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Q1 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View

Suppose $a , b$ denote the distinct real roots of the quadratic polynomial $x ^ { 2 } + 20 x - 2020$ and suppose $c , d$ denote the distinct complex roots of the quadratic polynomial $x ^ { 2 } - 20 x + 2020$. Then the value of
$$a c ( a - c ) + a d ( a - d ) + b c ( b - c ) + b d ( b - d )$$
is
(A) 0
(B) 8000
(C) 8080
(D) 16000

Q2 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View

If the function $f : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $f ( x ) = | x | ( x - \sin x )$, then which of the following statements is TRUE?
(A) $f$ is one-one, but NOT onto
(B) $f$ is onto, but NOT one-one
(C) $f$ is BOTH one-one and onto
(D) $f$ is NEITHER one-one NOR onto

Q3 Areas by integration View

Let the functions $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = e ^ { x - 1 } - e ^ { - | x - 1 | } \quad \text { and } \quad g ( x ) = \frac { 1 } { 2 } \left( e ^ { x - 1 } + e ^ { 1 - x } \right)$$
Then the area of the region in the first quadrant bounded by the curves $y = f ( x ) , y = g ( x )$ and $x = 0$ is
(A) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(B) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e - e ^ { - 1 } \right)$
(C) $( 2 - \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$
(D) $( 2 + \sqrt { 3 } ) + \frac { 1 } { 2 } \left( e + e ^ { - 1 } \right)$

Q4 Conic sections Eccentricity or Asymptote Computation View

Let $a , b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y ^ { 2 } = 4 \lambda x$, and suppose the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is
(A) $\frac { 1 } { \sqrt { 2 } }$
(B) $\frac { 1 } { 2 }$
(C) $\frac { 1 } { 3 }$
(D) $\frac { 2 } { 5 }$

Q5 Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View

Let $C _ { 1 }$ and $C _ { 2 }$ be two biased coins such that the probabilities of getting head in a single toss are $\frac { 2 } { 3 }$ and $\frac { 1 } { 3 }$, respectively. Suppose $\alpha$ is the number of heads that appear when $C _ { 1 }$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C _ { 2 }$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x ^ { 2 } - \alpha x + \beta$ are real and equal, is
(A) $\frac { 40 } { 81 }$
(B) $\frac { 20 } { 81 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 1 } { 4 }$

Q6 Applied differentiation Applied modeling with differentiation View

Consider all rectangles lying in the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq x \leq \frac { \pi } { 2 } \text { and } 0 \leq y \leq 2 \sin ( 2 x ) \right\}$$
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
(A) $\frac { 3 \pi } { 2 }$
(B) $\pi$
(C) $\frac { \pi } { 2 \sqrt { 3 } }$
(D) $\frac { \pi \sqrt { 3 } } { 2 }$

Q7 Chain Rule Piecewise Function Differentiability Analysis View

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = x ^ { 3 } - x ^ { 2 } + ( x - 1 ) \sin x$ and let $g : \mathbb { R } \rightarrow \mathbb { R }$ be an arbitrary function. Let $f g : \mathbb { R } \rightarrow \mathbb { R }$ be the product function defined by $( f g ) ( x ) = f ( x ) g ( x )$. Then which of the following statements is/are TRUE?
(A) If $g$ is continuous at $x = 1$, then $f g$ is differentiable at $x = 1$
(B) If $f g$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$
(C) If $g$ is differentiable at $x = 1$, then $f g$ is differentiable at $x = 1$
(D) If $f g$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$

Q8 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View

Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^ { - 1 } = \operatorname { adj } ( \operatorname { adj } M )$, then which of the following statements is/are ALWAYS TRUE?
(A) $M = I$
(B) $\operatorname { det } M = 1$
(C) $M ^ { 2 } = I$
(D) $( \operatorname { adj } M ) ^ { 2 } = I$

Q9 Complex numbers 2 Inequalities and Estimates for Complex Expressions View

Let $S$ be the set of all complex numbers $z$ satisfying $\left| z ^ { 2 } + z + 1 \right| = 1$. Then which of the following statements is/are TRUE?
(A) $\left| z + \frac { 1 } { 2 } \right| \leq \frac { 1 } { 2 }$ for all $z \in S$
(B) $| z | \leq 2$ for all $z \in S$
(C) $\left| z + \frac { 1 } { 2 } \right| \geq \frac { 1 } { 2 }$ for all $z \in S$
(D) The set $S$ has exactly four elements

Q10 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View

Let $x , y$ and $z$ be positive real numbers. Suppose $x , y$ and $z$ are the lengths of the sides of a triangle opposite to its angles $X , Y$ and $Z$, respectively. If
$$\tan \frac { X } { 2 } + \tan \frac { Z } { 2 } = \frac { 2 y } { x + y + z }$$
then which of the following statements is/are TRUE?
(A) $2 Y = X + Z$
(B) $Y = X + Z$
(C) $\tan \frac { X } { 2 } = \frac { x } { y + z }$
(D) $x ^ { 2 } + z ^ { 2 } - y ^ { 2 } = x z$

