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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 and be fixed positive real numbers. Let for . Then as increases,

(A) increases;

(B) decreases;

(C) increases first and then decreases;

(D) none of the above is necessarily true.

**Problem 2 :**

The number of polynomials of the form which are divisible by and where and belong to is

(A) 1 ;

(B)

(C) 11 ;

(D) 100 .

**Problem 3 :**

How many integers are there such that and the highest common factor of and 36 is

(A) 333

(B) 667

(C) 166

(D) 361

**Problem 4 :**

The value of where the summation is over all and such that is

(A) 1320 ;

(B) 2640 ;

(C) 3025

(D) none of the above.

**Problem 5 :**

Let be all the factors of a positive integer including 1 and . Suppose Then the value of.

(A) is ;

(B) is ;

(C) is

(D) cannot be computed.

**Problem 6 :**

The inequality is satisfied for real if and only if

(A)

(B)

(C)

(D)

**Problem 7 :**

In the Cartesian plane the equation 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 :**

is a variable point on a circle and is a fixed point on the outside of is a point in dividing it in the ratio where and are fixed. Then the locus of is

(A) a circle;

(B) an ellipse;

(C) a circle if and an ellipse otherwise;

(D) none of the above curves.

**Problem 9 :**

is a right-angled triangle with right angle at is a point on such that If and then is

(A)

(B)

(C)

(D)

**Problem 10 :**

The maximum value of the integral is attained

(A) exactly at two values of ;

(B) only at one value of which is positive;

(C) only at one value of which is negative;

(D) only at

**Problem 11 :**

Let be a function from a set to such that for all . Then

(A) is one-to-one but need not be onto;

(B) is onto but need not be one-to-one;

(C) is both one-to-one and onto;

(D) none of the above is necessarily true.

**Problem 12 :**

The value of the sum equals

(A)

(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 such that any 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 equals

(A) 51!;

(B) 2.5! ;

(C) ;

(D) .

**Problem 15 :**

The remainder obtained by dividing the polynominl by the polymomial is

(A) ;

(B)

(C)

(D) .

**Problem 16 :**

If three prime numbers, all greater than 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 denote the set of all positive integers and 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 , let denote the greatest integer such that The number of points in the open interval (-2,2) where is not continuous equals

(A) 5;

(B) 6

(C) 7;

(D) .

**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 and be a pair of intersecting lines in the plane. Then the locus of the points such that the distance of from is twice the distance of from is

(A) an ellipse;

(B) a parabola;

(C) a hyperbola;

(D) a pair of straight lines.

**Problem 21 :**

If where is a positive continuous function; then the value of is

(A) ;

(B) 4

(C) 2

(D) 1 .

**Problem 22 :**

The equations and 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 such that 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 and are relatively prime for

(A) any positive integer ;

(B) infinitely many, but not all, integers ;

(C) for finitely many integers ;

(D) no positive integer .

**Problem 25 :**

Let be two differentiable functions. If then the limit is

(A) 2

(B) 3

(C) 4

(D) 5

**Problem 26 :**

Let be continuous, for every and Then is

(A) ;

(B) ;

(C) ;

(D) .

**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)

(D)

**Problem 28 :**

The sum of the coefficients of the polynomial is

(A) 6

(B) 0

(C) 28

(D) 18

**Problem 29 :**

Let be a function. Then the number of functions such that for at least one is

(A) 11 ;

(B) 19

(C) 23

(D) 27 .

**Problem 30 :**

The polynomial 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 and be fixed positive real numbers. Let for . Then as increases,

(A) increases;

(B) decreases;

(C) increases first and then decreases;

(D) none of the above is necessarily true.

**Problem 2 :**

The number of polynomials of the form which are divisible by and where and belong to is

(A) 1 ;

(B)

(C) 11 ;

(D) 100 .

**Problem 3 :**

How many integers are there such that and the highest common factor of and 36 is

(A) 333

(B) 667

(C) 166

(D) 361

**Problem 4 :**

The value of where the summation is over all and such that is

(A) 1320 ;

(B) 2640 ;

(C) 3025

(D) none of the above.

**Problem 5 :**

Let be all the factors of a positive integer including 1 and . Suppose Then the value of.

(A) is ;

(B) is ;

(C) is

(D) cannot be computed.

**Problem 6 :**

The inequality is satisfied for real if and only if

(A)

(B)

(C)

(D)

**Problem 7 :**

In the Cartesian plane the equation 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 :**

is a variable point on a circle and is a fixed point on the outside of is a point in dividing it in the ratio where and are fixed. Then the locus of is

(A) a circle;

(B) an ellipse;

(C) a circle if and an ellipse otherwise;

(D) none of the above curves.

**Problem 9 :**

is a right-angled triangle with right angle at is a point on such that If and then is

(A)

(B)

(C)

(D)

**Problem 10 :**

The maximum value of the integral is attained

(A) exactly at two values of ;

(B) only at one value of which is positive;

(C) only at one value of which is negative;

(D) only at

**Problem 11 :**

Let be a function from a set to such that for all . Then

(A) is one-to-one but need not be onto;

(B) is onto but need not be one-to-one;

(C) is both one-to-one and onto;

(D) none of the above is necessarily true.

**Problem 12 :**

The value of the sum equals

(A)

(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 such that any 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 equals

(A) 51!;

(B) 2.5! ;

(C) ;

(D) .

**Problem 15 :**

The remainder obtained by dividing the polynominl by the polymomial is

(A) ;

(B)

(C)

(D) .

**Problem 16 :**

If three prime numbers, all greater than 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 denote the set of all positive integers and 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 , let denote the greatest integer such that The number of points in the open interval (-2,2) where is not continuous equals

(A) 5;

(B) 6

(C) 7;

(D) .

**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 and be a pair of intersecting lines in the plane. Then the locus of the points such that the distance of from is twice the distance of from is

(A) an ellipse;

(B) a parabola;

(C) a hyperbola;

(D) a pair of straight lines.

**Problem 21 :**

If where is a positive continuous function; then the value of is

(A) ;

(B) 4

(C) 2

(D) 1 .

**Problem 22 :**

The equations and 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 such that 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 and are relatively prime for

(A) any positive integer ;

(B) infinitely many, but not all, integers ;

(C) for finitely many integers ;

(D) no positive integer .

**Problem 25 :**

Let be two differentiable functions. If then the limit is

(A) 2

(B) 3

(C) 4

(D) 5

**Problem 26 :**

Let be continuous, for every and Then is

(A) ;

(B) ;

(C) ;

(D) .

**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)

(D)

**Problem 28 :**

The sum of the coefficients of the polynomial is

(A) 6

(B) 0

(C) 28

(D) 18

**Problem 29 :**

Let be a function. Then the number of functions such that for at least one is

(A) 11 ;

(B) 19

(C) 23

(D) 27 .

**Problem 30 :**

The polynomial 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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