This is a work in progress. Please come back for more solutions and discussions
Answer Key of Objective problems (Caution ... first please check the order of questions that we are using.)
Q1 | D | Q2 | A | Q3 | A |
Q4 | B | Q5 | B | Q6 | B |
Q7 | B | Q8 | D | Q9 | B |
Q10 | C | Q11 | D | Q12 | D |
Q13 | B | Q14 | D | Q15 | D |
Q16 | A | Q17 | C | Q18 | D |
Q19 | C | Q20 | B | Q21 | C |
Q22 | B | Q23 | A | Q24 | D |
Q25 | C | Q26 | B | Q27 | A |
Q28 | A | Q29 | A | Q30 | C |
Determine all integers such that every power of
has an odd number of digits.
Let and
be defined inductively by
(a) Show that for ,
and determine .
(b) Using (a) or otherwise, calculate
In a triangle , consider points
and
on
and
, respectively, and assume that they do not coincide with any of the vertices
. If the segments
and
intersect at
, consider the areas
of the quadrilateral
and the triangles
, respectively.
(a) Prove that .
(b) Determine in terms of
.
Let be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance,
has exactly 2023 positive integer factors
. Assume that no prime larger than 11 divides any of the
's. Show that there must be some perfect cube among the
's. You may use the fact that
There is a rectangular plot of size . This has to be covered by three types of tiles - red, blue and black. The red tiles are of size
, the blue tiles are of size
and the black tiles are of size
. Let
denote the number of ways this can be done. For example, clearly
because we can have either a red or a blue tile. Also,
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that for all
.
(b) Prove that for all
.
Here,
for integers .
Let be a sequence of real numbers defined as
and
Prove that for all
.
(a) Let be an integer. Prove that
can be written as a polynomial with integer coefficients in the variables
,
and
.
(b) Let , where
are real numbers such that
is an integral multiple of
. <br>Using (a) or otherwise, show that if
, then
for all positive integers
.
Let be a continuous function which is differentiable on
. Prove that either
is a linear function
or there exists
such that
.
The following notations are used in the question paper:
is the set of real numbers,
is the set of complex numbers,
is the set of integers,
for all
and
.
For a real number ,
if and only if
(A) >2
(B) -3<<1
(C) <-3 or 1<
<2
(D) -3<<1 or
>2
The number of consecutive zeroes adjacent to the digit in the unit's place of is
(A) 3
(B) 4
(C) 49
(D) 50
Consider a right-angled triangle whose hypotenuse
is of length 1. The bisector of
intersects
at
. If
is of length
, then what is the length of
?
(A)
(B)
(C)
(D)
Define a polynomial
for all , where the right hand side above is a determinant. Then the roots of
are of the form
(A) where
and
is a square root of
(B) where
are distinct.
(C) where
are all distinct.
(D) for some
.
Let be the set of those real numbers
for which the identity
is valid, and the quantities on both sides are finite. Then
(A) is the empty set.
(B) for all
.
(C) for all
.
(D) for all
Let
How many elements does have?
(A) 9.
(B) 1.
(C) 2.
(D) 3
How many numbers formed by rearranging the digits of 234578 are divisible by 55?
(A) 0
(B) 12
(C) 36
(D)
The limit
equals
(A) 0
(B) 1
(C) .
(D) .
Consider a triangle with vertices and
. Let
be the area of the triangle and
be the area of the circumcircle of the triangle. Then
equals
(A) .
(B) .
(C) .
(D) .
The value of
equals
(A) .
(C) .
(B) .
(D) .
For real numbers , consider the system of equations
If denotes the set of all real solutions
of the above system of equations, then the number of elements in
can never be
(A) 0
(B) 1
(C) 2
(D) 3
Let be continuous functions from
to itself,
and
If is the derivative of
, then for
,
(A) .
(B) .
(C) .
(D) .
Suppose and
are positive integers. If
and
are divided by 7 , then the respective remainders are 2 and 5. If
is divided by 7, then the remainder equals
(A) 0
(B) 1
(C) 2
(D) 3
Suppose is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) cannot have a local minimum.
(B) must have exactly one local minimum.
(C) must have at least two local minima.
(D) must have either a global maximum or a local minimum.
Suppose is such that the imaginary part of
is non-zero and
. Then
equals
(A) 0
(B) 1
(C) .
(D) .
