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# ISI B.Stat 2010 Objective Paper| problems & solutions

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

There are 8 balls numbered and 8 boxes numbered . The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is

(A) ;
(B) ;
(C) ;
(D) .

Problem 2:

Let and be two positive real numbers. For every integer , define

Then is equal to

(A)
(B)
(C)
(D) .

Problem 3:

Let be a function given by where and are fixed positive integers. Suppose that is one-one. Then

(A) both and must be odd;
(B) at least one of and must be odd;
(C) exactly one of and must be odd;
(D) neither nor can be odd.

Problem 4:

equals

(A) ;
(B) ;
(C) ;
(D)

Problem 5:

A circle is inscribed in a triangle with sides and centimetres. The radius of the circle in centimetres is

(A) ;
(B) ;
(C) ;
(D) none of the above.

Problem 6:

Let and be the angles of an acute angled triangle. Then the quantity

(A) can have any real value;
(B) is ;
(C) is ;
(D) none of the above.

Problem 7:

Let for . Then

(A) is differentiable at and
(B) is not differentiable at and ;
(C) is differentiable at but not differentiable at
(D) is not differentiable at but differentiable at .

Problem 8:

Consider a rectangular cardboard box of height breadth 4 and length 10 units. There is a lizard in one corner of the box and an insect in the corner which is farthest from . The length of the shortest path between the lizard and the insect along the surface of the box is

(A) units;
(B) units;
(C) units;
(D) units.

Problem 9:

Recall that, for any non-zero complex number which does not lie on the negative real axis, arg denotes the unique real number in such that

Let be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that satisfies the relations

Then can take

(A) any value in the set but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0) but none from outside.

Problem 10:

An aeroplane is moving in the air along a straight line path which passes through the points and , and makes an angle with the ground. Let be the position of an observer as shown in the figure below. When the plane is at the position its angle of elevation is and when it is at its angle of elevation is from the position of the observer. Moreover, the distances of the observer from the points and respectively are 1000 metres and metres.

Then is equal to

(A) ;
(B) ;
(C) ;
(D) .

Problem 11:

The sum of all even positive divisors of is

(A) ;
(B) ;
(C) ;
(D) .

Problem 12:

The equation has two real roots and If then the area under the curve between and is

(A) ;
(B)
(C)
(D) .

Problem 13:

The minimum value of subject to and is

(A)
(B) ;
(C) ;
(D) .

Problem 14:

The value of

is

(A)
(B) ;
(C) ;
(D) .

Problem 15:

For any real number , let denote the unique real number in such that . Then

(A) is equal to ;
(B) is equal to ;
(C) does not exist;
(D) none of the above

Problem 16:

Let be an integer. The number of primes which divide both and is

(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.

Problem 17:

The value of

is

(A) ;
(B) ;
(C) ;
(D) .

Problem 18:

A person standing at a point on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps comes back to the original position . Then the number of distinct paths that can take is

(A) ;
(B) ;
(C) ;
(D) .

Problem 19:

Consider the branch of the rectangular hyperbola in the first quadrant. Let be a fixed point on this curve. The locus of the mid-point of the line segment joining and an arbitrary point on the curve is part of

(A) a hyperbolas;

(B) a parabola;

(C) an ellipse;
(D) none of the above.

Problem 20:

The digit at the unit place of is

(A) ;
(B)
(C) ;
(D) .

Problem 21:

Let be the vertices of a regular polygon and be its sides. If

then the value of is

(A) ;
(B) ;
(C) ;
(D) .

Problem 22:

Suppose that and are two distinct numbers in the interval . If

then the value of is

(A) ;
(B) ;
(C)
(D)

Problem 23:

Consider the function Then has

(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.

Problem 24:

Consider the quadratic equation The number of pairs for which the equation has solutions of the form and for some is

(A)
(B) ;
(C) ;
(D) infinite.

Problem 25:

Let Then

(A)
(B)
(C)
(D) .

Problem 26:

Consider the following two curves on the interval (0,1) :

Then on

(A) lies above ;
(B) lies above
(C) and intersect at exactly one point;
(D) none of the above.

Problem 27:

Let be a real valued function on such that is twice differentiable. Suppose that vanishes only at 0 and is everywhere negative. Define a function by where Then

(A) has a local minima in ;
(B) has a local maxima in ;
(C) is monotonically increasing in ;
(D) is monotonically decreasing in .

Problem 28:

Consider the triangle with vertices and . Let denote the region enclosed by the above triangle. Consider the function defined by Then the range of is the interval

(A) ;
(B) ;
(C) ;
(D) .

Problem 29:

For every positive integer , let denote the integer closest to . Let The number of elements in is

(A) ;
(B)
(C) ;
(D) .

Problem 30:

Consider a square inscribed in a circle of radius Let and be two points on the (smaller) arcs and respectively, such that is a pentagon in which . If denotes the area of the pentagon then

(A) can not be equal to ;
(B) lies in the interval ;
(C) is greater than or equal to ;
(D) none of the above.

Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

Problem 1:

There are 8 balls numbered and 8 boxes numbered . The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is

(A) ;
(B) ;
(C) ;
(D) .

Problem 2:

Let and be two positive real numbers. For every integer , define

Then is equal to

(A)
(B)
(C)
(D) .

Problem 3:

Let be a function given by where and are fixed positive integers. Suppose that is one-one. Then

(A) both and must be odd;
(B) at least one of and must be odd;
(C) exactly one of and must be odd;
(D) neither nor can be odd.

Problem 4:

equals

(A) ;
(B) ;
(C) ;
(D)

Problem 5:

A circle is inscribed in a triangle with sides and centimetres. The radius of the circle in centimetres is

(A) ;
(B) ;
(C) ;
(D) none of the above.

Problem 6:

Let and be the angles of an acute angled triangle. Then the quantity

(A) can have any real value;
(B) is ;
(C) is ;
(D) none of the above.

Problem 7:

Let for . Then

(A) is differentiable at and
(B) is not differentiable at and ;
(C) is differentiable at but not differentiable at
(D) is not differentiable at but differentiable at .

Problem 8:

Consider a rectangular cardboard box of height breadth 4 and length 10 units. There is a lizard in one corner of the box and an insect in the corner which is farthest from . The length of the shortest path between the lizard and the insect along the surface of the box is

(A) units;
(B) units;
(C) units;
(D) units.

Problem 9:

Recall that, for any non-zero complex number which does not lie on the negative real axis, arg denotes the unique real number in such that

Let be any complex number such that its real and imaginary parts are both non-zero. Further, suppose that satisfies the relations

Then can take

(A) any value in the set but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0) but none from outside.

Problem 10:

An aeroplane is moving in the air along a straight line path which passes through the points and , and makes an angle with the ground. Let be the position of an observer as shown in the figure below. When the plane is at the position its angle of elevation is and when it is at its angle of elevation is from the position of the observer. Moreover, the distances of the observer from the points and respectively are 1000 metres and metres.

Then is equal to

(A) ;
(B) ;
(C) ;
(D) .

Problem 11:

The sum of all even positive divisors of is

(A) ;
(B) ;
(C) ;
(D) .

Problem 12:

The equation has two real roots and If then the area under the curve between and is

(A) ;
(B)
(C)
(D) .

Problem 13:

The minimum value of subject to and is

(A)
(B) ;
(C) ;
(D) .

Problem 14:

The value of

is

(A)
(B) ;
(C) ;
(D) .

Problem 15:

For any real number , let denote the unique real number in such that . Then

(A) is equal to ;
(B) is equal to ;
(C) does not exist;
(D) none of the above

Problem 16:

Let be an integer. The number of primes which divide both and is

(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.

Problem 17:

The value of

is

(A) ;
(B) ;
(C) ;
(D) .

Problem 18:

A person standing at a point on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps comes back to the original position . Then the number of distinct paths that can take is

(A) ;
(B) ;
(C) ;
(D) .

Problem 19:

Consider the branch of the rectangular hyperbola in the first quadrant. Let be a fixed point on this curve. The locus of the mid-point of the line segment joining and an arbitrary point on the curve is part of

(A) a hyperbolas;

(B) a parabola;

(C) an ellipse;
(D) none of the above.

Problem 20:

The digit at the unit place of is

(A) ;
(B)
(C) ;
(D) .

Problem 21:

Let be the vertices of a regular polygon and be its sides. If

then the value of is

(A) ;
(B) ;
(C) ;
(D) .

Problem 22:

Suppose that and are two distinct numbers in the interval . If

then the value of is

(A) ;
(B) ;
(C)
(D)

Problem 23:

Consider the function Then has

(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.

Problem 24:

Consider the quadratic equation The number of pairs for which the equation has solutions of the form and for some is

(A)
(B) ;
(C) ;
(D) infinite.

Problem 25:

Let Then

(A)
(B)
(C)
(D) .

Problem 26:

Consider the following two curves on the interval (0,1) :

Then on

(A) lies above ;
(B) lies above
(C) and intersect at exactly one point;
(D) none of the above.

Problem 27:

Let be a real valued function on such that is twice differentiable. Suppose that vanishes only at 0 and is everywhere negative. Define a function by where Then

(A) has a local minima in ;
(B) has a local maxima in ;
(C) is monotonically increasing in ;
(D) is monotonically decreasing in .

Problem 28:

Consider the triangle with vertices and . Let denote the region enclosed by the above triangle. Consider the function defined by Then the range of is the interval

(A) ;
(B) ;
(C) ;
(D) .

Problem 29:

For every positive integer , let denote the integer closest to . Let The number of elements in is

(A) ;
(B)
(C) ;
(D) .

Problem 30:

Consider a square inscribed in a circle of radius Let and be two points on the (smaller) arcs and respectively, such that is a pentagon in which . If denotes the area of the pentagon then

(A) can not be equal to ;
(B) lies in the interval ;
(C) is greater than or equal to ;
(D) none of the above.

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