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.
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
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) .
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
Download Pdf : ISI Entrance Paper 2012
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.
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
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) .
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
Download Pdf : ISI Entrance Paper 2012
Please publish the solutions to the problems
which one?
this one..plz publish the ans