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Q1 3 marks Differential equations Integral Equations Reducible to DEs View

Let $f : [ 1 , \infty ) \rightarrow \mathbb { R }$ be a differentiable function such that $f ( 1 ) = \frac { 1 } { 3 }$ and $3 \int _ { 1 } ^ { x } f ( t ) d t = x f ( x ) - \frac { x ^ { 3 } } { 3 } , x \in [ 1 , \infty )$. Let $e$ denote the base of the natural logarithm. Then the value of $f ( e )$ is
(A) $\frac { e ^ { 2 } + 4 } { 3 }$
(B) $\frac { \log _ { e } 4 + e } { 3 }$
(C) $\frac { 4 e ^ { 2 } } { 3 }$
(D) $\frac { e ^ { 2 } - 4 } { 3 }$

Q2 3 marks Probability Definitions Probability Using Set/Event Algebra View

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\frac { 1 } { 3 }$, then the probability that the experiment stops with head is
(A) $\frac { 1 } { 3 }$
(B) $\frac { 5 } { 21 }$
(C) $\frac { 4 } { 21 }$
(D) $\frac { 2 } { 7 }$

Q3 3 marks Standard trigonometric equations Inverse trigonometric equation View

For any $y \in \mathbb { R }$, let $\cot ^ { - 1 } ( y ) \in ( 0 , \pi )$ and $\tan ^ { - 1 } ( y ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the sum of all the solutions of the equation $\tan ^ { - 1 } \left( \frac { 6 y } { 9 - y ^ { 2 } } \right) + \cot ^ { - 1 } \left( \frac { 9 - y ^ { 2 } } { 6 y } \right) = \frac { 2 \pi } { 3 }$ for $0 < | y | < 3$, is equal to
(A) $2 \sqrt { 3 } - 3$
(B) $3 - 2 \sqrt { 3 }$
(C) $4 \sqrt { 3 } - 6$
(D) $6 - 4 \sqrt { 3 }$

Q4 3 marks Vectors 3D & Lines Section Division and Coordinate Computation View

Let the position vectors of the points $P , Q , R$ and $S$ be $\vec { a } = \hat { i } + 2 \hat { j } - 5 \hat { k } , \vec { b } = 3 \hat { i } + 6 \hat { j } + 3 \hat { k }$, $\vec { c } = \frac { 17 } { 5 } \hat { i } + \frac { 16 } { 5 } \hat { j } + 7 \hat { k }$ and $\vec { d } = 2 \hat { i } + \hat { j } + \hat { k }$, respectively. Then which of the following statements is true?
(A) The points $P , Q , R$ and $S$ are NOT coplanar
(B) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ internally in the ratio $5 : 4$
(C) $\frac { \vec { b } + 2 \vec { d } } { 3 }$ is the position vector of a point which divides $P R$ externally in the ratio $5 : 4$
(D) The square of the magnitude of the vector $\vec { b } \times \vec { d }$ is 95

Q5 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View

Let $M = \left( a _ { i j } \right) , i , j \in \{ 1,2,3 \}$, be the $3 \times 3$ matrix such that $a _ { i j } = 1$ if $j + 1$ is divisible by $i$, otherwise $a _ { i j } = 0$. Then which of the following statements is(are) true?
(A) $M$ is invertible
(B) There exists a nonzero column matrix $\left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right)$ such that $M \left( \begin{array} { l } a _ { 1 } \\ a _ { 2 } \\ a _ { 3 } \end{array} \right) = \left( \begin{array} { l } - a _ { 1 } \\ - a _ { 2 } \\ - a _ { 3 } \end{array} \right)$
(C) The set $\left\{ X \in \mathbb { R } ^ { 3 } : M X = \mathbf { 0 } \right\} \neq \{ \mathbf { 0 } \}$, where $\mathbf { 0 } = \left( \begin{array} { l } 0 \\ 0 \\ 0 \end{array} \right)$
(D) The matrix $( M - 2 I )$ is invertible, where $I$ is the $3 \times 3$ identity matrix

Q6 4 marks Curve Sketching Continuity and Differentiability of Special Functions View

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = [ 4 x ] \left( x - \frac { 1 } { 4 } \right) ^ { 2 } \left( x - \frac { 1 } { 2 } \right)$, where $[ x ]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
(A) The function $f$ is discontinuous exactly at one point in $( 0,1 )$
(B) There is exactly one point in $( 0,1 )$ at which the function $f$ is continuous but NOT differentiable
(C) The function $f$ is NOT differentiable at more than three points in $( 0,1 )$
(D) The minimum value of the function $f$ is $- \frac { 1 } { 512 }$

