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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
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2014
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2013
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2012
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2011
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2010
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2009
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2008
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2007
paper1 23 paper2 5
2017 paper1

18 maths questions

Q37 Conic sections Eccentricity or Asymptote Computation View
If $2x - y + 1 = 0$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{16} = 1$, then which of the following CANNOT be sides of a right angled triangle?
[A] $a, 4, 1$
[B] $a, 4, 2$
[C] $2a, 8, 1$
[D] $2a, 4, 1$
Q38 Circles Chord Length and Chord Properties View
If a chord, which is not a tangent, of the parabola $y^2 = 16x$ has the equation $2x + y = p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p$, $h$ and $k$?
[A] $p = -2, h = 2, k = -4$
[B] $p = -1, h = 1, k = -3$
[C] $p = 2, h = 3, k = -4$
[D] $p = 5, h = 4, k = -3$
Q39 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x) = x\cos(\pi(x + [x]))$ is discontinuous?
[A] $x = -1$
[B] $x = 0$
[C] $x = 1$
[D] $x = 2$
Q40 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f : \mathbb{R} \rightarrow (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$?
[A] $x^9 - f(x)$
[B] $x - \int_0^{\frac{\pi}{2} - x} f(t)\cos t\, dt$
[C] $e^x - \int_0^x f(t)\sin t\, dt$
[D] $f(x) + \int_0^{\frac{\pi}{2}} f(t)\sin t\, dt$
Q41 Matrices Diagonalizability and Similarity View
Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?
[A] $\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$
[B] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[C] $\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
[D] $\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]$
Q42 Complex Numbers Arithmetic Systems of Equations via Real and Imaginary Part Matching View
Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq 0$. If the complex number $z = x + iy$ satisfies $\operatorname{Im}\left(\frac{az + b}{z + 1}\right) = y$, then which of the following is(are) possible value(s) of $x$?
[A] $-1 + \sqrt{1 - y^2}$
[B] $-1 - \sqrt{1 - y^2}$
[C] $1 + \sqrt{1 + y^2}$
[D] $1 - \sqrt{1 + y^2}$
Q43 Conditional Probability Direct Conditional Probability Computation from Definitions View
Let $X$ and $Y$ be two events such that $P(X) = \frac{1}{3}$, $P(X \mid Y) = \frac{1}{2}$ and $P(Y \mid X) = \frac{2}{5}$. Then
[A] $P(Y) = \frac{4}{15}$
[B] $P(X' \mid Y) = \frac{1}{2}$
[C] $P(X \cap Y) = \frac{1}{5}$
[D] $P(X \cup Y) = \frac{2}{5}$
Q44 Circles Circle-Line Intersection and Point Conditions View
For how many values of $p$, the circle $x^2 + y^2 + 2x + 4y - p = 0$ and the coordinate axes have exactly three common points?
Q45 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$, $f\left(\frac{\pi}{2}\right) = 3$ and $f'(0) = 1$. If $$g(x) = \int_x^{\frac{\pi}{2}} \left[f'(t)\operatorname{cosec} t - \cot t\operatorname{cosec} t\, f(t)\right] dt$$ for $x \in \left(0, \frac{\pi}{2}\right]$, then $\lim_{x \rightarrow 0} g(x) =$
Q46 Simultaneous equations View
For a real number $\alpha$, if the system $$\left[\begin{array}{ccc}1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right] = \left[\begin{array}{r}1 \\ -1 \\ 1\end{array}\right]$$ of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 =$
Q47 Permutations & Arrangements Word Permutations with Repeated Letters View
Words of length 10 are formed using the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}, \mathrm{H}, \mathrm{I}, \mathrm{J}$. Let $x$ be the number of such words where no letter is repeated; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9x} =$
Q48 Sine and Cosine Rules Compute area of a triangle or related figure View
The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?
Q49 Circles Tangent Lines and Tangent Lengths View
Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
Column 1Column 2Column 3
(I) $x^2 + y^2 = a^2$(i) $my = m^2x + a$(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)$x^2 + a^2y^2 = a^2$(ii) $y = mx + a\sqrt{m^2+1}$(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)$y^2 = 4ax$(iii) $y = mx + \sqrt{a^2m^2 - 1}$(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)$x^2 - a^2y^2 = a^2$(iv) $y = mx + \sqrt{a^2m^2+1}$(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
[A] (I) (i) (P)
[B] (I) (ii) (Q)
[C] (II) (ii) (Q)
[D] (III) (i) (P)
Q50 Circles Tangent Lines and Tangent Lengths View
Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
Column 1Column 2Column 3
(I) $x^2 + y^2 = a^2$(i) $my = m^2x + a$(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)$x^2 + a^2y^2 = a^2$(ii) $y = mx + a\sqrt{m^2+1}$(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)$y^2 = 4ax$(iii) $y = mx + \sqrt{a^2m^2 - 1}$(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)$x^2 - a^2y^2 = a^2$(iv) $y = mx + \sqrt{a^2m^2+1}$(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?
[A] (I) (ii) (Q)
[B] (II) (iv) (R)
[C] (III) (i) (P)
[D] (III) (ii) (Q)
Q51 Conic sections Tangent and Normal Line Problems View
Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
Column 1Column 2Column 3
(I) $x^2 + y^2 = a^2$(i) $my = m^2x + a$(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)$x^2 + a^2y^2 = a^2$(ii) $y = mx + a\sqrt{m^2+1}$(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)$y^2 = 4ax$(iii) $y = mx + \sqrt{a^2m^2 - 1}$(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)$x^2 - a^2y^2 = a^2$(iv) $y = mx + \sqrt{a^2m^2+1}$(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

The tangent to a suitable conic (Column 1) at $\left(\sqrt{3}, \frac{1}{2}\right)$ is found to be $\sqrt{3}x + 2y = 4$, then which of the following options is the only CORRECT combination?
[A] (IV) (iii) (S)
[B] (IV) (iv) (S)
[C] (II) (iii) (R)
[D] (II) (iv) (R)
Q52 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.
Column 1Column 2Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$(i) $\lim_{x\to\infty} f(x) = 0$(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$(ii) $\lim_{x\to\infty} f(x) = -\infty$(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$(iii) $\lim_{x\to\infty} f'(x) = -\infty$(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$(iv) $\lim_{x\to\infty} f''(x) = 0$(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (i) (P)
[B] (II) (ii) (Q)
[C] (III) (iii) (R)
[D] (IV) (iv) (S)
Q53 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.
Column 1Column 2Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$(i) $\lim_{x\to\infty} f(x) = 0$(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$(ii) $\lim_{x\to\infty} f(x) = -\infty$(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$(iii) $\lim_{x\to\infty} f'(x) = -\infty$(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$(iv) $\lim_{x\to\infty} f''(x) = 0$(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (ii) (R)
[B] (II) (iii) (S)
[C] (III) (iv) (P)
[D] (IV) (i) (S)
Q54 Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.
Column 1Column 2Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$(i) $\lim_{x\to\infty} f(x) = 0$(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$(ii) $\lim_{x\to\infty} f(x) = -\infty$(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$(iii) $\lim_{x\to\infty} f'(x) = -\infty$(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$(iv) $\lim_{x\to\infty} f''(x) = 0$(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only INCORRECT combination?
[A] (I) (iii) (P)
[B] (II) (iv) (Q)
[C] (III) (i) (R)
[D] (II) (iii) (P)