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JEE Mathematics Solved Question Paper 2012

Multiple Choice Questions

11.
The sum of two positive numbers is given. If the sum of their cubes is minimum, then

they are equal

one is twice the other

they are unequal

one is thrice the other

they are equal

$Let x and y be two positive numbers ∴ According to question x + y = a \Rightarrow y = a - x Let z = x^{3} + y^{3} z = x^{3} + {(a - x)}^{3} . . . (i) On dtlferentiatmg Eq . (i) w . r . t .' x', we get \frac{dz}{dx} = 3 x^{2} + 3 {(a - x)}^{2} (- 1) = 3 [x^{2} - {(a - x)}^{2}] = 3 (x + a - x) (x - a + x) = 3 a (2 x - a) . . . (ii) For maxunum or minimum, \frac{dz}{dx} = 0 \Rightarrow 3 a (2 x - a) = 0 [∵ 3 a \neq 0] \Rightarrow 2 x - a = 0 \Rightarrow x = \frac{a}{2} On differentiating Eq . (ii) w . r . t' x', we get \frac{d^{2} z}{{dx}^{2}} = 6 a ∴ \frac{d^{2} z}{{dx}^{2}} > 0 ∴ z has a minimum value at x = \frac{a}{2} Therefore, the numbers are \frac{a}{2} and a - \frac{a}{2} = \frac{a}{2} .$

12.

If $\frac{3 x + 1}{(x - 1) (x + 3)} = \frac{A}{x - 1} + \frac{B}{x + 3}$ , $\sin^{- 1} (\frac{A}{B})$ is equal to

$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{6}$
$\frac{π}{4}$

13.

The perimeter of a sectoris a constant. Ifits area is to be maximum, then the sectorial angle is

$\frac{π^{c}}{6}$
$\frac{π^{c}}{4}$
4^c
2^c

14.

The characteristic equation of a matrix A is $λ^{3} - 5 λ^{2} - 3 λ + 2 = 0$ , then $|adj (A)|$ is equal to

9
25
$\frac{1}{2}$
4

15.

The number of real solutions of the equation $\tan^{- 1} (\sqrt{x (x + 1)}) + \sin^{- 1} (\sqrt{x^{2} + x + 1}) = \frac{π}{2}$ is

one
four
two
infinitely many

16.

If y = $\tan^{- 1} (\frac{1}{1 + x + x^{2}}) + \tan^{- 1} (\frac{1}{x^{2} + 2 x + 3}) + \tan^{- 1} (\frac{1}{x^{2} + 5 x + 7}) + . . . n$ terms, then y'(0) is