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2025 paper1

16 maths questions

Q1 3 marks Polynomial Division & Manipulation View
Let $\mathbb { R }$ denote the set of all real numbers. Let $a _ { i } , b _ { i } \in \mathbb { R }$ for $i \in \{ 1,2,3 \}$.
Define the functions $f : \mathbb { R } \rightarrow \mathbb { R } , g : \mathbb { R } \rightarrow \mathbb { R }$, and $h : \mathbb { R } \rightarrow \mathbb { R }$ by
$$\begin{aligned} & f ( x ) = a _ { 1 } + 10 x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } + x ^ { 4 } \\ & g ( x ) = b _ { 1 } + 3 x + b _ { 2 } x ^ { 2 } + b _ { 3 } x ^ { 3 } + x ^ { 4 } \\ & h ( x ) = f ( x + 1 ) - g ( x + 2 ) \end{aligned}$$
If $f ( x ) \neq g ( x )$ for every $x \in \mathbb { R }$, then the coefficient of $x ^ { 3 }$ in $h ( x )$ is
(A)8
(B)2
(C)-4
(D)-6
Q2 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
Three students $S _ { 1 } , S _ { 2 }$, and $S _ { 3 }$ are given a problem to solve. Consider the following events: $U$ : At least one of $S _ { 1 } , S _ { 2 }$, and $S _ { 3 }$ can solve the problem, $V : S _ { 1 }$ can solve the problem, given that neither $S _ { 2 }$ nor $S _ { 3 }$ can solve the problem, $W : S _ { 2 }$ can solve the problem and $S _ { 3 }$ cannot solve the problem, T: $S _ { 3 }$ can solve the problem.
For any event $E$, let $P ( E )$ denote the probability of $E$. If
$$P ( U ) = \frac { 1 } { 2 } , \quad P ( V ) = \frac { 1 } { 10 } , \quad \text { and } \quad P ( W ) = \frac { 1 } { 12 }$$
then $P ( T )$ is equal to
(A)$\frac { 13 } { 36 }$(B)$\frac { 1 } { 3 }$(C)$\frac { 19 } { 60 }$(D)$\frac { 1 } { 4 }$
Q3 3 marks Applied differentiation Properties of differentiable functions (abstract/theoretical) View
Let $\mathbb { R }$ denote the set of all real numbers. Define the function $f : \mathbb { R } \rightarrow \mathbb { R }$ by
$$f ( x ) = \left\{ \begin{array} { c c } 2 - 2 x ^ { 2 } - x ^ { 2 } \sin \frac { 1 } { x } & \text { if } x \neq 0 \\ 2 & \text { if } x = 0 \end{array} \right.$$
Then which one of the following statements is TRUE?
(A)The function $f$ is NOT differentiable at $x = 0$
(B)There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval ( $0 , \delta$ )
(C)For any positive real number $\delta$, the function $f$ is NOT an increasing function on the interval ( $- \delta , 0$ )
(D)$x = 0$ is a point of local minima of $f$
Q4 3 marks 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Consider the matrix
$$P = \left( \begin{array} { l l l } 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array} \right)$$
Let the transpose of a matrix $X$ be denoted by $X ^ { T }$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that
$$Q ^ { - 1 } = Q ^ { T } \text { and } P Q = Q P$$
is
(A)32(B)8(C)16(D)24
Q5 4 marks Vectors: Lines & Planes Multi-Step Geometric Modeling Problem View
Let $L _ { 1 }$ be the line of intersection of the planes given by the equations
$$2 x + 3 y + z = 4 \text { and } x + 2 y + z = 5 .$$
Let $L _ { 2 }$ be the line passing through the point $P ( 2 , - 1,3 )$ and parallel to $L _ { 1 }$. Let $M$ denote the plane given by the equation
$$2 x + y - 2 z = 6$$
Suppose that the line $L _ { 2 }$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$.
Then which of the following statements is (are) TRUE?
