# Indian Statistical Institute, ISI BStat & BMath 2018 Solutions & Hints

## ISI BMath & BSTAT 2018 Subjective Questions UGB along with hints and solutions and discussions

**Note: You must try the problems on your own before looking at the solutions. I have made the solutions so that you don’t have to join any paid program to get the solutions. After you have failed several times in your attempt to solve a question even with the hints provided, then only look at the solution of that problem. If you cannot solve a single question on your own then this exam is not for you. And if you like my work then please do like my videos and subscribe me on youtube to get more such videos in the future.Â **

**Problem 1.**

Find all pairs with real, satisfying the equations:

### Topic: Triginometry

Difficulty level: Very Easy

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 1 UGB Solution & Discussion Video:**

#### Full Solution

**Problem 2.**

Suppose that and are two chords of a circle intersecting at a point It is given that cm and cm. Moreover, the

area of the triangle is cm Find the area of the triangle

### Topic: Calculus: Geometry

Difficulty Level: Very Easy

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 2 UGB Solution & Discussion Video:**

## Diagram

#### Hint 2

#### Hint 3

#### Full Solution

Consider triangles and Consider angles and They are subtended by the same chord and are located on the same side of the chord (since and intersect). So, Angles and are vertical, so Triangles and are similar by the equality of two angles. Coefficiant of similarity is found from the ratio of sides: the side is opposite to the angle the side is opposite to the angle So, the coefficient of similarity is The area of the triangle is times bigger than the area of the triangle area of

**Problem 3.**

Let be a continuous function such that for all and

for all

Show that is a constant function.

### Topic: Calculus: Functions, Limits

Difficulty Level: Medium

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 3 UGB Solution & Discussion Video:**

**Problem 4.**

Let be a continuous function such that for all

Show that the function defined by the equation

is a constant function.

### Topic: Calculus: Differentiation, Definite Integral

Difficulty Level: Medium

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 4 UGB Solution & Discussion Video:**

#### Hint 1

Try to see if you have all the conditions satisfied to apply Leibniz integral rule.

If you don’t know what Leibniz integral rule is then read: Leibntz Integral rule.

#### Hints 3

#### Full Solution

**Problem 5.**

Let be a differentiable function such that its derivative is a continuous function. Moreover, assume that for all ,

Define a sequence of real numbers by:

Prove that there exists a positive real number such that for all

### Topic: Calculus: Differentiation, Continuity

Difficulty Level: Medium

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 5 UGB Solution & Discussion Video:**

#### Full Solution

**Problem 6.**

Let be real numbers such that for all there

exist triangles of side lengths Prove that the triangles are

isosceles.

### Topic: Geometry, Inequality, Limits

Difficulty Level: Easy

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 6 UGB Solution & Discussion Video:**

#### Full Solution

**Problem 7.**

Let be such that

Prove that

(i) is odd,

(ii) is divisible by 4,

(iii) is divisible by

### Topic: Number Theory

Difficulty Level: First two parts: Very Easy, Third Part: Medium

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 7 UGB Solution & Discussion Video:**

#### Hint 3

#### Full Solution

(i) Substitute in the equation

It follows that is odd, hence is odd.

(ii) Since is odd, write we get

So,

One of the numbers is even, so the product is divisible by and is divisible by 4.

(iii) Substitute . We must prove that is divisible by Expand

is odd, so

The second summand is divisible by so it is enough to check that is divisible by From part (ii) we have so From the initial equation

Then

expand

is even, so

and the expression is also divisible by

**Problem 8.**

Let . Let be an matrix such that

for all Suppose that

Show that is a multiple of

### Topic: Number Theory

Difficulty Level: Very Hard

**Indian Statistical Institute, ISI Subjective BStat & BMath 2018 Problem 8 UGB Solution & Discussion Video:**

#### Hint 2

#### Hint 3

#### Full Solution

Applying the definition of the matrix with we get that for all

Since we can choose such that Then

Consider following quantities:

is the number of indices such that

is the number of indices such that

is the number of indices such that

is the number of indices such that

Clearly,

Further,

is the number of indices such that

is the number of indices such that

is the number of indices such that

is the number of indices such that

Using these quantities we calculate sums in (2):

We get following relations:

Adding first and second we get

Adding first and third we get

Adding second and third we get

It follows that is a multiple of

If there is any typing error please do let us know by commenting here.

Thank you

Fractions