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JEE Mathematics Solved Question Paper 2002

Multiple Choice Questions

81.

A random variable X takes the values 0, 1 and 2. If P(X = 1) = P(X = 2) and P(X = 0) = 0.4, then the mean of the random variable X is

82.

The acute angle between the two lines whose direction ratios are given by l + m - n = 0 and $l^{2} + m^{2} + n^{2} = 0, is$

0
$\frac{π}{6}$
$\frac{π}{4}$
$\frac{π}{3}$

83.
The direction ratios of normal to the plane passing through (0, 0, 1), (0, 1, 2) and (1, 0, 3) are

(2, 1, - 1)

(1, 0, 1)

(0, 0, - 1)

(1, 0, 0)

(2, 1, - 1)

Equation of the plane passing through (0, 0, 1), (0, 1, 2) and (1, 0, 3) is

$|\begin{matrix} x & y & z & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 3 & 1 \end{matrix}| = 0 Applying C_{3} \to C_{3} - C_{4} \Rightarrow |\begin{matrix} x & y & z - 1 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 2 & 1 \end{matrix}| = 0 \Rightarrow |\begin{matrix} x & y & z - 1 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix}| = 0 Applying C_{2} \to C_{2} - C_{3} |\begin{matrix} x & y - z + 1 & z - 1 \\ 0 & 0 & 1 \\ 1 & - 2 & 2 \end{matrix}| = 0 \Rightarrow - |\begin{matrix} x & y - z + 1 \\ 1 & - 2 \end{matrix}| = 0 \Rightarrow - [- 2 x - (y - z + 1)] = 0 \Rightarrow 2 x + y - z = 0 The direction of the normal are (2, 1, - 1)$

84.

If P = (0, 1, 0), Q = (0, 1, 0), then the projection of PQ on the plane x + y + z = 3 is

2
$\sqrt{2}$
3
$\sqrt{3}$

85.

In the space the equation by + cz + d = 0 represents a plane perpendicular to the

YOZ-plane
ZOX-plane
XOY-plane
None of these

86.

A plane x passes through the point (1, 1, 1). If b, c, a are the direction ratios of a normal to the plane, where a, b, c (a < b < c) are the factors of 2001, then the equation of plane is

29x + 31y + 3z = 63
23x + 29y - 29z = 23
23x + 29y + 3z = 55
31x + 37y + 3z = 71

87.

If the plane 7x + 11y + 13z = 3003 meets the co-ordinate axes in A, B, C, then the centroid of the $∆ ABC$ is

(143, 91, 77)
(143, 77, 91)
(91, 143, 77)
(143, 66, 91)

88.

$\int \frac{dx}{1 - \cos (x) - \sin (x)} is equal to$

$\log (|\begin{matrix} 1 + \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \tan (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 - \cot (\frac{x}{2}) \end{matrix}|) + c$
$\log (|\begin{matrix} 1 + \tan (\frac{x}{2}) \end{matrix}|) + c$

89.

$\int \frac{dx}{7 + 5 \cos (x)} is equal to$

$\frac{1}{\sqrt{3}} \tan^{- 1} (\frac{1}{\sqrt{3}} \tan \frac{x}{2}) + c$
$\frac{1}{\sqrt{6}} \tan^{- 1} (\frac{1}{\sqrt{6}} \tan \frac{x}{2}) + c$
$\frac{1}{7} \tan^{- 1} (\tan \frac{x}{2}) + c$
$\frac{1}{4} \tan^{- 1} (\tan \frac{x}{2}) + c$

90.

$\int \frac{3^{x} dx}{\sqrt{9^{x} - 1}} is equal to$

$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 3^{x} - \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (9)} \log (|\begin{matrix} 3^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$
$\frac{1}{\log (3)} \log (|\begin{matrix} 9^{x} + \sqrt{9^{x} - 1} \end{matrix}|) + c$