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# B.Math 2009 Objective Paper| Problems & Solutions

Problem 1:

The domain of definition of $f(x)=-\log \left(x^{2}-2 x-3\right)$ is

(a) $(0, \infty)$
(b) $(-\infty,-1)$
(c) $(-\infty,-1) \cup(3, \infty)$
(d) $(-\infty,-3) \cup(1, \infty)$

Problem 2:

$A B C$ is a right-angled triangle with the right angle at B. If $A B=7$ and $B C=24$, then the length of the perpendicular from $B$ to $A C$ is

(a) $12.2$
(b) $6.72$
(c) $7.2$
(d) $3.36$

Problem 3:

If the points $\mathbf{z_{1}}$ and $\mathbf{z_{2}}$ are on the circles $|\mathbf{z}|=2$ and $|\mathbf{z}|=3$ respectively and the angle included between these vectors is $60^{\circ}$, then $\left|\left(\mathbf{z_{1}}+\mathbf{z_{2}}\right) /\left(\mathbf{z_{1}}-\mathbf{z_{2}}\right)\right|$ equals

(a) $\sqrt{(19 / 7)}$
(b) $\sqrt{19}$
(c) $\sqrt{7}$
(d) $\sqrt{133}$

Problem 4:

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ be positive integers such that $\log \mathrm{a}(\mathbf{b})=\mathbf{3 / 2}$ and
$\log (\mathrm{d})=5 / 4 .$ If $\mathrm{a}-\mathrm{c}=9$, then $b-d$ equals

(a) 55

(b) 23
(c) 89
(d) 93

Problem 5 :

$1-x-e^{-x}>0$ for :
(a) All $\mathbf{x} \in \mathbf{R}$.
(b) No $\mathbf{x} \in \mathbf{R}$.
(c) $x>0$.
(d) $x<0 .$

Problem 6 :

If $P(x)=a x^{2}+b x+c$ and $Q(x)=-a x^{2}+b x+c$ where $a c \neq 0$, then the equation $\mathbf{P}(\mathbf{x}) Q(\mathbf{x})=0$ has :
(a) Only real roots.
(b) No real roots.
(c) At least two real roots.
(d) Exactly two real roots.

Problem 7:

$\lim \mid \sqrt{\left(x^{2}+x\right)-x \mid}$ as x $\rightarrow \infty$ is equal to
(a) $1 / 2$
(b) 0
(c) $\infty$
(d) 2

Problem 8:

$\mathrm{lim}\left(\mathrm{n} / 2^{\mathrm{n}}\right) \Sigma \sin \left(\mathrm{jn} / 2^{n}\right)$ where $j$ runs from 1 to $2^{n}$ as $n \rightarrow \infty$ is equal
(a) 0
(b) $\Pi$
(c) 2
(d) 1

Problem 9:

Let f: $\mathbf{R} \rightarrow \mathbf{R}$ is given by $\mathbf{f}(\mathbf{x})=\mathbf{x}(\mathbf{x}-\mathbf{1})(\mathbf{x}+\mathbf{1})$. Then,
(a) $f$ is $1-1$ and onto
(b) $\mathbf{f}$ is neither $\mathbf{1}-\mathbf{1}$ nor onto
(c) $f$ is $1-1$ but not onto
(d) $f$ is onto but not $1-1$

Problem 10:

The last digit of $22^{22}$ is :
(a) $2$
(b) $4$
(c) $6$
(d) $0$

Problem 11:

The average scores of $10$ students in a test is $25$. The lowest score is $20$. Then the highest score is at most
(a) $100$
(b) $30$
(c) $70$
(d) $75$

Problem 12:

The coefficient of $t^{3}$ in the expansion of $\{(1-t^{6}) /(1-t)\}^{3}$ is
(a) $10$
(b) $12$
(c) $8$
(d) $9$

problem 13:

Let $p_{n}(x), n \geq 0$ be polynomials defined by $p_{0}(x)=1, p_{1}(x)=x$ and

${P_{n}}(x)=x p_{n-1}(x)- p_{n-2}(x)$ for $n \geq 2$. Then $\mathbf{p}_{10}(\mathbf{x})$ equals

(a) $0$
(b) $10$
(c) $1$
(d) $-1$

Problem 14:

Suppose $A, B$ are matrices satisfying $A B+B A=0$. Then $A^{2} B^{5}$ is equal to
(a) 0
(b) $\mathrm{B}^{2} \mathrm{~A}^{5}$
(c) $-\mathrm{B}^{2} \mathrm{~A}^{5}$
(d) $A B$

Problem 15:

The number of terms in the expansion of $(x+y+z+w)^{2 0 0 9}$
is

(a) $2009_\mathrm{C_{4}}$

(b) $2013 _\mathrm{C_{4}}$
(c) $2012_\mathrm{ C_{3}}$
(d) $(2010)^{4}$

Problem 16:

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are positive real numbers satisfying $\mathbf{a b}+\mathbf{b c}+\mathbf{c a}=\mathbf{1 2}$,

then the maximum value of $\mathrm{abc}$ is
(a) 8
(b) 9
(c) 6
(d) 12

Problem 17:

If at least $90$ percent students in a class are good in sports, and at least $80$ percent are good in music and at least $70$ percent are good in studies, then the percentage of students who are good in all three is at least
(a) $25$
(b) $40$
(c) $20$
(d) $50$

Problem 18:

If $\cot \{\sin ^{-1} \sqrt{(13 / 17)}\}=\sin (\tan ^{-1} \theta).$, then $\theta$ is

(a) $2 / \sqrt{17}$
(b) $\sqrt{(} 13 / 17)$
(c) $\sqrt{(2 / \sqrt{13})}$
(d) 2/3

Problem 19:

Let $f(t)=(t+1) /(t-1) .$ Then $f(f(2010))$ equals
(a) $2011 / 2009$
(b) 2010
(c) $2010 / 2009$
(d) None of the above

Problem 20:

If each side of a cube is increased by $60 \%$, then the surface area of the cube increased by
(a) $156 \%$
(b) $160 \%$
(c) $120 \%$
(d) $240 \%$

Problem 21:

If $\mathbf{a}>\mathbf{2}$, then

(a) $\log {e}(a)+\log {a}(10)<0$

(b) $\log {e}(a)+\log {a}(10)>0$
(c) $e^{a}<1$
(d) None of the above is true.

Problem 22:

The number of complex numbers w such that $|\mathbf{w}|=1$ and imaginary part of $\mathrm{w}^{4}$ is 0 , is
(a) 4
(b) 2
(c) 8
(d) Infinite

Problem 23:

Let $f(x)=csin(x)$ for all $x \in R .$ Suppose $f(x)=\sum f(x+k n) / 2^{k}$
(summation is running from $\mathbf{k}=1$ to $\mathbf{k}=\infty)$ for all $\mathbf{x} \in \mathbf{R}$. Then
(a) $c=1$
(b) $c=0$
(c) $c<0$
(d) $c=-1$

Problem 24:

The number of points at which the function $f(x)=\max (1+x, 1-$ $x$ ) if $x<0$ and $f(x)=\min \left(1+x, 1+x^{2}\right)$ if $x \geq 0$ is not differentiable, is
(a) 1
(b) 0
(c) 2
(d) None of the above.

Problem 25:

The greatest value of function $f(x)=\sin ^{2}(x) \cos (x)$

(a) $2 / 3 \sqrt{3}$
(b) $\sqrt{(2 / 3)}$
(c) $2 / 9$
(d) $\sqrt{2} / 3 \sqrt{3}$

problem 26:

Let $g(t)=\int\left(x^{2}+1\right)^{10} d x$ (integration running from $-10$ to $\left.t\right)$ for all $\mathbf{t} \geq-10$. Then
(a) $g$ is not differentiable.
(b) $\mathbf{g}$ is constant.
(c) $\mathbf{g}$ is increasing in $(-10, \infty)$.
(d) $\mathbf{g}$ is decreasing in $(-10, \infty)$.

Problem 27:

Let $\mathbf{p}(\mathbf{x})$ be a continuous function which is positive for all $\mathbf{x}$ and $\int p(x) d x=c \int p{(x+4) / 2} d x$ (first integration is running from 2 to 3 and second integration running from 0 to 2). Then
(a) $c=4$
(b) $c=1 / 2$
(c) $c=1 / 4$
(d) $c=2$

Problem 28:

Let $\mathrm{f}:[0,1] \rightarrow(1, \infty)$ be a continuous function. Let $g(x)=1 / x$ for $x>0$. Then, the equation $f(x)=g(x)$ has
(a) No solution.
(b) All points in $(0,1]$ as solutions.
(c) At least one solution.
(d) None of the above.

Problem 29:

Let $0 \leq \theta, \Phi<2 n$ be two angles. Then the equation $\sin \theta+\sin \phi=\cos \theta+\cos \Phi$
(a) Determines $\Theta$ uniquely in terms of $\Phi$
(b) Gives two value of $\Theta$ for each value of $\boldsymbol{\Phi}$
(c) Gives more than two values of $\Theta$ for each value of $\Phi$
(d) None of the above.

Problem 30:

Ten players are to pay a tennis tournament. The number of pairings for the first round is

(a) $10 ! / 2^{5} 5 !$
(b) $2^{10}$
(c) $10_{C _{2}}$
(d) $10_{\mathrm{P}_{2}}$

Problem 1:

The domain of definition of $f(x)=-\log \left(x^{2}-2 x-3\right)$ is

(a) $(0, \infty)$
(b) $(-\infty,-1)$
(c) $(-\infty,-1) \cup(3, \infty)$
(d) $(-\infty,-3) \cup(1, \infty)$

Problem 2:

$A B C$ is a right-angled triangle with the right angle at B. If $A B=7$ and $B C=24$, then the length of the perpendicular from $B$ to $A C$ is

(a) $12.2$
(b) $6.72$
(c) $7.2$
(d) $3.36$

Problem 3:

If the points $\mathbf{z_{1}}$ and $\mathbf{z_{2}}$ are on the circles $|\mathbf{z}|=2$ and $|\mathbf{z}|=3$ respectively and the angle included between these vectors is $60^{\circ}$, then $\left|\left(\mathbf{z_{1}}+\mathbf{z_{2}}\right) /\left(\mathbf{z_{1}}-\mathbf{z_{2}}\right)\right|$ equals

(a) $\sqrt{(19 / 7)}$
(b) $\sqrt{19}$
(c) $\sqrt{7}$
(d) $\sqrt{133}$

Problem 4:

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ be positive integers such that $\log \mathrm{a}(\mathbf{b})=\mathbf{3 / 2}$ and
$\log (\mathrm{d})=5 / 4 .$ If $\mathrm{a}-\mathrm{c}=9$, then $b-d$ equals

(a) 55

(b) 23
(c) 89
(d) 93

Problem 5 :

$1-x-e^{-x}>0$ for :
(a) All $\mathbf{x} \in \mathbf{R}$.
(b) No $\mathbf{x} \in \mathbf{R}$.
(c) $x>0$.
(d) $x<0 .$

Problem 6 :

If $P(x)=a x^{2}+b x+c$ and $Q(x)=-a x^{2}+b x+c$ where $a c \neq 0$, then the equation $\mathbf{P}(\mathbf{x}) Q(\mathbf{x})=0$ has :
(a) Only real roots.
(b) No real roots.
(c) At least two real roots.
(d) Exactly two real roots.

Problem 7:

$\lim \mid \sqrt{\left(x^{2}+x\right)-x \mid}$ as x $\rightarrow \infty$ is equal to
(a) $1 / 2$
(b) 0
(c) $\infty$
(d) 2

Problem 8:

$\mathrm{lim}\left(\mathrm{n} / 2^{\mathrm{n}}\right) \Sigma \sin \left(\mathrm{jn} / 2^{n}\right)$ where $j$ runs from 1 to $2^{n}$ as $n \rightarrow \infty$ is equal
(a) 0
(b) $\Pi$
(c) 2
(d) 1

Problem 9:

Let f: $\mathbf{R} \rightarrow \mathbf{R}$ is given by $\mathbf{f}(\mathbf{x})=\mathbf{x}(\mathbf{x}-\mathbf{1})(\mathbf{x}+\mathbf{1})$. Then,
(a) $f$ is $1-1$ and onto
(b) $\mathbf{f}$ is neither $\mathbf{1}-\mathbf{1}$ nor onto
(c) $f$ is $1-1$ but not onto
(d) $f$ is onto but not $1-1$

Problem 10:

The last digit of $22^{22}$ is :
(a) $2$
(b) $4$
(c) $6$
(d) $0$

Problem 11:

The average scores of $10$ students in a test is $25$. The lowest score is $20$. Then the highest score is at most
(a) $100$
(b) $30$
(c) $70$
(d) $75$

Problem 12:

The coefficient of $t^{3}$ in the expansion of $\{(1-t^{6}) /(1-t)\}^{3}$ is
(a) $10$
(b) $12$
(c) $8$
(d) $9$

problem 13:

Let $p_{n}(x), n \geq 0$ be polynomials defined by $p_{0}(x)=1, p_{1}(x)=x$ and

${P_{n}}(x)=x p_{n-1}(x)- p_{n-2}(x)$ for $n \geq 2$. Then $\mathbf{p}_{10}(\mathbf{x})$ equals

(a) $0$
(b) $10$
(c) $1$
(d) $-1$

Problem 14:

Suppose $A, B$ are matrices satisfying $A B+B A=0$. Then $A^{2} B^{5}$ is equal to
(a) 0
(b) $\mathrm{B}^{2} \mathrm{~A}^{5}$
(c) $-\mathrm{B}^{2} \mathrm{~A}^{5}$
(d) $A B$

Problem 15:

The number of terms in the expansion of $(x+y+z+w)^{2 0 0 9}$
is

(a) $2009_\mathrm{C_{4}}$

(b) $2013 _\mathrm{C_{4}}$
(c) $2012_\mathrm{ C_{3}}$
(d) $(2010)^{4}$

Problem 16:

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are positive real numbers satisfying $\mathbf{a b}+\mathbf{b c}+\mathbf{c a}=\mathbf{1 2}$,

then the maximum value of $\mathrm{abc}$ is
(a) 8
(b) 9
(c) 6
(d) 12

Problem 17:

If at least $90$ percent students in a class are good in sports, and at least $80$ percent are good in music and at least $70$ percent are good in studies, then the percentage of students who are good in all three is at least
(a) $25$
(b) $40$
(c) $20$
(d) $50$

Problem 18:

If $\cot \{\sin ^{-1} \sqrt{(13 / 17)}\}=\sin (\tan ^{-1} \theta).$, then $\theta$ is

(a) $2 / \sqrt{17}$
(b) $\sqrt{(} 13 / 17)$
(c) $\sqrt{(2 / \sqrt{13})}$
(d) 2/3

Problem 19:

Let $f(t)=(t+1) /(t-1) .$ Then $f(f(2010))$ equals
(a) $2011 / 2009$
(b) 2010
(c) $2010 / 2009$
(d) None of the above

Problem 20:

If each side of a cube is increased by $60 \%$, then the surface area of the cube increased by
(a) $156 \%$
(b) $160 \%$
(c) $120 \%$
(d) $240 \%$

Problem 21:

If $\mathbf{a}>\mathbf{2}$, then

(a) $\log {e}(a)+\log {a}(10)<0$

(b) $\log {e}(a)+\log {a}(10)>0$
(c) $e^{a}<1$
(d) None of the above is true.

Problem 22:

The number of complex numbers w such that $|\mathbf{w}|=1$ and imaginary part of $\mathrm{w}^{4}$ is 0 , is
(a) 4
(b) 2
(c) 8
(d) Infinite

Problem 23:

Let $f(x)=csin(x)$ for all $x \in R .$ Suppose $f(x)=\sum f(x+k n) / 2^{k}$
(summation is running from $\mathbf{k}=1$ to $\mathbf{k}=\infty)$ for all $\mathbf{x} \in \mathbf{R}$. Then
(a) $c=1$
(b) $c=0$
(c) $c<0$
(d) $c=-1$

Problem 24:

The number of points at which the function $f(x)=\max (1+x, 1-$ $x$ ) if $x<0$ and $f(x)=\min \left(1+x, 1+x^{2}\right)$ if $x \geq 0$ is not differentiable, is
(a) 1
(b) 0
(c) 2
(d) None of the above.

Problem 25:

The greatest value of function $f(x)=\sin ^{2}(x) \cos (x)$

(a) $2 / 3 \sqrt{3}$
(b) $\sqrt{(2 / 3)}$
(c) $2 / 9$
(d) $\sqrt{2} / 3 \sqrt{3}$

problem 26:

Let $g(t)=\int\left(x^{2}+1\right)^{10} d x$ (integration running from $-10$ to $\left.t\right)$ for all $\mathbf{t} \geq-10$. Then
(a) $g$ is not differentiable.
(b) $\mathbf{g}$ is constant.
(c) $\mathbf{g}$ is increasing in $(-10, \infty)$.
(d) $\mathbf{g}$ is decreasing in $(-10, \infty)$.

Problem 27:

Let $\mathbf{p}(\mathbf{x})$ be a continuous function which is positive for all $\mathbf{x}$ and $\int p(x) d x=c \int p{(x+4) / 2} d x$ (first integration is running from 2 to 3 and second integration running from 0 to 2). Then
(a) $c=4$
(b) $c=1 / 2$
(c) $c=1 / 4$
(d) $c=2$

Problem 28:

Let $\mathrm{f}:[0,1] \rightarrow(1, \infty)$ be a continuous function. Let $g(x)=1 / x$ for $x>0$. Then, the equation $f(x)=g(x)$ has
(a) No solution.
(b) All points in $(0,1]$ as solutions.
(c) At least one solution.
(d) None of the above.

Problem 29:

Let $0 \leq \theta, \Phi<2 n$ be two angles. Then the equation $\sin \theta+\sin \phi=\cos \theta+\cos \Phi$
(a) Determines $\Theta$ uniquely in terms of $\Phi$
(b) Gives two value of $\Theta$ for each value of $\boldsymbol{\Phi}$
(c) Gives more than two values of $\Theta$ for each value of $\Phi$
(d) None of the above.

Problem 30:

Ten players are to pay a tennis tournament. The number of pairings for the first round is

(a) $10 ! / 2^{5} 5 !$
(b) $2^{10}$
(c) $10_{C _{2}}$
(d) $10_{\mathrm{P}_{2}}$

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