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Q1 Inequalities Solve Polynomial/Rational Inequality for Solution Set View

For a real number $x$, $$x ^ { 3 } - 7 x + 6 > 0$$ if and only if
(A) $x > 2$.
(B) $- 3 < x < 1$.
(C) $x < - 3$ or $1 < x < 2$.
(D) $- 3 < x < 1$ or $x > 2$.

Q2 Matrices Determinant and Rank Computation View

Define a polynomial $f ( x )$ by $$f ( x ) = \left| \begin{array} { l l l } 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{array} \right|$$ for all $x \in \mathbb { R }$, where the right hand side above is a determinant. Then the roots of $f ( x )$ are of the form
(A) $\alpha , \beta \pm i \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R } , \gamma \neq 0$ and $i$ is a square root of $- 1$.
(B) $\alpha , \alpha , \beta$ where $\alpha , \beta \in \mathbb { R }$ are distinct.
(C) $\alpha , \beta , \gamma$ where $\alpha , \beta , \gamma \in \mathbb { R }$ are all distinct.
(D) $\alpha , \alpha , \alpha$ for some $\alpha \in \mathbb { R }$.

Q3 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then
(A) $S$ is the empty set.
(B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$.
(C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$.
(D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.

Q4 Number Theory Divisibility and Divisor Analysis View

The number of consecutive zeroes adjacent to the digit in the unit's place of $401 ^ { 50 }$ is
(A) 3.
(B) 4.
(C) 49.
(D) 50.

Q5 Sine and Cosine Rules Multi-step composite figure problem View

Consider a right angled triangle $\triangle A B C$ whose hypotenuse $A C$ is of length 1. The bisector of $\angle A C B$ intersects $A B$ at $D$. If $B C$ is of length $x$, then what is the length of $CD$?
(A) $\sqrt { \frac { 2 x ^ { 2 } } { 1 + x } }$
(B) $\frac { 1 } { \sqrt { 2 + 2 x } }$
(C) $\sqrt { \frac { x } { 1 + x } }$
(D) $\frac { x } { \sqrt { 1 - x ^ { 2 } } }$

Q6 Circles Area and Geometric Measurement Involving Circles View

Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals
(A) $\frac { \pi } { 2 }$.
(B) $\frac { 5 \pi } { 4 }$.
(C) $\frac { 3 } { \sqrt { 2 } } \pi$.
(D) $2 \pi$.

Q7 Chain Rule Chain Rule Combined with Fundamental Theorem of Calculus View

Let $f , g$ be continuous functions from $[ 0 , \infty )$ to itself, $$h ( x ) = \int _ { 2 ^ { x } } ^ { 3 ^ { x } } f ( t ) d t , x > 0$$ and $$F ( x ) = \int _ { 0 } ^ { h ( x ) } g ( t ) d t , x > 0$$ If $F ^ { \prime }$ is the derivative of $F$, then for $x > 0$,
(A) $F ^ { \prime } ( x ) = g ( h ( x ) )$.
(B) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ f \left( 3 ^ { x } \right) - f \left( 2 ^ { x } \right) \right]$.
(C) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ x 3 ^ { x - 1 } f \left( 3 ^ { x } \right) - x 2 ^ { x - 1 } f \left( 2 ^ { x } \right) \right]$.
(D) $F ^ { \prime } ( x ) = g ( h ( x ) ) \left[ 3 ^ { x } f \left( 3 ^ { x } \right) \ln 3 - 2 ^ { x } f \left( 2 ^ { x } \right) \ln 2 \right]$.

Q8 Permutations & Arrangements Forming Numbers with Digit Constraints View

How many numbers formed by rearranging the digits of 234578 are divisible by 55?
(A) 0
(B) 12
(C) 36
(D) 72

Q9 Polar coordinates View

Let $$S = \left\{ \left( \theta \sin \frac { \pi \theta } { 1 + \theta } , \frac { 1 } { \theta } \cos \frac { \pi \theta } { 1 + \theta } \right) : \theta \in \mathbb { R } , \theta > 0 \right\}$$ and $$T = \left\{ ( x , y ) : x \in \mathbb { R } , y \in \mathbb { R } , x y = \frac { 1 } { 2 } \right\}$$ How many elements does $S \cap T$ have?
(A) 0
(B) 1
(C) 2
(D) 3

Q10 Taylor series Limit evaluation using series expansion or exponential asymptotics View

The limit $$\lim _ { n \rightarrow \infty } n ^ { - \frac { 3 } { 2 } } \left( ( n + 1 ) ^ { ( n + 1 ) } ( n + 2 ) ^ { ( n + 2 ) } \ldots ( 2 n ) ^ { ( 2 n ) } \right) ^ { \frac { 1 } { n ^ { 2 } } }$$ equals
(A) 0.
(B) 1.
(C) $e ^ { - \frac { 1 } { 4 } }$.
(D) $4 e ^ { - \frac { 3 } { 4 } }$.

