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Q1 Addition & Double Angle Formulae Trigonometric Equation Solving via Identities View

Suppose, for some $\theta \in \left[ 0 , \frac { \pi } { 2 } \right] , \frac { \cos 3 \theta } { \cos \theta } = \frac { 1 } { 3 }$. Then $( \cot 3 \theta ) \tan \theta$ equals
(A) $\frac { 1 } { 2 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 1 } { 8 }$
(D) $\frac { 1 } { 7 }$

Q2 Number Theory Properties of Integer Sequences and Digit Analysis View

Any positive real number $x$ can be expanded as $x = a _ { n } \cdot 2 ^ { n } + a _ { n - 1 } \cdot 2 ^ { n - 1 } + \cdots + a _ { 1 } \cdot 2 ^ { 1 } + a _ { 0 } \cdot 2 ^ { 0 } + a _ { - 1 } \cdot 2 ^ { - 1 } + a _ { - 2 } \cdot 2 ^ { - 2 } + \cdots$, for some $n \geq 0$, where each $a _ { i } \in \{ 0,1 \}$. In the above-described expansion of 21.1875, the smallest positive integer $k$ such that $a _ { - k } \neq 0$ is:
(A) 3
(B) 2
(C) 1
(D) 4

Q3 Proof Computation of a Limit, Value, or Explicit Formula View

Amongst all polynomials $p ( x ) = c _ { 0 } + c _ { 1 } x + \cdots + c _ { 10 } x ^ { 10 }$ with real coefficients satisfying $| p ( x ) | \leq | x |$ for all $x \in [ - 1,1 ]$, what is the maximum possible value of $\left( 2 c _ { 0 } + c _ { 1 } \right) ^ { 10 }$ ?
(A) $4 ^ { 10 }$
(B) $3 ^ { 10 }$
(C) $2 ^ { 10 }$
(D) 1

Q4 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View

The locus of points $z$ in the complex plane satisfying $z ^ { 2 } + | z | ^ { 2 } = 0$ is
(A) a straight line
(B) a pair of straight lines
(C) a circle
(D) a parabola

Q5 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

Let $A$ and $B$ be two $3 \times 3$ matrices such that $( A + B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$. Which of the following must be true?
(A) $A$ and $B$ are zero matrices.
(B) $A B$ is the zero matrix.
(C) $( A - B ) ^ { 2 } = A ^ { 2 } - B ^ { 2 }$
(D) $( A - B ) ^ { 2 } = A ^ { 2 } + B ^ { 2 }$

Q6 Sequences and series, recurrence and convergence Direct term computation from recurrence View

Let $\mathbb { Z }$ denote the set of integers. Let $f : \mathbb { Z } \rightarrow \mathbb { Z }$ be such that $f ( x ) f ( y ) = f ( x + y ) + f ( x - y )$ for all $x , y \in \mathbb { Z }$. If $f ( 1 ) = 3$, then $f ( 7 )$ equals
(A) 840
(B) 844
(C) 843
(D) 842

Q7 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

The sides of a regular hexagon $A B C D E F$ is extended by doubling them to form a bigger hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$ as in the figure below. Then the ratio of the areas of the bigger to the smaller hexagon is:
(A) $\sqrt { 3 }$
(B) 3
(C) $2 \sqrt { 3 }$
(D) 4

Q8 Combinations & Selection Counting Functions or Mappings with Constraints View

Let $( n _ { 1 } , n _ { 2 } , \cdots , n _ { 12 } )$ be a permutation of the numbers $1,2 , \cdots , 12$. The number of arrangements with $$n _ { 1 } > n _ { 2 } > n _ { 3 } > n _ { 4 } > n _ { 5 } > n _ { 6 }$$ and $$n _ { 6 } < n _ { 7 } < n _ { 8 } < n _ { 9 } < n _ { 10 } < n _ { 11 } < n _ { 12 }$$ equals:
(A) $\binom { 12 } { 5 }$
(B) $\binom { 12 } { 6 }$
(C) $\binom { 11 } { 6 }$
(D) $\frac { 11 ! } { 2 }$

Q9 Number Theory Modular Arithmetic Computation View

Suppose the numbers 71, 104 and 159 leave the same remainder $r$ when divided by a certain number $N > 1$. Then, the value of $3 N + 4 r$ must equal:
(A) 53
(B) 48
(C) 37
(D) 23

Q10 Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View

In how many ways can we choose $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ from the set $\{ 1,2 , \ldots , 30 \}$ such that $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in arithmetic progression?
(A) 135
(B) 145
(C) 155
(D) 165

Q11 Modulus function Optimisation of sums of absolute values View

What is the minimum value of the function $| x - 3 | + | x + 2 | + | x + 1 | + | x |$ for real $x$?
(A) 3
(B) 5
(C) 6
(D) 8

Q12 Stationary points and optimisation Find absolute extrema on a closed interval or domain View

If $x , y$ are positive real numbers such that $3 x + 4 y < 72$, then the maximum possible value of $12 x y ( 72 - 3 x - 4 y )$ is:
(A) 12240
(B) 13824
(C) 10656
(D) 8640

Q13 Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View

A straight road has walls on both sides of height 8 feet and 4 feet respectively. Two ladders are placed from the top of one wall to the foot of the other as in the figure below. What is the height (in feet) of the maximum clearance $x$ below the ladders?
(A) 3
(B) $2 \sqrt { 2 }$
(C) $\frac { 8 } { 3 }$
(D) $2 \sqrt { 3 }$

Q14 Differential equations Integral Equations Reducible to DEs View

Consider a differentiable function $u : [ 0,1 ] \rightarrow \mathbb { R }$. Assume the function $u$ satisfies $$u ( a ) = \frac { 1 } { 2 r } \int _ { a - r } ^ { a + r } u ( x ) d x , \quad \text{for all } a \in ( 0,1 ) \text{ and all } r < \min ( a , 1 - a ).$$ Which of the following four statements must be true?
(A) $u$ attains its maximum but not its minimum on the set $\{ 0,1 \}$.
(B) $u$ attains its minimum but not maximum on the set $\{ 0,1 \}$.
(C) If $u$ attains either its maximum or its minimum on the set $\{ 0,1 \}$, then it must be constant.
(D) $u$ attains both its maximum and its minimum on the set $\{ 0,1 \}$.

Q15 Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View

In the figure below, $A B C D$ is a square and $\triangle C E F$ is a triangle with given sides inscribed as in the figure. Find the length $B E$.
(A) $\frac { 13 } { \sqrt { 17 } }$
(B) $\frac { 14 } { \sqrt { 17 } }$
(C) $\frac { 15 } { \sqrt { 17 } }$
(D) $\frac { 16 } { \sqrt { 17 } }$

Q16 Conic sections Tangent and Normal Line Problems View

Let $y = x + c _ { 1 } , y = x + c _ { 2 }$ be the two tangents to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 1$. What is the value of $\left| c _ { 1 } - c _ { 2 } \right|$?
(A) $\sqrt { 2 }$
(B) $\sqrt { 5 }$
(C) $\frac { \sqrt { 5 } } { 2 }$
(D) 1

Q17 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

For $n \in \mathbb { N }$, let $a _ { n }$ be defined as $$a _ { n } = \int _ { 0 } ^ { n } \frac { 1 } { 1 + n x ^ { 2 } } d x$$ Then $\lim _ { n \rightarrow \infty } a _ { n }$
(A) equals 0
(B) equals $\frac { \pi } { 4 }$
(C) equals $\frac { \pi } { 2 }$
(D) does not exist

Q18 Roots of polynomials Existence or counting of roots with specified properties View

Let $p$ and $q$ be two non-zero polynomials such that the degree of $p$ is less than or equal to the degree of $q$, and $p ( a ) q ( a ) = 0$ for $a = 0,1,2 , \ldots , 10$. Which of the following must be true?
(A) degree of $q \neq 10$
(B) degree of $p \neq 10$
(C) degree of $q \neq 5$
(D) degree of $p \neq 5$

Q19 Number Theory Modular Arithmetic Computation View

The number of positive integers $n$ less than or equal to 22 such that 7 divides $n ^ { 5 } + 4 n ^ { 4 } + 3 n ^ { 3 } + 2022$ is
(A) 7
(B) 8
(C) 9
(D) 10

Q20 Number Theory Combinatorial Number Theory and Counting View

A $3 \times 3$ magic square is a $3 \times 3$ rectangular array of positive integers such that the sum of the three numbers in any row, any column or any of the two major diagonals, is the same. For the following incomplete magic square

27	36

31

the column sum is
(A) 90
(B) 96
(C) 94
(D) 99

Q21 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View

Let $1 , \omega , \omega ^ { 2 }$ be the cube roots of unity. Then the product $$\left( 1 - \omega + \omega ^ { 2 } \right) \left( 1 - \omega ^ { 2 } + \omega ^ { 2 ^ { 2 } } \right) \left( 1 - \omega ^ { 2 ^ { 2 } } + \omega ^ { 2 ^ { 3 } } \right) \cdots \left( 1 - \omega ^ { 2 ^ { 9 } } + \omega ^ { 2 ^ { 10 } } \right)$$ is equal to:
(A) $2 ^ { 10 }$
(B) $3 ^ { 10 }$
(C) $2 ^ { 10 } \omega$
(D) $3 ^ { 10 } \omega ^ { 2 }$

Q22 Principle of Inclusion/Exclusion View

In a class of 45 students, three students can write well using either hand. The number of students who can write well only with the right hand is 24 more than the number of those who write well only with the left hand. Then, the number of students who can write well with the right hand is:
(A) 33
(B) 36
(C) 39
(D) 41

Q23 Number Theory Linear Diophantine Equations View

The number of triples $( a , b , c )$ of positive integers satisfying the equation $$\frac { 1 } { a } + \frac { 1 } { b } + \frac { 1 } { c } = 1 + \frac { 2 } { a b c }$$ and such that $a < b < c$, equals:
(A) 3
(B) 2
(C) 1
(D) 0

Q24 Stationary points and optimisation Find critical points and classify extrema of a given function View

The function $x ^ { 2 } \log _ { e } x$ in the interval $( 0,2 )$ has:
(A) exactly one point of local maximum and no points of local minimum.
(B) exactly one point of local minimum and no points of local maximum.
(C) points of local maximum as well as local minimum.
(D) neither a point of local maximum nor a point of local minimum.

Q25 Sine and Cosine Rules Circumradius or incircle radius computation View

A triangle has sides of lengths $\sqrt { 5 } , 2 \sqrt { 2 } , \sqrt { 3 }$ units. Then, the radius of its inscribed circle is:
(A) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 2 }$
(B) $\frac { \sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 } } { 3 }$
(C) $\sqrt { 5 } + \sqrt { 3 } + 2 \sqrt { 2 }$
(D) $\frac { \sqrt { 5 } + \sqrt { 3 } - 2 \sqrt { 2 } } { 2 }$

Q26 Combinations & Selection Combinatorial Probability View

An urn contains 30 balls out of which one is special. If 6 of these balls are taken out at random, what is the probability that the special ball is chosen?
(A) $\frac { 1 } { 30 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 5 }$
(D) $\frac { 1 } { 15 }$

Q27 Inequalities Optimization Subject to an Algebraic Constraint View

If $x _ { 1 } > x _ { 2 } > \cdots > x _ { 10 }$ are real numbers, what is the least possible value of $$\left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 1 } - x _ { 2 } } \right) \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 2 } - x _ { 3 } } \right) \cdots \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 9 } - x _ { 10 } } \right) ?$$ (A) $10 ^ { 10 }$
(B) $10 ^ { 9 }$
(C) $9 ^ { 9 }$
(D) $9 ^ { 10 }$

Q28 Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View

Two ships are approaching a port along straight routes at constant velocities. Initially, the two ships and the port formed an equilateral triangle. After the second ship travelled 80 km, the triangle became right-angled. When the first ship reaches the port, the second ship was still 120 km from the port. Find the initial distance of the ships from the port.
(A) 240 km
(B) 300 km
(C) 360 km
(D) 180 km

Q29 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

In the following diagram, four triangles and their sides are given. Areas of three of them are also given. Find the area $x$ of the remaining triangle. The four triangles have areas 4, 5, $x$, and 13 respectively.
(A) 12
(B) 13
(C) 14
(D) 15

Q30 Inequalities Optimization Subject to an Algebraic Constraint View

The range of values that the function $$f ( x ) = \frac { x ^ { 2 } + 2 x + 4 } { 2 x ^ { 2 } + 4 x + 9 }$$ takes as $x$ varies over all real numbers in the domain of $f$ is:
(A) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 1 } { 2 }$
(B) $\frac { 3 } { 7 } \leq f ( x ) < \frac { 1 } { 2 }$
(C) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 4 } { 9 }$
(D) $\frac { 3 } { 7 } \leq f ( x ) \leq \frac { 1 } { 2 }$

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