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JEE Mathematics Solved Question Paper 2007

Multiple Choice Questions

31.

If $xsin (\frac{x}{y}) dy = [ysin (\frac{y}{x}) - x] dx$ and $y (1) = \frac{π}{2}$ , then the value of $\cos (\frac{y}{x})$ is equal to :

x
$\frac{1}{x}$
$\log (x)$
e^x

32.

The differential equation of the system of all circles of radius r in the xy plane is :

${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$
${[1 + {(\frac{dy}{dx})}^{3}]}^{2} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$
${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{2}$
${[1 + {(\frac{dy}{dx})}^{2}]}^{3} = r^{2} {(\frac{d^{2} y}{{dx}^{2}})}^{3}$

33.

The general solution of the differential equation

$\frac{d^{2} y}{{dx}^{2}} + 2 \frac{d y}{d x} + y = 2 e^{3 x}$ is given by

$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{- 3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{- x} + \frac{e^{3 x}}{8}$
$y = (c_{1} + c_{2} x) e^{x} + \frac{e^{- 3 x}}{8}$

34.

The solution of the differential ydx + (x - y³)dy = 0 is:

xy = $\frac{1}{3} y^{3} + c$
xy = y⁴ + c
y⁴ = 4xy + c
4y = y³ + c

35.

If the probability density function of a random variable X is f(x) = $\frac{x}{2}$ in $0 \leq x \leq 2$ , then $P (X > 1.5 | x > 1)$ is equal to :

$\frac{7}{16}$
$\frac{3}{4}$
$\frac{7}{12}$
$\frac{21}{64}$

36.

If X follows a binomial distribution with parameters n = 100 and p = $\frac{1}{3}$ then P(X = r) 3 is maximum when r is equal to

16
32
33
none of these

37.

If $\vec{b}$ is a unit vector, then $(\vec{a} . \vec{b}) \vec{b} + \vec{b} (\vec{a} \times \vec{b})$ is :

${|\vec{a}|}^{2} \vec{b}$
$|\vec{a} . \vec{b}| \vec{a}$
$\vec{a}$
$\vec{b}$

38.

If $θ$ is the angle between the lines AB and AC where A, B and C are the three points with coordinates (1, 2, - 1), (2, 0, 3), (3, - 1, 2) respectively, then $\sqrt{462} \cos (θ)$ is equal to

39.
Let the pairs $\vec{a}, \vec{b} and \vec{c}, \vec{d}$ each determine a plane. Then the planes are parallel, if :

$(\vec{a} \times \vec{c}) and (\vec{b} \times \vec{d}) = \vec{0}$

$(\vec{a} \times \vec{c}) . (\vec{b} \times \vec{d}) = 0$

$(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0}$

$(\vec{a} \times \vec{b}) . (\vec{c} \times \vec{d}) = 0$

$(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0}$

Since, $\vec{a} and \vec{b}$ are coplanar, therefore $\vec{a} \times \vec{b}$ is a vector perpendicular to the plane containing $\vec{a} and \vec{b}$ . Similarly, $\vec{c} \times \vec{d}$ is a vector perpendicular to the plane containing $\vec{c} and \vec{d}$ . Thus, the two planes will be parallel if their normals, i.e., $(\vec{a} \times \vec{b}) and (\vec{c} \times \vec{d})$ are parallel.