List of top Mathematics Questions

Consider the sample space \( \Omega = \{(x, y): x, y \in \{1, 2, 3, 4\} \} \) where each outcome is equally likely. Let \( A = \{ x \geq 2 \} \) and \( B = \{ y > x \} \) be two events. Then which of the following is NOT true?
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 value of \( \frac{d}{dx} \left( \int 2 \sin x \sin x^2 e^{t^2} \right) \) at \( t = \pi \) is:
The slope of the normal line to the curve \( x = t^2 + 3t - 8 \) and \( y = t^2 - 2t - 5 \) at the point \( (2, -1) \) is
The number of all even integers between 99 and 999 which are not multiples of 3 and 5 is
Let \( \mathbf{a} = e\hat{i} + 3\hat{j} + 4\hat{k}, \mathbf{b} = \hat{i} + 2\hat{j} - \hat{k}, \mathbf{c} = 3\hat{i} + \hat{j} + \lambda \hat{k} \) be the co-terminal edges of a parallelepiped whose volume is 5 units. Then the value of \( \lambda \) is:

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: 
 

A circle with its center in the first quadrant touches both the coordinate axes and the line \( x - y - 2 = 0 \). Then the area of the circle is:
The scores of students in a national level examination are normally distributed with a mean of 500 and a standard deviation of 100. If the value of the cumulative distribution of the standard normal random variable at 0.5 is 0.691, then the probability that a randomly selected student scored between 450 and 500 is:

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

The number of accidents per week in a town follows a Poisson distribution with mean 2. If the probability that there are three accidents in two weeks time is \( k e^{-6} \), then the value of \( k \) is:

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

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $|\vec{a} + \vec{b}| = 1$, then the value of $|\vec{a} \times \vec{b}|$ is:
 

The curve \( y = \frac{x}{1 + x \tan{x}} \) attains maxima at:
If \( 8^{x-1} = \left( \frac{1}{4} \right)^x \), then the value of \[ \frac{1}{\log_{x+14} x - \log_{x+15}} \div \frac{1}{\log_{1-x} 4 - \log_{1-x} 5} \] is

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:
 

The value of the integral \[ \int_0^{\frac{\pi}{2}} \frac{1 + 2 \cos x}{(2 + \cos x)^2} \, dx \] lies in the interval:
The ratio of the ages of A and B is 5:3. After 6 years, their ages will be in the ratio 6:4. What is the current age of A?
Let the polished side of the mirror be along the line \[ \frac{x}{1} = \frac{1 - y}{2} = \frac{2z - 4}{6}. \] A point \( P(1, 6, 3) \), some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point \( P \) and its image.
Find : \[ I = \int \frac{x + \sin x}{1 + \cos x} \, dx \]

Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:

A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?
A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
If sin θ + cos θ = √2, what is the value of sin θ · cos θ?

Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$, and $y = 0$.