – P(F)) = 2521 – P(E) – P(F) + P(E) P(F) = 2523P(E) + P(F) – P(E) P(F) = 25From (1) & (2)12P(E) P(F) = 257and P(E) + P(F) = 5so eitherP(E) = 54 , P(F) = 53 and P(E) = 53 , P(F) = 54… (1)... (2)b − x52. Let f : (0, 1) → R be defined by f ( x)= , 1 − bxwhere b is a constant such that 0 < b < 1. Then(A) f is not invertible on (0, 1)(B) f ≠ f –1 1on (0, 1) & f '(b) =f '(0)(C) f = f –1 1on (0, 1) and f '(b) =f '(0)(D) f –1 is differentiable on (0, 1)Ans. [A, B]Sol. f : (0, 1) → Rb − xf(x) = ∀ b ∈ (0, 1)1−bx2b −1f ′(x) = = (–) ve2(1 − bx)XtraEdge for IIT-JEE 93 MAY 2011