Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then, (A) $a _ { n } < b _ { n }$ for all $n > 1$ (B) $a _ { n } > b _ { n }$ for all $n > 1$ (C) $a _ { n } = b _ { n }$ for infinitely many $n$ (D) None of the above
The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is (A) 399900 (B) 399960 (C) 390000 (D) 360000
$z _ { 1 } , z _ { 2 }$ are two complex numbers with $z _ { 2 } \neq 0$ and $z _ { 1 } \neq z _ { 2 }$ and satisfying $\left| \frac { z _ { 1 } + z _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| = 1$. Then $\frac { z _ { 1 } } { z _ { 2 } }$ is (A) real and negative (B) real and positive (C) purely imaginary (D) none of the above need to be true always
The set of all real numbers $x$ satisfying the inequality $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is (A) the interval $[ 2 , \infty )$ (B) the interval $[ 0 , \infty )$ (C) the interval $[ - 1 , \infty )$ (D) none of the above
The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is (A) $a b$ (B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$ (C) $\frac { ( a + b ) ^ { 2 } } { 2 }$ (D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$
Let $A$ be the fixed point $(0,4)$ and $B$ be a moving point $(2t, 0)$. Let $M$ be the mid-point of $A B$ and let the perpendicular bisector of $A B$ meet the $y$-axis at $R$. The locus of the mid-point $P$ of $M R$ is (A) $y + x ^ { 2 } = 2$ (B) $x ^ { 2 } + ( y - 2 ) ^ { 2 } = 1 / 4$ (C) $( y - 2 ) ^ { 2 } - x ^ { 2 } = 1 / 4$ (D) none of the above
The sides of a triangle are given to be $x ^ { 2 } + x + 1, 2 x + 1$ and $x ^ { 2 } - 1$. Then the largest of the three angles of the triangle is (A) $75 ^ { \circ }$ (B) $\left( \frac { x } { x + 1 } \pi \right)$ radians (C) $120 ^ { \circ }$ (D) $135 ^ { \circ }$
Two poles, $A B$ of length two metres and $C D$ of length twenty metres are erected vertically with bases at $B$ and $D$. The two poles are at a distance not less than twenty metres. It is observed that $\tan \angle A C B = 2 / 77$. The distance between the two poles is (A) $72 m$ (B) 68 m (C) 24 m (D) 24.27 m
If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to (A) $30 ^ { \circ }$ (B) $90 ^ { \circ }$ (C) $45 ^ { \circ }$ (D) none of the above
In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$ (A) never changes sign (B) changes sign only once (C) changes sign more than once, but finitely many times (D) changes sign infinitely many times
The limit $$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$$ (A) does not exist (B) exists and equals 0 (C) exists and equals $2 / 3$ (D) exists and equals 1
Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is equal to (A) $f _ { n } ( x )$ (B) $f _ { n } ( x ) f _ { n - 1 } ( x )$ (C) $f _ { n } ( x ) f _ { n - 1 } ( x ) \cdots f _ { 1 } ( x )$ (D) $f _ { n + 1 } ( x ) f _ { n } ( x ) \cdots f _ { 1 } ( x ) e ^ { x }$
If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then (A) $A = 0 , B = 0$ (B) $A = 0 , B = - 2$ (C) $A = 1 , B = 1$ (D) $A = 1 , B = 0$
A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is (A) 30 (B) 60 (C) $30 \sqrt { 3.3 }$ (D) $20 \sqrt { 33 }$
If $b = \int _ { 0 } ^ { 1 } \frac { e ^ { t } } { t + 1 } d t$ then $\int _ { a - 1 } ^ { a } \frac { e ^ { - t } } { t - a - 1 } d t$ is (A) $b e ^ { a }$ (B) $b e ^ { - a }$ (C) $- b e ^ { - a }$ (D) $- b e ^ { a }$
In the triangle $A B C$, the angle $\angle B A C$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $A B C$ is (A) obtuse angled (B) right angled (C) acute angled but not equilateral (D) equilateral
Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle? (A) $4 n - 2$ (B) $4 n$ (C) $8 n - 4$ (D) $8 n - 2$
A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is (A) 2.4 (B) 3 (C) 3.6 (D) 12
An isosceles triangle with base 6 cms. and base angles $30 ^ { \circ }$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is (A) $3 \sqrt { 3 } / 2$ (B) $\sqrt { 3 } / 2$ (C) $\sqrt { 3 }$ (D) $4 / \sqrt { 3 }$
Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is (A) $2 ^ { n }$ (B) $3 ^ { n }$ (C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$ (D) $n !$
The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is (A) 60 (B) 50 (C) 35 (D) 30
Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i$-th box, $i = 1, 2, 3$. Then the number of ways in which the balls can be chosen such that $n _ { 1 } < n _ { 2 } < n _ { 3 }$ is (A) 120 (B) 130 (C) 150 (D) 160
Let $a$ be a real number. The number of distinct solutions $(x, y)$ of the system of equations $( x - a ) ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } = y ^ { 2 }$, can only be (A) $0, 1, 2, 3, 4$ or 5 (B) 0, 1 or 3 (C) $0, 1, 2$ or 4 (D) $0, 2, 3$, or 4
The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e ^ { - x }$ is: (A) $1 / e$ (B) 1 (C) $1 / 2$ (D) $e$
Suppose $a, b$ and $n$ are positive integers, all greater than one. If $a ^ { n } + b ^ { n }$ is prime, what can you say about $n$? (A) The integer $n$ must be 2 (B) The integer $n$ need not be 2, but must be a power of 2 (C) The integer $n$ need not be a power of 2, but must be even (D) None of the above is necessarily true
The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by (A) $\theta = 0$ (B) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer (C) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer (D) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer
Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as shown in the graphs.
