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Q1 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

The number of subsets of $\{ 1,2,3 , \ldots , 10 \}$ having an odd number of elements is
(A) 1024
(B) 512
(C) 256
(D) 50 .

Q2 Differentiating Transcendental Functions Piecewise function analysis with transcendental components View

For the function on the real line $\mathbb { R }$ given by $f ( x ) = | x | + | x + 1 | + e ^ { x }$, which of the following is true ?
(A) It is differentiable everywhere.
(B) It is differentiable everywhere except at $x = 0$ and $x = - 1$.
(C) It is differentiable everywhere except at $x = 1 / 2$.
(D) It is differentiable everywhere except at $x = - 1 / 2$.

Q3 Chain Rule Derivative of Inverse Functions View

If $f , g$ are real-valued differentiable functions on the real line $\mathbb { R }$ such that $f ( g ( x ) ) = x$ and $f ^ { \prime } ( x ) = 1 + ( f ( x ) ) ^ { 2 }$, then $g ^ { \prime } ( x )$ equals
(A) $\frac { 1 } { 1 + x ^ { 2 } }$
(B) $1 + x ^ { 2 }$
(C) $\frac { 1 } { 1 + x ^ { 4 } }$
(D) $1 + x ^ { 4 }$.

Q4 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of real solutions of $e ^ { x } = \sin ( x )$ is
(A) 0
(B) 1
(C) 2
(D) infinite.

Q5 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View

What is the limit of $\sum _ { k = 1 } ^ { n } \frac { e ^ { - k / n } } { n }$ as $n$ tends to $\infty$ ?
(A) The limit does not exist.
(B) $\infty$
(C) $1 - e ^ { - 1 }$
(D) $e ^ { - 0.5 }$

Q6 Combinations & Selection Partitioning into Teams or Groups View

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?
(A) $\frac { 64 ! } { 32 ! 2 ^ { 32 } }$
(B) $\binom { 64 } { 2 } \binom { 62 } { 2 } \cdots \binom { 4 } { 2 } \binom { 2 } { 2 }$
(C) $\frac { 64 ! } { 32 ! 32 ! }$
(D) $\frac { 64 ! } { 2 ^ { 64 } }$

Q7 Sequences and Series Estimation or Bounding of a Sum View

The integral part of $\sum _ { n = 2 } ^ { 9999 } \frac { 1 } { \sqrt { n } }$ equals
(A) 196
(B) 197
(C) 198
(D) 199 .

Q8 Sequences and Series Recurrence Relations and Sequence Properties View

Let $a _ { n }$ be the number of subsets of $\{ 1,2 , \ldots , n \}$ that do not contain any two consecutive numbers. Then
(A) $a _ { n } = a _ { n - 1 } + a _ { n - 2 }$
(B) $a _ { n } = 2 a _ { n - 1 }$
(C) $a _ { n } = a _ { n - 1 } - a _ { n - 2 }$
(D) $a _ { n } = a _ { n - 1 } + 2 a _ { n - 2 }$.

Q9 Number Theory Combinatorial Number Theory and Counting View

There are 128 numbers $1,2 , \ldots , 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3 ) and delete 4 . Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left ?
(A) 1
(B) 63
(C) 127
(D) None of the above.

Q10 Complex Numbers Argand & Loci Modulus Inequalities and Triangle Inequality Applications View

Let $z$ and $w$ be complex numbers lying on the circles of radii 2 and 3 respectively, with centre ( 0,0 ). If the angle between the corresponding vectors is 60 degrees, then the value of $| z + w | / | z - w |$ is:
(A) $\frac { \sqrt { 19 } } { \sqrt { 7 } }$
(B) $\frac { \sqrt { 7 } } { \sqrt { 19 } }$
(C) $\frac { \sqrt { 12 } } { \sqrt { 7 } }$
(D) $\frac { \sqrt { 7 } } { \sqrt { 12 } }$.

Q11 Circles Area and Geometric Measurement Involving Circles View

Two vertices of a square lie on a circle of radius $r$ and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is
(A) $\frac { 3 r } { 2 }$
(B) $\frac { 4 r } { 3 }$
(C) $\frac { 6 r } { 5 }$
(D) $\frac { 8 r } { 5 }$.

Q12 Modulus function Counting solutions satisfying modulus conditions View

For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Then the number of real solutions of $| 2 x - [ x ] | = 4$ is
(A) 4
(B) 3
(C) 2
(D) 1 .

