Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

JEE Mathematics Solved Question Paper 2010

Multiple Choice Questions

71.

$If \int (1 - \cos (x)) \csc^{2} (x) dx = f (x) + c, then f (x) is equal to$

$\tan (\frac{x}{2})$
$\cot (\frac{x}{2})$
$2 \tan (\frac{x}{2})$
$\frac{1}{2} \tan (\frac{x}{2})$

72.
$If I_{n} = \int_{0}^{\frac{n^{} π}{4}} \tan (x) dx, then I_{2} + I_{4}, I_{3} + I_{5}, I_{4} + I_{6}, . . ., are in$

arithmatic progression

geometric progression

harmonic progression

arithmetic-geometric progression

harmonic progression

$I_{n} = \int_{0}^{\frac{π}{4}} \tan^{n} (x) dx We have, I_{r + 2} = \int_{0}^{\frac{π}{4}} \tan^{r + 2} (x) dx = \int_{0}^{\frac{π}{4}} \tan^{r} (x) . \tan^{2} (x) and I_{r} = \int_{0}^{\frac{π}{4}} \tan^{r} (x) dx Then, I_{r} + I_{r + 2} = \int_{0}^{\frac{π}{4}} \tan^{r} (x) dx + \int_{0}^{\frac{π}{4}} \tan^{r} (x) . \tan^{2} (x) = \int_{0}^{\frac{π}{4}} \tan^{r} (x) (1 + \tan^{2} (x)) dx = \int_{0}^{\frac{π}{4}} \tan^{r} (x) \sec^{2} (x) dx [Put t = \tan (x) \Rightarrow dt = \sec^{2} (x) dx] = \int_{0}^{1} t^{r} dt \Rightarrow {[\frac{t^{r + 1}}{r + 1}]}^{1}_{0} = \frac{1}{r + 1} So, I_{r} + I_{r + 2} = \frac{1}{r + 1} ie, I_{2} + I_{4} = \frac{1}{3} I_{3} + I_{5} = \frac{1}{4} I_{4} + I_{6} = \frac{1}{5} which are clearly in HP .$

73.

The area (in square units) of the region enclosed by the two circles x² + y² = 1 and (x - 1)² + y² = 1 is

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

74.

The values of a function f(x) at different values of x are as follows

x 0 1 2 3 4 5

f(x) 2 3 6 11 18 27

Then, the approximate area (in square units) bounded by the curve y = f(x) and x-axis between x = 0 and 5, using the Trapezoidal rule, is

50
75
52.5
62.5

75.

$The solution of \tan (y) \frac{dy}{dx} = \sin (x + y) + \sin (x - y) is$

sec(y) = 2cos(x) + c
sec(y) = - 2cos(x) + c
tan(y) = - 2cos(x) + c
sec²(y) = - 2cos(x) + c

76.

A family of curves has the differential equation $xy \frac{dy}{dx} = 2 y^{2} - x^{2}$ . Then, the family of curves is

$y^{2} = {cx}^{2} + x^{3}$
$y^{2} = {cx}^{4} + x^{3}$
$y^{2} = x + {cx}^{4}$
$y^{2} = x^{2} + {cx}^{4}$

77.

The length of the common chord of the circles of radii 15 and 20, whose centres are 25 unit of distance apart, is

78.

Let M be the foot of the perpendicular from a point P on the parabola y = 8(x - 3) onto its directrix and let S be the focus ofthe parabola. If $∆$ SPM is an equilateral triangle, then P is equal to