Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2008.
Problem 1 :
Let $a, b$ and $c$ be fixed positive real numbers. Let $u_{n}=\frac{n^{2} a}{b+n^{2} c}$ for $n \geq 1$. Then as $n$ increases,
(A) $u_{n}$ increases;
(B) $u_{n}$ decreases;
(C) $u_{n}$ increases first and then decreases;
(D) none of the above is necessarily true.
Problem 2 :
The number of polynomials of the form $x^{3}+a x^{2}+b x+c$ which are divisible by $x^{2}+1$ and where $a, b$ and $c$ belong to ${1,2, \ldots, 10},$ is
(A) 1 ;
(B) $10 ;$
(C) 11 ;
(D) 100 .
Problem 3 :
How many integers $n$ are there such that $1 \leq n \leq 1000$ and the highest common factor of $n$ and 36 is $1 ?$
(A) 333
(B) 667
(C) 166
(D) 361
Problem 4 :
The value of $\Sigma i j,$ where the summation is over all $i$ and $j$ such that $1 \leq i, j \leq 10$ is
(A) 1320 ;
(B) 2640 ;
(C) 3025
(D) none of the above.
Problem 5 :
Let $d_{1}, d_{2}, \ldots, d_{k}$ be all the factors of a positive integer $n$ including 1 and $n$. Suppose $d_{1}+d_{2}+\cdots+d_{k}=72 .$ Then the value of.
$\frac{1}{d_{1}}+\frac{1}{d_{2}}+\cdots+\frac{1}{d_{k}}$
(A) is $\frac{k^{2}}{72}$;
(B) is $\frac{72}{k}$;
(C) is $\frac{72}{n}$
(D) cannot be computed.
Problem 6 :
The inequality $\sqrt{x+6} \geq x$ is satisfied for real $x$ if and only if
(A) $-3 \leq x \leq 3$
(B) $-2 \leq x \leq 3$
(C) $-6 \leq x \leq 3$
(D) $0 \leq x \leq 6$
Problem 7 :
In the Cartesian plane 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.
Problem 8 :
$P$ is a variable point on a circle $C$ and $Q$ is a fixed point on the outside of $C .$ $R$ is a point in $P Q$ dividing it in the ratio $p: q,$ where $p>0$ and $q>0$ are fixed. Then the locus of $R$ is
(A) a circle;
(B) an ellipse;
(C) a circle if $p=q$ and an ellipse otherwise;
(D) none of the above curves.
Problem 9 :
$A B C$ is a right-angled triangle with right angle at $B . D$ is a point on $A C$ such that $\angle A B D=45^{\circ} .$ If $A C=6 \mathrm{~cm}$ and $A D=2 \mathrm{~cm}$ then $A B$ is
(A) $\frac{6}{\sqrt{5}} \mathrm{~cm}$
(B) $3 \sqrt{2} \mathrm{~cm}$
(C) $\frac{12}{\sqrt{5}} \mathrm{~cm} ;$
(D) $2 \mathrm{~cm}$
Problem 10 :
The maximum value of the integral $\int_{a-1}^{a+1} \frac{1}{1+x^{8}} d x$ is attained
(A) exactly at two values of $a$;
(B) only at one value of $a$ which is positive;
(C) only at one value of $a$ which is negative;
(D) only at $a=0$
Problem 11 :
Let $f$ be a function from a set $X$ to $X$ such that $f(f(x))=x$ for all $x \in X$. Then
(A) $f$ is one-to-one but need not be onto;
(B) $f$ is onto but need not be one-to-one;
(C) $f$ is both one-to-one and onto;
(D) none of the above is necessarily true.
Problem 12 :
The value of the sum $\cos \frac{2 \pi}{1000}+\cos \frac{.4 \pi}{1000}+\cdots+\cos \frac{1998 \pi}{1000}$ equals
(A) $-1 ;$
(B) 0
(C) 1
(D) an irrational number.
Problem 13 :
A box contains 100 balls of different colours: 28 red, 17 blue, 21 green, 10 white, 12 yellow and 12 black. The smallest number $n$ such that any $n$ balls drawn from the box will contain at least 15 balls of the same colour is
(A) 73
(B) 77
(C) 81
(D) 85
Problem 14 :
The sum $(1 \cdot 1 !)+(2 \cdot 2 !)+(3 \cdot 3!)+\cdots+(50 \cdot 50!)$ equals
(A) 51!;
(B) 2.5! ;
(C) $51!-1$;
(D) $51!+1$.
