List of practice Questions

Let the lines \[ 3x - 4y - \alpha = 0, \quad 8x - 11y - 33 = 0, \quad 2x - 3y + \lambda = 0 \] be concurrent. If the image of the point $ (1, 2) $ in the line \[ 2x - 3y + \lambda = 0 \text{ is } \left( \frac{57}{13}, \frac{-40}{13} \right), \text{ then } |\alpha \lambda| \text{ is equal to:} \]

The interior angles of a polygon with $ n $ sides, are in an A.P. with common difference 6°. If the largest interior angle of the polygon is 219°, then $ n $ is equal to:

Let $ x = x(y) $ be the solution of the differential equation $ y^2 dx + \left( x - \frac{1}{y} \right) dy = 0 $. If $ x(1) = 1 $, then $ x\left( \frac{1}{2} \right) $ is :

For the function $ f(x) = \ln(x^2 + 1) $, what is the second derivative of $ f(x) $?

The perpendicular distance of the line $ \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z + 3}{2} $ from the point $ P(2, -10, 1) $ is:

The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are $ \mu $ and $ \sigma $ respectively, then $ 10(\mu + \sigma) $ is equal to:

The integral $ \int_{0}^{\pi} \frac{8x dx}{4 \cos^2 x + \sin^2 x} $ is equal to

Let the length of a latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be 10. If its eccentricity is $ e $, and the minimum value of the function $ f(t) = t^2 + t + \frac{11}{12} $, where $ t \in \mathbb{R} $, then $ a^2 + b^2 $ is equal to:

Let the points $A(a,-1,2)$, $B(1,b,-4)$, $C(-1,1,c)$ and $D(1,-2,8)$ be the vertices of a parallelogram $ABCD$. Then its area is equal to:

Number of solutions in the interval $ [-2\pi, 2\pi] $ for the equation: \[ 2\sqrt{2} \cos^2 \theta + (2 - \sqrt{6}) \cos \theta - \sqrt{3} = 0 \]

\[ \sum_{k=1}^{n} \left( \alpha^k + \frac{1}{\alpha^k} \right)^2 = 20, \quad \alpha \text{ is one of the roots of } x^2 + x + 1 = 0, \text{ then } n = ? \]

Let $ y = y(x) $ be the solution curve of the differential equation $ x(x^2 + e^x) \, dy + \left( e^x(x - 2) y - x^3 \right) \, dx = 0, \quad x>0, $ passing through the point $ (1, 0) $. Then $ y(2) $ is equal to:

If the system of equations: $$ \begin{aligned} 3x + y + \beta z &= 3 \\2x + \alpha y + z &= 2 \\x + 2y + z &= 4 \end{aligned} $$ has infinitely many solutions, then the value of $ 22\beta - 9\alpha $ is:

If all the words, with or without meaning, made using all the letters of the word ``RANCHI'' are arranged as in a dictionary, then the word at $560^{\text{th}}$ position is:

Let \[ f(x)= \begin{cases} 3x, & x<0,\\ 1+x+[x], & 0\le x\le 2,\\ 5, & x>2, \end{cases} \] where $[x]$ denotes the greatest integer function. If $\alpha$ and $\beta$ are the number of points in $\mathbb{R}$ where $f$ is not continuous and is not differentiable respectively, then $\alpha+\beta$ equals:

Let {an}ⁿ⁼⁰_∞be a sequence such that a₀=a₁=0 and a_n+2=3a_n+1−2a_n+1,∀ n≥0. Then a₂₅a₂₃−2a₂₅a₂₂−2a₂₃a₂₄+4a₂₂a₂₄is equal to

Let $ A = [a_{ij}] $ be a square matrix of order 2 with entries either 0 or 1. Let $ E $ be the event that $ A $ is an invertible matrix. Then the probability $ P(E) $ is:

Let $f: [0, 3] \to A$ be defined by $f(x) = 2x^3 - 15x^2 + 36x + 7$ and $g: [0, \infty) \to B$ be defined by $g(x) = \frac{x}{x^{2025} + 1}$. If both functions are onto and $S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \}$, then $n(S)$ is equal to:

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability that A wins if A makes the first throw is:

Two equal sides of an isosceles triangle are along $ -x + 2y = 4 $ and $ x + y = 4 $. If $ m $ is the slope of its third side, then the sum of all possible distinct values of $ m $ is:

Let $ A = \{1, 2, 3, \dots, 10\} $ and $ B = \left\{ \frac{m}{n} : m, n \in A, m <n \text{ and } \gcd(m, n) = 1 \right\} $. Then $ n(B) $ is equal to :

The area of the region, inside the circle $(x-2\sqrt{3})^2 + y^2 = 12$ and outside the parabola $y^2 = 2\sqrt{3}x$ is:

The value of $ (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) $ is:

If the image of the point $ (4, 4, 3) $ in the line $ \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} $ is $ (a, \beta, \gamma) $, then $ a + \beta + \gamma $ is equal to:

Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k} $. Let $ L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in {R} $ and $ L_2 : \vec{r} = (\hat{j} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} $ be two lines. If the line $ L_3 $ passes through the point of intersection of $ L_1 $ and $ L_2 $, and is parallel to $ \vec{a} + \vec{b} $, then $ L_3 $ passes through the point: