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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

37.
$\int_{0}^{2} [x^{2}] dx$ is equal to

$2 - \sqrt{2}$

$2 + \sqrt{2}$

$\sqrt{2} - 1$

$- \sqrt{2} - \sqrt{3} + 5$

$- \sqrt{2} - \sqrt{3} + 5$

$\int_{0}^{2} [x^{2}] dx = \int_{0}^{1} [x^{2}] dx + \int_{1}^{\sqrt{2}} [x^{2}] dx + \int_{\sqrt{2}}^{\sqrt{3}} [x^{2}] dx + \int_{\sqrt{3}}^{2} [x^{2}] dx = \int_{0}^{1} 0 dx + \int_{1}^{\sqrt{2}} 1 dx + \int_{\sqrt{2}}^{\sqrt{3}} 2 dx + \int_{\sqrt{3}}^{2} 3 dx = {[x]}_{1}^{\sqrt{2}} + 2 {[x]}_{\sqrt{2}}^{\sqrt{3}} + 3 {[x]}_{\sqrt{3}}^{2} = \sqrt{2} - 1 + 2 (\sqrt{3} - \sqrt{2}) + 3 (2 - \sqrt{3}) = \sqrt{2} - 1 + 2 \sqrt{3} - 2 \sqrt{2} + 6 - 3 \sqrt{3} = - \sqrt{2} - \sqrt{3} + 5$