(A)The length of the line segment $PQ$ is $9 \sqrt { 3 }$
(B)The length of the line segment $QR$ is 15
(C)The area of $\triangle PQR$ is $\frac { 3 } { 2 } \sqrt { 234 }$
(D)The acute angle between the line segments $PQ$ and $PR$ is $\cos ^ { - 1 } \left( \frac { 1 } { 2 \sqrt { 3 } } \right)$
Q6 4 marks Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $\mathbb { N }$ denote the set of all natural numbers, and $\mathbb { Z }$ denote the set of all integers. Consider the functions $f : \mathbb { N } \rightarrow \mathbb { Z }$ and $g : \mathbb { Z } \rightarrow \mathbb { N }$ defined by
$$f ( n ) = \begin{cases} ( n + 1 ) / 2 & \text { if } n \text { is odd } \\ ( 4 - n ) / 2 & \text { if } n \text { is even } \end{cases}$$
and
$$g ( n ) = \begin{cases} 3 + 2 n & \text { if } n \geq 0 \\ - 2 n & \text { if } n < 0 \end{cases}$$
Define $( g \circ f ) ( n ) = g ( f ( n ) )$ for all $n \in \mathbb { N }$, and $( f \circ g ) ( n ) = f ( g ( n ) )$ for all $n \in \mathbb { Z }$.
Then which of the following statements is (are) TRUE?
(A)$g \circ f$ is NOT one-one and $g \circ f$ is NOT onto
(B)$f \circ g$ is NOT one-one but $f \circ g$ is onto
(C)$g$ is one-one and $g$ is onto
(D)$f$ is NOT one-one but $f$ is onto
Q7 4 marks Circles Circle Identification and Classification View
Let $\mathbb { R }$ denote the set of all real numbers. Let $z _ { 1 } = 1 + 2 i$ and $z _ { 2 } = 3 i$ be two complex numbers, where $i = \sqrt { - 1 }$. Let
$$S = \left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : \left| x + i y - z _ { 1 } \right| = 2 \left| x + i y - z _ { 2 } \right| \right\}$$
Then which of the following statements is (are) TRUE?
(A)$S$ is a circle with centre $\left( - \frac { 1 } { 3 } , \frac { 10 } { 3 } \right)$
(B)$S$ is a circle with centre $\left( \frac { 1 } { 3 } , \frac { 8 } { 3 } \right)$
(C)$S$ is a circle with radius $\frac { \sqrt { 2 } } { 3 }$
(D)$S$ is a circle with radius $\frac { 2 \sqrt { 2 } } { 3 }$
Q8 4 marks Permutations & Arrangements Counting Functions with Constraints View
Let the set of all relations $R$ on the set $\{ a , b , c , d , e , f \}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal { S }$.
Then the number of elements in $\mathcal { S }$ is $\_\_\_\_$ .
Q9 4 marks Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow { MN }$ denote the vector from $M$ to $N$, and $\overrightarrow { 0 }$ denote the zero vector. Let $P , Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that
$$\overrightarrow { SP } + 5 \overrightarrow { SQ } + 6 \overrightarrow { SR } = \overrightarrow { 0 }$$
Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of
$$\frac { \text { length of the line segment } EF } { \text { length of the line segment } ES }$$
is $\_\_\_\_$ .
Q10 4 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
Let $S$ be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in $S$, but 0210222 is NOT in $S$.
Then the number of elements $x$ in $S$ such that at least one of the digits 0 and 1 appears exactly twice in $x$, is equal to $\_\_\_\_$ .
Q11 4 marks Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View
Let $\alpha$ and $\beta$ be the real numbers such that
$$\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \left( \frac { \alpha } { 2 } \int _ { 0 } ^ { x } \frac { 1 } { 1 - t ^ { 2 } } d t + \beta x \cos x \right) = 2$$
Then the value of $\alpha + \beta$ is $\_\_\_\_$ .
Q12 4 marks Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function such that $f ( x ) > 0$ for all $x \in \mathbb { R }$, and $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in \mathbb { R }$.
