Mathematics Previous Year Questions 2025 (2025–2025)

375 Mathematics previous-year questions spanning 2025–2025.

Mathematics 2025 · Three Dimensional Geometry

If the image of the point $\mathrm{P}(1, 0, 3)$ in the line joining the points $\mathrm{A}(4, 7, 1)$ and $\mathrm{B}(3, 5, 3)$ is $\mathrm{Q}(\alpha, \beta, \gamma)$, then $\alpha + \beta + \gamma$ is equal to

Mathematics 2025 · Differential Equations

Let $f:[1,\infty) \to [2,\infty)$ be a differentiable function. If $10\int_{1}^{x}f(t)\mathrm{d}t = 5xf(x) - x^{5} - 9$ for all $x \geq 1$, then the value of $f(3)$ is:

Mathematics 2025 · Sequences and Series

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

Mathematics 2025 · Relations and Functions

Let $\mathrm{A} = \{1, 2, 3, \dots, 100\}$ and $\mathrm{R}$ be a relation on $\mathrm{A}$ such that $\mathrm{R} = \{(a, b) : a = 2b + 1\}$. Let $(a_1, a_2), (a_2, a_3), (a_3, a_4), \dots, (a_k, a_{k+1})$ be a sequence of $k$ elements of $\mathrm{R}$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer $k$, for which such a sequence exists, is equal to:

Mathematics 2025 · Conic Sections

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

Mathematics 2025 · Three Dimensional Geometry

The line $\mathrm{L}_1$ is parallel to the vector $\vec{\mathrm{a}} = -3\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$ and passes through the point $(7, 6, 2)$ and the line $\mathrm{L}_2$ is parallel to the vector $\vec{\mathrm{b}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}$ and passes through the point $(5, 3, 4)$. The shortest distance between the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ is:

Mathematics 2025 · Integral Calculus

Let $(a, b)$ be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_{a}^{b} \frac{9x^2}{1 + 5^x} \, dx$ is equal to:

Mathematics 2025 · Matrices and Determinants

If the system of equations $$\begin{aligned} 2x + \lambda y + 3z &= 5 \\ 3x + 2y - z &= 7 \\ 4x + 5y + \mu z &= 9 \end{aligned}$$ has infinitely many solutions, then $(\lambda^2 + \mu^2)$ is equal to:

Mathematics 2025 · Trigonometric Functions

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

Mathematics 2025 · Probability

Given three identical bags each containing 10 balls, whose colours are as follows: $$\begin{array}{|l|l|l|l|} \hline & \textbf{Red} & \textbf{Blue} & \textbf{Green} \\ \hline \textbf{Bag I} & 3 & 2 & 5 \\ \hline \textbf{Bag II} & 4 & 3 & 3 \\ \hline \textbf{Bag III} & 5 & 1 & 4 \\ \hline \end{array}$$ A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is $p$ and if the ball is Green, the probability that it is from bag III is $q$, then the value of $\left(\frac{1}{p} + \frac{1}{q}\right)$ is:

Mathematics 2025 · Statistics

If the mean and the variance of $6, 4, a, 8, b, 12, 10, 13$ are $9$ and $9.25$ respectively, then $a + b + ab$ is equal to :

Mathematics 2025 · Relations and Functions

If the domain of the function $$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$$ is $(a, b)$, then $(1 + a)^2 + b^2$ is equal to:

Mathematics 2025 · Integral Calculus

$4\int_{0}^{1}\left(\frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}}\right)\mathrm{d}x - 3\log_{e}\left(\sqrt{3}\right)$ is equal to:

Mathematics 2025 · Limits, Continuity and Differentiability

If $\lim_{x\to 0}\frac{\cos(2x) + a\cos(4x) - b}{x^4}$ is finite, then $(a + b)$ is equal to:

Mathematics 2025 · Binomial Theorem

If $\sum_{\mathrm{r} = 0}^{10}\left(\frac{10^{\mathrm{r} + 1} - 1}{10^{\mathrm{r}}}\right) \cdot \binom{11}{r + 1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$, then $\alpha$ is equal to:

Mathematics 2025 · Permutations and Combinations

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is: {{IMG1}}

Mathematics 2025 · Conic Sections

Let the point $\mathrm{P}$ of the focal chord $\mathrm{PQ}$ of the parabola $\mathrm{y}^2 = 16\mathrm{x}$ be $(1, -4)$. If the focus of the parabola divides the chord $\mathrm{PQ}$ in the ratio $\mathrm{m} : \mathrm{n}$, $\gcd(\mathrm{m}, \mathrm{n}) = 1$, then $\mathrm{m}^2 + \mathrm{n}^2$ is equal to:

Mathematics 2025 · Vector Algebra

Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{b} \times \vec{c} = \vec{d}$, then $|\vec{a} \cdot \vec{d}|$ is equal to:

Mathematics 2025 · Straight Lines

Let the area of the triangle formed by a straight Line $L: x + by + c = 0$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^\circ$ with the positive $x$-axis, then the value of $b^2 + c^2$ is:

Mathematics 2025 · Matrices and Determinants

Let A be a $3 \times 3$ real matrix such that $\mathrm{A}^2 (\mathrm{A} - 2\mathrm{I}) - 4(\mathrm{A} - \mathrm{I}) = \mathrm{O}$, where I and O are the identity and null matrices, respectively. If $\mathrm{A}^5 = \alpha \mathrm{A}^2 + \beta \mathrm{A} + \gamma \mathrm{I}$, where $\alpha, \beta$ and $\gamma$ are real constants, then $\alpha + \beta + \gamma$ is equal to:
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