Q13 Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $f , g$ be differentiable functions on the real line $\mathbb { R }$ with $f ( 0 ) > g ( 0 )$. Assume that the set $M = \{ t \in \mathbb { R } \mid f ( t ) = g ( t ) \}$ is non-empty and that $f ^ { \prime } ( t ) \geq g ^ { \prime } ( t )$ for all $t \in M$. Then which of the following is necessarily true?
(A) If $t \in M$, then $t < 0$.
(B) For any $t \in M , f ^ { \prime } ( t ) > g ^ { \prime } ( t )$.
(C) For any $t \notin M , f ( t ) > g ( t )$.
(D) None of the above.

Q14 Sequences and Series Recurrence Relations and Sequence Properties View

Consider the sequence $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 , \ldots$ obtained by writing one 1 , two 2's, three 3's and so on. What is the $2020 ^ { \text {th} }$ term in the sequence?
(A) 62
(B) 63
(C) 64
(D) 65

Q15 Combinations & Selection Counting Functions or Mappings with Constraints View

Let $A = \left\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { 50 } \right\}$ and $B = \left\{ y _ { 1 } , y _ { 2 } , \ldots , y _ { 20 } \right\}$ be two sets of real numbers. What is the total number of functions $f : A \rightarrow B$ such that $f$ is onto and $f \left( x _ { 1 } \right) \leq f \left( x _ { 2 } \right) \leq \cdots \leq f \left( x _ { 50 } \right)$ ?
(A) $\binom { 49 } { 19 }$
(B) $\binom { 49 } { 20 }$
(C) $\binom { 50 } { 19 }$
(D) $\binom { 50 } { 20 }$

Q16 Complex Numbers Arithmetic Modulus Computation View

The number of complex roots of the polynomial $z ^ { 5 } - z ^ { 4 } - 1$ which have modulus 1 is
(A) 0
(B) 1
(C) 2
(D) more than 2 .

Q17 Curve Sketching Number of Solutions / Roots via Curve Analysis View

The number of real roots of the polynomial $$p ( x ) = \left( x ^ { 2020 } + 2020 x ^ { 2 } + 2020 \right) \left( x ^ { 3 } - 2020 \right) \left( x ^ { 2 } - 2020 \right)$$ is
(A) 2
(B) 3
(C) 2023
(D) 2025 .

Q18 Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View

Which of the following is the sum of an infinite geometric sequence whose terms come from the set $\left\{ 1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots , \frac { 1 } { 2 ^ { n } } , \ldots \right\}$ ?
(A) $\frac { 1 } { 5 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 1 } { 9 }$
(D) $\frac { 1 } { 11 }$

Q19 Discriminant and conditions for roots Nature of roots given coefficient constraints View

If $a , b , c$ are distinct odd natural numbers, then the number of rational roots of the polynomial $a x ^ { 2 } + b x + c$
(A) must be 0 .
(B) must be 1 .
(C) must be 2 .
(D) cannot be determined from the given data.

Q20 Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Let $A , B , C$ be finite subsets of the plane such that $A \cap B , B \cap C$ and $C \cap A$ are all empty. Let $S = A \cup B \cup C$. Assume that no three points of $S$ are collinear and also assume that each of $A , B$ and $C$ has at least 3 points. Which of the following statements is always true?
(A) There exists a triangle having a vertex from each of $A , B , C$ that does not contain any point of $S$ in its interior.
(B) Any triangle having a vertex from each of $A , B , C$ must contain a point of $S$ in its interior.
(C) There exists a triangle having a vertex from each of $A , B , C$ that contains all the remaining points of $S$ in its interior.
(D) There exist 2 triangles, both having a vertex from each of $A , B , C$ such that the two triangles do not intersect.

Q21 Conditional Probability Bayes' Theorem with Diagnostic/Screening Test View

Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:

For people who really do have the allergy, the test says ``Yes'' $90 \%$ of the time.
For people who do not have the allergy, the test says ``Yes'' $15 \%$ of the time.

If $2 \%$ of the population has the allergy and Shubhaangi's test says ``Yes'', then the chances that Shubhaangi does really have the allergy are
(A) $1 / 9$
(B) $6 / 55$
(C) $1 / 11$
(D) cannot be determined from the given data.

