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# ISI B.Stat & B.Math Paper 2012 Objective| Problems & Solutions

Here, you will find all the questions of ISI Entrance Paper 2012 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

### Multiple-Choice Test

Problem 1:

A rod of length 3 rests on a wall as follows:

is a point on such that . If the rod slides along the wall, then the locus of lies on

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

Problem 2:

Consider the equation . How many solutions exist such that and are positive integer?

(A) None
(B) Exactly two
(C) More than two but finitely many
(D) Infinitely many.

Discussion:

Problem 3:

Consider the functions (where is the base of natural logarithm ). The graphs of the functions intersect

(A) once in (0,1) and never in

(B) once in (0,1) and once in

(C) once in (0,1) and once in

(D) more than twice in .

Problem 4:

Consider the sequence

Then the limit of as is
(A)
(B)
(C)
(D) .

Problem 5:

Suppose that is any complex number which is not equal to any of where is a complex cube root of unity. Then

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

Problem 6:

Consider all functions which are one-one, onto and satisfy the following property:
if is odd then is even, .
The number of such functions is
(A)
(B)
(C)
(D) .

Problem 7:

A function is defined by

Then
(A) is not continuous
(B) is differentiable but is not continuous
(C) is continuous but does not exist
(D) is differentiable and is continuous.

Problem 8:

The last digit of is
(A)
(B)
(C)
(D) .

Problem 9:

Consider the function . Then

(A) maximum of is attained inside the interval
(B) minimum of is
(C) maximum of is
(D) is a decreasing function in .

Problem 10:

A particle moves in the plane in such a way that the angle between the two tangents drawn from to the curve is always The locus of is
(A) a parabola
(B) a circle
(C) an ellipse
(D) a straight line.

Problem 11:

Let be given by . Then

(A) has a local minima at but no local maximum
(B) has a local maximum at but no local minima
(C) has a local minima at and a local maximum at
(D) none of the above is true.

Problem 12:

The number of triples of positive integers satisfying is

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

Problem 13:

Let be a fixed real number greater than The locus of satisfying is
(A) parabola
(B) ellipse
(C) hyperbola
(D) not a conic.

Problem 14:

Which of the following is closest to the graph of

Problem 15:

Consider the function given by . Then

(A) is one-one but not onto
(B) is onto but not one-one
(C) is neither one-one nor onto
(D) is both one-one and onto.

Problem 16:

Consider a real valued continuous function satisfying for all Let . Define , provided the limit exists. Then

(A) is defined only for
(B) is defined only when is an integer
(C) is defined for all and is independent of
(D) none of the above is true.

Problem 17:

Consider the sequence . Then the integer part of equals
(A)
(B)
(C)
(D) .

Problem 18:

Let and Then the minimum value of is

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

Problem 19:

What is the limit of as ?
(A)
(B)
(C)
(D) .

Problem 20:

Consider the function The function
(A) is zero at but is increasing near
(B) has a zero in
(C) has two zeros in (-1,0)
(D) has exactly one local minimum in (-1,0) .

Problem 21:

Consider a sequence of 10 A's and 8 B's placed in a row. By a run we mean one or more letters of the same type placed side by side. Here is an arrangement of 's and 's which contains 4 runs of and 4 runs of

AAABBABBBAABAAAABB

In how many ways can 's and 's be arranged in a row so that there are 4 runs of and 4 runs of
(A)
(B)
(C)
(D) .

Problem 22:

Suppose is a fixed positive integer and . Then

(A) is differentiable everywhere only when is even
(B) is differentiable everywhere except at 0 if is odd
(C) is differentiable everywhere
(D) none of the above is true.

Problem 23:

The line with cuts the axis and axis at points and respectively. Then the equation of the circle having as diameter is
(A)
(B)
(C)
(D) .

Problem 24:

Let and consider the sequence

Then is

(A) for any
(B) for any
(C) or depending on what is
(D) or depending on what is.

