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Explore the Back-StoryHere are the problems and their corresponding solutions from B.Math Hons Objective Admission Test 2007.

**Problem 1 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 2 :**

Consider the surface defined by . If we cut the surface by the plane given by the equation then we obtain a

(A) hyperbola;

(B) circle;

(C) parabola;

(D) pair of straight lines.

**Problem 3 :**

Let be real numbers. The number of real solutions of the system of equations and is

(A) at most ;

(B) at most ;

(C) at least ;

(D) at least .

**Problem 4 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 5 :**

Let be a degree five polynomial with real coefficients. Then the number of real roots of must be

(A) ;

(B) or ;

(C) or or ;

(D) none of the above.

**Problem 6 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 7 :**

Let square root signs). Then equals

(A) ;

(B) ;

(C) ;

(D) .

**Problem 8 :**

Let be the sequence whose th term is the sum of the digits of the natural number . For example, etc. The minimum such that is

(A) ;

(B) ;

(C) ;

(D) none of the above.

**Problem 9 :**

(A) is ;

(B) is ;

(C) is ;

(D) does not exist.

**Problem 10 :**

Let be a subset of the real line. Let be a non-empty subset of such that for all in their product is also in Then can have

(A) only one element;

(B) at most elements;

(C) more than 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) ;

(B) ;

(C) ;

(D) .

**Problem 12 :**

Let and for all natural number , define Then, as , the number of prime factors of

(A) goes to infinity;

(B) goes to a finite limit;

(C) oscillates boundedly;

(D) oscillates unboundedly.

**Problem 13 :**

Suppose that the equation has a rational solution. If are integers then

(A) at least one of is even;

(B) all of are even;

(C) at most one of is odd;

(D) all of are odd.

**Problem 14 :**

Let . The number of functions such that for all is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 15 :**

Let be a function defined by . Then

(A) is one-one but not onto;

(B) is onto but not one-one;

(C) is both one-one and onto;

(D) is neither one-one nor onto.

**Problem 16 :**

Define the sequence by Then

is

(A) ;

(B) ;

(C) ;

(D) infinite.

**Problem 17 :**

Let be the circle of radius 1 around 0 in the complex plane and be a fixed point on . Then the number of ordered pairs of points on such that is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 18 :**

The number of real solutions of is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 19 :**

The set of complex numbers such that 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 :**

is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 21 :**

The number of rational roots of the polynomial is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 22 :**

equals

(A) ;

(B) ;

(C) ;

(D) .

**Problem 23 :**

Let be a natural number and let Then

(A) ;

(B) ;

(C) ;

(D) none of these numbers.

**Problem 24 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 25 :**

Let be a right angled triangle in the plane with area . Then the maximum area of a rectangle inside is

(A) ;

(B) ;

(C) ;

(D) .

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 steps if we take one or two steps at a time is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 2 :**

Consider the surface defined by . If we cut the surface by the plane given by the equation then we obtain a

(A) hyperbola;

(B) circle;

(C) parabola;

(D) pair of straight lines.

**Problem 3 :**

Let be real numbers. The number of real solutions of the system of equations and is

(A) at most ;

(B) at most ;

(C) at least ;

(D) at least .

**Problem 4 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 5 :**

Let be a degree five polynomial with real coefficients. Then the number of real roots of must be

(A) ;

(B) or ;

(C) or or ;

(D) none of the above.

**Problem 6 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 7 :**

Let square root signs). Then equals

(A) ;

(B) ;

(C) ;

(D) .

**Problem 8 :**

Let be the sequence whose th term is the sum of the digits of the natural number . For example, etc. The minimum such that is

(A) ;

(B) ;

(C) ;

(D) none of the above.

**Problem 9 :**

(A) is ;

(B) is ;

(C) is ;

(D) does not exist.

**Problem 10 :**

Let be a subset of the real line. Let be a non-empty subset of such that for all in their product is also in Then can have

(A) only one element;

(B) at most elements;

(C) more than 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) ;

(B) ;

(C) ;

(D) .

**Problem 12 :**

Let and for all natural number , define Then, as , the number of prime factors of

(A) goes to infinity;

(B) goes to a finite limit;

(C) oscillates boundedly;

(D) oscillates unboundedly.

**Problem 13 :**

Suppose that the equation has a rational solution. If are integers then

(A) at least one of is even;

(B) all of are even;

(C) at most one of is odd;

(D) all of are odd.

**Problem 14 :**

Let . The number of functions such that for all is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 15 :**

Let be a function defined by . Then

(A) is one-one but not onto;

(B) is onto but not one-one;

(C) is both one-one and onto;

(D) is neither one-one nor onto.

**Problem 16 :**

Define the sequence by Then

is

(A) ;

(B) ;

(C) ;

(D) infinite.

**Problem 17 :**

Let be the circle of radius 1 around 0 in the complex plane and be a fixed point on . Then the number of ordered pairs of points on such that is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 18 :**

The number of real solutions of is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 19 :**

The set of complex numbers such that 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 :**

is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 21 :**

The number of rational roots of the polynomial is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 22 :**

equals

(A) ;

(B) ;

(C) ;

(D) .

**Problem 23 :**

Let be a natural number and let Then

(A) ;

(B) ;

(C) ;

(D) none of these numbers.

**Problem 24 :**

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

(A) ;

(B) ;

(C) ;

(D) .

**Problem 25 :**

Let be a right angled triangle in the plane with area . Then the maximum area of a rectangle inside is

(A) ;

(B) ;

(C) ;

(D) .

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