Q7 4 marks Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $S$ be the set of all twice differentiable functions $f$ from $\mathbb { R }$ to $\mathbb { R }$ such that $\frac { d ^ { 2 } f } { d x ^ { 2 } } ( x ) > 0$ for all $x \in ( - 1,1 )$. For $f \in S$, let $X _ { f }$ be the number of points $x \in ( - 1,1 )$ for which $f ( x ) = x$. Then which of the following statements is(are) true?
(A) There exists a function $f \in S$ such that $X _ { f } = 0$
(B) For every function $f \in S$, we have $X _ { f } \leq 2$
(C) There exists a function $f \in S$ such that $X _ { f } = 2$
(D) There does NOT exist any function $f$ in $S$ such that $X _ { f } = 1$

Q8 4 marks Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View

For $x \in \mathbb { R }$, let $\tan ^ { - 1 } ( x ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the minimum value of the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { x \tan ^ { - 1 } x } \frac { e ^ { ( t - \cos t ) } } { 1 + t ^ { 2023 } } d t$ is

Q9 4 marks First order differential equations (integrating factor) View

For $x \in \mathbb { R }$, let $y ( x )$ be a solution of the differential equation $$\left( x ^ { 2 } - 5 \right) \frac { d y } { d x } - 2 x y = - 2 x \left( x ^ { 2 } - 5 \right) ^ { 2 }$$ such that $y ( 2 ) = 7$. Then the maximum value of the function $y ( x )$ is

Q10 4 marks Permutations & Arrangements Probability via Permutation Counting View

Let $X$ be the set of all five digit numbers formed using $1,2,2,2,4,4,0$. For example, 22240 is in $X$ while 02244 and 44422 are not in $X$. Suppose that each element of $X$ has an equal chance of being chosen. Let $p$ be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of $38p$ is equal to

Q11 4 marks Complex numbers 2 Roots of Unity and Cyclotomic Properties View

Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots , A _ { 8 }$ be the vertices of a regular octagon that lie on a circle of radius 2. Let $P$ be a point on the circle and let $P A _ { i }$ denote the distance between the points $P$ and $A _ { i }$ for $i = 1,2 , \ldots , 8$. If $P$ varies over the circle, then the maximum value of the product $P A _ { 1 } \cdot P A _ { 2 } \cdots P A _ { 8 }$, is

Q12 4 marks Matrices Determinant and Rank Computation View

Let $R = \left\{ \left( \begin{array} { c c c } a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{array} \right) : a , b , c , d \in \{ 0,3,5,7,11,13,17,19 \} \right\}$. Then the number of invertible matrices in $R$ is

Q13 4 marks Circles Tangent Lines and Tangent Lengths View

Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is

Q14 3 marks Sine and Cosine Rules Multi-step composite figure problem View

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac { \pi } { 2 }$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.
Let $a$ be the area of the triangle $ABC$. Then the value of $( 64a ) ^ { 2 }$ is

Q15 3 marks Sine and Cosine Rules Circumradius or incircle radius computation View

Consider an obtuse angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac { \pi } { 2 }$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius 1.
Then the inradius of the triangle $ABC$ is

Q16 3 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

Consider the $6 \times 6$ square in the figure. Let $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$ be the points of intersections (dots in the picture) in some order. We say that $A _ { i }$ and $A _ { j }$ are friends if they are adjacent along a row or along a column. Assume that each point $A _ { i }$ has an equal chance of being chosen.
Let $p _ { i }$ be the probability that a randomly chosen point has $i$ many friends, $i = 0,1,2,3,4$. Let $X$ be a random variable such that for $i = 0,1,2,3,4$, the probability $P ( X = i ) = p _ { i }$. Then the value of $7 E ( X )$ is

Q17 3 marks Probability Definitions Finite Equally-Likely Probability Computation View

Consider the $6 \times 6$ square in the figure. Let $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$ be the points of intersections (dots in the picture) in some order. We say that $A _ { i }$ and $A _ { j }$ are friends if they are adjacent along a row or along a column. Assume that each point $A _ { i }$ has an equal chance of being chosen.
Two distinct points are chosen randomly out of the points $A _ { 1 } , A _ { 2 } , \ldots , A _ { 49 }$. Let $p$ be the probability that they are friends. Then the value of $7p$ is

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