How Cheenta works to ensure student success?
Explore the Back-Story

B.Math 2008 Objective Paper| Problems & Solutions

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] .

Some Useful Links:

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] .

Some Useful Links:

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
Menu
Trial
Whatsapp
magic-wandrockethighlight