Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{\text{th}}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to (1) $3 + \sqrt{3}$ (2) 4 (3) $4 - \sqrt{3}$ (4) 3
Q2
Vectors 3D & LinesDistance from a Point to a Line (Show/Compute)View
Let in a $\triangle ABC$, the length of the side $AC$ be 6, the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3} = \frac{y-7}{2} = \frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle ABC$ is: (1) 17 (2) 21 (3) 56 (4) 42
Q3
Conic sectionsEccentricity or Asymptote ComputationView
Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is (1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$ (2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$ (3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$ (4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$
Q4
MatricesLinear System and Inverse ExistenceView
If the system of equations $$2x - y + z = 4$$ $$5x + \lambda y + 3z = 12$$ $$100x - 47y + \mu z = 212$$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to (1) 57 (2) 59 (3) 55 (4) 56
Q5
Binomial Theorem (positive integer n)Determine Parameters from Conditions on Coefficients or TermsView
For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is: (1) 20 (2) 10 (3) 35 (4) 70
Q6
Solving quadratics and applicationsPolynomial identity or factoring to simplify a given expressionView
The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x-4)(x-5) = 3$, is equal to (1) 14 (2) 21 (3) 28 (4) 7
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 (1) 5 (2) $5\sqrt{5}$ (3) $5\sqrt{6}$ (4) 10
Q8
Straight Lines & Coordinate GeometryCollinearity and ConcurrencyView
Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to (1) 84 (2) 113 (3) 91 (4) 101
Q9
Complex numbers 2Complex Function Evaluation and Algebraic ManipulationView
If $\alpha$ and $\beta$ are the roots of the equation $2z^2 - 3z - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \operatorname{Re}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{15} + \beta^{15}}\right)$ is equal to (1) 441 (2) 398 (3) 312 (4) 409
For a statistical data $x_1, x_2, \ldots, x_{10}$ 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 (1) 9 (2) 5 (3) 7 (4) 4
Q11
Areas Between CurvesArea Involving Piecewise or Composite FunctionsView
The area of the region $\left\{(x, y) : x^2 + 4x + 2 \leq y \leq |x+2|\right\}$ is equal to (1) 7 (2) 5 (3) $24/5$ (4) $20/3$
Q12
Arithmetic Sequences and SeriesCompute Partial Sum of an Arithmetic SequenceView
Let $S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026 S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is (1) 20 (2) 90 (3) 45 (4) 25
Q13
Composite & Inverse FunctionsRecover a Function from a Composition or Functional EquationView
Let $f : \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If $\lim_{x \rightarrow 0}\left(\frac{1}{\alpha x} + f(x)\right) = \beta$; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to (1) 5 (2) 3 (3) 4 (4) 6
Q14
Indefinite & Definite IntegralsDefinite Integral Evaluation (Computational)View
If $I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m, n > 0$, then $I(9, 14) + I(10, 13)$ is (1) $I(19, 27)$ (2) $I(9, 1)$ (3) $I(1, 13)$ (4) $I(9, 13)$
$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability that $A$ wins if $A$ makes the first throw, is (1) $\frac{8}{17}$ (2) $\frac{9}{19}$ (3) $\frac{9}{17}$ (4) $\frac{8}{19}$
Q16
Indefinite & Definite IntegralsIntegral Equation with Symmetry or SubstitutionView
Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)$ is equal to (1) 92 (2) 118 (3) 102 (4) 108
Q17
Differential equationsSolving Separable DEs with Initial ConditionsView
Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1+x^2}\right)dx + \left(1+x^2\right)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to (1) $\sqrt{\frac{15}{2}}$ (2) $\frac{5\sqrt{3}}{2}$ (3) $2\sqrt{2}$ (4) $\sqrt{\frac{14}{3}}$
$\lim_{x \rightarrow 0} \operatorname{cosec} x \left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is: (1) 0 (2) $\frac{1}{\sqrt{15}}$ (3) $\frac{1}{2\sqrt{5}}$ (4) $-\frac{1}{2\sqrt{5}}$
Q19
Stationary points and optimisationGeometric or applied optimisation problemView
Consider the region $R = \left\{(x, y) : x \leq y \leq 9 - \frac{11}{3}x^2,\, x \geq 0\right\}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is: (1) $\frac{730}{119}$ (2) $\frac{625}{111}$ (3) $\frac{821}{123}$ (4) $\frac{567}{121}$
Q20
Vectors 3D & LinesVector Algebra and Triple Product ComputationView
Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to (1) $\sqrt{\frac{11}{6}}$ (2) $\frac{1}{3\sqrt{2}}$ (3) 16 (4) 18
Q21
Number TheoryCombinatorial Number Theory and CountingView
Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is $\underline{\hspace{2cm}}$.
Q22
Standard trigonometric equationsInverse trigonometric equationView
If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1}\alpha) + \operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2 + \beta$ is $\underline{\hspace{2cm}}$.
Let $A$ be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix},\quad A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix},$$ and $\det(\operatorname{adj}(2(A+I))) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is $\underline{\hspace{2cm}}$.
Q24
Differential equationsIntegral Equations Reducible to DEsView
Let $f$ be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt$, $x \geq 0$. Then $f(2)$ is equal to $\underline{\hspace{2cm}}$.
Q25
Number TheoryCombinatorial Number Theory and CountingView
The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is $\underline{\hspace{2cm}}$.