Problem 15 :
The remainder $R(x)$ obtained by dividing the polynominl $x^{100}$ by the polymomial $x^{2}-3 x+2$ is
(A) $2^{100}-1$;
(B) $\left(2^{100}-1\right) x-2\left(2^{99}-1\right)$
(C) $2^{100} x-3 \cdot 2^{100}$
(D) $\left(2^{100}-1\right) x+2\left(2^{99}-1\right)$.
Problem 16 :
If three prime numbers, all greater than $3,$ are in A.P., then their common difference
(A) must be divisible by 2 but not, necessarily by 3
(B) must be divisible by 3 but not necessorily by 2 ;
(C) must be divisible by both 2 nnd 3
(D) must not be divisible by any of 2 and 3 .
Problem 17 :
Let $P$ denote the set of all positive integers and $S=\{(x, y) \in P \times P: x^{2}-y^{2}=666\}$ Then S
(A) is an empty set;
(B) contains exactly one element;
(C) contains exactly two elements;
(D) contains more than two elements.
Problem 18 :
For any real number $x$, let $[x]$ denote the greatest integer $m$ such that $m \leq x$ The number of points in the open interval (-2,2) where $f(x)=\left[x^{2}-1\right]$ is not continuous equals
(A) 5;
(B) 6
(C) 7;
(D) $\infty$.
Problem 19 :
The equation log 3 x - log x 3=2 has
(A) no real solution;
(B) exactly one real solution;
(C) exactly two real solutions;
(D) infinitely many real solutions.
Problem 20 :
Let $l_{1}$ and $l_{2}$ be a pair of intersecting lines in the plane. Then the locus of the points $P$ such that the distance of $P$ from $l_{1}$ is twice the distance of $P$ from $l_{2}$ is
(A) an ellipse;
(B) a parabola;
(C) a hyperbola;
(D) a pair of straight lines.
Problem 21 :
If $c \int_{0}^{1} x f(2 x) d x=\int_{0}^{2} t f(t) d t,$ where $f$ is a positive continuous function; then the value of $c$ is
(A) $\frac{1}{2}$;
(B) 4
(C) 2
(D) 1 .
Problem 22 :
The equations $x^{3}+2 x^{2}+2 x+1=0$ and $x^{200}+x^{130}+1=0$ have
(A) exactly one common root ;
(B) no common root;
(C) exactly three common roots:
(D) exactly two common roots.
Problem 23 :
The set of complex numbers $z$ such that $z(1-z)$ is a real number forms
(A) a line and circle;
(B) a pair of lines;
(C) a line and a parabola;
(D) a line and a hyperbola.
Problem 24 :
The numbers $12 n+1$ and $30 n+2$ are relatively prime for
(A) any positive integer $n$;
(B) infinitely many, but not all, integers $n$;
(C) for finitely many integers $n$;
(D) no positive integer $n$.
Problem 25 :
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two differentiable functions. If $f(a)=2, f^{\prime}(a)=1, g(a)=$ $-1, g^{\prime}(a)=2,$ then the limit is
$\lim _{x \rightarrow a} \frac{g(x) f(a)-g(a) f(x)}{x-a}$
(A) 2
(B) 3
(C) 4
(D) 5
Problem 26 :
Let $f:(-1,1) \rightarrow(-1,1)$ be continuous, $f(x)=f\left(x^{2}\right)$ for every $x$ and $f(0)=\frac{1}{2}$ Then $f\left(\frac{1}{4}\right)$ is
(A) $\frac{1}{2}$;
(B) $\sqrt{\frac{3}{2}}$;
(C) $\frac{3}{\sqrt{2}}$;
(D) $\frac{\sqrt{2}}{3}$.
Problem 27 :
The number of ways in which a team of 6 members containing at least 2 lefthanders can be formed from 7 right-handers and 4 left-handers is:
(A) 210 ;
(B) 371
(C) $\left(\begin{array}{c} 11\\ 6\end{array}\right) .$
(D) $\left(\begin{array}{c} 11\\ 2\end{array}\right) .$
Problem 28 :
The sum of the coefficients of the polynomial $(x-1)^{2}(x-2)^{4}(x-3)^{6}$ is
(A) 6
(B) 0
(C) 28
(D) 18
Problem 29 :
Let $f:{1,2,3} \rightarrow{1,2,3}$ be a function. Then the number of functions $g:{1,2,3} \rightarrow{1,2,3}$ such that $f(x)=g(x)$ for at least one $x \in{1,2,3}$ is
(A) 11 ;
(B) 19
(C) 23
(D) 27 .
Problem 30 :
The polynomial $p(x)=x^{4}-4 x^{2}+1$ has
(A) no roots in the interval [0,3] ;
(B) exactly one root in the interval [0,3] ;
(C) exactly two roots in the interval [0,3]
(D) more than two roots in the interval [0,3] .
Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2008.
Problem 1 :
Let $a, b$ and $c$ be fixed positive real numbers. Let $u_{n}=\frac{n^{2} a}{b+n^{2} c}$ for $n \geq 1$. Then as $n$ increases,
(A) $u_{n}$ increases;
(B) $u_{n}$ decreases;
(C) $u_{n}$ increases first and then decreases;
(D) none of the above is necessarily true.
Problem 2 :
The number of polynomials of the form $x^{3}+a x^{2}+b x+c$ which are divisible by $x^{2}+1$ and where $a, b$ and $c$ belong to ${1,2, \ldots, 10},$ is
(A) 1 ;
(B) $10 ;$
(C) 11 ;
(D) 100 .
Problem 3 :
How many integers $n$ are there such that $1 \leq n \leq 1000$ and the highest common factor of $n$ and 36 is $1 ?$
(A) 333
(B) 667
(C) 166
(D) 361
Problem 4 :
The value of $\Sigma i j,$ where the summation is over all $i$ and $j$ such that $1 \leq i, j \leq 10$ is
(A) 1320 ;
(B) 2640 ;
(C) 3025
(D) none of the above.
Problem 5 :
Let $d_{1}, d_{2}, \ldots, d_{k}$ be all the factors of a positive integer $n$ including 1 and $n$. Suppose $d_{1}+d_{2}+\cdots+d_{k}=72 .$ Then the value of.
$\frac{1}{d_{1}}+\frac{1}{d_{2}}+\cdots+\frac{1}{d_{k}}$
(A) is $\frac{k^{2}}{72}$;
(B) is $\frac{72}{k}$;
(C) is $\frac{72}{n}$
(D) cannot be computed.
Problem 6 :
The inequality $\sqrt{x+6} \geq x$ is satisfied for real $x$ if and only if
(A) $-3 \leq x \leq 3$
(B) $-2 \leq x \leq 3$
(C) $-6 \leq x \leq 3$
(D) $0 \leq x \leq 6$
Problem 7 :
In the Cartesian plane 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.
Problem 8 :
$P$ is a variable point on a circle $C$ and $Q$ is a fixed point on the outside of $C .$ $R$ is a point in $P Q$ dividing it in the ratio $p: q,$ where $p>0$ and $q>0$ are fixed. Then the locus of $R$ is
(A) a circle;
(B) an ellipse;
(C) a circle if $p=q$ and an ellipse otherwise;
(D) none of the above curves.
Problem 9 :
$A B C$ is a right-angled triangle with right angle at $B . D$ is a point on $A C$ such that $\angle A B D=45^{\circ} .$ If $A C=6 \mathrm{~cm}$ and $A D=2 \mathrm{~cm}$ then $A B$ is
(A) $\frac{6}{\sqrt{5}} \mathrm{~cm}$
(B) $3 \sqrt{2} \mathrm{~cm}$
(C) $\frac{12}{\sqrt{5}} \mathrm{~cm} ;$
(D) $2 \mathrm{~cm}$
Problem 10 :
The maximum value of the integral $\int_{a-1}^{a+1} \frac{1}{1+x^{8}} d x$ is attained
(A) exactly at two values of $a$;
(B) only at one value of $a$ which is positive;
(C) only at one value of $a$ which is negative;
(D) only at $a=0$
Problem 11 :
Let $f$ be a function from a set $X$ to $X$ such that $f(f(x))=x$ for all $x \in X$. Then
(A) $f$ is one-to-one but need not be onto;
(B) $f$ is onto but need not be one-to-one;
(C) $f$ is both one-to-one and onto;
(D) none of the above is necessarily true.
Problem 12 :
The value of the sum $\cos \frac{2 \pi}{1000}+\cos \frac{.4 \pi}{1000}+\cdots+\cos \frac{1998 \pi}{1000}$ equals
(A) $-1 ;$
(B) 0
(C) 1
(D) an irrational number.
Problem 13 :
A box contains 100 balls of different colours: 28 red, 17 blue, 21 green, 10 white, 12 yellow and 12 black. The smallest number $n$ such that any $n$ balls drawn from the box will contain at least 15 balls of the same colour is
(A) 73
(B) 77
(C) 81
(D) 85
Problem 14 :
The sum $(1 \cdot 1 !)+(2 \cdot 2 !)+(3 \cdot 3!)+\cdots+(50 \cdot 50!)$ equals
(A) 51!;
(B) 2.5! ;
(C) $51!-1$;
(D) $51!+1$.