Q22 Reciprocal Trig & Identities View

If $\sin \left( \tan ^ { - 1 } ( x ) \right) = \cot \left( \sin ^ { - 1 } \left( \sqrt { \frac { 13 } { 17 } } \right) \right)$ then $x$ is
(A) $\frac { 4 } { 17 }$
(B) $\frac { 2 } { 3 }$
(C) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 ^ { 2 } + 13 ^ { 2 } } }$
(D) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 \times 13 } }$.

Q23 Permutations & Arrangements Dictionary Order / Rank of a Permutation View

If the word PERMUTE is permuted in all possible ways and the different resulting words are written down in alphabetical order (also known as dictionary order), irrespective of whether the word has meaning or not, then the $720 ^ { \text {th} }$ word would be:
(A) EEMPRTU
(B) EUTRPME
(C) UTRPMEE
(D) MEETPUR.

Q24 Areas by integration View

The points $( 4,7 , - 1 ) , ( 1,2 , - 1 ) , ( - 1 , - 2 , - 1 )$ and $( 2,3 , - 1 )$ in $\mathbb { R } ^ { 3 }$ are the vertices of a
(A) rectangle which is not a square.
(B) rhombus.
(C) parallelogram which is not a rectangle.
(D) trapezium which is not a parallelogram.

Q25 Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $f ( x ) , g ( x )$ be functions on the real line $\mathbb { R }$ such that both $f ( x ) + g ( x )$ and $f ( x ) g ( x )$ are differentiable. Which of the following is FALSE ?
(A) $f ( x ) ^ { 2 } + g ( x ) ^ { 2 }$ is necessarily differentiable.
(B) $f ( x )$ is differentiable if and only if $g ( x )$ is differentiable.
(C) $f ( x )$ and $g ( x )$ are necessarily continuous.
(D) If $f ( x ) > g ( x )$ for all $x \in \mathbb { R }$, then $f ( x )$ is differentiable.

Q26 Stationary points and optimisation Geometric or applied optimisation problem View

Let $S$ be the set consisting of all those real numbers that can be written as $p - 2 a$ where $p$ and $a$ are the perimeter and area of a right-angled triangle having base length 1 . Then $S$ is
(A) $( 2 , \infty )$
(B) $( 1 , \infty )$
(C) $( 0 , \infty )$
(D) the real line $\mathbb { R }$.

Q27 Combinations & Selection Subset Counting with Set-Theoretic Conditions View

Let $S = \{ 1,2 , \ldots , n \}$. For any non-empty subset $A$ of $S$, let $l ( A )$ denote the largest number in $A$. If $f ( n ) = \sum _ { A \subseteq S } l ( A )$, that is, $f ( n )$ is the sum of the numbers $l ( A )$ while $A$ ranges over all the nonempty subsets of $S$, then $f ( n )$ is
(A) $2 ^ { n } ( n + 1 )$
(B) $2 ^ { n } ( n + 1 ) - 1$
(C) $2 ^ { n } ( n - 1 )$
(D) $2 ^ { n } ( n - 1 ) + 1$.

Q28 Areas by integration View

The area of the region in the plane $\mathbb { R } ^ { 2 }$ given by points $( x , y )$ satisfying $| y | \leq 1$ and $x ^ { 2 } + y ^ { 2 } \leq 2$ is
(A) $\pi + 1$
(B) $2 \pi - 2$
(C) $\pi + 2$
(D) $2 \pi - 1$.

Q29 Binomial Distribution Compute Expectation of a Binomial Sum (Algebraic Evaluation) View

Let $n$ be a positive integer and $t \in ( 0,1 )$. Then $\sum _ { r = 0 } ^ { n } r \binom { n } { r } t ^ { r } ( 1 - t ) ^ { n - r }$ equals
(A) $n t$
(B) $( n - 1 ) ( 1 - t )$
(C) $n t + ( n - 1 ) ( 1 - t )$
(D) $\left( n ^ { 2 } - 2 n + 2 \right) t$.

Q30 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View

For any real number $x$, let $[ x ]$ be the greatest integer $m$ such that $m \leq x$. Then the number of points of discontinuity of the function $g ( x ) = \left[ x ^ { 2 } - 2 \right]$ on the interval $( - 3,3 )$ is
(A) 5
(B) 9
(C) 13
(D) 16 .

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