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Papers (38)
2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
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2017
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2016
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2015
paper1 20 paper2 20
2014
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2013
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2012
paper1 1 paper2 2
2011
paper1 2 paper2 9
2010
paper1 28 paper2 19
2009
paper1 19 paper2 19
2008
paper1 23 paper2 22
2007
paper1 23 paper2 5
2009 paper2

19 maths questions

Q20 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
If the sum of first $n$ terms of an A.P. is $cn^{2}$, then the sum of squares of these $n$ terms is
(A) $\frac{n\left(4n^{2}-1\right)c^{2}}{6}$
(B) $\frac{n\left(4n^{2}+1\right)c^{2}}{3}$
(C) $\frac{n\left(4n^{2}-1\right)c^{2}}{3}$
(D) $\frac{n\left(4n^{2}+1\right)c^{2}}{6}$
Q21 Vectors 3D & Lines Line-Plane Intersection View
A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $$2x+y+z=9$$ at point $Q$. The length of the line segment $PQ$ equals
(A) 1
(B) $\sqrt{2}$
(C) $\sqrt{3}$
(D) 2
Q22 Conic sections Locus and Trajectory Derivation View
The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$
Q23 Straight Lines & Coordinate Geometry Locus Determination View
The locus of the orthocentre of the triangle formed by the lines $$\begin{aligned} &(1+p)x-py+p(1+p)=0\\ &(1+q)x-qy+q(1+q)=0 \end{aligned}$$ and $y=0$, where $p\neq q$, is
(A) a hyperbola
(B) a parabola
(C) an ellipse
(D) a straight line
Q24 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
If $$I_{n}=\int_{-\pi}^{\pi}\frac{\sin nx}{\left(1+\pi^{x}\right)\sin x}dx,\quad n=0,1,2,\ldots,$$ then
(A) $I_{n}=I_{n+2}$
(B) $\sum_{m=1}^{10}I_{2m+1}=10\pi$
(C) $\sum_{m=1}^{10}I_{2m}=0$
(D) $I_{n}=I_{n+1}$
Q25 Conic sections Equation Determination from Geometric Conditions View
An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
(A) Equation of ellipse is $x^{2}+2y^{2}=2$
(B) The foci of ellipse are $(\pm1,0)$
(C) Equation of ellipse is $x^{2}+2y^{2}=4$
(D) The foci of ellipse are $(\pm\sqrt{2},0)$
Q26 Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$
Q27 Conic sections Locus and Trajectory Derivation View
The tangent $PT$ and the normal $PN$ to the parabola $y^{2}=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose
(A) vertex is $\left(\frac{2a}{3},0\right)$
(B) directrix is $x=0$
(C) latus rectum is $\frac{2a}{3}$
(D) focus is $(a,0)$
Q28 Trigonometric equations in context View
For $0<\theta<\frac{\pi}{2}$, the solution(s) of $$\sum_{m=1}^{6}\operatorname{cosec}\left(\theta+\frac{(m-1)\pi}{4}\right)\operatorname{cosec}\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$ is(are)
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{12}$
(D) $\frac{5\pi}{12}$
Q29 Vectors: Cross Product & Distances View
Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) Root(s) of the equation $$2\sin^{2}\theta+\sin^{2}2\theta=2$$ (B) Points of discontinuity of the function $$f(x)=\left[\frac{6x}{\pi}\right]\cos\left[\frac{3x}{\pi}\right],$$ where $[y]$ denotes the largest integer less than or equal to $y$
(C) Volume of the parallelopiped with its edges represented by the vectors $$\hat{i}+\hat{j},\quad\hat{i}+2\hat{j}\text{ and }\hat{i}+\hat{j}+\pi\hat{k}$$ (D) Angle between vectors $\vec{a}$ and $\vec{b}$ where $\vec{a},\vec{b}$ and $\vec{c}$ are unit vectors satisfying $$\vec{a}+\vec{b}+\sqrt{3}\vec{c}=\overrightarrow{0}$$
Column II
(p) $\frac{\pi}{6}$
(q) $\frac{\pi}{4}$
(r) $\frac{\pi}{3}$
(s) $\frac{\pi}{2}$
(t) $\pi$
Q30 Differential equations Solving Separable DEs with Initial Conditions View
Match the statements/expressions given in Column I with the values given in Column II.
Column I
(A) The number of solutions of the equation $$xe^{\sin x}-\cos x=0$$ in the interval $\left(0,\frac{\pi}{2}\right)$
(B) Value(s) of $k$ for which the planes $kx+4y+z=0,4x+ky+2z=0$ and $2x+2y+z=0$ intersect in a straight line
(C) Value(s) of $k$ for which $$|x-1|+|x-2|+|x+1|+|x+2|=4k$$ has integer solution(s)
(D) If $$y^{\prime}=y+1\text{ and }y(0)=1$$ then value(s) of $y(\ln2)$
Column II
(p) 1
(q) 2
(r) 3
(s) 4
(t) 5
Q31 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
The maximum value of the function $f(x)=2x^{3}-15x^{2}+36x-48$ on the set $A=\left\{x\mid x^{2}+20\leq9x\right\}$ is
Q32 Matrices Linear System and Inverse Existence View
Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations: $$\begin{array}{r} 3x-y-z=0\\ -3x+z=0\\ -3x+2y+z=0 \end{array}$$ Then the number of such points for which $x^{2}+y^{2}+z^{2}\leq100$ is
Q33 Sine and Cosine Rules Ambiguous case and triangle existence/uniqueness View
Let $ABC$ and $ABC^{\prime}$ be two non-congruent triangles with sides $AB=4$, $AC=AC^{\prime}=2\sqrt{2}$ and angle $B=30^{\circ}$. The absolute value of the difference between the areas of these triangles is
Q34 Stationary points and optimisation Determine parameters from given extremum conditions View
Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $$\lim_{x\rightarrow0}\left(1+\frac{p(x)}{x^{2}}\right)=2.$$ Then the value of $p(2)$ is
Q35 Differential equations Integral Equations Reducible to DEs View
Let $f:\mathbf{R}\rightarrow\mathbf{R}$ be a continuous function which satisfies $$f(x)=\int_{0}^{x}f(t)\,dt.$$ Then the value of $f(\ln5)$ is
Q36 Circles Circles Tangent to Each Other or to Axes View
The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$, then the radius of the circle $C$ is
Q37 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
The smallest value of $k$, for which both the roots of the equation $$x^{2}-8kx+16\left(k^{2}-k+1\right)=0$$ are real, distinct and have values at least 4, is
Q38 Composite & Inverse Functions Derivative of an Inverse Function View
If the function $f(x)=x^{3}+e^{\frac{x}{2}}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is