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JEE Mathematics Solved Question Paper 2007

Multiple Choice Questions

The equation of the tangent to the curve $y = {(1 + x)}^{y} + \sin^{- 1} (\sin^{2} (x))$

x - y + 1 = 0
x + y + 1 = 0
2x - y + 1 = 0
x + 2y + 2 = 0

If $[\begin{matrix} 2 & 1 \\ 3 & 2 \end{matrix}] A [\begin{matrix} - 3 & 2 \\ 5 & - 3 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , then A is equal to :

$- [\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}]$
$[\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}]$
$[\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix}]$

The point on the curve y = 2x² - 6x - 4 at which the tangent is parallel to the x-axis, is :

$(\frac{3}{2}, \frac{13}{2})$
$(- \frac{5}{2}, - \frac{17}{2})$
$(\frac{3}{2}, \frac{17}{2})$
$(\frac{3}{2}, - \frac{17}{2})$

4.
The value of $|\begin{matrix} \cos (x - a) & \cos (x + a) & \cos (x) \\ \sin (x + a) & \sin (x - a) & \sin (x) \\ \cos (a) \tan (x) & \cos (a) \cot (x) & \csc (2 x) \end{matrix}|$ is equal to :

1

$\sin (a) \cos (a)$

0

$\sin (x) \cos (x)$

$|\begin{matrix} \cos (x - a) & \cos (x + a) & \cos (x) \\ \sin (x + a) & \sin (x - a) & \sin (x) \\ \cos (a) \tan (x) & \cos (a) \cot (x) & \csc (2 x) \end{matrix}| Applying C_{1} + C_{2}, = |\begin{matrix} \cos (x - a) + \cos (x + a) & \cos (x + a) & \cos (x) \\ \sin (x + a) + \sin (x - a) & \sin (x - a) & \sin (x) \\ \cos (a) (\tan (x) + \cot (x)) & \cos (a) \cot (x) & \csc (2 x) \end{matrix}| = |\begin{matrix} 2 \cos (x) \cos (a) & \cos (x + a) & \cos (x) \\ 2 \sin (x) \cos (a) & \sin (x - a) & \sin (x) \\ \cos (a) (\frac{\tan^{2} (x) + 1}{\tan (x)}) & \cos (a) \cot (x) & \csc (2 x) \end{matrix}| = 2 \cos (a) |\begin{matrix} \cos (x) & \cos (x + a) & \cos (x) \\ \sin (x) & \sin (x - a) & \sin (x) \\ \frac{\tan^{2} (x) + 1}{2 \tan (x)} & \cos (a) \cot (x) & \csc (2 x) \end{matrix}| = 2 \cos (a) |\begin{matrix} \cos (x) & \cos (x + a) & \cos (x) \\ \sin (x) & \sin (x - a) & \sin (x) \\ \csc (2 x) & \cos (a) \cot (x) & \csc (2 x) \end{matrix}| = 2 \cos (a) \times 0 = 0$

Let f be twice differentiable function such that f"(x) = - f(x) and f'(x) = g(x), h(x) = {f(x)}² + {g(x)}². If h(5) = 11, then h(10) is equal to :

Suppose A is a matrix of order 3 and $B = |A| A^{- 1}$ . If $|A| = - 5$ , then $|B|$ is equal to :

A differentiable function f(x) is defined for all x > 0 and satisfies f(x³) = 4x⁴ for all x > 0. The value of f'(8) is :

$\frac{16}{3}$
$\frac{32}{3}$
$\frac{16 \sqrt{2}}{3}$
$\frac{32 \sqrt{2}}{3}$

The value of $\cos [2 \tan^{- 1} (- 7)]$ is :

$\frac{49}{50}$
$- \frac{49}{50}$
$\frac{24}{25}$
$- \frac{24}{25}$

If $|\begin{matrix} 2 & - 3 i & 1 \\ 3 & 3 i & - 1 \\ 4 & 3 & i \end{matrix}|$ = x + iy, then :

x = 3, y = 1
x = 2, y = 5
x = 0, y = 0
x = 1, y = 1

10.

The value of the $|\begin{matrix} {}^{10}C_{4} & {}^{10}C_{5} & {}^{11}C_{m} \\ {}^{11}C_{6} & {}^{11}C_{7} & {}^{12}C_{m + 2} \\ {}^{12}C_{8} & {}^{12}C_{9} & {}^{13}C_{m + 4} \end{matrix}| = 0$ when m is equal to :