Problem 25:

If then
(A)
(B)
(C)
(D)

Problem 26:

Consider a cardboard box in the shape of a prism as shown below. The length of the prism is 5 . The two triangular faces and are congruent and isosceles with side lengths 2,2,3 . The shortest distance between and along the surface of the prism is
(A)
(B)
(C)
(D)

Problem 27:

Assume the following inequalities for positive integer . The integer part of

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

Problem 28:

Consider the sets defined by the inequalities

Then
(A)
(B)
(C) each of the sets and is non-empty
(D) none of the above is true.

Problem 29:

The number is

(A) strictly larger than

(B) strictly larger than but strictly smaller than

(C) less than or equal to

(D) equal to

Problem 30:

If the roots of the equation are in geometric progression then
(A)
(B)
(C)
(D) .

### B.Stat. (Hons.) & B.Math. (Hons.) Admission Test: 2012

Problem 1:

Let be the angles of a triangle.
(1) Prove that

(2) Using (1) or otherwise prove that

Problem 2:

Let be s real number. Consider the function

(i) Determine the values of for which is continuous at all .
(ii) Determine the values of for which is differentiable at all .

Problem 3:

Write the set of all positive integers in triangular array as

Find the row number and column number where occurs. For example appears in the third row and second column.

Problem 4:

Show that the polynomial has no real root.

Problem 5:

Let be a natural number with digits consisting entirely of 6 's and 0 's. Prove that is not the square of a natural number.

Problem 6:

Let .
(i) Show that amongst the triangles with base and perimeter the maximum area is obtained when the other two sides have equal length .
(ii) Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter the square has maximum area.

Problem 7:

Let . Consider two circles with radii and and centers and respectively with . Let be the center of any circle in the crescent shaped region between the two circles and tangent to both (See figure below). Determine the locus of as its circle traverses through region maintaining tangency.

Problem 8:

Let , and .For a function , a subset is said be invariant under if for all . Note that the empty set and are invariant for all . Let be the number of subsets of invariant under .
(i) Show that there is a function such that .
(ii) Further show that for any such that there is a function such that

Here, you will find all the questions of ISI Entrance Paper 2012 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

### Multiple-Choice Test

Problem 1:

A rod of length 3 rests on a wall as follows:

is a point on such that . If the rod slides along the wall, then the locus of lies on

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

Problem 2:

Consider the equation . How many solutions exist such that and are positive integer?

(A) None
(B) Exactly two
(C) More than two but finitely many
(D) Infinitely many.

Discussion:

Problem 3:

Consider the functions (where is the base of natural logarithm ). The graphs of the functions intersect

(A) once in (0,1) and never in

(B) once in (0,1) and once in

(C) once in (0,1) and once in

(D) more than twice in .

Problem 4:

Consider the sequence

Then the limit of as is
(A)
(B)
(C)
(D) .

Problem 5:

Suppose that is any complex number which is not equal to any of where is a complex cube root of unity. Then

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

Problem 6:

Consider all functions which are one-one, onto and satisfy the following property:
if is odd then is even, .
The number of such functions is
(A)
(B)
(C)
(D) .

Problem 7:

A function is defined by

Then
(A) is not continuous
(B) is differentiable but is not continuous
(C) is continuous but does not exist
(D) is differentiable and is continuous.

Problem 8:

The last digit of is
(A)
(B)
(C)
(D) .

Problem 9:

Consider the function . Then

(A) maximum of is attained inside the interval
(B) minimum of is
(C) maximum of is
(D) is a decreasing function in .

Problem 10:

A particle moves in the plane in such a way that the angle between the two tangents drawn from to the curve is always The locus of is
(A) a parabola
(B) a circle
(C) an ellipse
(D) a straight line.

Problem 11:

Let be given by . Then

(A) has a local minima at but no local maximum
(B) has a local maximum at but no local minima
(C) has a local minima at and a local maximum at
(D) none of the above is true.

Problem 12:

The number of triples of positive integers satisfying is

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

Problem 13:

Let be a fixed real number greater than The locus of satisfying is
(A) parabola
(B) ellipse
(C) hyperbola
(D) not a conic.