Q11 Number Theory Modular Arithmetic Computation View

Suppose $x$ and $y$ are positive integers. If $4 x + 3 y$ and $2 x + 4 y$ are divided by 7, then the respective remainders are 2 and 5. If $11 x + 5 y$ is divided by 7, then the remainder equals
(A) 0.
(B) 1.
(C) 2.
(D) 3.

Q12 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View

The value of $$\sum _ { k = 0 } ^ { 202 } ( - 1 ) ^ { k } \binom { 202 } { k } \cos \left( \frac { k \pi } { 3 } \right)$$ equals
(A) $\quad \sin \left( \frac { 202 } { 3 } \pi \right)$.
(B) $- \sin \left( \frac { 202 } { 3 } \pi \right)$.
(C) $\quad \cos \left( \frac { 202 } { 3 } \pi \right)$.
(D) $\cos ^ { 202 } \left( \frac { \pi } { 3 } \right)$.

Q13 Circles Intersection of Circles or Circle with Conic View

For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned} a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\ a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0 \end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be
(A) 0.
(B) 1.
(C) 2.
(D) 3.

Q14 Differentiation from First Principles View

The limit $$\lim _ { x \rightarrow 0 } \frac { 1 } { x } \left( \cos ( x ) + \cos \left( \frac { 1 } { x } \right) - \cos ( x ) \cos \left( \frac { 1 } { x } \right) - 1 \right)$$ (A) equals 0.
(B) equals $\frac { 1 } { 2 }$.
(C) equals 1.
(D) does not exist.

Q15 Number Theory Divisibility and Divisor Analysis View

Let $n$ be a positive integer having 27 divisors including 1 and $n$, which are denoted by $d _ { 1 } , \ldots , d _ { 27 }$. Then the product of $d _ { 1 } , d _ { 2 } , \ldots , d _ { 27 }$ equals
(A) $n ^ { 13 }$.
(B) $n ^ { 14 }$.
(C) $n ^ { \frac { 27 } { 2 } }$.
(D) $27 n$.

Q16 Stationary points and optimisation Find critical points and classify extrema of a given function View

Suppose $F : \mathbb { R } \rightarrow \mathbb { R }$ is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) $F$ cannot have a local minimum.
(B) $F$ must have exactly one local minimum.
(C) $F$ must have at least two local minima.
(D) $F$ must have either a global maximum or a local minimum.

Q17 Complex numbers 2 Roots of Unity and Cyclotomic Properties View

Suppose $z \in \mathbb { C }$ is such that the imaginary part of $z$ is non-zero and $z ^ { 25 } = 1$. Then $$\sum _ { k = 0 } ^ { 2023 } z ^ { k }$$ equals
(A) 0.
(B) 1.
(C) $- 1 - z ^ { 24 }$.
(D) $- z ^ { 24 }$.

Q18 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable one-to-one function. If $f ( 2 ) = 2 , f ( 3 ) = - 8$ and $$\int _ { 2 } ^ { 3 } f ( x ) d x = - 3$$ then $$\int _ { - 8 } ^ { 2 } f ^ { - 1 } ( x ) d x$$ equals
(A) $- 25$.
(B) $25$.
(C) $- 31$.
(D) $31$.

Q19 Differential equations Integral Equations Reducible to DEs View

If $f : [ 0 , \infty ) \rightarrow \mathbb { R }$ is a continuous function such that $$f ( x ) + \ln 2 \int _ { 0 } ^ { x } f ( t ) d t = 1 , x \geq 0$$ then for all $x \geq 0$,
(A) $f ( x ) = e ^ { x } \ln 2$.
(B) $f ( x ) = e ^ { - x } \ln 2$.
(C) $f ( x ) = 2 ^ { x }$.
(D) $f ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$.

Q20 Number Theory Congruence Reasoning and Parity Arguments View

If $[ x ]$ denotes the largest integer less than or equal to $x$, then $$\left[ ( 9 + \sqrt { 80 } ) ^ { 20 } \right]$$ equals
(A) $( 9 + \sqrt { 80 } ) ^ { 20 } - ( 9 - \sqrt { 80 } ) ^ { 20 }$.
(B) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 20$.
(C) $( 9 + \sqrt { 80 } ) ^ { 20 } + ( 9 - \sqrt { 80 } ) ^ { 20 } - 1$.
(D) $( 9 - \sqrt { 80 } ) ^ { 20 }$.

Q21 Sequences and series, recurrence and convergence Multiple-choice on sequence properties View

The limit $$\lim _ { n \rightarrow \infty } \left( 2 ^ { - 2 ^ { n + 1 } } + 2 ^ { - 2 ^ { n - 1 } } \right) ^ { 2 ^ { - n } }$$ equals
(A) 1.
(B) $\frac { 1 } { \sqrt { 2 } }$.
(C) 0.
(D) $\frac { 1 } { 4 }$.

