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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let be the circle . The point is

(A) inside but not the centre of ;

(B) outside ;

(C) on :

(D) the centre of .

**Problem 2:**

The number of distinct real roots of the equation

is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 3:**

The set of complex numbers satisfying the equation

represents, in the Argand plane,

(A) a straight line;

(B) a pair of intersecting straight lines;

(C) a pair of distinct parallel straight lines;

(D) a point.

**Problem 4:**

Let be the set and the subset The number of subsets of such that is

(A) ;

(B)

(C)

(D) .

**Problem 5:**

The number of triplets of integers such that and are sides of a triangle with perimeter is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 6:**

Suppose and are three numbers in G.P. If the equations and have a common root, then and are in

(A) A.P.

(B) G.P.;

(C) H.P.;

(D) none of the above.

**Problem 7:**

The number of solutions of the equation is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 8:**

Suppose has two real roots and with If and are the roots of then the equation has

(A) one positive and one negative root;

(B) two distinct positive roots;

(C) two distinct negative roots:

(D) no real roots.

**Problem 9:**

Suppose is a quadrilateral such that and If is the point of intersection of and then the value of is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 10:**

Let be the set of all real numbers. The function defined by

(A) one-to-one, but not onto:

(B) one-to-one and onto;

(C) onto, but not one-to-one;

(D) neither one-to-one nor onto.

**Problem 11:**

The angles of a convex pentagon are in A.P. Then, the minimum possible value of the smallest angle is

(A)

(B)

(C) ;

(D) .

**Problem 12:**

The number of points lying on the circle such that the quadratic equation has real roots, is

(A) infinite;

(B) ;

(C) ;

(D)

**Problem 13:**

Let be the point and be a point on the -axis such that has slope Then the locus of the midpoint of as varies over all real values, is

(A)

(B)

(C)

(D)

**Problem 14**:

Suppose and Which of the following statements is true?

(A) for all

(B) for all

(C) There exist such that ;

(D) None of the above.

**Problem 15:**

A triangle has a fixed base . If then the locus of the vertex is

(A) a circle whose centre is the midpoint of ;

(B) a circle whose centre is on the line but not the midpoint of ;

(C) a straight line;

(D) none of the above.

**Problem 16:**

Suppose for some . Then

(A) ;

(B) ;

(C) ;

(D) .

**Problem 17:**

Let The set of all values taken by as varies over the interval is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 18:**

is a digit number. All the digits except the th from the right are If is divisible by then the unknown digit is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 19:**

If , then the -th derivative of equals

(A) ;

(B) ;

(C)

(D) .

**Problem 20:**

Suppose . The maximum value of the integral

over all possible values of and is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 21:**

For any the value of lies between

(A) and ;

(B) and

(C) and ;

(D) none of the above.

**Problem 22:**

Let denote a cube root of unity which is not equal to Then the number of distinct elements in the set

is

(A) ;

(B) ;

(C) ;

(D) infinite.

**Problem 23:**

The graph of a function defined on (0,2) with the property is as follows:

The graph of the derivative of the above function looks like:

**Problem 24:**

The value of the integral

(A) is less than ;

(B) is equal to ;

(C) lies in the interval ;

(D) is greater than .

**Problem 25:**

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is metres and the radius of the base is metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is

(A) metres;

(B) metres;

(C) metres;

(D) metres.

**Problem 26:**

For each positive integer , define a function on as follows:

Then, the value of is

(A) :

(B) :

(C)

(D)

**Problem 27:**

The limit

(A) does not exist;

(B) equals ;

(C) equals ;

(D) equals .

**Problem 28:**

Let be the set of all points such that Given a point in the plane, let be the point in which is closest to . Then the points for which are

(A) all points with ;

(B) all points with and ;

(C) all points with and ;

(D) all points with and .

**Problem 29:**

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players and play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, has 8 points and has 4 points. It is also known that won against . In the next set of games and win their games against and respectively. If and move to the final round, the final scores of and are, respectively,

(A) and ;

(B) and

(C) and ;

(D) and .

**Problem 30:**

The number of ways in which one can select six distinct integers from the set such that no two consecutive integers are selected, is

(A) ;

(B) ;

(C) ;

(D) .

Here, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let be the circle . The point is

(A) inside but not the centre of ;

(B) outside ;

(C) on :

(D) the centre of .

**Problem 2:**

The number of distinct real roots of the equation

is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 3:**

The set of complex numbers satisfying the equation

represents, in the Argand plane,

(A) a straight line;

(B) a pair of intersecting straight lines;

(C) a pair of distinct parallel straight lines;

(D) a point.

**Problem 4:**

Let be the set and the subset The number of subsets of such that is

(A) ;

(B)

(C)

(D) .

**Problem 5:**

The number of triplets of integers such that and are sides of a triangle with perimeter is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 6:**

Suppose and are three numbers in G.P. If the equations and have a common root, then and are in

(A) A.P.

(B) G.P.;

(C) H.P.;

(D) none of the above.

**Problem 7:**

The number of solutions of the equation is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 8:**

Suppose has two real roots and with If and are the roots of then the equation has

(A) one positive and one negative root;

(B) two distinct positive roots;

(C) two distinct negative roots:

(D) no real roots.

**Problem 9:**

Suppose is a quadrilateral such that and If is the point of intersection of and then the value of is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 10:**

Let be the set of all real numbers. The function defined by

(A) one-to-one, but not onto:

(B) one-to-one and onto;

(C) onto, but not one-to-one;

(D) neither one-to-one nor onto.

**Problem 11:**

The angles of a convex pentagon are in A.P. Then, the minimum possible value of the smallest angle is

(A)

(B)

(C) ;

(D) .

**Problem 12:**

The number of points lying on the circle such that the quadratic equation has real roots, is

(A) infinite;

(B) ;

(C) ;

(D)

**Problem 13:**

Let be the point and be a point on the -axis such that has slope Then the locus of the midpoint of as varies over all real values, is

(A)

(B)

(C)

(D)

**Problem 14**:

Suppose and Which of the following statements is true?

(A) for all

(B) for all

(C) There exist such that ;

(D) None of the above.

**Problem 15:**

A triangle has a fixed base . If then the locus of the vertex is

(A) a circle whose centre is the midpoint of ;

(B) a circle whose centre is on the line but not the midpoint of ;

(C) a straight line;

(D) none of the above.

**Problem 16:**

Suppose for some . Then

(A) ;

(B) ;

(C) ;

(D) .

**Problem 17:**

Let The set of all values taken by as varies over the interval is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 18:**

is a digit number. All the digits except the th from the right are If is divisible by then the unknown digit is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 19:**

If , then the -th derivative of equals

(A) ;

(B) ;

(C)

(D) .

**Problem 20:**

Suppose . The maximum value of the integral

over all possible values of and is

(A) ;

(B) ;

(C) ;

(D) .

**Problem 21:**

For any the value of lies between

(A) and ;

(B) and

(C) and ;

(D) none of the above.

**Problem 22:**

Let denote a cube root of unity which is not equal to Then the number of distinct elements in the set

is

(A) ;

(B) ;

(C) ;

(D) infinite.

**Problem 23:**

The graph of a function defined on (0,2) with the property is as follows:

The graph of the derivative of the above function looks like:

**Problem 24:**

The value of the integral

(A) is less than ;

(B) is equal to ;

(C) lies in the interval ;

(D) is greater than .

**Problem 25:**

A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is metres and the radius of the base is metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is

(A) metres;

(B) metres;

(C) metres;

(D) metres.

**Problem 26:**

For each positive integer , define a function on as follows:

Then, the value of is

(A) :

(B) :

(C)

(D)

**Problem 27:**

The limit

(A) does not exist;

(B) equals ;

(C) equals ;

(D) equals .

**Problem 28:**

Let be the set of all points such that Given a point in the plane, let be the point in which is closest to . Then the points for which are

(A) all points with ;

(B) all points with and ;

(C) all points with and ;

(D) all points with and .

**Problem 29:**

In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players and play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points, has 8 points and has 4 points. It is also known that won against . In the next set of games and win their games against and respectively. If and move to the final round, the final scores of and are, respectively,

(A) and ;

(B) and

(C) and ;

(D) and .

**Problem 30:**

The number of ways in which one can select six distinct integers from the set such that no two consecutive integers are selected, is

(A) ;

(B) ;

(C) ;

(D) .

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