List of practice Questions

In a group of 3 girls and 4 boys, there are two boys $ B_1 $ and $ B_2 $. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $ B_1 $ and $ B_2 $ are not adjacent to each other, is:

Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6 \], then f(1) is equal to:

Let P be the foot of the perpendicular from the point $ (1, 2, 2) $ on the line \[ \frac{x-1}{1} = \frac{y + 1}{-1} = \frac{z - 2}{2} \] Let the line $ \mathbf{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k})$, $ \lambda \in \mathbb{R} $, intersect the line $L$ at $Q$. Then $ 2(PQ)^2 $ is equal to:

The number of integral terms in the binomial expansion of \[ \left(11^{\frac{1}{2}} + 17^{\frac{1}{8}}\right)^{1024} \] is:

For a statistical data $ x_1, x_2, \dots, x_{10} $ of 10 values, a student obtained the mean as 5.5 and \[ \sum_{i=1}^{10} x_i^2 = 371. \] He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively.
The variance of the corrected data is:

Let the line passing through the points $ (-1, 2, 1) $ and parallel to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \] intersect the line \[ \frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1} \] at the point P. Then the distance of P from the point Q(4, -5, 1) is:

Let a curve $ y = f(x) $ pass through the points $ (0,5) $ and $ (\log 2, k) $. If the curve satisfies the differential equation: \[ 2(3+y)e^{2x}dx - (7+e^{2x})dy = 0, \] then $ k $ is equal to:

Evaluate the following limit: \[ \lim_{x \to \infty} \frac{(2x^2 - 3x + 5) \left( 3x - 1 \right)^{x/2}}{(3x^2 + 5x + 4) \sqrt{(3x + 2)^x}}. \] The value of the limit is:

The sum of all rational numbers in $ (2 + \sqrt{3})^8 $ is:

The sum 1 + 3 + 11 + 25 + 45 + 71 + ... upto 20 terms, is equal to

If $ z_1, z_2, z_3 \in \mathbb{C} $ are the vertices of an equilateral triangle, whose centroid is $ z_0 $, then $ \sum_{k=1}^{3} (z_k - z_0)^2 $ is equal to

Let ABCD be a trapezium whose vertices lie on the parabola $ y^2 = 4x $. Let the sides AD and BC of the trapezium be parallel to the y-axis. If the diagonal AC is of length $ \frac{25}{4} $ and it passes through the point $ (1, 0) $, then the area of ABCD is:

Let R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set $ \{1, 2, 3, 4\} $. Then the minimum number of elements needed to be added in $ R $ so that $ R $ becomes an equivalence relation, is:

If the system of equations \[ (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \] \[ \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \] \[ (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \] has infinitely many solutions, then $ \lambda^2 + \lambda $ is equal to:

Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that
\[\sum_{i=1}^{10} (x_i - 2) = 30, \quad \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \quad \beta > 2\]
and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of
\[ 2(x_1 - 1) + 4\beta, 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta\]
then $\frac{\beta \mu}{\sigma^2}$ is equal to:

A line passing through the point $ P(a, 0) $ makes an acute angle $ \alpha $ with the positive $ x $-axis. Let this line be rotated about the point $ P $ through an angle $ \frac{\alpha}{2} $ in the clock-wise direction. If in the new position, the slope of the line is $ 2 - \sqrt{3} $ and its distance from the origin is $ \frac{1}{\sqrt{2}} $, then the value of $ 3a^2 \tan^2 \alpha - 2\sqrt{3} $ is

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

Let the area of the region $$ \{(x, y) : 2y \leq x^2 + 3, \quad y + |x| \leq 3, \quad y \geq |x-1|\} $$ be $ A $. Then $ 6A $ is equal to:

Bag $ B_1 $ contains 6 white and 4 blue balls, Bag $ B_2 $ contains 4 white and 6 blue balls, and Bag $ B_3 $ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $ B_2 $ is:

Let $ f : \mathbb{R} \to \mathbb{R} $ be a twice-differentiable function such that $ f(2) = 1 $. If $ F(x) = x f(x) $ for all $ x \in \mathbb{R} $, and the integrals $ \int_0^2 x F'(x) \, dx = 6 $ and $ \int_0^2 x^2 F''(x) \, dx = 40 $, then $ F'(2) + \int_0^2 F(x) \, dx $ is equal to:

If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{3/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$, where C is the constant of integration, then $g\left( \frac{1}{2} \right)$equals:

If $ \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] $, then the number of solutions of the equation \[ \sqrt{3} \csc^2 \theta - 2 (\sqrt{3} - 1) \csc \theta - 4 = 0 \] is:

The integral $ \int_0^\pi \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx $ is equal to:

Let $ A = [a_{ij}] $ be a $3 \times 3 $ matrix such that: $$ A = \begin{bmatrix} 0 & 0 & 4 \\ 1 & 0 & 0 \\ 0 & 1 & 3 \\ \end{bmatrix}, A^{-1} = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 3 & 0 \\ 2 & 1 & 0 \end{bmatrix}. $$ Then $ a_{23} $ equals: