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# B.Math 2007 Objective Paper| Problems & Solutions

Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2007.

Problem 1 :

The number of ways of going up $7$ steps if we take one or two steps at a time is

(A) $19$ ;
(B) $20$;
(C) $21$ ;
(D) $22$ .

Problem 2 :

Consider the surface defined by $x^{2}+2 y^{2}-5 z^{2}=0$. If we cut the surface by the plane given by the equation $x=z,$ then we obtain a

(A) hyperbola;
(B) circle;
(C) parabola;
(D) pair of straight lines.

Problem 3 :

Let $a, b$ be real numbers. The number of real solutions of the system of equations $x+y=a$ and $x y=b$ is

(A) at most $1$ ;
(B) at most $2$;
(C) at least $1$;
(D) at least $2$.

Problem 4 :

If a fair coin is tossed $100$ times, then the probability of getting at least one head is

(A) $\frac{100}{2^{100}}$;
(B) $\frac{99}{100}$;
(C) $1-\frac{1}{100 !}$;
(D) $1-\frac{1}{2^{100}}$.

Problem 5 :

Let $f(x)$ be a degree five polynomial with real coefficients. Then the number of real roots of $f$ must be

(A) $1$;
(B) $2$ or $4$;
(C) $1$ or $3$ or $5$ ;
(D) none of the above.

Problem 6 :

The number of ways in which $3$ girls and $2$ boys can sit on a bench so that no two girls are adjacent is

(A) $6$ ;
(B) $12$ ;
(C) $32$ ;
(D) $120$ .

Problem 7 :

Let $R_{n}=2+\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}\left(n\right.$ square root signs). Then $\lim _{n \rightarrow \infty} R_{n}$ equals

(A) $4$;
(B) $8$;
(C) $16$ ;
(D) $e^{2}$.

Problem 8 :

Let $a_{n}$ be the sequence whose $n$ th term is the sum of the digits of the natural number $9 n$. For example, $a_{1}=9, a_{11}=18$ etc. The minimum $m$ such that $a_{m}=81$ is

(A) $110111112$ ;
(B) $119111113$ ;
(C) $111111111$;
(D) none of the above.

Problem 9 :

$\lim _{n \rightarrow \infty}\left[2 \log (3 n)-\log \left(n^{2}+1\right)\right]$

(A) is $0$ ;
(B) is $2 \log 3$;
(C) is $4 \log 6$;
(D) does not exist.

Problem 10 :

Let $S=\{x \in \mathbb{R}|1 \leq| x \mid \leq 100\}$ be a subset of the real line. Let $M$ be a non-empty subset of $\bar{S}$ such that for all $x, y$ in $M,$ their product $x y$ is also in $M .$ Then $M$ can have

(A) only one element;
(B) at most $2$ elements;
(C) more than $2$ but only finitely many elements;
(D) infinitely many elements.

Problem 11:

An astronaut lands on a planet and meets a native of the planet. She asks the native "How many days do you have in your year?" He answers "It is the sum of the souares of three consecutive natural numbers but it is also the sum of the sauares of the next two numbers". The answer to the astronaut's question is

(A) $365$;
(B) $1095$ ;
(C) $30000$ ;
(D) $10^{10}$.

Problem 12 :

Let $a_{1}=2$ and for all natural number $n$, define $a_{n+1}=a_{n}\left(a_{n}+1\right) .$ Then, as $n \rightarrow \infty$, the number of prime factors of $a_{n}$

(A) goes to infinity;
(B) goes to a finite limit;
(C) oscillates boundedly;
(D) oscillates unboundedly.

Problem 13 :

Suppose that the equation $a x^{2}+b x+c=0$ has a rational solution. If $a, b, c$ are integers then

(A) at least one of $a, b, c$ is even;
(B) all of $a, b, c$ are even;
(C) at most one of $a, b, c$ is odd;
(D) all of $a, b, c$ are odd.

Problem 14 :

Let $S={1,2,3,4}$. The number of functions $f: S \rightarrow S$ such that $f(i) \leq 2 i$ for all $i \in S$ is

(A) $32$ ;
(B) $64$;
(C) $128$;
(D) $256$ .

Problem 15 :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$. Then

(A) $f$ is one-one but not onto;
(B) $f$ is onto but not one-one;
(C) $f$ is both one-one and onto;
(D) $f$ is neither one-one nor onto.

Problem 16 :

Define the sequence $\{a_{n}\}$ by $a_{1}=1, a_{2}=\frac{e}{2}, a_{3}=\frac{e^{2}}{4}, a_{4}=\frac{e^{3}}{8}, \ldots$ Then
$\lim_ {n \rightarrow \infty} a_{n}$ is

(A) $0$;
(B) $1$;
(C) $e^{e}$;
(D) infinite.

Problem 17 :

Let $C$ be the circle of radius 1 around 0 in the complex plane and $z_{0}$ be a fixed point on $C$. Then the number of ordered pairs $\left(z_{1}, z_{2}\right)$ of points on $C$ such that $z_{0}+z_{1}+z_{2}=0$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 18 :

The number of real solutions of $e^{x}+x^{2}=\sin x$ is

(A) $0$;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 19 :

The set of complex numbers $z$ such that $|z+1| \leq|z-1|$ is the half plane

(A) of complex numbers that lie above the real axis;
(B) of complex numbers that lie below the real axis;
(C) of complex numbers that lie left of the imaginary axis;
(D) of complex numbers that lie right of the imaginary axis.

Problem 20 :

$\lim _{x \rightarrow 0} \cos (\sin x)$ is

(A) $-1$ ;
(B) $0$;
(C) $1 / 2$;
(D) $1$ .

Problem 21 :

The number of rational roots of the polynomial $x^{3}-3 x-1$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $3$.

Problem 22 :

$\sum_{n=1}^{\infty} \frac{n^{2}}{n !}$ equals

(A) $e$;
(B) $2 e$;
(C) $e^{2}$;
(D) $\infty$.

Problem 23 :

Let $n>1$ be a natural number and let $A=\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) .$ Then

(A) $A^{n}=I d$;
(B) $A^{n^{2}+1}=I d$;
(C) $A^{n^{4}+1}=I d$;
(D) none of these numbers.

Problem 24 :

The number of ways of breaking a stick of length $n>1$ into $n$ pieces of unit length (at each step break one of the pieces with length $>1$ into two pieces of integer lengths) is

(A) $(n-1) !$;
(B) $n !-1$;
(C) $2^{n-2}$;
(D) $2^{n-1}-1$.

Problem 25 :

Let $A B C$ be a right angled triangle in the plane with area $s$. Then the maximum area of a rectangle inside $A B C$ is

(A) $s / 4$;
(B) $s / 3$;
(C) $s / 2$;
(D) $s$.

## Some Useful Links:

Here are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2007.

Problem 1 :

The number of ways of going up $7$ steps if we take one or two steps at a time is

(A) $19$ ;
(B) $20$;
(C) $21$ ;
(D) $22$ .

Problem 2 :

Consider the surface defined by $x^{2}+2 y^{2}-5 z^{2}=0$. If we cut the surface by the plane given by the equation $x=z,$ then we obtain a

(A) hyperbola;
(B) circle;
(C) parabola;
(D) pair of straight lines.

Problem 3 :

Let $a, b$ be real numbers. The number of real solutions of the system of equations $x+y=a$ and $x y=b$ is

(A) at most $1$ ;
(B) at most $2$;
(C) at least $1$;
(D) at least $2$.

Problem 4 :

If a fair coin is tossed $100$ times, then the probability of getting at least one head is

(A) $\frac{100}{2^{100}}$;
(B) $\frac{99}{100}$;
(C) $1-\frac{1}{100 !}$;
(D) $1-\frac{1}{2^{100}}$.

Problem 5 :

Let $f(x)$ be a degree five polynomial with real coefficients. Then the number of real roots of $f$ must be

(A) $1$;
(B) $2$ or $4$;
(C) $1$ or $3$ or $5$ ;
(D) none of the above.

Problem 6 :

The number of ways in which $3$ girls and $2$ boys can sit on a bench so that no two girls are adjacent is

(A) $6$ ;
(B) $12$ ;
(C) $32$ ;
(D) $120$ .

Problem 7 :

Let $R_{n}=2+\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}\left(n\right.$ square root signs). Then $\lim _{n \rightarrow \infty} R_{n}$ equals

(A) $4$;
(B) $8$;
(C) $16$ ;
(D) $e^{2}$.

Problem 8 :

Let $a_{n}$ be the sequence whose $n$ th term is the sum of the digits of the natural number $9 n$. For example, $a_{1}=9, a_{11}=18$ etc. The minimum $m$ such that $a_{m}=81$ is

(A) $110111112$ ;
(B) $119111113$ ;
(C) $111111111$;
(D) none of the above.

Problem 9 :

$\lim _{n \rightarrow \infty}\left[2 \log (3 n)-\log \left(n^{2}+1\right)\right]$

(A) is $0$ ;
(B) is $2 \log 3$;
(C) is $4 \log 6$;
(D) does not exist.

