List of practice Questions

Let $ A = (1, 2, 3, \dots, 20) $. Let $ R \subseteq A \times A $ such that $ R = \{(x, y) : y = 2x - 7 \} $. Then the number of elements in $ R $ is equal to:

If $ x, y, z $ $\text{ are the three cube roots of 27, then the determinant of the matrix}$ \[ \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix} \] $\text{is:}$

$ \text{A tower subtends angles a, 2a, and 3a respectively at points A, B, and C, which are lying on a horizontal line through the foot of the tower. Then }$ $ \frac{AB}{BC} $ $\text{ is equal to:}$

Let $\vec{a}$ and $\vec{c}$ be unit vectors such that the angle between them is $\cos^{-1} \left( \frac{1}{4} \right)$. If $\vec{b} = 2\vec{c} + \lambda \vec{a}$. Where $\lambda > 0$ and $|\vec{b}| = 4$, then $\lambda$ is equal to:

Let the line $\frac{x}{4} + \frac{y}{2} = 1$ meet the x-axis and y-axis at A and B, respectively. M is the midpoint of side AB, and M' is the image of the point M across the line $x + y = 1$. Let the point P lie on the line $x + y = 1$ such that $\Delta ABP$ is an isosceles triangle with $AP = BP$. Then the distance between M' and P is:

Let E and F be two events such that $ P(E) > 0 $ and $ P(F) > 0 $. Which one of the following is NOT equivalent to the condition that $ P(E) = P(E \cap F) $?

If $ \alpha $ and $ \beta $ are the two roots of the equation $ x^2 + ax + b = 0 $, $ ab \neq 0 $, then the quadratic equation with roots $ \frac{1}{\alpha^3 + \alpha} $ and $ \frac{1}{\beta^3 + \beta} $ is

The area enclosed between the curve $ y = \sin x, y = \cos x $, $\text{ for }$ $ 0 \leq x \leq \frac{\pi}{2} $ $\text{ is:}$

The length of the projection of $ \mathbf{a} = 2\hat{i} + 3\hat{j} + \hat{k} $ $\text{ on }$ $ \mathbf{b} = -2\hat{i} + \hat{j} + 2\hat{k} $ $\text{ is equal to:}$

Let $ g: \mathbb{R} \to \mathbb{R} $ and $ h: \mathbb{R} \to \mathbb{R} $ be two functions such that $ h(x) = s \cdot g \cdot n(g(x)) $, where $ \mathbb{R} $ denotes the set of all real numbers and $ s $ and $ n $ stand for the signum and the sine functions respectively. Then, select which of the following is not true?

A ball is thrown vertically upwards with a speed of $ 20 \, \text{m/s} $. What is the maximum height reached by the ball? Assume the acceleration due to gravity is $ g = 9.8 \, \text{m/s}^2 $.

If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are three vectors such that $\vec{a} \times \vec{b} = \vec{c}$, $\vec{a} \cdot \vec{c} = 2$ and $\vec{b} \cdot \vec{c} = 1$. If $|\vec{b}| = 1$, then the value of $|\vec{a}|$ is:

Number of permutations of the letters of the word BANGLORE such that the string ANGLE appears together in all permutations, is:

The circle $ x^2 + y^2 + ax + by + \gamma = 0 $ is the image of the circle $ x^2 + y^2 - 6x - 10y + 30 = 0 $ across the line $ 3x + y = 2 $. The value of $ [\alpha + \beta + \gamma] $ is (where $ [.] $ represents the floor function)

How many triangles are there in the figure given below?

What is the next number in each of the following 3 sequences?
8, 17, 33, 67, 133, 1?

In a certain code language, the word "REASON" is encoded as TDDQSK. Using the same rule, how is the word PRAYER coded?

At an IIM entrepreneurship summit, two young founders, Karan and Deepak, introduced their startup. In their quirky opening, they said "The product of our ages is 240. And just like in our startup strategy, twice Deepak's age is 4 years more than Karan's age. How old was Deepak two years ago?"

A research team is studying the effects of three different fertilizers (X, Y, and Z) on the growth of a specific plant species. They have four experimental plots with different fertilizer combinations: Plot 1: Fertilizer X only
Plot 2: Fertilizer Y only
Plot 3: Fertilizer X and Fertilizer Z only
Plot 4: Fertilizer Y and Fertilizer Z only

After a period of observation, they note the following: \[\begin{array}{rl} \bullet & \text{Plants in plots with Fertilizer X showed significantly increased height compared to a control group (no fertilizer).} \\ \bullet & \text{Plants in plots with Fertilizer Y showed a slightly increased leaf area compared to the control group.} \\ \bullet & \text{Plants in Plot 3 (X and Z) showed no significant difference in height compared to the control group.} \\ \bullet & \text{Plants in Plot 4 (Y and Z) showed significantly increased height compared to the control group.} \\ \end{array}\] Based on these observations, which of the following conclusions is best supported?

In a certain code language, 'SUN' is written as 'RWK' and 'MOON' is written as 'LQLK'. Following the same pattern, how will 'STAR' be written?

Five friends A, B, C, D, and E are sitting in a row. The following conditions hold:
A is sitting next to B.
C is sitting at one of the ends.
D is sitting between A and E.
E is sitting to the right of D.
Who is sitting at the second position?

Statements: All apples are fruits. All fruits are tasty.

Conclusions: 1. All apples are tasty. 2. Some tasty things are apples.

Average Cost (AC) is defined as the sum of Average Fixed Cost (AFC) and Average Variable Cost (AVC), and dumping is defined as selling a product at a price less than AC but more than AVC. A company in India, suddenly, found that the demand for its product "ZOOM" has fallen to 60% of the output produced in that financial year. As a result, the company must sell 40% of the produced output in a foreign market. If it decides to "dump" 40% of its output in a neighbouring country (by reducing the price by 20%), what would be the objective of its 'dumping strategy', among the following? (The company has set a profit margin of 10% of AC while fixing the price of its product for sale in India).

Complete the next series:
\[ \begin{array}{ccc} 2 & 8 & 12 \\ \hline 8 & 40 & 45 \\ \hline 40 & 240 & 246 \\ \hline --- & --- & --- \\ \hline \end{array} \]

A body of mass 0.2 kg is attached to a light string of length 1 m and revolved in a vertical circle. What is the minimum speed at the lowest point so that the body can complete the circular motion? (Take $ g = 10 \, \text{m/s}^2 $)