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JEE Mathematics Solved Question Paper 2015

Multiple Choice Questions

71.
$\int e^{x} \frac{x^{2} + 1}{{(x + 1)}^{2}} dx$ is equal to

$\frac{e^{x}}{x + 1} + C$

$\frac{- e^{x}}{x + 1} + C$

$e^{x} (\frac{x - 1}{x + 1}) + C$

$e^{x} (\frac{x + 1}{x - 1}) + C$

$e^{x} (\frac{x - 1}{x + 1}) + C$

$Let I = \int e^{x} \frac{x^{2} + 1}{{(x + 1)}^{2}} dx = \int e^{x} (\frac{x^{2} - 1 + 2}{{(x + 1)}^{2}}) dx = \int e^{x} (\frac{x^{2} - 1}{{(x + 1)}^{2}} + \frac{2}{{(x + 1)}^{2}}) dx = \int e^{x} (\frac{x - 1}{x + 1} + \frac{2}{{(x + 1)}^{2}}) dx Let f (x) = \frac{x - 1}{x + 1} \Rightarrow f' (x) = \frac{(1) (x + 1) - (x - 1) (1)}{{(x + 1)}^{2}} = \frac{2}{{(x + 1)}^{2}} ∴ I = e^{x} (\frac{x - 1}{x + 1}) + C [∵ e^{x} \{f (x) + f' (x)\} dx = e^{x} f (x) + C]$

72.

$\int \frac{x + 1}{x (1 + {xe}^{x})} dx$ is equal to

$\log |\frac{1 + {xe}^{x}}{{xe}^{x}}| + C$
$\log |\frac{{xe}^{x}}{1 + {xe}^{x}}| + C$
$\log |{xe}^{x} (1 + {xe}^{x})| + C$
$\log |1 + {xe}^{x}| + C$

73.

$\int \frac{f (x) g' (x) - f' (x) g (x)}{f (x) g (x)} [logg (x) - \log (f (x))] dx$ is equal to

$\log [\frac{g (x)}{f (x)}] + C$
$\frac{1}{2} {[\log (\frac{g (x)}{f (x)})]}^{2} + C$
$\frac{g (x)}{f (x)} [\log (\frac{g (x)}{f (x)})] + C$
$[\log (\frac{g (x)}{f (x)})] - \frac{g (x)}{f (x)} + C$

74.

$\int_{0}^{\frac{π}{4}} \frac{\sin (x) + \cos (x)}{3 + \sin (2 x)} dx = ?$

$\frac{1}{2} \log (3)$
$\log (2)$
log(3)
$\frac{1}{4} \log (3)$

75.

$\int_{- 1}^{1} \frac{\sqrt{1 + x + x^{2}} - \sqrt{1 - x + x^{2}}}{\sqrt{1 + x + x^{2}} + \sqrt{1 - x + x^{2}}} dx = ?$

$\frac{3 π}{2}$
$\frac{π}{2}$
0
- 1

76.

$The area of the region described by \{(x, y) / x^{2} + y \leq 1 and y \leq 1 - x\} is$

$\frac{π}{2} - \frac{2}{3}$
$\frac{π}{2} + \frac{2}{3}$
$\frac{π}{2} + \frac{4}{3}$
$\frac{π}{2} - \frac{4}{3}$

77.

$The solution of \frac{dy}{dx} + \frac{1}{x} = \frac{e^{y}}{x^{2}} is$

$2 x = (1 + {Cx}^{2}) e^{y}$
$x = (1 + {Cx}^{2}) e^{y}$
$2 x^{2} = (1 + {Cx}^{2}) e^{- y}$
$x^{2} = (1 + {Cx}^{2}) e^{- y}$

78.

The differential equation $\frac{dy}{dx} = \frac{1}{ax + by + c}$ ,where a, b, c are all non-zero real numbers, is

linear in y
linear in x
linear in both x and y
homogeneous equation

79.

$Let D be the domain of a twice differentiable function f . For all x \in D, f " (x) + f (x) = 0 and f (x) = \int g (x) dx + constant . If h (x) = {\{f (x)\}}^{2} + {\{g (x)\}}^{2} and h (0) = 5, then h (2015) - h (2014) is equal to$

80.

The foci of the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{b^{2}} = 1 and the hyperbola \frac{x^{2}}{144} - \frac{y^{2}}{81} = \frac{1}{25}$ coincide. Then, the value of b² is