Problem 10 :

Let $S=\{x \in \mathbb{R}|1 \leq| x \mid \leq 100\}$ be a subset of the real line. Let $M$ be a non-empty subset of $\bar{S}$ such that for all $x, y$ in $M,$ their product $x y$ is also in $M .$ Then $M$ can have

(A) only one element;
(B) at most $2$ elements;
(C) more than $2$ but only finitely many elements;
(D) infinitely many elements.

Problem 11:

An astronaut lands on a planet and meets a native of the planet. She asks the native "How many days do you have in your year?" He answers "It is the sum of the souares of three consecutive natural numbers but it is also the sum of the sauares of the next two numbers". The answer to the astronaut's question is

(A) $365$;
(B) $1095$ ;
(C) $30000$ ;
(D) $10^{10}$.

Problem 12 :

Let $a_{1}=2$ and for all natural number $n$, define $a_{n+1}=a_{n}\left(a_{n}+1\right) .$ Then, as $n \rightarrow \infty$, the number of prime factors of $a_{n}$

(A) goes to infinity;
(B) goes to a finite limit;
(C) oscillates boundedly;
(D) oscillates unboundedly.

Problem 13 :

Suppose that the equation $a x^{2}+b x+c=0$ has a rational solution. If $a, b, c$ are integers then

(A) at least one of $a, b, c$ is even;
(B) all of $a, b, c$ are even;
(C) at most one of $a, b, c$ is odd;
(D) all of $a, b, c$ are odd.

Problem 14 :

Let $S={1,2,3,4}$. The number of functions $f: S \rightarrow S$ such that $f(i) \leq 2 i$ for all $i \in S$ is

(A) $32$ ;
(B) $64$;
(C) $128$;
(D) $256$ .

Problem 15 :

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=x^{2}-\frac{x^{2}}{1+x^{2}}$. Then

(A) $f$ is one-one but not onto;
(B) $f$ is onto but not one-one;
(C) $f$ is both one-one and onto;
(D) $f$ is neither one-one nor onto.

Problem 16 :

Define the sequence $\{a_{n}\}$ by $a_{1}=1, a_{2}=\frac{e}{2}, a_{3}=\frac{e^{2}}{4}, a_{4}=\frac{e^{3}}{8}, \ldots$ Then
$\lim_ {n \rightarrow \infty} a_{n}$ is

(A) $0$;
(B) $1$;
(C) $e^{e}$;
(D) infinite.

Problem 17 :

Let $C$ be the circle of radius 1 around 0 in the complex plane and $z_{0}$ be a fixed point on $C$. Then the number of ordered pairs $\left(z_{1}, z_{2}\right)$ of points on $C$ such that $z_{0}+z_{1}+z_{2}=0$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 18 :

The number of real solutions of $e^{x}+x^{2}=\sin x$ is

(A) $0$;
(B) $1$;
(C) $2$;
(D) $\infty$.

Problem 19 :

The set of complex numbers $z$ such that $|z+1| \leq|z-1|$ is the half plane

(A) of complex numbers that lie above the real axis;
(B) of complex numbers that lie below the real axis;
(C) of complex numbers that lie left of the imaginary axis;
(D) of complex numbers that lie right of the imaginary axis.

Problem 20 :

$\lim _{x \rightarrow 0} \cos (\sin x)$ is

(A) $-1$ ;
(B) $0$;
(C) $1 / 2$;
(D) $1$ .

Problem 21 :

The number of rational roots of the polynomial $x^{3}-3 x-1$ is

(A) $0$ ;
(B) $1$;
(C) $2$;
(D) $3$.

Problem 22 :

$\sum_{n=1}^{\infty} \frac{n^{2}}{n !}$ equals

(A) $e$;
(B) $2 e$;
(C) $e^{2}$;
(D) $\infty$.

Problem 23 :

Let $n>1$ be a natural number and let $A=\left(\begin{array}{cc}1 & n \\ 0 & 1\end{array}\right) .$ Then

(A) $A^{n}=I d$;
(B) $A^{n^{2}+1}=I d$;
(C) $A^{n^{4}+1}=I d$;
(D) none of these numbers.

Problem 24 :

The number of ways of breaking a stick of length $n>1$ into $n$ pieces of unit length (at each step break one of the pieces with length $>1$ into two pieces of integer lengths) is

(A) $(n-1) !$;
(B) $n !-1$;
(C) $2^{n-2}$;
(D) $2^{n-1}-1$.

Problem 25 :

Let $A B C$ be a right angled triangle in the plane with area $s$. Then the maximum area of a rectangle inside $A B C$ is

(A) $s / 4$;
(B) $s / 3$;
(C) $s / 2$;
(D) $s$.

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