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JEE Mathematics Solved Question Paper 2007

Multiple Choice Questions

81.

The value of the integral $\int_{- \frac{π}{4}}^{\frac{π}{4}} \sin (- 4 x) d x$ is

$- \frac{8}{3}$
$\frac{3}{2}$
$\frac{8}{3}$
None of these

82.

The solution of the differential equation $xdy - ydx = \sqrt{x^{2} + y^{2}} dx$ is

$x + \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$y - \sqrt{x^{2} + y^{2}} = {cx}^{2}$
$x - \sqrt{x^{2} + y^{2}} = cx$
$y + \sqrt{x^{2} + y^{2}} = {cx}^{2}$

83.

The differential equation of all non-vertical lines in a plane is

$\frac{d^{2} y}{d x^{2}} = 0$
$\frac{d^{2} x}{d y^{2}} = 0$
$\frac{d y}{d x} = 0$
$\frac{d x}{d y} = 0$

84.

$\int {(\frac{x + 2}{x + 4})}^{2} e^{x} dx$ is equal to

$e^{x} (\frac{x}{x + 4}) + C$
$e^{x} (\frac{x + 2}{x + 4}) + C$
$e^{x} (\frac{x - 2}{x + 4}) + C$
$e^{x} (\frac{2 {xe}^{x}}{x + 4}) + C$

85.

The value of $\int_{3}^{5} \frac{x^{2}}{x^{2} - 4} d x$

$2 - \log_{e} (\frac{15}{7})$
$2 + \log_{e} (\frac{15}{7})$
2 + 4 $\log_{e} (3) - 4 \log_{e} (7) + 4 \log_{e} (5)$
$2 - \tan^{- 1} (\frac{5}{7})$

86.
$\int_{0}^{\infty} \frac{d x}{{(x + \sqrt{x^{2} + 1})}^{3}}$ is equal to

$\frac{3}{8}$

$\frac{1}{8}$

$\frac{- 3}{8}$

None of these

$\frac{3}{8}$

$Let I = \int_{0}^{\infty} \frac{d x}{{(x + \sqrt{x^{2} + 1})}^{3}} Put x = \tan (θ) \Rightarrow dx = \sec^{2} (θ) dθ ∴ I = \int_{0}^{\frac{π}{2}} \frac{\sec^{2} (θ) d θ}{{(\tan (θ) + \sqrt{\tan^{2} (θ) + 1})}^{3}} = \int_{0}^{\frac{π}{2}} \frac{\sec^{2} (θ) d θ}{{(\tan (θ) + \sec (θ))}^{3}} = \int_{0}^{\frac{π}{2}} \frac{\frac{1}{\cos^{2} (θ)}}{(\frac{\sin (θ)}{\cos (θ)} + \frac{1}{\cos (θ)})} d θ = \int_{0}^{\frac{π}{2}} \frac{\cos (θ)}{{(\sin (θ) + 1)}^{3}} d θ = {[- \frac{1}{2 {(\sin (θ) + 1)}^{2}}]}_{0}^{\frac{π}{2}} I = \frac{- 1}{8} + \frac{1}{2} = \frac{3}{8}$