Problem 14:

Which of the following is closest to the graph of

Problem 15:

Consider the function given by . Then

(A) is one-one but not onto
(B) is onto but not one-one
(C) is neither one-one nor onto
(D) is both one-one and onto.

Problem 16:

Consider a real valued continuous function satisfying for all Let . Define , provided the limit exists. Then

(A) is defined only for
(B) is defined only when is an integer
(C) is defined for all and is independent of
(D) none of the above is true.

Problem 17:

Consider the sequence . Then the integer part of equals
(A)
(B)
(C)
(D) .

Problem 18:

Let and Then the minimum value of is

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

Problem 19:

What is the limit of as ?
(A)
(B)
(C)
(D) .

Problem 20:

Consider the function The function
(A) is zero at but is increasing near
(B) has a zero in
(C) has two zeros in (-1,0)
(D) has exactly one local minimum in (-1,0) .

Problem 21:

Consider a sequence of 10 A's and 8 B's placed in a row. By a run we mean one or more letters of the same type placed side by side. Here is an arrangement of 's and 's which contains 4 runs of and 4 runs of

AAABBABBBAABAAAABB

In how many ways can 's and 's be arranged in a row so that there are 4 runs of and 4 runs of
(A)
(B)
(C)
(D) .

Problem 22:

Suppose is a fixed positive integer and . Then

(A) is differentiable everywhere only when is even
(B) is differentiable everywhere except at 0 if is odd
(C) is differentiable everywhere
(D) none of the above is true.

Problem 23:

The line with cuts the axis and axis at points and respectively. Then the equation of the circle having as diameter is
(A)
(B)
(C)
(D) .

Problem 24:

Let and consider the sequence

Then is

(A) for any
(B) for any
(C) or depending on what is
(D) or depending on what is.

Problem 25:

If then
(A)
(B)
(C)
(D)

Problem 26:

Consider a cardboard box in the shape of a prism as shown below. The length of the prism is 5 . The two triangular faces and are congruent and isosceles with side lengths 2,2,3 . The shortest distance between and along the surface of the prism is
(A)
(B)
(C)
(D)

Problem 27:

Assume the following inequalities for positive integer . The integer part of

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

Problem 28:

Consider the sets defined by the inequalities

Then
(A)
(B)
(C) each of the sets and is non-empty
(D) none of the above is true.

Problem 29:

The number is

(A) strictly larger than

(B) strictly larger than but strictly smaller than

(C) less than or equal to

(D) equal to

Problem 30:

If the roots of the equation are in geometric progression then
(A)
(B)
(C)
(D) .

### B.Stat. (Hons.) & B.Math. (Hons.) Admission Test: 2012

Problem 1:

Let be the angles of a triangle.
(1) Prove that

(2) Using (1) or otherwise prove that

Problem 2:

Let be s real number. Consider the function

(i) Determine the values of for which is continuous at all .
(ii) Determine the values of for which is differentiable at all .

Problem 3:

Write the set of all positive integers in triangular array as

Find the row number and column number where occurs. For example appears in the third row and second column.

Problem 4:

Show that the polynomial has no real root.

Problem 5:

Let be a natural number with digits consisting entirely of 6 's and 0 's. Prove that is not the square of a natural number.

Problem 6:

Let .
(i) Show that amongst the triangles with base and perimeter the maximum area is obtained when the other two sides have equal length .
(ii) Using the result (i) or otherwise show that amongst the quadrilateral of given perimeter the square has maximum area.

Problem 7:

Let . Consider two circles with radii and and centers and respectively with . Let be the center of any circle in the crescent shaped region between the two circles and tangent to both (See figure below). Determine the locus of as its circle traverses through region maintaining tangency.

Problem 8:

Let , and .For a function , a subset is said be invariant under if for all . Note that the empty set and are invariant for all . Let be the number of subsets of invariant under .
(i) Show that there is a function such that .
(ii) Further show that for any such that there is a function such that

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