The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if (A) $\alpha \geq 2$ (B) $\alpha < 2$ (C) $\alpha < - 1$ (D) $\alpha > 2$
Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is (A) $36 ^ { \circ }$ (B) $29 ^ { \circ }$ (C) $7 ^ { \circ }$ (D) $25 ^ { \circ }$
A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number? (A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$ (B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$ (C) $\binom { 10 } { 3 } 2 ^ { 7 }$ (D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$
Let $P$ be a point on the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ which does not lie on the axes. If the normal at the point $P$ intersects the major and minor axes at $C$ and $D$ respectively, then the ratio $P C : P D$ equals (A) 2 (B) $1 / 2$ (C) 4 (D) $1 / 4$
The set of complex numbers $z$ satisfying the equation $$( 3 + 7 i ) z + ( 10 - 2 i ) \bar { z } + 100 = 0$$ represents, in the complex plane, (A) a straight line (B) a pair of intersecting straight lines (C) a pair of distinct parallel straight lines (D) a point
The number of triplets $(a, b, c)$ of integers such that $a < b < c$ and $a, b, c$ are sides of a triangle with perimeter 21 is (A) 7 (B) 8 (C) 11 (D) 12
Suppose $a, b$ and $c$ are three numbers in G.P. If the equations $a x ^ { 2 } + 2 b x + c = 0$ and $d x ^ { 2 } + 2 e x + f = 0$ have a common root, then $\frac { d } { a } , \frac { e } { b }$ and $\frac { f } { c }$ are in (A) A.P. (B) G.P. (C) H.P. (D) none of the above
Suppose $A B C D$ is a quadrilateral such that $\angle B A C = 50 ^ { \circ } , \angle C A D = 60 ^ { \circ } , \angle C B D = 30 ^ { \circ }$ and $\angle B D C = 25 ^ { \circ }$. If $E$ is the point of intersection of $A C$ and $B D$, then the value of $\angle A E B$ is (A) $75 ^ { \circ }$ (B) $85 ^ { \circ }$ (C) $95 ^ { \circ }$ (D) $110 ^ { \circ }$
Let $\mathbb { R }$ be the set of all real numbers. The function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x - 5$ is (A) one-to-one, but not onto (B) one-to-one and onto (C) onto, but not one-to-one (D) neither one-to-one nor onto
Let $L$ be the point $(t, 2)$ and $M$ be a point on the $y$-axis such that $L M$ has slope $-t$. Then the locus of the midpoint of $L M$, as $t$ varies over all real values, is (A) $y = 2 + 2 x ^ { 2 }$ (B) $y = 1 + x ^ { 2 }$ (C) $y = 2 - 2 x ^ { 2 }$ (D) $y = 1 - x ^ { 2 }$
Let $f : ( 0, 2 ) \cup ( 4, 6 ) \rightarrow \mathbb { R }$ be a differentiable function. Suppose also that $f ^ { \prime \prime } ( x ) = 1$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$. Which of the following is ALWAYS true? (A) $f$ is increasing (B) $f$ is one-to-one (C) $f ( x ) = x$ for all $x \in ( 0, 2 ) \cup ( 4, 6 )$ (D) $f ( 5.5 ) - f ( 4.5 ) = f ( 1.5 ) - f ( 0.5 )$
A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is (A) a circle whose centre is the midpoint of $B C$ (B) a circle whose centre is on the line $B C$ but not the midpoint of $B C$ (C) a straight line (D) none of the above
Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the mid-point of the line segment $P Q$, then the locus of $R$ is (A) a circle (B) an ellipse (C) a line segment (D) segment of a parabola
$N$ is a 50 digit number. All the digits except the 26th from the right are 1. If $N$ is divisible by 13, then the unknown digit is (A) 1 (B) 3 (C) 7 (D) 9
Suppose $a < b$. The maximum value of the integral $$\int _ { a } ^ { b } \left( \frac { 3 } { 4 } - x - x ^ { 2 } \right) d x$$ over all possible values of $a$ and $b$ is (A) $\frac { 3 } { 4 }$ (B) $\frac { 4 } { 3 }$ (C) $\frac { 3 } { 2 }$ (D) $\frac { 2 } { 3 }$
