List of practice Questions

Let {an}_n≥1 be a sequence of non-zero real numbers. Then which one of the following statements is true ?

Let f : \(\R→\R\) be the function defined by
\(f(x)=\begin{cases} \lim\limits_{h \rightarrow0}\frac{(x+h)\sin(\frac{1}{x}+h)-x\sin\frac{1}{x}}{h} & x \ne 0\\ 0, & x=0\end{cases}\)
Then which one of the following statements is NOT true ?

Find the probability of getting the sum as a perfect square number when two dice are thrown together.

The total number of 3-digit numbers, the sum of whose digits is even, is equal to:

The variance of the data \( 2, 4, 6, 8, 10 \) is:

An urn contains five balls. Two balls are drawn and found to be white. The probability that all the balls are white is:

How many students have got 60% or more marks in all the subjects?

What was the ratio between the ages of P and Q four years ago? I. The ratio between the present ages of P and Q is 3 : 4. II. The ratio between the present ages of Q and R is 4 : 5.

What was the cost price of the suitcase purchased by Samir? I. Samir got a 25 percent concession on the labelled price. II. Samir sold the suitcase for Rs. 2000 with 25 percent profit on the labelled price.

Let f : \(\R → \R\) be the function defined by
\(f(x)=\text{det}\begin{pmatrix}1+x & 9 & 9 \\ 9 & 1+x & 9 \\ 9 & 9 & 1+x \end{pmatrix}\)
Then the maximum value of f on the interval [9, 10] equals

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: