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Q1 Permutations & Arrangements Distribution of Objects into Bins/Groups View

You are given a $4 \times 4$ chessboard, and asked to fill it with five $3 \times 1$ pieces and one $1 \times 1$ piece. Then, over all such fillings, the number of squares that can be occupied by the $1 \times 1$ piece is
(A) 4
(B) 8
(C) 12
(D) 16 .

Q2 Conditional Probability Bayes' Theorem with Production/Source Identification View

A brand called Jogger's Pride produces pairs of shoes in three different units that are named $U _ { 1 } , U _ { 2 }$ and $U _ { 3 }$. These units produce $10 \% , 30 \% , 60 \%$ of the total output of the brand with the chance that a pair of shoes being defective is $20 \% , 40 \% , 10 \%$ respectively. If a randomly selected pair of shoes from the combined output is found to be defective, then what is the chance that the pair was manufactured in the unit $U _ { 3 }$?
(A) $30 \%$
(B) $15 \%$
(C) $\frac { 3 } { 5 } \times 100 \%$
(D) Cannot be determined from the given data.

Q3 Sine and Cosine Rules Compute area of a triangle or related figure View

Consider a paper in the shape of an equilateral triangle $A B C$ with circumcenter $O$ and perimeter 9 units. If we fold the paper in such a way that each of the vertices $A , B , C$ gets identified with $O$, then the area of the resulting shape in square units is:
(A) $\frac { 3 \sqrt { 3 } } { 4 }$
(B) $\frac { 4 } { \sqrt { 3 } }$
(C) $\frac { 3 \sqrt { 3 } } { 2 }$
(D) $3 \sqrt { 3 }$.

Q4 Combinations & Selection Geometric Combinatorics View

Let $P$ be a regular twelve-sided polygon. The number of right-angled triangles formed by the vertices of $P$ is
(A) 60
(B) 120
(C) 160
(D) 220 .

Q5 Arithmetic Sequences and Series Summation of Derived Sequence from AP View

If the $n$ terms $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ are in arithmetic progression with increment $r$, then the difference between the mean of their squares and the square of their mean is
(A) $\frac { r ^ { 2 } \left( ( n - 1 ) ^ { 2 } - 1 \right) } { 12 }$
(B) $\frac { r ^ { 2 } } { 12 }$
(C) $\frac { r ^ { 2 } \left( n ^ { 2 } - 1 \right) } { 12 }$
(D) $\frac { n ^ { 2 } - 1 } { 12 }$

Q6 Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View

A father wants to distribute a certain sum of money between his daughter and son in such a way that if both of them invest their shares in the scheme that offers compound interest at $\frac { 25 } { 3 } \%$ per annum, for $t$ and $t + 2$ years respectively, then the two shares grow to become equal. If the son's share was rupees 4320, then the total money distributed by the father was
(A) rupees 9360
(B) rupees 9390
(C) rupees 16, 590
(D) rupees 16, 640.

Q7 Modulus function Solving inequalities involving modulus View

Q8 Complex Numbers Arithmetic Modulus Computation View

For each natural number $k$, choose a complex number $z _ { k }$ with $\left| z _ { k } \right| = 1$ and denote by $a _ { k }$ the area of the triangle formed by $z _ { k } , i z _ { k } , z _ { k } + i z _ { k }$. Then, which of the following is true for the series below?
$$\sum _ { k = 1 } ^ { \infty } \left( a _ { k } \right) ^ { k }$$
(A) It converges only if every $z _ { k }$ lies in the same quadrant.
(B) It always diverges.
(C) It always converges.
(D) none of the above.

Q9 Differential equations Verification that a Function Satisfies a DE View

The function $y = e ^ { k x }$ satisfies
$$\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } \right) \left( \frac { d y } { d x } - y \right) = y \frac { d y } { d x }$$
for
(A) exactly one value of $k$.
(B) two distinct values of $k$.
(C) three distinct values of $k$.
(D) infinitely many values of $k$.

Q10 Simultaneous equations View

For a real number $\theta$, consider the following simultaneous equations:
$$\begin{aligned} & \cos ( \theta ) x - \sin ( \theta ) y = 1 \\ & \sin ( \theta ) x + \cos ( \theta ) y = 2 \end{aligned}$$
The number of solutions of these equations in $x$ and $y$ is
(A) 0
(B) 1
(C) infinite for some values of $\theta$
(D) finite only when $\theta = \frac { m \pi } { n }$ for integers $m$, and $n \neq 0$.

Q11 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

Q12 Number Theory Congruence Reasoning and Parity Arguments View

A particle is allowed to move in the $XY$-plane by choosing any one of the two jumps:

move two units to right and one unit up, i.e., $( a , b ) \mapsto ( a + 2 , b + 1 )$ or
move two units up and one unit to right, i.e., $( a , b ) \mapsto ( a + 1 , b + 2 )$.

Let $P = ( 30,63 )$ and $Q = ( 100,100 )$. If the particle starts at the origin, then
(A) $P$ is reachable but not $Q$.
(B) $Q$ is reachable but not $P$.
(C) both $P$ and $Q$ are reachable.
(D) neither $P$ nor $Q$ is reachable.

Q13 Curve Sketching Identifying the Correct Graph of a Function View

For a real polynomial in one variable $P$, let $Z ( P )$ denote the locus of points ( $x , y$ ) in the plane such that $P ( x ) + P ( y ) = 0$. Then,
(A) there exist polynomials $Q _ { 1 }$ and $Q _ { 2 }$ such that $Z \left( Q _ { 1 } \right)$ is a circle and $Z \left( Q _ { 2 } \right)$ is a parabola.
(B) there does not exist any polynomial $Q$ such that $Z ( Q )$ is a circle or a parabola.
(C) there exists a polynomial $Q$ such that $Z ( Q )$ is a circle but there does not exist any polynomial $P$ such that $Z ( P )$ is a parabola.
(D) there exists a polynomial $Q$ such that $Z ( Q )$ is a parabola but there does not exist any polynomial $P$ such that $Z ( P )$ is a circle.

Q14 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View

Let $P ( X ) = X ^ { 4 } + a _ { 3 } X ^ { 3 } + a _ { 2 } X ^ { 2 } + a _ { 1 } X + a _ { 0 }$ be a polynomial in $X$ with real coefficients. Assume that
$$P ( 0 ) = 1 , P ( 1 ) = 2 , P ( 2 ) = 3 , \text { and } P ( 3 ) = 4 .$$
Then, the value of $P ( 4 )$ is
(A) 5
(B) 24
(C) 29
(D) not determinable from the given data.

Q15 Stationary points and optimisation Find critical points and classify extrema of a given function View

Let $f$ be a real-valued differentiable function defined on the real line $\mathbb { R }$ such that its derivative $f ^ { \prime }$ is zero at exactly two distinct real numbers $\alpha$ and $\beta$. Then,
(A) $\alpha$ and $\beta$ are points of local maxima of the function $f$.
(B) $\alpha$ and $\beta$ are points of local minima of the function $f$.
(C) one must be a point of local maximum and the other must be a point of local minimum of $f$.
(D) given data is insufficient to conclude about either of them being local extrema points.

Q16 Permutations & Arrangements Distribution of Objects into Bins/Groups View

A school allowed the students of a class to go to swim during the days March 11th to March 15, 2019. The minimum number of students the class should have had that ensures that at least two of them went to swim on the same set of dates is:
(A) 6
(B) 32
(C) 33
(D) 121 .

Q17 Number Theory Linear Diophantine Equations View

Let $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ be positive integers such that
$$\sum _ { i = 1 } ^ { 4 } \frac { 1 } { a _ { i } } = \frac { 11 } { 6 }$$
Then, $a _ { 4 } - a _ { 2 }$ equals
(A) 11
(B) 10
(C) 9
(D) 8 .