Problem 15 :
The remainder $R(x)$ obtained by dividing the polynominl $x^{100}$ by the polymomial $x^{2}-3 x+2$ is
(A) $2^{100}-1$;
(B) $\left(2^{100}-1\right) x-2\left(2^{99}-1\right)$
(C) $2^{100} x-3 \cdot 2^{100}$
(D) $\left(2^{100}-1\right) x+2\left(2^{99}-1\right)$.
Problem 16 :
If three prime numbers, all greater than $3,$ are in A.P., then their common difference
(A) must be divisible by 2 but not, necessarily by 3
(B) must be divisible by 3 but not necessorily by 2 ;
(C) must be divisible by both 2 nnd 3
(D) must not be divisible by any of 2 and 3 .
Problem 17 :
Let $P$ denote the set of all positive integers and $S=\{(x, y) \in P \times P: x^{2}-y^{2}=666\}$ Then S
(A) is an empty set;
(B) contains exactly one element;
(C) contains exactly two elements;
(D) contains more than two elements.
Problem 18 :
For any real number $x$, let $[x]$ denote the greatest integer $m$ such that $m \leq x$ The number of points in the open interval (-2,2) where $f(x)=\left[x^{2}-1\right]$ is not continuous equals
(A) 5;
(B) 6
(C) 7;
(D) $\infty$.
Problem 19 :
The equation log 3 x - log x 3=2 has
(A) no real solution;
(B) exactly one real solution;
(C) exactly two real solutions;
(D) infinitely many real solutions.
Problem 20 :
Let $l_{1}$ and $l_{2}$ be a pair of intersecting lines in the plane. Then the locus of the points $P$ such that the distance of $P$ from $l_{1}$ is twice the distance of $P$ from $l_{2}$ is
(A) an ellipse;
(B) a parabola;
(C) a hyperbola;
(D) a pair of straight lines.
Problem 21 :
If $c \int_{0}^{1} x f(2 x) d x=\int_{0}^{2} t f(t) d t,$ where $f$ is a positive continuous function; then the value of $c$ is
(A) $\frac{1}{2}$;
(B) 4
(C) 2
(D) 1 .
Problem 22 :
The equations $x^{3}+2 x^{2}+2 x+1=0$ and $x^{200}+x^{130}+1=0$ have
(A) exactly one common root ;
(B) no common root;
(C) exactly three common roots:
(D) exactly two common roots.
Problem 23 :
The set of complex numbers $z$ such that $z(1-z)$ is a real number forms
(A) a line and circle;
(B) a pair of lines;
(C) a line and a parabola;
(D) a line and a hyperbola.
Problem 24 :
The numbers $12 n+1$ and $30 n+2$ are relatively prime for
(A) any positive integer $n$;
(B) infinitely many, but not all, integers $n$;
(C) for finitely many integers $n$;
(D) no positive integer $n$.
Problem 25 :
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two differentiable functions. If $f(a)=2, f^{\prime}(a)=1, g(a)=$ $-1, g^{\prime}(a)=2,$ then the limit is
$\lim _{x \rightarrow a} \frac{g(x) f(a)-g(a) f(x)}{x-a}$
(A) 2
(B) 3
(C) 4
(D) 5
Problem 26 :
Let $f:(-1,1) \rightarrow(-1,1)$ be continuous, $f(x)=f\left(x^{2}\right)$ for every $x$ and $f(0)=\frac{1}{2}$ Then $f\left(\frac{1}{4}\right)$ is
(A) $\frac{1}{2}$;
(B) $\sqrt{\frac{3}{2}}$;
(C) $\frac{3}{\sqrt{2}}$;
(D) $\frac{\sqrt{2}}{3}$.
Problem 27 :
The number of ways in which a team of 6 members containing at least 2 lefthanders can be formed from 7 right-handers and 4 left-handers is:
(A) 210 ;
(B) 371
(C) $\left(\begin{array}{c} 11\\ 6\end{array}\right) .$
(D) $\left(\begin{array}{c} 11\\ 2\end{array}\right) .$
Problem 28 :
The sum of the coefficients of the polynomial $(x-1)^{2}(x-2)^{4}(x-3)^{6}$ is
(A) 6
(B) 0
(C) 28
(D) 18
Problem 29 :
Let $f:{1,2,3} \rightarrow{1,2,3}$ be a function. Then the number of functions $g:{1,2,3} \rightarrow{1,2,3}$ such that $f(x)=g(x)$ for at least one $x \in{1,2,3}$ is
(A) 11 ;
(B) 19
(C) 23
(D) 27 .
Problem 30 :
The polynomial $p(x)=x^{4}-4 x^{2}+1$ has
(A) no roots in the interval [0,3] ;
(B) exactly one root in the interval [0,3] ;
(C) exactly two roots in the interval [0,3]
(D) more than two roots in the interval [0,3] .
