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Q1 Exponential Equations & Modelling Solve Exponential Equation for Unknown Variable View

The sum of all the solutions of $2 + \log _ { 2 } ( x - 2 ) = \log _ { ( x - 2 ) } 8$ in the interval $( 2 , \infty )$ is
(A) $\frac { 35 } { 8 }$.
(B) 5 .
(C) $\frac { 49 } { 8 }$.
(D) $\frac { 55 } { 8 }$.

Q2 Sequences and Series Evaluation of a Finite or Infinite Sum View

The value of $$1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \cdots + \frac { 1 } { 1 + 2 + 3 + \cdots 2021 }$$ is
(A) $\frac { 2021 } { 1010 }$.
(B) $\frac { 2021 } { 1011 }$.
(C) $\frac { 2021 } { 1012 }$.
(D) $\frac { 2021 } { 1013 }$.

Q3 Number Theory GCD, LCM, and Coprimality View

The number of ways one can express $2 ^ { 2 } 3 ^ { 3 } 5 ^ { 5 } 7 ^ { 7 }$ as a product of two numbers $a$ and $b$, where $\operatorname { gcd } ( a , b ) = 1$, and $1 < a < b$, is
(A) 5 .
(B) 6 .
(C) 7 .
(D) 8 .

Q4 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that $$f ( x + 1 ) = \frac { 1 } { 2 } f ( x ) \text { for all } x \in \mathbb { R } ,$$ and let $a _ { n } = \int _ { 0 } ^ { n } f ( x ) d x$ for all integers $n \geq 1$. Then:
(A) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $\int _ { 0 } ^ { 1 } f ( x ) d x$.
(B) $\lim _ { n \rightarrow \infty } a _ { n }$ does not exist.
(C) $\lim _ { n \rightarrow \infty } a _ { n }$ exists if and only if $\left| \int _ { 0 } ^ { 1 } f ( x ) d x \right| < 1$.
(D) $\lim _ { n \rightarrow \infty } a _ { n }$ exists and equals $2 \int _ { 0 } ^ { 1 } f ( x ) d x$.

Q5 Conic sections Optimization on Conics View

Let $a , b , c , d > 0$, be any real numbers. Then the maximum possible value of $c x + d y$, over all points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, must be
(A) $\sqrt { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } }$.
(B) $\sqrt { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } }$.
(C) $\sqrt { \frac { a ^ { 2 } c ^ { 2 } + b ^ { 2 } d ^ { 2 } } { a ^ { 2 } + b ^ { 2 } } }$.
(D) $\sqrt { \frac { a ^ { 2 } b ^ { 2 } + c ^ { 2 } d ^ { 2 } } { c ^ { 2 } + d ^ { 2 } } }$.

Q6 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View

Let $f ( x ) = \sin x + \alpha x , x \in \mathbb { R }$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha > 1$ or $\alpha < - 1$.
(B) $\alpha \geq 1$ or $\alpha \leq - 1$.
(C) $\alpha \geq 1$ or $\alpha < - 1$.
(D) $\alpha > 1$ or $\alpha \leq - 1$.

Q7 Volumes of Revolution Volume of a Region Defined by Inequalities in 3D View

The volume of the region $S = \{ ( x , y , z ) : | x | + 2 | y | + 3 | z | \leq 6 \}$ is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 6 .

Q8 Applied differentiation Convexity and inflection point analysis View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a twice differentiable function such that $\frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } }$ is positive for all $x \in \mathbb { R }$, and suppose $f ( 0 ) = 1 , f ( 1 ) = 4$. Which of the following is not a possible value of $f ( 2 )$ ?
(A) 7 .
(B) 8 .
(C) 9 .
(D) 10 .

Q9 Standard Integrals and Reverse Chain Rule Limit Involving an Integral (FTC Application) View

Let $$f ( x ) = e ^ { - | x | } , x \in \mathbb { R }$$ and $$g ( \theta ) = \int _ { - 1 } ^ { 1 } f \left( \frac { x } { \theta } \right) d x , \theta \neq 0$$ Then, $$\lim _ { \theta \rightarrow 0 } \frac { g ( \theta ) } { \theta }$$ (A) equals 0 .
(B) equals $+ \infty$.
(C) equals 2 .
(D) does not exist.

Q10 Circles Area and Geometric Measurement Involving Circles View

Consider the curves $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0,9 x ^ { 2 } + 4 y ^ { 2 } - 900 = 0$ and $y ^ { 2 } - 6 y - 6 x + 51 = 0$. The maximum number of disjoint regions into which these curves divide the $XY$-plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .

Q11 Combinations & Selection Combinatorial Probability View

A box has 13 distinct pairs of socks. Let $p _ { r }$ denote the probability of having at least one matching pair among a bunch of $r$ socks drawn at random from the box. If $r _ { 0 }$ is the maximum possible value of $r$ such that $p _ { r } < 1$, then the value of $p _ { r _ { 0 } }$ is
(A) $1 - \frac { 12 } { { } ^ { 26 } C _ { 12 } }$.
(B) $1 - \frac { 13 } { { } ^ { 26 } C _ { 13 } }$.
(C) $1 - \frac { 2 ^ { 13 } } { { } ^ { 26 } C _ { 13 } }$.
(D) $1 - \frac { 2 ^ { 12 } } { { } ^ { 26 } C _ { 12 } }$.

Q12 Complex Numbers Argand & Loci Circle Equation and Properties via Complex Number Manipulation View

Consider the following two subsets of $\mathbb { C }$ : $$A = \left\{ \frac { 1 } { z } : | z | = 2 \right\} \text { and } B = \left\{ \frac { 1 } { z } : | z - 1 | = 2 \right\} .$$ Then
(A) $A$ is a circle, but $B$ is not a circle.
(B) $B$ is a circle, but $A$ is not a circle.
(C) $A$ and $B$ are both circles.
(D) Neither $A$ nor $B$ is a circle.

Q13 Proof Characterization or Determination of a Set or Class View

Let $a , b , c$ and $d$ be four non-negative real numbers where $a + b + c + d = 1$. The number of different ways one can choose these numbers such that $a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + d ^ { 2 } = \max \{ a , b , c , d \}$ is
(A) 1 .
(B) 5 .
(C) 11 .
(D) 15 .

Q14 Differential equations Qualitative Analysis of DE Solutions View

Suppose $f ( x )$ is a twice differentiable function on $[ a , b ]$ such that $$f ( a ) = 0 = f ( b )$$ and $$x ^ { 2 } \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } + 4 x \frac { d f ( x ) } { d x } + 2 f ( x ) > 0 \text { for all } x \in ( a , b )$$ Then,
(A) $f$ is negative for all $x \in ( a , b )$.
(B) $f$ is positive for all $x \in ( a , b )$.
(C) $f ( x ) = 0$ for exactly one $x \in ( a , b )$.
(D) $f ( x ) = 0$ for at least two $x \in ( a , b )$.

Q15 Stationary points and optimisation Count or characterize roots using extremum values View

The polynomial $x ^ { 4 } + 4 x + c = 0$ has at least one real root if and only if
(A) $c < 2$.
(B) $c \leq 2$.
(C) $c < 3$.
(D) $c \leq 3$.

Q16 Permutations & Arrangements Circular Arrangement View

The number of different ways to colour the vertices of a square $PQRS$ using one or more colours from the set \{Red, Blue, Green, Yellow\}, such that no two adjacent vertices have the same colour is
(A) 36 .
(B) 48 .
(C) 72 .
(D) 84 .

Q17 Number Theory GCD, LCM, and Coprimality View

Define $a = p ^ { 3 } + p ^ { 2 } + p + 11$ and $b = p ^ { 2 } + 1$, where $p$ is any prime number. Let $d = \operatorname { gcd } ( a , b )$. Then the set of possible values of $d$ is
(A) $\{ 1,2,5 \}$.
(B) $\{ 2,5,10 \}$.
(C) $\{ 1,5,10 \}$.
(D) $\{ 1,2,10 \}$.

Q18 Matrices Determinant and Rank Computation View

Consider all $2 \times 2$ matrices whose entries are distinct and taken from the set $\{ 1,2,3,4 \}$. The sum of determinants of all such matrices is
(A) 24 .
(B) 10 .
(C) 12 .
(D) 0 .

Q19 Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be any twice differentiable function such that its second derivative is continuous and $$\frac { d f ( x ) } { d x } \neq 0 \text { for all } x \neq 0$$ If $$\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 2 } } = \pi$$ then
(A) for all $x \neq 0 , \quad f ( x ) > f ( 0 )$.
(B) for all $x \neq 0 , \quad f ( x ) < f ( 0 )$.
(C) for all $x , \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } > 0$.
(D) for all $x , \frac { d ^ { 2 } f ( x ) } { d x ^ { 2 } } < 0$.

Q20 Number Theory Quadratic Diophantine Equations and Perfect Squares View

The number of all integer solutions of the equation $x ^ { 2 } + y ^ { 2 } + x - y = 2021$ is
(A) 5 .
(B) 7 .
(C) 1 .
(D) 0 .

Q21 Roots of polynomials Multiplicity and derivative analysis of roots View

The number of different values of $a$ for which the equation $x ^ { 3 } - x + a = 0$ has two identical real roots is
(A) 0 .
(B) 1 .
(C) 2 .
(D) 3 .