Q22 Circles Area and Geometric Measurement Involving Circles View

In the following figure, $O A B$ is a quarter-circle. The unshaded region is a circle to which $O A$ and $C D$ are tangents. If $C D$ is of length 10 and is parallel to $O A$, then the area of the shaded region in the above figure equals
(A) $25 \pi$.
(B) $50 \pi$.
(C) $75 \pi$.
(D) $100 \pi$.

Q23 Permutations & Arrangements Combinatorial Proof or Identity Derivation View

Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?
(A) 3
(B) 4
(C) 5
(D) 6

Q24 Factor & Remainder Theorem Divisibility and Factor Determination View

The polynomial $x ^ { 10 } + x ^ { 5 } + 1$ is divisible by
(A) $x ^ { 2 } + x + 1$.
(B) $x ^ { 2 } - x + 1$.
(C) $x ^ { 2 } + 1$.
(D) $x ^ { 5 } - 1$.

Q25 Discriminant and conditions for roots Quadratic sandwiching and coefficient determination View

Suppose $a , b , c \in \mathbb { R }$ and $$f ( x ) = a x ^ { 2 } + b x + c , x \in \mathbb { R } .$$ If $0 \leq f ( x ) \leq ( x - 1 ) ^ { 2 }$ for all $x$, and $f ( 3 ) = 2$, then
(A) $a = \frac { 1 } { 2 } , b = - 1 , c = \frac { 1 } { 2 }$.
(B) $a = \frac { 1 } { 3 } , b = - \frac { 1 } { 3 } , c = 0$.
(C) $a = \frac { 2 } { 3 } , b = - \frac { 5 } { 3 } , c = 1$.
(D) $a = \frac { 3 } { 4 } , b = - 2 , c = \frac { 5 } { 4 }$.

Q26 Straight Lines & Coordinate Geometry Locus Determination View

The straight line $O A$ lies in the second quadrant of the $(x, y)$-plane and makes an angle $\theta$ with the negative half of the $x$-axis, where $0 < \theta < \frac { \pi } { 2 }$. The line segment $C D$ of length 1 slides on the $(x, y)$-plane in such a way that $C$ is always on $O A$ and $D$ on the positive side of the $x$-axis. The locus of the mid-point of $C D$ is
(A) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$.
(B) $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 } + \cot ^ { 2 } \theta$.
(C) $x ^ { 2 } + 4 x y \cot \theta + y ^ { 2 } = \frac { 1 } { 4 }$.
(D) $x ^ { 2 } + y ^ { 2 } \left( 1 + 4 \cot ^ { 2 } \theta \right) = \frac { 1 } { 4 }$.

Q27 Stationary points and optimisation Find critical points and classify extrema of a given function View

Suppose that $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ where $a , b , c , d$ are real numbers with $a \neq 0$. The equation $f ( x ) = 0$ has exactly two distinct real solutions. If $f ^ { \prime } ( x )$ is the derivative of $f ( x )$, then which of the following is a possible graph of $f ^ { \prime } ( x )$?
(A), (B), (C), (D) [graphs as provided in the figure]

Q28 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

Consider the function $f : \mathbb { C } \rightarrow \mathbb { C }$ defined by $$f ( a + i b ) = e ^ { a } ( \cos b + i \sin b ) , a , b \in \mathbb { R }$$ where $i$ is a square root of $-1$. Then
(A) $f$ is one-to-one and onto.
(B) $f$ is one-to-one but not onto.
(C) $f$ is onto but not one-to-one.
(D) $f$ is neither one-to-one nor onto.

Q29 Number Theory Combinatorial Number Theory and Counting View

Suppose $f : \mathbb { Z } \rightarrow \mathbb { Z }$ is a non-decreasing function. Consider the following two cases: $$\begin{aligned} & \text { Case 1. } f ( 0 ) = 2 , f ( 10 ) = 8 \\ & \text { Case 2. } f ( 0 ) = - 2 , f ( 10 ) = 12 \end{aligned}$$ In which of the above cases it is necessarily true that there exists an $n$ with $f ( n ) = n$?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.

Q30 Combinations & Selection Counting Functions or Mappings with Constraints View

How many functions $f : \{ 1,2 , \ldots , 10 \} \rightarrow \{ 1 , \ldots , 2000 \}$, which satisfy $$f ( i + 1 ) - f ( i ) \geq 20 , \text { for all } 1 \leq i \leq 9 ,$$ are there?
(A) $10 ! \binom { 1829 } { 10 }$
(B) $11 ! \binom { 1830 } { 11 }$
(C) $\binom { 1829 } { 10 }$
(D) $\binom { 1830 } { 11 }$

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