The limit
(A) equals 0
(B) equals ,
(C) equals 1
(D) does not exist.
Let be a positive integer having 27 divisors including 1 and
, which are denoted by
. Then the product of
equals
(A) .
(B) .
(C) ,
(D) .
If is a continuous function such that
then for all ,
(A) .
(B) .
(C) .
(D) .
If denotes the largest integer less than or equal to
, then
equals
(A) .
(B) .
(C) .
(D) .
Let be a twice differentiable one-to-one function. If
and
then
equals
(A) -25
(B) 25
(C) -31
(D) 31
Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?
(A) 3
(B) 4
(C) 5
(D) 6
The limit
equals
(A) 1
(B)
(C) 0
(D)
In the following figure, is a quarter-circle. The unshaded region is a circle to which
and
are tangents.
If is of length 10 and is parallel to
, then the area of the shaded region in the above figure equals
(A) .
(B) .
(C)
(D) .
Consider the function defined by
where is a square root of -1. Then
(A) is one-to-one and onto.
(B) is one-to-one but not onto.
(C) is onto but not one-to-one.
(D) is neither one-to-one nor onto.
Suppose is a non-decreasing function. Consider the following two cases:
Case 1. ,
Case 2. .
In which of the above cases it is necessarily true that there exists an with
?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.
Suppose that where
are real numbers with
. The equation
has exactly two distinct real solutions. If
is the derivative of
, then which of the following is a possible graph of
?
Suppose and
If for all
, and
, then
(A) .
(B) .
(C) .
(D) .
The polynomial is divisible by
(A) .
(B) .
(C) .
(D) .
As in the following figure, the straight line lies in the second quadrant of the
-plane and makes an angle
with the negative half of the
-axis, where
.
The line segment of length 1 slides on the
-plane in such a way that
is always on
and
on the positive side of the
-axis. The locus of the mid-point of
is
(A) .
(B) .
(C) .
(D) .
Q30. How many functions , which satisfy
are there?
(A)
(B)
(C)
(D)
This is a work in progress. Please come back for more solutions and discussions
Answer Key of Objective problems (Caution ... first please check the order of questions that we are using.)
Q1 | D | Q2 | A | Q3 | A |
Q4 | B | Q5 | B | Q6 | B |
Q7 | B | Q8 | D | Q9 | B |
Q10 | C | Q11 | D | Q12 | D |
Q13 | B | Q14 | D | Q15 | D |
Q16 | A | Q17 | C | Q18 | D |
Q19 | C | Q20 | B | Q21 | C |
Q22 | B | Q23 | A | Q24 | D |
Q25 | C | Q26 | B | Q27 | A |
Q28 | A | Q29 | A | Q30 | C |
Determine all integers such that every power of
has an odd number of digits.
Let and
be defined inductively by
(a) Show that for ,
and determine .
(b) Using (a) or otherwise, calculate
In a triangle , consider points
and
on
and
, respectively, and assume that they do not coincide with any of the vertices
. If the segments
and
intersect at
, consider the areas
of the quadrilateral
and the triangles
, respectively.
(a) Prove that .
(b) Determine in terms of
.
Let be distinct natural numbers each of which has exactly 2023 positive integer factors. For instance,
has exactly 2023 positive integer factors
. Assume that no prime larger than 11 divides any of the
's. Show that there must be some perfect cube among the
's. You may use the fact that
There is a rectangular plot of size . This has to be covered by three types of tiles - red, blue and black. The red tiles are of size
, the blue tiles are of size
and the black tiles are of size
. Let
denote the number of ways this can be done. For example, clearly
because we can have either a red or a blue tile. Also,
since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that for all
.
(b) Prove that for all
.
Here,
for integers .
Let be a sequence of real numbers defined as
and
Prove that for all
.
(a) Let be an integer. Prove that
can be written as a polynomial with integer coefficients in the variables
,
and
.
(b) Let , where
are real numbers such that
is an integral multiple of
. <br>Using (a) or otherwise, show that if
, then
for all positive integers
.
Let be a continuous function which is differentiable on
. Prove that either
is a linear function
or there exists
such that
.
The following notations are used in the question paper:
is the set of real numbers,
is the set of complex numbers,
is the set of integers,
for all
and
.
