Let $x$ be an irrational number. If $a , b , c$ and $d$ are rational numbers such that $\frac { a x + b } { c x + d }$ is a rational number, which of the following must be true? (A) $a d = b c$ (B) $a c = b d$. (C) $a b = c d$. (D) $a = d = 0$
Let $z = x + i y$ be a complex number, which satisfies the equation $( z + \bar { z } ) z = 2 + 4 i$. Then (A) $y = \pm 2$. (B) $x = \pm 2$. (C) $x = \pm 3$. (D) $y = \pm 1$.
Let $a , b$ and $c$ be three real numbers. Then the equation $\frac { 1 } { x - a } + \frac { 1 } { x - b } + \frac { 1 } { x - c } = 0$ (A) always have real roots. (B) can have real or complex roots depending on the values of $a , b$ and $c$. (C) always have real and equal roots. (D) always have real roots, which are not necessarily equal.
Let $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is (A) 1 (B) $2 ^ { 4 }$ (C) $2 ^ { 5 }$ (D) $2 ^ { 9 }$
Suppose that the equations $x ^ { 2 } + b x + c a = 0$ and $x ^ { 2 } + c x + a b = 0$ have exactly one common non-zero root. Then (A) $a + b + c = 0$. (B) the two roots which are not common must necessarily be real. (C) the two roots which are not common may not be real. (D) the two roots which are not common are either both real or both not real.
Let $K$ be the set of all points $(x , y)$ such that $| x | + | y | \leq 1$. Given a point $A$ in the plane, let $F _ { A }$ be the point in $K$ which is closest to $A$. Then the points $A$ for which $F _ { A } = ( 1,0 )$ are (A) all points $A = ( x , y )$ with $x \geq 1$. (B) all points $A = ( x , y )$ with $x \geq y + 1$ and $x \geq 1 - y$. (C) all points $A = ( x , y )$ with $x \geq 1$ and $y = 0$. (D) all points $A = ( x , y )$ with $x \geq 0$ and $y = 0$.
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.
For $k \geq 1$, the value of $\binom { n } { 0 } + \binom { n + 1 } { 1 } + \binom { n + 2 } { 2 } + \cdots + \binom { n + k } { k }$ equals (a) $\binom { n + k + 1 } { n + k }$. (B) $( n + k + 1 ) \binom { n + k } { n + 1 }$. (C) $\binom { n + k + 1 } { n + 1 }$. (D) $\binom { n + k + 1 } { n }$.
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 Argand 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.
Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true? (a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$. (b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$. (c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$. (d) None of the above.
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 $N$ be a 50 digit number. All the digits except the 26th one 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) $3 / 4$. (B) $4 / 3$. (C) $3 / 2$. (D) $2 / 3$.