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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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