Let $\omega$ denote a cube root of unity which is not equal to 1. Then the number of distinct elements in the set $$\left\{ \left( 1 + \omega + \omega ^ { 2 } + \cdots + \omega ^ { n } \right) ^ { m } \quad : \quad m , n = 1, 2, 3, \cdots \right\}$$ is (A) 4 (B) 5 (C) 7 (D) infinite
The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2 (B) is equal to 2 (C) lies in the interval $( 2, 3 )$ (D) is greater than 3
The number of ways in which one can select six distinct integers from the set $\{ 1, 2, 3, \cdots, 49 \}$, such that no two consecutive integers are selected, is (A) $\binom { 49 } { 6 } - 5 \binom { 48 } { 5 }$ (B) $\binom { 43 } { 6 }$ (C) $\binom { 25 } { 6 }$ (D) $\binom { 44 } { 6 }$
Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is (A) $r \operatorname { cosec } \frac { \pi } { n }$ (B) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$ (C) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$ (D) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$
If $n$ is a positive integer such that $8 n + 1$ is a perfect square, then (A) $n$ must be odd (B) $n$ cannot be a perfect square (C) $2 n$ cannot be a perfect square (D) none of the above
Let $\mathbb { C }$ denote the set of all complex numbers. Define $$\begin{aligned}
& A = \{ ( z , w ) \mid z , w \in \mathbb { C } \text { and } | z | = | w | \} \\
& B = \left\{ ( z , w ) \mid z , w \in \mathbb { C } , \text { and } z ^ { 2 } = w ^ { 2 } \right\}
\end{aligned}$$ Then, (A) $A = B$ (B) $A \subset B$ and $A \neq B$ (C) $B \subset A$ and $B \neq A$ (D) none of the above
Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. (A) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ (B) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ Then (C) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$ (D) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$
If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then (A) $f ( x )$ is an increasing function on the real line (B) $f ( x )$ is a decreasing function on the real line (C) $f ( x )$ is increasing on $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$ (D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$
The number of roots of the equation $x ^ { 2 } + \sin ^ { 2 } x = 1$ in the closed interval $\left[ 0 , \frac { \pi } { 2 } \right]$ is (A) 0 (B) 1 (C) 2 (D) 3
The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is (A) $m < \frac { 1 } { 4 }$ (B) $m \geq 0$ (C) $0 \leq m \leq \frac { 1 } { 4 }$ (D) $0 \leq m < \frac { 1 } { 4 }$
For any integer $n \geq 1$, define $a _ { n } = \frac { 1000 ^ { n } } { n ! }$. Then the sequence $\left\{ a _ { n } \right\}$ (A) does not have a maximum (B) attains maximum at exactly one value of $n$ (C) attains maximum at exactly two values of $n$ (D) attains maximum for infinitely many values of $n$
The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents (A) a circle (B) a circle and a pair of straight lines (C) a rectangular hyperbola (D) a pair of straight lines
A subset $W$ of the set of real numbers is called a ring if it contains 1 and if for all $a , b \in W$, the numbers $a - b$ and $a b$ are also in $W$. Let $S = \left\{ \left. \frac { m } { 2 ^ { n } } \right\rvert\, m , n \text{ integers} \right\}$ and $T = \left\{ \left. \frac { p } { q } \right\rvert\, p , q \text{ integers}, q \text{ odd} \right\}$. Then (A) neither $S$ nor $T$ is a ring (B) $S$ is a ring, $T$ is not a ring (C) $T$ is a ring, $S$ is not a ring (D) both $S$ and $T$ are rings