Q11 Vectors 3D & Lines Parametric Representation of a Line View

Let $L _ { 1 }$ and $L _ { 2 }$ be the following straight lines.
$$L _ { 1 } : \frac { x - 1 } { 1 } = \frac { y } { - 1 } = \frac { z - 1 } { 3 } \text { and } L _ { 2 } : \frac { x - 1 } { - 3 } = \frac { y } { - 1 } = \frac { z - 1 } { 1 } .$$
Suppose the straight line
$$L : \frac { x - \alpha } { l } = \frac { y - 1 } { m } = \frac { z - \gamma } { - 2 }$$
lies in the plane containing $L _ { 1 }$ and $L _ { 2 }$, and passes through the point of intersection of $L _ { 1 }$ and $L _ { 2 }$. If the line $L$ bisects the acute angle between the lines $L _ { 1 }$ and $L _ { 2 }$, then which of the following statements is/are TRUE?
(A) $\alpha - \gamma = 3$
(B) $l + m = 2$
(C) $\alpha - \gamma = 1$
(D) $l + m = 0$

Q12 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Which of the following inequalities is/are TRUE?
(A) $\int _ { 0 } ^ { 1 } x \cos x \, d x \geq \frac { 3 } { 8 }$
(B) $\int _ { 0 } ^ { 1 } x \sin x \, d x \geq \frac { 3 } { 10 }$
(C) $\int _ { 0 } ^ { 1 } x ^ { 2 } \cos x \, d x \geq \frac { 1 } { 2 }$
(D) $\int _ { 0 } ^ { 1 } x ^ { 2 } \sin x \, d x \geq \frac { 2 } { 9 }$

Q13 Laws of Logarithms Optimize a Logarithmic Expression View

Let $m$ be the minimum possible value of $\log _ { 3 } \left( 3 ^ { y _ { 1 } } + 3 ^ { y _ { 2 } } + 3 ^ { y _ { 3 } } \right)$, where $y _ { 1 } , y _ { 2 } , y _ { 3 }$ are real numbers for which $y _ { 1 } + y _ { 2 } + y _ { 3 } = 9$. Let $M$ be the maximum possible value of ( $\log _ { 3 } x _ { 1 } + \log _ { 3 } x _ { 2 } + \log _ { 3 } x _ { 3 }$ ), where $x _ { 1 } , x _ { 2 } , x _ { 3 }$ are positive real numbers for which $x _ { 1 } + x _ { 2 } + x _ { 3 } = 9$. Then the value of $\log _ { 2 } \left( m ^ { 3 } \right) + \log _ { 3 } \left( M ^ { 2 } \right)$ is

Q14 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b _ { 1 } , b _ { 2 } , b _ { 3 } , \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a _ { 1 } = b _ { 1 } = c$, then the number of all possible values of $c$, for which the equality
$$2 \left( a _ { 1 } + a _ { 2 } + \cdots + a _ { n } \right) = b _ { 1 } + b _ { 2 } + \cdots + b _ { n }$$
holds for some positive integer $n$, is $\_\_\_\_$

Q15 Trig Graphs & Exact Values View

Let $f : [ 0,2 ] \rightarrow \mathbb { R }$ be the function defined by
$$f ( x ) = ( 3 - \sin ( 2 \pi x ) ) \sin \left( \pi x - \frac { \pi } { 4 } \right) - \sin \left( 3 \pi x + \frac { \pi } { 4 } \right)$$
If $\alpha , \beta \in [ 0,2 ]$ are such that $\{ x \in [ 0,2 ] : f ( x ) \geq 0 \} = [ \alpha , \beta ]$, then the value of $\beta - \alpha$ is $\_\_\_\_$

Q16 Vectors Introduction & 2D Dot Product Computation View

In a triangle $P Q R$, let $\vec { a } = \overrightarrow { Q R } , \vec { b } = \overrightarrow { R P }$ and $\vec { c } = \overrightarrow { P Q }$. If
$$| \vec { a } | = 3 , \quad | \vec { b } | = 4 \quad \text { and } \quad \frac { \vec { a } \cdot ( \vec { c } - \vec { b } ) } { \vec { c } \cdot ( \vec { a } - \vec { b } ) } = \frac { | \vec { a } | } { | \vec { a } | + | \vec { b } | }$$
then the value of $| \vec { a } \times \vec { b } | ^ { 2 }$ is $\_\_\_\_$

Q17 Factor & Remainder Theorem Root Interlacing and Sign Conditions View

For a polynomial $g ( x )$ with real coefficients, let $m _ { g }$ denote the number of distinct real roots of $g ( x )$. Suppose $S$ is the set of polynomials with real coefficients defined by
$$S = \left\{ \left( x ^ { 2 } - 1 \right) ^ { 2 } \left( a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } \right) : a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } \in \mathbb { R } \right\}$$
For a polynomial $f$, let $f ^ { \prime }$ and $f ^ { \prime \prime }$ denote its first and second order derivatives, respectively. Then the minimum possible value of ( $m _ { f ^ { \prime } } + m _ { f ^ { \prime \prime } }$ ), where $f \in S$, is $\_\_\_\_$

Q18 Laws of Logarithms Determine Parameters of a Logarithmic Function View

Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right hand limit
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { ( 1 - x ) ^ { \frac { 1 } { x } } - e ^ { - 1 } } { x ^ { a } }$$
is equal to a nonzero real number, is $\_\_\_\_$

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