List of top Mathematics Questions

Let $ S $ denote the locus of the midpoints of those chords of the parabola $ y^2 = x $, such that the area of the region enclosed between the parabola and the chord is $ \frac{4}{3} $. Let $ \mathcal{R} $ denote the region lying in the first quadrant, enclosed by the parabola $ y^2 = x $, the curve $ S $, and the lines $ x = 1 $ and $ x = 4 $.
Then which of the following statements is (are) TRUE?

Let $ \vec{w} = \hat{i} + \hat{j} - 2\hat{k} $, and $ \vec{u} $ and $ \vec{v} $ be two vectors, such that $ \vec{u} \times \vec{v} = \vec{w} $ and $ \vec{v} \times \vec{w} = \vec{u} $. Let $ \alpha, \beta, \gamma $, and $ t $ be real numbers such that: $$ \vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}, $$ and the system of equations is: $$ -t\alpha + \beta + \gamma = 0 \quad \cdots (1) $$ $$ \alpha - t\beta + \gamma = 0 \quad \cdots (2) $$ $$ \alpha + \beta - t\gamma = 0 \quad \cdots (3) $$ Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I

[(P)] $ |\vec{v}|^2 $ is equal to
[(Q)] If $ \alpha = \sqrt{3} $, then $ \gamma^2 $ is equal to
[(R)] If $ \alpha = \sqrt{3} $, then $ (\beta + \gamma)^2 $ is equal to
[(S)] If $ \alpha = \sqrt{2} $, then $ t + 3 $ is equal to

List-II

[(1)] 0
[(2)] 1
[(3)] 2
[(4)] 3
[(5)]

Consider the following frequency distribution:

Value	4	5	8	9	6	12	11
Frequency	5	$ f_1 $	$ f_2 $	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6.

For the given frequency distribution, let:

$ \alpha $ denote the mean deviation about the mean
$ \beta $ denote the mean deviation about the median
$ \sigma^2 $ denote the variance

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I

(P) $ 7f_1 + 9f_2 $ is equal to
(Q) $ 19\alpha $ is equal to
(R) $ 19\beta $ is equal to
(S) $ 19\sigma^2 $ is equal to

List-II

(1) 146
(2) 47
(3) 48
(4) 145
(5) 55

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ z_1 = 1 + 2i $ and $ z_2 = 3i $ be two complex numbers, where $ i = \sqrt{-1} $. Let $$ S = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + iy - z_1| = 2|x + iy - z_2|\}. $$ Then which of the following statements is (are) TRUE?

Let $ \mathbb{N} $ denote the set of all natural numbers, and $ \mathbb{Z} $ denote the set of all integers. Consider the functions $ f : \mathbb{N} \to \mathbb{Z} $ and $ g : \mathbb{Z} \to \mathbb{N} $ defined by $$ f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{4 - n}{2}, & \text{if } n \text{ is even} \end{cases} \quad \text{and} \quad g(n) = \begin{cases} 3 + 2n, & \text{if } n \ge 0 \\ -2n, & \text{if } n<0 \end{cases} $$ Define $ (g \circ f)(n) = g(f(n)) $ for all $ n \in \mathbb{N} $, and $ (f \circ g)(n) = f(g(n)) $ for all $ n \in \mathbb{Z} $. Then which of the following statements is (are) TRUE?

Consider the matrix $$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$ Let the transpose of a matrix $ X $ be denoted by $ X^T $. Then the number of $ 3 \times 3 $ invertible matrices $ Q $ with integer entries, such that $$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$ is:

Let $ \mathbb{R} $ denote the set of all real numbers. Define the function $ f: \mathbb{R} \to \mathbb{R} $ by $$ f(x) = \begin{cases} 2 - 2x^2 - x^2 \sin\left(\frac{1}{x}\right), & \text{if } x \ne 0, \\ 2, & \text{if } x = 0. \end{cases} $$ Then which one of the following statements is TRUE?

Let $ \mathbb{R} $ denote the set of all real numbers. For a real number $ x $, let $ \lfloor x \rfloor $ denote the greatest integer less than or equal to $ x $. Let $ n $ denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I

[(P)] The minimum value of $ n $ for which the function $$ f(x) = \left\lfloor \frac{10x^3 - 45x^2 + 60x + 35}{n} \right\rfloor $$ is continuous on the interval $ [1, 2] $, is
[(Q)] The minimum value of $ n $ for which $$ g(x) = (2n^2 - 13n - 15)(x^3 + 3x), $$ $ x \in \mathbb{R} $, is an increasing function on $ \mathbb{R} $, is
[(R)] The smallest natural number $ n $ which is greater than 5, such that $ x = 3 $ is a point of local minima of $$ h(x) = (x^2 - 9)^n (x^2 + 2x + 3), $$ is
[(S)] Number of $ x_0 \in \mathbb{R} $ such that $$ l(x) = \sum_{k=0}^{4} \left( \sin |x - k| + \cos \left| x - k + \frac{1}{2} \right| \right) $$ is not differentiable at $ x_0 $, is

List-II

[(1)] 8
[(2)] 9
[(3)] 5
[(4)] 6
[(5)] 10

Options:

For any two points $ M $ and $ N $ in the $ XY $-plane, let $ \overrightarrow{MN} $ denote the vector from $ M $ to $ N $, and $ \vec{0} $ denote the zero vector. Let $ P, Q $, and $ R $ be three distinct points in the $ XY $-plane. Let $ S $ be a point inside the triangle $ \Delta PQR $ such that $$ \overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}. $$ Let $ E $ and $ F $ be the mid-points of the sides $ PR $ and $ QR $, respectively. Then the value of $$ \frac{\text{length of the line segment } EF}{\text{length of the line segment } ES} $$ is __________.