Q22 Number Theory Quadratic Diophantine Equations and Perfect Squares View

For a positive integer $n$, the equation $$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$ does not have a solution if and only if
(A) $n = 2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .

Q23 Quadratic trigonometric equations View

For $0 \leq x < 2 \pi$, the number of solutions of the equation $$\sin ^ { 2 } x + 2 \cos ^ { 2 } x + 3 \sin x \cos x = 0$$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .

Q24 Exponential Functions Functional Equation with Exponentials View

Let $f : \mathbb { R } \rightarrow [ 0 , \infty )$ be a continuous function such that $$f ( x + y ) = f ( x ) f ( y )$$ for all $x , y \in \mathbb { R }$. Suppose that $f$ is differentiable at $x = 1$ and $$\left. \frac { d f ( x ) } { d x } \right| _ { x = 1 } = 2$$ Then, the value of $f ( 1 ) \log _ { e } f ( 1 )$ is
(A) $e$.
(B) 2.
(C) $\log _ { e } 2$.
(D) 1 .

Q25 Addition & Double Angle Formulae Telescoping Sum of Trigonometric Terms View

The expression $$\sum _ { k = 0 } ^ { 10 } 2 ^ { k } \tan \left( 2 ^ { k } \right)$$ equals
(A) $\cot 1 + 2 ^ { 11 } \cot \left( 2 ^ { 11 } \right)$.
(B) $\cot 1 - 2 ^ { 10 } \cot \left( 2 ^ { 10 } \right)$.
(C) $\cot 1 + 2 ^ { 10 } \cot \left( 2 ^ { 10 } \right)$.
(D) $\cot 1 - 2 ^ { 11 } \cot \left( 2 ^ { 11 } \right)$.

Q26 Differentiating Transcendental Functions Piecewise function analysis with transcendental components View

Define $f : \mathbb { R } \rightarrow \mathbb { R }$ by $$f ( x ) = \begin{cases} ( 1 - \cos x ) \sin \left( \frac { 1 } { x } \right) , & x \neq 0 \\ 0 , & x = 0 \end{cases}$$ Then,
(A) $f$ is discontinuous.
(B) $f$ is continuous but not differentiable.
(C) $f$ is differentiable and its derivative is discontinuous.
(D) $f$ is differentiable and its derivative is continuous.

Q27 Trig Proofs Extremal Value of Trigonometric Expression View

If the maximum and minimum values of $\sin ^ { 6 } x + \cos ^ { 6 } x$, as $x$ takes all real values, are $a$ and $b$, respectively, then $a - b$ equals
(A) $\frac { 1 } { 2 }$.
(B) $\frac { 2 } { 3 }$.
(C) $\frac { 3 } { 4 }$.
(D) 1 .

Q28 Circles Optimization on a Circle View

If two real numbers $x$ and $y$ satisfy $( x + 5 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = 196$, then the minimum possible value of $x ^ { 2 } + 2 x + y ^ { 2 } - 4 y$ is
(A) $271 - 112 \sqrt { 5 }$.
(B) $14 - 4 \sqrt { 5 }$.
(C) $276 - 112 \sqrt { 5 }$.
(D) $9 - 4 \sqrt { 5 }$.

Q29 Sequences and Series Limit Evaluation Involving Sequences View

Let us denote the fractional part of a real number $x$ by $\{ x \}$ (note: $\{ x \} = x - [ x ]$ where $[ x ]$ is the integer part of $x$ ). Then, $$\lim _ { n \rightarrow \infty } \left\{ ( 3 + 2 \sqrt { 2 } ) ^ { n } \right\}$$ (A) equals 0 .
(B) equals 1 .
(C) equals $\frac { 1 } { 2 }$.
(D) does not exist.

Q30 Chain Rule Iterated/Nested Exponential Differentiation View

Let $$\begin{gathered} p ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 2 x , x \in \mathbb { R } \\ f _ { 0 } ( x ) = \begin{cases} \int _ { 0 } ^ { x } p ( t ) d t , & x \geq 0 \\ - \int _ { x } ^ { 0 } p ( t ) d t , & x < 0 \end{cases} \\ f _ { 1 } ( x ) = e ^ { f _ { 0 } ( x ) } , \quad f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) } , \quad \ldots \quad , f _ { n } ( x ) = e ^ { f _ { n - 1 } ( x ) } \end{gathered}$$ How many roots does the equation $\frac { d f _ { n } ( x ) } { d x } = 0$ have in the interval $( - \infty , \infty ) ?$
(A) 1 .
(B) 3 .
(C) $n + 3$.
(D) $3n$.

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