Let the real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { 50 }$ be in an arithmetic progression. If $f \left( a _ { 31 } \right) = 64 f \left( a _ { 25 } \right)$, and
$$\sum _ { i = 1 } ^ { 50 } f \left( a _ { i } \right) = 3 \left( 2 ^ { 25 } + 1 \right)$$
then the value of
$$\sum _ { i = 6 } ^ { 30 } f \left( a _ { i } \right)$$
is $\_\_\_\_$ .
Q13 4 marks Differential equations Solving Separable DEs with Initial Conditions View
For all $x > 0$, let $y _ { 1 } ( x ) , y _ { 2 } ( x )$, and $y _ { 3 } ( x )$ be the functions satisfying
$$\begin{aligned} & \frac { d y _ { 1 } } { d x } - ( \sin x ) ^ { 2 } y _ { 1 } = 0 , \quad y _ { 1 } ( 1 ) = 5 \\ & \frac { d y _ { 2 } } { d x } - ( \cos x ) ^ { 2 } y _ { 2 } = 0 , \quad y _ { 2 } ( 1 ) = \frac { 1 } { 3 } \\ & \frac { d y _ { 3 } } { d x } - \left( \frac { 2 - x ^ { 3 } } { x ^ { 3 } } \right) y _ { 3 } = 0 , \quad y _ { 3 } ( 1 ) = \frac { 3 } { 5 e } \end{aligned}$$
respectively. Then
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { y _ { 1 } ( x ) y _ { 2 } ( x ) y _ { 3 } ( x ) + 2 x } { e ^ { 3 x } \sin x }$$
is equal to $\_\_\_\_$ .
Q14 4 marks Measures of Location and Spread View
Consider the following frequency distribution:
Value458961211
Frequency5$f _ { 1 }$$f _ { 2 }$2113

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let $\alpha$ denote the mean deviation about the mean, $\beta$ denote the mean deviation about the median, and $\sigma ^ { 2 }$ denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I
(P) $7 f _ { 1 } + 9 f _ { 2 }$ is equal to (Q) $19 \alpha$ is equal to (R) $19 \beta$ is equal to (S) $19 \sigma ^ { 2 }$ is equal to
List-II
(1) 146
(2) 47
(3) 48
(4) 145
(5) 55
(A)$( \mathrm { P } ) \rightarrow ( 5 )$$( \mathrm { Q } ) \rightarrow ( 3 )$$( \mathrm { R } ) \rightarrow ( 2 )$$( \mathrm { S } ) \rightarrow ( 4 )$
(B)$( \mathrm { P } ) \rightarrow ( 5 )$$( \mathrm { Q } ) \rightarrow ( 2 )$$( \mathrm { R } ) \rightarrow ( 3 )$$( \mathrm { S } ) \rightarrow ( 1 )$
(C)$( \mathrm { P } ) \rightarrow ( 5 )$$( \mathrm { Q } ) \rightarrow ( 3 )$$( \mathrm { R } ) \rightarrow ( 2 )$$( \mathrm { S } ) \rightarrow ( 1 )$
(D)$( \mathrm { P } ) \rightarrow ( 3 )$$( \mathrm { Q } ) \rightarrow ( 2 )$$( \mathrm { R } ) \rightarrow ( 5 )$$( \mathrm { S } ) \rightarrow ( 4 )$
Q15 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
Let $\mathbb { R }$ denote the set of all real numbers. For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I
(P) The minimum value of $n$ for which the function
$$f ( x ) = \left[ \frac { 10 x ^ { 3 } - 45 x ^ { 2 } + 60 x + 35 } { n } \right]$$
is continuous on the interval $[ 1,2 ]$, is (Q) The minimum value of $n$ for which