For a real number ,
if and only if
(A) >2
(B) -3<<1
(C) <-3 or 1<
<2
(D) -3<<1 or
>2
The number of consecutive zeroes adjacent to the digit in the unit's place of is
(A) 3
(B) 4
(C) 49
(D) 50
Consider a right-angled triangle whose hypotenuse
is of length 1. The bisector of
intersects
at
. If
is of length
, then what is the length of
?
(A)
(B)
(C)
(D)
Define a polynomial
for all , where the right hand side above is a determinant. Then the roots of
are of the form
(A) where
and
is a square root of
(B) where
are distinct.
(C) where
are all distinct.
(D) for some
.
Let be the set of those real numbers
for which the identity
is valid, and the quantities on both sides are finite. Then
(A) is the empty set.
(B) for all
.
(C) for all
.
(D) for all
Let
How many elements does have?
(A) 9.
(B) 1.
(C) 2.
(D) 3
How many numbers formed by rearranging the digits of 234578 are divisible by 55?
(A) 0
(B) 12
(C) 36
(D)
The limit
equals
(A) 0
(B) 1
(C) .
(D) .
Consider a triangle with vertices and
. Let
be the area of the triangle and
be the area of the circumcircle of the triangle. Then
equals
(A) .
(B) .
(C) .
(D) .
The value of
equals
(A) .
(C) .
(B) .
(D) .
For real numbers , consider the system of equations
If denotes the set of all real solutions
of the above system of equations, then the number of elements in
can never be
(A) 0
(B) 1
(C) 2
(D) 3
Let be continuous functions from
to itself,
and
If is the derivative of
, then for
,
(A) .
(B) .
(C) .
(D) .
Suppose and
are positive integers. If
and
are divided by 7 , then the respective remainders are 2 and 5. If
is divided by 7, then the remainder equals
(A) 0
(B) 1
(C) 2
(D) 3
Suppose is a continuous function which has exactly one local maximum. Then which of the following is true?
(A) cannot have a local minimum.
(B) must have exactly one local minimum.
(C) must have at least two local minima.
(D) must have either a global maximum or a local minimum.
Suppose is such that the imaginary part of
is non-zero and
. Then
equals
(A) 0
(B) 1
(C) .
(D) .
The limit
(A) equals 0
(B) equals ,
(C) equals 1
(D) does not exist.
Let be a positive integer having 27 divisors including 1 and
, which are denoted by
. Then the product of
equals
(A) .
(B) .
(C) ,
(D) .
If is a continuous function such that
then for all ,
(A) .
(B) .
(C) .
(D) .
If denotes the largest integer less than or equal to
, then
equals
(A) .
(B) .
(C) .
(D) .
Let be a twice differentiable one-to-one function. If
and
then
equals
(A) -25
(B) 25
(C) -31
(D) 31
Three left brackets and three right brackets have to be arranged in such a way that if the brackets are serially counted from the left, then the number of right brackets counted is always less than or equal to the number of left brackets counted. In how many ways can this be done?
(A) 3
(B) 4
(C) 5
(D) 6
The limit
equals
(A) 1
(B)
(C) 0
(D)
In the following figure, is a quarter-circle. The unshaded region is a circle to which
and
are tangents.
If is of length 10 and is parallel to
, then the area of the shaded region in the above figure equals
(A) .
(B) .
(C)
(D) .
Consider the function defined by
where is a square root of -1. Then
(A) is one-to-one and onto.
(B) is one-to-one but not onto.
(C) is onto but not one-to-one.
(D) is neither one-to-one nor onto.
Suppose is a non-decreasing function. Consider the following two cases:
Case 1. ,
Case 2. .
In which of the above cases it is necessarily true that there exists an with
?
(A) In both cases.
(B) In neither case.
(C) In Case 1. but not necessarily in Case 2.
(D) In Case 2. but not necessarily in Case 1.
Suppose that where
are real numbers with
. The equation
has exactly two distinct real solutions. If
is the derivative of
, then which of the following is a possible graph of
?
Suppose and
If for all
, and
, then
(A) .
(B) .
(C) .
(D) .
The polynomial is divisible by
(A) .
(B) .
(C) .
(D) .
As in the following figure, the straight line lies in the second quadrant of the
-plane and makes an angle
with the negative half of the
-axis, where
.
The line segment of length 1 slides on the
-plane in such a way that
is always on
and
on the positive side of the
-axis. The locus of the mid-point of
is
(A) .
(B) .
(C) .
(D) .
Q30. How many functions , which satisfy
are there?
(A)
(B)
(C)
(D)