List of practice Questions

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x+y=3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :
The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :
Let \(*, \square \in \{\wedge, \vee\}\) be such that the Boolean expression \((p * \sim q) \implies (p \square q)\) is a tautology. Then :
Let \( A \) be the set of all points \( (\alpha, \beta) \) such that the area of triangle formed by the points \( (5, 6), (3, 2) \) and \( (\alpha, \beta) \) is 12 square units. Then the least possible length of a line segment joining the origin to a point in \( A \), is :
The number of solutions of the equation \(32^{\tan^2 x} + 32^{\sec^2 x} = 81\), \(0 \le x \le \frac{\pi}{4}\) is :
Let A be a set of all 4-digit natural numbers whose exactly one digit is 7. Then the probability that a randomly chosen element of A leaves remainder 2 when divided by 5 is :
Let C be the set of all complex numbers. Let
$S_1 = \{z \in C : |z-2| \le 1\}$ and
$S_2 = \{z \in C : z(1+i) + \bar{z}(1-i) \ge 4\}$.
Then, the maximum value of $|z-\frac{5}{2}|^2$ for $z \in S_1 \cap S_2$ is equal to :
An angle of intersection of the curves, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and \( x^2 + y^2 = ab, a>b \), is :
Let \( f \) be any continuous function on \( [0, 2] \) and twice differentiable on \( (0, 2) \). If \( f(0) = 0, f(1) = 1 \) and \( f(2) = 2 \), then :
Let \(A\) and \(B\) be independent events such that \(P(A) = p, P(B) = 2p\). The largest value of \(p\), for which \(P(\text{exactly one of } A, B \text{ occurs}) = \frac{5{9}\), is :}
\(\csc 18^\circ\) is a root of the equation :
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set\{P, Q\} is equal to :
Two tangents are drawn from the point P(-1, 1) to the circle $x^2 + y^2 - 2x - 6y + 6 = 0$. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :
The area (in sq. units) of the region, given by the set $\{(x, y) \in R \times R \mid x \ge 0, 2x^2 \le y \le 4 - 2x\}$ is :
Let 9 distinct balls be distributed among 4 boxes, $B_1, B_2, B_3$ and $B_4$. If the probability that $B_3$ contains exactly 3 balls is $k \left(\frac{3}{4}\right)^9$ then $k$ lies in the set :
The sum of the series \(\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \dots + \frac{2^{100}}{x^{2^{100}}+1}\) when \(x = 2\) is :
If the matrix \( A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix} \) satisfies \( A(A^3 + 3I) = 2I \), then the value of \( K \) is :
If \( y \frac{dy}{dx} = x \left[ \frac{\phi(y^2/x^2)}{\phi'(y^2/x^2)} + \frac{y^2}{x^2} \right], x>0, \phi>0, \) and \( y(1) = -1 \), then \( \phi\left(\frac{y^2}{4}\right) \) is equal to :
If \( 0 < \theta, \phi < \frac{\pi}{2} \), \( x = \sum_{n=0}^{\infty} \cos^{2n} \theta \), \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \) and \( z = \sum_{n=0}^{\infty} \cos^{2n} \theta \cdot \sin^{2n} \phi \) then :
The equation of the line through the point (0, 1, 2) and perpendicular to the line (x-1)/2 = (y+1)/3 = (z-1)/(-2) is:
In a triangle ABC, if |BC| = 8, |CA| = 7, |AB| = 10, then the projection of the vector AB on AC is equal to :
Define a relation R over a class of n × n real matrices A and B as "ARB iff there exists a non-singular matrix P such that P A P⁻¹ = B". Then which of the following is true ?
Let the mean and variance of the frequency distribution be 6 and 6.8 respectively.
x : \( x_1 = 2,\; x_2 = 6,\; x_3 = 8,\; x_4 = 9 \)
f : \( 4,\; 4,\; \alpha,\; \beta \)
If \( x_3 \) is changed from 8 to 7, then the mean for the new data will be:
The Boolean expression $(p \implies q) \wedge (q \implies \sim p)$ is equivalent to :
cosec[2cot$^{-1}$(5) + cos$^{-1}$($\frac{4}{5}$)] is equal to :