List of practice Questions

Let origin be the centre, $(\pm 3, 0)$ the foci and $\dfrac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$

The locus of a variable point whose chord of contact w.r.t. the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ subtends a right angle at the origin is

Starting from the point $A(-3,4)$, a moving object touches $2x + y - 7 = 0$ at $B$ and reaches $C(0,1)$. If the object travels along the shortest path, the distance between $A$ and $B$ is

The locus of centers of the circles, possessing the same area and having $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ as their common tangent, is

The mean deviation about the mean for the following data:
5, 6, 7, 8, 6.9, 13, 12, 15

If 6 is the mean of a Poisson distribution, then $P(X \geq 3) =$

The point of intersection of the lines \[ \vec{r} = 2\vec{b} + t(6\vec{c} - \vec{a}) \quad \text{and} \quad \vec{r} = \vec{a} + s(\vec{b} - 3\vec{c}) \text{ is:} \]

Given vectors $3\vec{a} - 5\vec{b}$ and $2\vec{a} + \vec{b}$ are mutually perpendicular, and so are $\vec{a} + 4\vec{b}$ and $-\vec{a} + \vec{b}$. Find the acute angle between $\vec{a}$ and $\vec{b}$:

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices of a triangle $ABC$. Through the vertices, lines are drawn parallel to the sides to form the triangle $A'B'C'$. Then the centroid of $\Delta A'B'C'$ is

In a triangle ABC, $a = 2,\ b = 3,\ c = 4$, then $\tan\left(\dfrac{A}{2}\right) = $ ?

The number of real roots of the equation \[ \frac{\sqrt{x}}{\sqrt{1 - x}} + \frac{\sqrt{1 - x}}{\sqrt{x}} = \frac{13}{6} \] is:

If \[ \frac{x^4 + 24x^2 + 28}{(x^2 + 1)^3} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{(x^2 + 1)^2} + \frac{Ex + F}{(x^2 + 1)^3} \] then the value of $A + B + C + D + E + F =$ ?

Let $A = \begin{bmatrix} 5 & \sin^2\theta & \cos^2\theta \\ -\sin^2\theta & -5 & 1 \\ \cos^2\theta & 1 & 5 \end{bmatrix}$, then the maximum value of $\det(A)$ is:

If $\dfrac{x - 1}{3 + i} + \dfrac{y - 1}{3 - i} = i$, then the true statement among the following is:

If the identity $\cos^4\theta = a\cos4\theta + b\cos2\theta + c$ holds for some $a, b, c \in \mathbb{Q}$, then $(a, b, c) =$ ?

The number of integer solutions of the equation $|1 - i|^x = 2^x$ is:

Given $a_i^2 + b_i^2 + c_i^2 = 1$ and $a_i a_j + b_i b_j + c_i c_j = 0$ for $i \neq j$, and
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} \] Then $\det(AA^T) = $ ?

If $A = \frac{1}{7} \begin{bmatrix} 3 & -2 & 6 \\-6 & -3 & 2 \\ -2 & 6 & 3 \end{bmatrix}$, then $A^{-1} = $ ?

If $A = \begin{bmatrix} \alpha^2 & 5 \\ 5 & -\alpha \end{bmatrix}$ and $\det(A^{10}) = 1024$, then $\alpha =$ ?

The range of the real valued function \[ f(x) = \sqrt{\frac{x^2 + 2x + 8}{x^2 + 2x + 4}} \] is