Let the set of all relations $ R $ on the set $ \{a, b, c, d, e, f\} $, such that $ R $ is reflexive and symmetric, and $ R $ contains exactly 10 elements, be denoted by $ S $. Then the number of elements in $ S $ is ___.

Let $ \mathbb{R} $ denote the set of all real numbers. Let $ f: \mathbb{R} \to \mathbb{R} $ be a function such that $ f(x)>0 $ for all $ x \in \mathbb{R} $, and $ f(x + y) = f(x)f(y) $ for all $ x, y \in \mathbb{R} $.
Let the real numbers $ a_1, a_2, \ldots, a_{50} $ be in an arithmetic progression. If $ f(a_{31}) = 64f(a_{25}) $, and $$ \sum_{i=1}^{50} f(a_i) = 3(2^{25} + 1), $$ then the value of $$ \sum_{i=6}^{30} f(a_i) $$ is __________.

Let $ \alpha $ and $ \beta $ be the real numbers such that $$ \lim_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int_{0}^{x} \frac{1}{1 - t^2} \, dt + \beta x \cos x \right) = 2. $$ Then the value of $ \alpha + \beta $ is __________.

Let $ S $ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $ S $, but 0210222 is NOT in $ S $.
Then the number of elements $ x $ in $ S $ such that at least one of the digits 0 and 1 appears exactly twice in $ x $, is equal to __________.

For all $ x>0 $, let $ y_1(x), y_2(x), y_3(x) $ be the functions satisfying $$ \frac{dy_1}{dx} - (\sin x)^2 y_1 = 0, \quad y_1(1) = 5, $$ $$ \frac{dy_2}{dx} - (\cos x)^2 y_2 = 0, \quad y_2(1) = \frac{1}{3}, $$ $$ \frac{dy_3}{dx} - \left(\frac{2 - x^3}{x^3}\right) y_3 = 0, \quad y_3(1) = \frac{3}{5e}, $$ respectively. Then $$ \lim_{x \to 0^+} \frac{y_1(x)y_2(x)y_3(x) + 2x}{e^{3x} \sin x} $$ is equal to __________.

Let $ A $ be a $ 3 \times 3 $ matrix such that $ X^T AX = 0 $ for all nonzero $ 3 \times 1 $ matrices $ X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} $. \[ A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] If $ \det(\text{adj}(2(A + I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} $, $ \alpha, \beta, \gamma \in \mathbb{N} $, then $ \alpha^2 + \beta^2 + \gamma^2 $ is:

Evaluate the definite integral: \[ I = \int_{\frac{1}{2}}^{2} \frac{\tan^{-1} x \, dx}{2x^2 - 3x + 2} \]

Let \[ f(x) = \begin{cases} \frac{7}{2}, & x = 0 \\ \frac{6x + \sin x}{2x + \sin x}, & x \neq 0 \end{cases} \] Then, which of the following statements are correct?

Let \[ \left(1 + \frac{2}{5}x\right)^{23} = \sum_{i=0}^{23} a_i x^i \] If the maximum value of $a_i$ occurs at $i = r$, find the value of $r$.

Let a complex number $z$ satisfy the equation \[ |z - z_1| = 2 |z - z_2|,\quad \text{where } z_1 = 1 + 2i,\ z_2 = 3i \] Then choose the correct options:

If \[ \alpha = \frac{1}{\sin 60^\circ \sin 61^\circ} + \frac{1}{\sin 62^\circ \sin 63^\circ} + \dots + \frac{1}{\sin 118^\circ \sin 119^\circ} \] Then find the value of $\dfrac{\csc^2 1^\circ}{\alpha^2}$

Given the differential equation: \[ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad \text{with } y(1) = 0 \] Find the value of $\frac{y^2(e)}{y(e^2)}$.

Evaluate the limit: \[ \lim_{x \to 0} \frac{\frac{\alpha}{2} \int_0^x \frac{dt}{1 - t^2} + \beta x \cos x}{x^3} = \gamma \]

The derivative of $\sin\left(\tan^{-1} e^{2x}\right)$ with respect to $x$ 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 { or } x \in B \} $, then $ n(S) $ is equal to:

Solve for $ a $ and $ b $ given the equations: \[ \sin x + \sin y = a, \quad \cos x + \cos y = b, \quad x + y = \frac{2\pi}{3} \]