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Explore the Back-StoryHere 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] .

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