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 $x = - 1 / 2$.
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) $1 + x ^ { 4 }$.
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) $e ^ { - 0.5 }$
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) $a _ { n } = a _ { n - 1 } + 2 a _ { n - 2 }$.
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) $\frac { \sqrt { 7 } } { \sqrt { 12 } }$.
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) $\frac { 8 r } { 5 }$.
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 .
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.
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
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) $\binom { 50 } { 20 }$
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 .
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.
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.
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.
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.
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) $2 \pi - 1$.
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) $\left( n ^ { 2 } - 2 n + 2 \right) t$.