$$g ( x ) = \left( 2 n ^ { 2 } - 13 n - 15 \right) \left( x ^ { 3 } + 3 x \right)$$
$x \in \mathbb { R }$, is an increasing function on $\mathbb { R }$, is (R) The smallest natural number $n$ which is greater than 5, such that $x = 3$ is a point of local minima of
$$h ( x ) = \left( x ^ { 2 } - 9 \right) ^ { n } \left( x ^ { 2 } + 2 x + 3 \right) ,$$
is (S) Number of $x _ { 0 } \in \mathbb { R }$ such that
$$l ( x ) = \sum _ { k = 0 } ^ { 4 } \left( \sin | x - k | + \cos \left| x - k + \frac { 1 } { 2 } \right| \right) ,$$
$x \in \mathbb { R }$, is NOT differentiable at $x _ { 0 }$, is
List-II
(1) 8
(2) 9
(3) 5
(4) 6
(5) 10
(A)$( \mathrm { P } ) \rightarrow ( 1 )$$( \mathrm { Q } ) \rightarrow ( 3 )$$( \mathrm { R } ) \rightarrow ( 2 )$$( \mathrm { S } ) \rightarrow ( 5 )$
(B)$( \mathrm { P } ) \rightarrow ( 2 )$$( \mathrm { Q } ) \rightarrow ( 1 )$$( \mathrm { R } ) \rightarrow ( 4 )$$( \mathrm { S } ) \rightarrow ( 3 )$
(C)$( \mathrm { P } ) \rightarrow ( 5 )$$( \mathrm { Q } ) \rightarrow ( 1 )$$( \mathrm { R } ) \rightarrow ( 4 )$$( \mathrm { S } ) \rightarrow ( 3 )$
(D)$( \mathrm { P } ) \rightarrow ( 2 )$$( \mathrm { Q } ) \rightarrow ( 3 )$$( \mathrm { R } ) \rightarrow ( 1 )$$( \mathrm { S } ) \rightarrow ( 5 )$
Q16 4 marks Vector Product and Surfaces View
Let $\vec { w } = \hat { \imath } + \hat { \jmath } - 2 \hat { k }$, and $\vec { u }$ and $\vec { v }$ be two vectors, such that $\vec { u } \times \vec { v } = \vec { w }$ and $\vec { v } \times \vec { w } = \vec { u }$. Let $\alpha , \beta , \gamma$, and $t$ be real numbers such that $\vec { u } = \alpha \hat { \imath } + \beta \hat { \jmath } + \gamma \hat { k } , \quad - t \alpha + \beta + \gamma = 0 , \quad \alpha - t \beta + \gamma = 0 , \quad$ and $\alpha + \beta - t \gamma = 0$.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
List-I
(P) $| \vec { v } | ^ { 2 }$ is equal to (Q) If $\alpha = \sqrt { 3 }$, then $\gamma ^ { 2 }$ is equal to (R) If $\alpha = \sqrt { 3 }$, then $( \beta + \gamma ) ^ { 2 }$ is equal to (S) If $\alpha = \sqrt { 2 }$, then $t + 3$ is equal to
List-II
(1) 0
(2) 1
(3) 2
(4) 3
(5) 5
(A)$( \mathrm { P } ) \rightarrow ( 2 )$$( \mathrm { Q } ) \rightarrow ( 1 )$$( \mathrm { R } ) \rightarrow ( 4 )$$( \mathrm { S } ) \rightarrow ( 5 )$
(B)$( \mathrm { P } ) \rightarrow ( 2 )$$( \mathrm { Q } ) \rightarrow ( 4 )$$( \mathrm { R } ) \rightarrow ( 3 )$$( \mathrm { S } ) \rightarrow ( 5 )$
(C)$( \mathrm { P } ) \rightarrow ( 2 )$$( \mathrm { Q } ) \rightarrow ( 1 )$$( \mathrm { R } ) \rightarrow ( 4 )$$( \mathrm { S } ) \rightarrow ( 3 )$
(D)$( \mathrm { P } ) \rightarrow ( 5 )$$( \mathrm { Q } ) \rightarrow ( 4 )$$( \mathrm { R } ) \rightarrow ( 1 )$$( \mathrm { S } ) \rightarrow ( 3 )$