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JEE Mathematics Solved Question Paper 2012

Multiple Choice Questions

61.

Let y = $(\frac{3^{x} - 1}{3^{x} + 1}) \sin (x) + \log_{e} (1 + x), x > - 1$ . Then, at x = 0, $\frac{dy}{dx}$ equals

62.

Maximum value of the function f(x) = $\frac{x}{8} + \frac{2}{x}$ on the interval [1, 6] is

1
$\frac{9}{8}$
$\frac{13}{12}$
$\frac{17}{8}$

63.

For $- \frac{π}{2} < x < \frac{3 π}{2}$ , the avlue of $\frac{d}{dx} (\tan^{- 1} (\frac{\cos (x)}{1 + \sin (x)}))$ is equal to

$\frac{1}{2}$
- $\frac{1}{2}$
1
$\frac{\sin (x)}{{(1 + \sin (x))}^{2}}$

64.

If f is a real-valued differentiable function such that f(x)f' (x) < 0 for all real x, then

f(x) must be an increasing function
f(x) must be a decreasing function
$|f (x)|$ must be an increasing function
$|f (x)|$ must be a decreasing function

65.

Rolle's theorem is applicable in the interval [- 2, 2] for the function

f(x) = x³
f(x) = 4x⁴
f(x) = 2x³ + 3
f(x) = $π |x|$

66.
The solution of $25 \frac{d^{2} y}{d x^{2}} - 10 \frac{d y}{d x} + y = 0$ , y(0) = 1, y(1) = $2 e^{\frac{1}{5}}$ is

$y = e^{5 x} + e^{- 5 x}$

$y = (1 + x) e^{5 x}$

$y = (1 + x) e^{\frac{x}{5}}$

$y = (1 + x) e^{\frac{- x}{5}}$

$y = (1 + x) e^{\frac{x}{5}}$

Let y = e^mx be the solution of given differential equation,

$\Rightarrow \frac{d y}{d x} = {me}^{mx} \Rightarrow \frac{d^{2} y}{{dx}^{2}} = m^{2} e^{mx} ∴ 25 \frac{d^{2} y}{{dx}^{2}} - 10 \frac{d y}{d x} + y = 0 25 m^{2} e^{mx} - 10 {me}^{mx} + e^{mx} = 0 \Rightarrow e^{mx} (25 m^{2} - 10 m + 1) = 0 \Rightarrow Auxiliary equation \Rightarrow 25 m^{2} - 10 m + 1 = 0 e^{mx} \neq 0 \Rightarrow {(5 m)}^{2} - 2 (5 m) \times 1 + 1 = 0 \Rightarrow {(5 m - 1)}^{2} = 0 \Rightarrow m = \frac{1}{5}, \frac{1}{5}$

Since, roots are real and euqal.

$∴ General solution is y = (c_{1} + c_{2} x) e^{\frac{x}{5}} . . . (i) y (0) = 1 \Rightarrow c_{1} = 1 y (1) = 2 e^{\frac{1}{5}} \Rightarrow 2 e^{\frac{e}{5}} = (c_{1} + c_{2}) e^{\frac{1}{5}} \Rightarrow c_{1} + c_{2} = 2 \Rightarrow c_{1} = 1$

Putting the value of c₁ and c₂ in Eq. (i), we get particular solution

y = ${(1 + x)}^{\frac{x}{5}}$

67.

The system of linear equations $λ$ x + y + z = 3, x - y - 2z = 6, - x + y + z = µ has

infinite number of solutions for $λ \neq$ -1 and all µ
infinite number of solutions for $λ$ = - 1 and $μ$ = 3
no solution for $λ \neq - 1$
unique solution for $λ$ = - 1 and $μ$ = 3

68.

If f(x) and g(x) are twice differentiable functions on (0, 3) satisfying f"(x) = g''(c), f'(1) = 4g'(D) = 6, f(2) = 3, g(2) = 9, then f(1) - g(1) is

69.

Two coins are available, one fair and the other two headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probability $\frac{3}{4}$ . Given that the outcome is head, 4 the probability that the two headed coin was chosen, is