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JEE Mathematics Solved Question Paper 2016

Multiple Choice Questions

51.

Suppose f(x) = x(x + 3)(x - 2), x $\in$ [- 1, 4]. Then, a value of c in (- 1, 4) satisfying f'(c) = 10 is

2
$\frac{5}{2}$
3
$\frac{7}{2}$

52.
The area included between the parabola $y = \frac{x^{2}}{4 a} and the curve y = \frac{8 a^{3}}{(x^{2} + 4 a^{2})} is$

$a^{2} (2 π + \frac{2}{3})$

$a^{2} (2 π - \frac{8}{3})$

$a^{2} (π + \frac{4}{3})$

$a^{2} (2 π - \frac{4}{3})$

$a^{2} (2 π - \frac{4}{3})$

$For point of intersection, equate both equations y = \frac{x^{2}}{4 a} and the curve y = \frac{8 a^{3}}{(x^{2} + 4 a^{2})} \Rightarrow x^{2} (x^{2} + 4 a^{2}) = 4 a (8 a^{3}) \Rightarrow x^{4} + 4 a^{2} x^{2} = 32 a^{4} \Rightarrow x^{4} + 4 a^{2} x^{2} - 32 a^{4} = 0 \Rightarrow x^{2} (x^{2} + 8 a^{2}) - 4 a^{2} (x^{2} + 8 a^{2}) = 0 \Rightarrow (x^{2} + 8 a^{2}) (x^{2} - 4 a^{2}) = 0 \Rightarrow x^{2} - 4 a^{2} = 0 or x^{2} + 8 a^{2} = 0 \Rightarrow x^{2} = 4 a^{2} or x^{2} = - 8 a^{2} ∴ x^{2} = 4 a^{2}$

$\Rightarrow x = \pm 2 a \Rightarrow x = 2 a Hence, we take limits from 0 to 2 a . ∴ Area between 2 graphs = 2 \times \int_{0}^{2 a} [\frac{8 a^{3}}{(x^{2} + 4 a^{2})} - \frac{x^{2}}{4 a}] dx = 2 \times [\int_{0}^{2 a} \frac{8 a^{3}}{(x^{2} + 4 a^{2})} dx - \int_{0}^{2 a} \frac{x^{2}}{4 a} dx] = 2 \times [8 a^{3} \times \int_{0}^{2 a} \frac{1}{(x^{2} + 4 a^{2})} dx - \frac{1}{4 a} \int_{0}^{2 a} x^{2} dx] = 2 \times [8 a^{3} {[\frac{1}{2 a} \tan^{- 1} (\frac{x}{2 a})]}_{0}^{2 a} - \frac{1}{4 a} {[\frac{x^{3}}{3}]}_{0}^{2 a}] = 2 \times \{\frac{8 a^{3}}{2 a} \tan^{- 1} (\frac{2 a}{2 a}) - \tan^{- 1} (\frac{0}{2 a}) - \frac{1}{4 a} [\frac{{(2 a)}^{3}}{3} - \frac{{(0)}^{3}}{3}]\}$

$= 2 \times \{4 a^{2} \tan^{- 1} (1) - \tan^{- 1} (0) - \frac{1}{4 a} (\frac{8 a^{3}}{3} - 0)\} = 2 \times \{4 a^{2} (\frac{π}{4} - 0) - \frac{1}{4 a} (\frac{8 a^{3}}{3})\} = 2 \times \{4 a^{2} \times \frac{π}{4} - \frac{1}{4 a} \times \frac{8 a^{3}}{3}\} = 2 \times \{a^{2} π - \frac{2 a^{2}}{3}\} = 2 \{a^{2} (π - \frac{2}{3})\} = 2 a^{2} (π - \frac{2}{3}) = a^{2} (2 π - \frac{4}{3})$

53.

If a, b, c are distinct positive real numbers, then the value of the determinant $|\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| is$