May 2011 - Career Point

21/ x45. If lim [1 + xl n(1 + b )] = 2b sin 2 θ, b > 0 andAns.Sol.x→0θ∈(– π, π], then the value of θ is -(A)(C)π± (B)4π± (D)6π±3π±2[D]lim (1 + x ln(1 + b 2 )] 1/x = 2b sin 2 θ b > 0;x →0ln(1+b )1 ⎞22 x ln(1+b ) ⎟)]x⎟⎛⎜lim [1 x n (1 b→0⎜ + l +⎝2ln(1+b )e = 2b sin 2 θ1 + b 2 = 2b sin 2 θ2 sin 2 1θ = b + b⎟⎠RHS = b + b1 ≥ 2 as b > 0But LHS = 2 sin 2 θ ≤ 2Only possibility2 sin 2 θ = 2sin 2 θ = 1θ = ± 2π2θ ∈ (–π, π)= 2b sin 2 θ46. The circle passing through the point (–1, 0) andtouching the y-axis at (0, 2) also passes throughthe point -⎛ 3 ⎞(A) ⎜ − , 0 ⎟⎝ 2 ⎠⎞(C) ⎜⎛ 3 5 − , ⎟⎝ 2 2 ⎠⎛ 5 ⎞(B) ⎜ − , 2 ⎟⎝ 2 ⎠(D) (– 4, 0)Ans. [D]Sol. ∴ (h – 0) 2 + (2 –2) 2 = (h + 1) 2 + (2 – 0) 2h 2 = h 2 + 1 + 2h + 4(h, 2)(–1, 0)5h = −2Equation of circle is(0, 2)2⎛ 5 ⎞⎜ x + ⎟⎠ + (y –2) 2 ⎛ 5 ⎞= ⎜ – − 0⎟ ⎝ 2⎝ 2 ⎠x 2 +25 + 5x + y 2 + 4 – 4y =42254x 2 + y 2 + 5x – 4y + 4 = 0from given points only point (– 4, 0) satisfies thisequation.47. Let ω ≠ 1 be a cube root of unity and S be the setof all non-singular matrices of the form⎡ 1 a b⎤⎢ ⎥⎢ω 1 c⎥, where each of a, b and c is either⎢2⎣ωω 1⎥⎦ω or ω 2 . Then the number of distinct matrices inthe set S is :(A) 2 (B) 6(C) 4 (D) 8Ans. [A]Sol.1ωω2a1ωbc1≠ 0(1 – ωc) – a (ω – ω 2 c) + b(ω 2 – ω 2 ) ≠ 01 – ωc – aω + acω 2 ≠ 0(1 – ωc) – aω (1 – ωc) ≠ 0(1 – ωc) (1 – aω) ≠ 0c ≠ ω 2 & a ≠ ω 2 & b = ω or ω 2(a, b, c) ≡ (ω, ω, ω) or (ω, ω 2 , ω)48. A value of b for which the equationsx 2 + bx – 1 = 0, x 2 + x + b = 0have one root in common is -(A) − 2(B) − i 3(C) i 5(D) 2Ans. [B]Sol. x 2 + bx –1 = 0 … (i)x 2 + x + b = 0… (ii)b + 1(i) – (ii) we get x =b – 1Put this value in (i)2⎛ b + 1⎞⎛ b + 1⎞⎜ ⎟⎠ + b ⎜ ⎟ –1 = 0⎝ b –1 ⎝ b –1 ⎠⇒ b 3 + 3b = 0⇒ b(b 2 + 3) = 0⇒ b = 0 or b = ± i 3XtraEdge for IIT-JEE 92 MAY 2011