Q18 Permutations & Arrangements Distribution of Objects into Bins/Groups View

Three children and two adults want to cross a river using a rowing boat. The boat can carry no more than a single adult or, in case no adult is in the boat, a maximum of two children. The least number of times the boat needs to cross the river to transport all five people is:
(A) 9
(B) 11
(C) 13
(D) 15 .

Q19 Matrices Determinant and Rank Computation View

Let $M$ be a $3 \times 3$ matrix with all entries being 0 or 1 . Then, all possible values for $\operatorname { det } ( M )$ are
(A) $0 , \pm 1$
(B) $0 , \pm 1 , \pm 2$
(C) $0 , \pm 1 , \pm 3$
(D) $0 , \pm 1 , \pm 2 , \pm 3$.

Q20 Circles Inscribed/Circumscribed Circle Computations View

In the following picture, $A B C$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the mid-point of $B C$. If $A B = A C = 15$ and $B C = 10$, then $O P$ equals:
(A) $\frac { \sqrt { 5 } } { \sqrt { 2 } }$
(B) $\frac { 5 } { \sqrt { 2 } }$
(C) $2 \sqrt { 5 }$
(D) $5 \sqrt { 2 }$.

Q21 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View

For every real number $x \neq - 1$, let $f ( x ) = \frac { x } { x + 1 }$. Write $f _ { 1 } ( x ) = f ( x )$ and for $n \geq 2 , f _ { n } ( x ) = f \left( f _ { n - 1 } ( x ) \right)$. Then,
$$f _ { 1 } ( - 2 ) \cdot f _ { 2 } ( - 2 ) \cdots \cdots f _ { n } ( - 2 )$$
must equal
(A) $\frac { 2 ^ { n } } { 1 \cdot 3 \cdot 5 \cdots \cdot ( 2 n - 1 ) }$
(B) 1
(C) $\frac { 1 } { 2 } \binom { 2 n } { n }$
(D) $\binom { 2 n } { n }$.

Q22 Binomial Theorem (positive integer n) Extract Coefficients Using Roots of Unity or Substitution Filter View

Let the integers $a _ { i }$ for $0 \leq i \leq 54$ be defined by the equation
$$\left( 1 + X + X ^ { 2 } \right) ^ { 27 } = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { 54 } X ^ { 54 }$$
Then, $a _ { 0 } + a _ { 3 } + a _ { 6 } + a _ { 9 } + \cdots + a _ { 54 }$ equals
(A) $3 ^ { 26 }$
(B) $3 ^ { 27 }$
(C) $3 ^ { 28 }$
(D) $3 ^ { 29 }$.

Q23 Number Theory Combinatorial Number Theory and Counting View

An examination has 20 questions. For each question the marks that can be obtained are either $-1$ or $0$ or $4$. Let $S$ be the set of possible total marks that a student can score in the examination. Then, the number of elements in $S$ is
(A) 93
(B) 94
(C) 95
(D) 96 .

Q24 Circles Inscribed/Circumscribed Circle Computations View

Chords $A B$ and $C D$ of a circle intersect at right angle at the point $P$. If the lengths of $A P , P B , C P , P D$ are $2,6,3,4$ units respectively, then the radius of the circle is:
(A) 4
(B) $\frac { \sqrt { 65 } } { 2 }$
(C) $\frac { \sqrt { 66 } } { 2 }$
(D) $\frac { \sqrt { 67 } } { 2 }$

Q25 Standard trigonometric equations Locus or solution set characterization of a trigonometric relation View

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

Q26 Number Theory Quadratic Diophantine Equations and Perfect Squares View

The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3

Q27 Number Theory Modular Arithmetic Computation View

Let $a \geq b \geq c \geq 0$ be integers such that $2 ^ { a } + 2 ^ { b } - 2 ^ { c } = 144$. Then, $a + b - c$ equals:
(A) 7
(B) 8
(C) 9
(D) 10 .

Q28 Roots of polynomials Existence or counting of roots with specified properties View

The number of integers $n$ for which the cubic equation $X ^ { 3 } - X + n = 0$ has 3 distinct integer solutions is:
(A) 0
(B) 1
(C) 2
(D) infinite.

Q29 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

Q30 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

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