List of practice Questions

Let A and B be two events such that 0 < P(A) < 1 and 0 < P(B) < 1. Then which one of the following statements is NOT true ?
If M(t), t ∈ \(\R\), is the moment generating function of a random variable, then which one of the following is NOT the moment generating function of any random variable ?
The equations of the lines which cut off an intercept 1 from the y-axis and are equally inclined to the axes are:

A parabola has the origin as its focus and the line \( x = 2 \)  as the directrix. Then the vertex of the parabola is at:

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3, 1) and has eccentricity \( \sqrt{\frac{2}{5}} \) is:

If the system of linear equations \[ x + ky + 3z = 0, \quad 3x + ky - 2z = 0, \quad 2x + 4y - 3z = 0 \] has a non-zero solution  \( (x, y, z) \), then \( \frac{xz}{y^2} \) is equal to:

Kailash faces towards north. Turnings to his right, he walks 25 metres. He then turns to his left and walks 30 metres. Next, he moves 25 metres to his right. He then turns to the right again and walks 55 metres. Finally, he turns to the right and moves 40 metres. In which direction is he now from his starting point?
The solution set of the inequality \[ 37 - (3x + 5) \geq 9x - 8(x - 3) { is:} \]
The distance between the parallel lines \[ 3x - 4y + 7 = 0 \quad {and} \quad 3x - 4y + 5 = 0 { is } \frac{a}{b}. { Value of } a + b { is:} \]

The relationship between \( a \) and \( b \) so that the function \( f(x) \) defined by

\[ f(x) = \begin{cases} ax + 1, & \text{if } x \leq 3 \\ bx + 3, & \text{if } x > 3 \end{cases} \]

is continuous at \( x = 3 \), is:

The local minimum value of the function \[ f(x) = 3 + |x|, \quad x \in \mathbb{R} \] is:

The point \( (t^2 + 2t + 5, 2t^2 + t - 2) \) { lies on the line} \( x + y = 2 \) { for:}

The coordinates of the point which divides the line segment joining the points \( (2, -1, 3) \) and \( (4, 3, 1) \) internally in the ratio \( 3:4 \) are:

The function \( f(x) \) is given by:

\[ f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & \text{for } x \neq 0 \\ 0, & \text{for } x = 0 \end{cases} \]

For the parabola \( y^2 = -12x \), the equation of the directrix is  \( x = a \). The value of  \( a \) is:

The principal value of \(\sin^{-1} \left( \sin \frac{5\pi}{3} \right)\) is:

The function \( f(x) \) is given by:

\[ f(x) = \begin{cases} x \lfloor x \rfloor & \text{if } 0 \leq x < 2 \\ (x - 1)x & \text{if } 2 \leq x < 3 \end{cases} \]

The function is:

If \( \alpha, \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then \[ \frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b} = \]

The value of the definite integral

\[ \int_0^{\frac{\pi}{2}} \log(\tan x) \, dx \]

is:

The following determinant is equal to:

\[ \begin{vmatrix} \sin^2 x & \cos^2 x & 1 \\ \cos^2 x & \sin^2 x & 1 \\ -10 & 12 & 2 \end{vmatrix} \]
Let X be a random variable having binomial distribution with parameters n(> 1) and p(0 < p < 1). Then \(E(\frac{1}{1+X})\) equals
Let (X, Y) be a random vector having the joint probability density function
\(f(x,y)=\begin{cases} \frac{\sqrt2}{\sqrt{\pi}}e^{-2x}e^{-\frac{(y-x)^2}{2}}, & 0 <x< ∞,-∞<y<∞\\ 0,& \text{otherwise} \end{cases}\)
Then E(Y) equals
Let X1 and X2 be two independent and identically distributed discrete random variables having the probability mass function
\(f(x) = \begin{cases} (\frac{1}{2})^x, & x=1,2,3,... \\   0, & \text{otherwise. }  \end{cases}\)
Then P(min{X1 , X2} ≥ 5) equals
Let X1 , X2 , … , Xn(n ≥ 2) be a random sample from Exp(\(\frac{1}{\theta}\)) distribution, where θ > 0 is unknown. If \(\overline{X}=\frac{1}{n}\sum^n_{i=1}X_i\), then which one of the following statements is NOT true ?
Let X1,X2 , … , Xn(n ≥ 3) be a random sample from a N(μ,σ2) distribution, where μ ∈ \(\R\) and σ > 0 are both unknown. Then which one of the following is a simple null hypothesis ?