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JEE Mathematics Solved Question Paper 2004

Multiple Choice Questions

31.

If a and b are unit vectors and is the angle between them, then the value of is

$\frac{1}{2} |a + b|$
$\frac{1}{2} |a - b|$
$\frac{|a - b|}{|a + b|}$
$\frac{|a + b|}{|a - b|}$

32.

$\int {xsec}^{2} (x) dx$ is equal to

$xtan (x) + \log (\sec (x)) + c$
$\frac{x^{2}}{2} \sec (x) + \log (\cos (x)) + c$
$xtan (x) + \log (\cos (x)) + c$
$\tan (x) + \log (\cos (x)) + c$

33.

$\int \frac{t}{e^{3 t^{2}}} dt$ is equal to

$\frac{1}{6} e^{3 t^{2}} + c$
$- \frac{1}{6} e^{3 t^{2}} + c$
$\frac{1}{6} e^{- 3 t^{2}} + c$
$- \frac{1}{6} e^{- 3 t^{2}} + c$

34.

$\int_{0}^{π} \log (\sin^{2} (x)) dx$ is equal to

$2 {πlog}_{e} (\frac{1}{2})$
${πlog}_{e} (2)$
$\frac{π}{2} \log_{e} (\frac{1}{2})$
None of these

35.

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

$\frac{1}{n} \log (\frac{x^{n}}{x^{n} + 1}) + c$
$\frac{1}{n} \log (\frac{x^{n} + 1}{x^{n}}) + c$
$\log (\frac{x^{n}}{x^{n} + 1}) + c$
None of these

36.
The area bounded by the curves y² - x = 0 and y - x² = 0, is

7/3 sq unit

1/3 sq unit

5/3 sq unit

1 sq unit

1/3 sq unit

$The equations of given curves are y^{2} - x = 0 . . . (i) and y - x^{2} = 0 . . . (ii) On solving Eqs . (i) and (ii), we get x^{4} - x = 0 \Rightarrow x (x^{3} - 1) = 0 \Rightarrow x = 1, 0 ∴ Points of intersection of curves are (0, 0) and (1, 1) . ∴ Required area = \int_{0}^{1} \sqrt{x} dx - \int_{0}^{1} x^{2} dx = {[\frac{2}{3} x^{3 / 2}]}_{0}^{1} - {[\frac{x^{3}}{3}]}_{0}^{1} = \frac{2}{3} - \frac{1}{3} = \frac{1}{3} sq unit$