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# ISI BStat - BMath Entrance 2023 problems and solutions.

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.)

### Subjective

###### Problem 1

Determine all integers such that every power of has an odd number of digits.

###### Problem 2

Let and be defined inductively by

(a) Show that for ,

an=cosθn for some 0<θn<π2

and determine .

(b) Using (a) or otherwise, calculate

###### Problem 3

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 .

###### Problem 4

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

###### Problem 5

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 .

###### Problem 6

Let be a sequence of real numbers defined as and

Prove that for all .

###### Problem 7

(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 .

###### Problem 8

Let be a continuous function which is differentiable on . Prove that either is a linear function or there exists such that .

### Objective

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 .

###### Problem 1

For a real number ,

if and only if

(A) >2

(B) -3<<1

(C) <-3 or 1<<2

(D) -3<<1 or >2

###### Problem 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

###### Problem 3

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)

###### Problem 4

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 .

###### Problem 5

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

###### Problem 6

Let

How many elements does have?

(A) 9.

(B) 1.

(C) 2.

(D) 3

###### Problem 7

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) .

###### Problem 9

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) .

###### Problem 11

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

###### Problem 12

Let be continuous functions from to itself,

and

If is the derivative of , then for ,

(A) .

(B) .

(C) .

(D) .

###### Problem 13

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

###### Problem 14

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.

###### Problem 15

Suppose is such that the imaginary part of is non-zero and . Then

equals

(A) 0

(B) 1

(C) .

(D) .

###### Problem 16

The limit

(A) equals 0

(B) equals ,

(C) equals 1

(D) does not exist.

###### Problem 17

Let be a positive integer having 27 divisors including 1 and , which are denoted by . Then the product of equals

(A) .

(B) .

(C) ,

(D) .

###### Problem 18

If is a continuous function such that

then for all ,

(A) .

(B) .

(C) .

(D) .

###### Problem 19

If denotes the largest integer less than or equal to , then

equals

(A) .

(B) .

(C) .

(D) .

###### Problem 20

Let be a twice differentiable one-to-one function. If and

then

equals

(A) -25

(B) 25

(C) -31

(D) 31

###### Problem 21

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)

###### Problem 23

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) .

###### Problem 24

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.

###### Problem 25

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.

###### Problem 26

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 ?

###### Problem 27

Suppose and

If for all , and , then

(A) .

(B) .

(C) .

(D) .

###### Problem 28

The polynomial is divisible by

(A) .

(B) .

(C) .

(D) .

###### Problem 29

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.)

### Subjective

###### Problem 1

Determine all integers such that every power of has an odd number of digits.

###### Problem 2

Let and be defined inductively by

(a) Show that for ,

an=cosθn for some 0<θn<π2

and determine .

(b) Using (a) or otherwise, calculate

###### Problem 3

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 .

###### Problem 4

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

###### Problem 5

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 .

###### Problem 6

Let be a sequence of real numbers defined as and

Prove that for all .

###### Problem 7

(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 .

###### Problem 8

Let be a continuous function which is differentiable on . Prove that either is a linear function or there exists such that .

### Objective

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 .

###### Problem 1

For a real number ,

if and only if

(A) >2

(B) -3<<1

(C) <-3 or 1<<2

(D) -3<<1 or >2

###### Problem 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

###### Problem 3

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)

###### Problem 4

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 .

###### Problem 5

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

###### Problem 6

Let

How many elements does have?

(A) 9.

(B) 1.

(C) 2.

(D) 3

###### Problem 7

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) .

###### Problem 9

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) .

###### Problem 11

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

###### Problem 12

Let be continuous functions from to itself,

and

If is the derivative of , then for ,

(A) .

(B) .

(C) .

(D) .

###### Problem 13

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

###### Problem 14

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.

###### Problem 15

Suppose is such that the imaginary part of is non-zero and . Then

equals

(A) 0

(B) 1

(C) .

(D) .

###### Problem 16

The limit

(A) equals 0

(B) equals ,

(C) equals 1

(D) does not exist.

###### Problem 17

Let be a positive integer having 27 divisors including 1 and , which are denoted by . Then the product of equals

(A) .

(B) .

(C) ,

(D) .

###### Problem 18

If is a continuous function such that

then for all ,

(A) .

(B) .

(C) .

(D) .

###### Problem 19

If denotes the largest integer less than or equal to , then

equals

(A) .

(B) .

(C) .

(D) .

###### Problem 20

Let be a twice differentiable one-to-one function. If and

then

equals

(A) -25

(B) 25

(C) -31

(D) 31

###### Problem 21

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)

###### Problem 23

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) .

###### Problem 24

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.

###### Problem 25

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.

###### Problem 26

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 ?

###### Problem 27

Suppose and

If for all , and , then

(A) .

(B) .

(C) .

(D) .

###### Problem 28

The polynomial is divisible by

(A) .

(B) .

(C) .

(D) .

###### Problem 29

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)

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