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JEE Mathematics Solved Question Paper 2010

Multiple Choice Questions

41.

The vector equation of the straight line is

$\vec{r} = (\hat{i} - \hat{j} + 3 \hat{k}) + λ (3 \hat{i} - 2 \hat{j} - \hat{k})$
$\vec{r} = (3 \hat{i} - 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 3 \hat{k})$
$\vec{r} = (3 \hat{i} + 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 3 \hat{k})$

42.

The probability distribution of a random variable X is given as

x - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

P(X = x) p 2p 3p 4p 5p 7p 8p 9p 10p 11p 12p

Then, the value of p is

$\frac{1}{72}$
$\frac{3}{73}$
$\frac{5}{72}$
$\frac{1}{74}$

43.

The angle between the curves, y = x and y² - x = 0 at the point (1, 1) is

$\frac{π}{2}$
$\tan^{- 1} (\frac{4}{3})$
$\frac{π}{3}$
$\tan^{- 1} (\frac{3}{4})$

44.

If $\int \frac{x + 2}{2 x^{2} + 6 x + 5} dx$ = P $\int \frac{4 x + 6}{2 x^{2} + 6 x + 5} dx$ + $\frac{1}{2} \int \frac{dx}{2 x^{2} + 6 x + 5}$ , then the values of P is

$\frac{1}{3}$
$\frac{1}{2}$
$\frac{1}{4}$
2

45.

$\int (x + 1) {(x + 2)}^{7} (x + 3) dx$ is equal to

$\frac{{(x + 2)}^{10}}{10} - \frac{{(x + 2)}^{8}}{8} + c$
$\frac{{(x + 1)}^{10}}{2} - \frac{{(x + 2)}^{8}}{8} - \frac{{(x + 3)}^{2}}{2} + c$
$\frac{{(x + 2)}^{10}}{10} + c$
$\frac{{(x + 1)}^{2}}{2} + \frac{{(x + 2)}^{8}}{8} + \frac{{(x + 3)}^{2}}{2} + c$

46.

$\int (x^{2} + 1) \sqrt{x + 1} dx$ is equal to

$\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + 2 \frac{{(x + 1)}^{\frac{3}{2}}}{3} + c$
$2 [\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + 2 \frac{{(x + 1)}^{\frac{3}{2}}}{3}] + c$
$\frac{{(x + 1)}^{\frac{7}{2}}}{7} - 2 \frac{{(x + 1)}^{\frac{5}{2}}}{5} + c$
${(x + 1)}^{\frac{7}{2}} + {(x + 1)}^{\frac{5}{2}} + {(x + 1)}^{\frac{3}{2}} + c$

47.

$\int \frac{1 + x}{x + e^{- x}} dx$ is equal to

$\log |(x - e^{- x})|$
$\log |(x + e^{- x})|$
$\log |(1 + {xe}^{x})| + c$
${(1 + {xe}^{x})}^{2}$ + c

48.

$\int \frac{\log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx$ is equal to

${[\log (x + \sqrt{1 + x^{2}})]}^{2} + c$
$xlog (x + \sqrt{1 + x^{2}}) + c$
$\frac{1}{2} \log (x + \sqrt{1 + x^{2}}) + c$
$\frac{x}{2} \log (x + \sqrt{1 + x^{2}}) + c$

49.

$\int \frac{dx}{\sqrt{1 - e^{2 x}}}$ is equal to

$\log |e^{- x} + \sqrt{e^{- 2 x} - 1}| + c$
$\log |e^{x} + \sqrt{e^{2 x} - 1}| + c$
$- \log |e^{- x} + \sqrt{e^{- 2 x} - 1}| + c$
$- \log |e^{- 2 x} + \sqrt{e^{- 2 x} - 1}| + c$

50.
$\int \frac{\cos (x) + xsin (x)}{x^{2} \cos (x)} dx$ is equal to

$\log |\frac{\sin (x)}{1 + \cos (x)}| + c$

$\log |\frac{\sin (x)}{x + \cos (x)}| + c$

$\log |\frac{2 \sin (x)}{x + \cos (x)}| + c$

$\log |\frac{x}{x + \cos (x)}| + c$

$\log |\frac{x}{x + \cos (x)}| + c$

$\frac{\cos (x) + xsin (x)}{x^{2} \cos (x)} = \frac{(x + \cos (x)) - x + xsin (x)}{x (x + \cos (x))} = \frac{1}{x} - \frac{(1 - \sin (x))}{x + \cos (x)} ∴ \int \frac{\cos (x) + xsin (x)}{x^{2} \cos (x)} dx = \int (\frac{1}{x} - \frac{(1 - \sin (x))}{x + \cos (x)}) dx = \log (x) - \log (x + \cos (x)) + c = \log (\frac{x}{x + \cos (x)}) + c$