Select Page
##### Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2018. Subjective Problem no. 2.

Plane Geometry

5.5 out of 10

##### Suggested Book

‘Challenge and Thrill of Pre-College Mathematics’ by V,Krishnamurthy, C.R.Pranesachar, ect.

Do you really need a hint? Try it first!

$$PQ$$ and $$RS$$ are two chords of the circle $$C$$ , intersecting at the point $$O$$. See figure: click here.

Given $$PO=3$$ cm

$$SO=4$$ cm

$$[\triangle POR]= 7 cm^2$$.

From the triangles $$POS$$ and $$QOS$$ we have,

$$\angle POR=\angle SOQ$$ [Opposite angles]

$$\angle SRP=\angle SQO$$ [Angle on the same semi-circle $$STP$$]

$$\angle QSO= \angle OPR$$ [Angle on the same semi-circle $$ST’P$$]

Therefore the $$\triangle POR$$ and $$\triangle SOQ$$ are similar triangles .

$$\frac{[\triangle POR]}{OP^2}=\frac{[\triangle SOQ]}{SO^2}.$$

$$\Rightarrow [\triangle SOQ]=\frac{SO^2}{PO^2}\cdot [\triangle POR]$$=$$\frac{4^2}{3^2}\cdot 7=12\frac{4}{9}$$.(Ans.)

# I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

# Similar Problem

## The Product of Digits, ISI Entrance 2017, Subjective Solution to problem – 5.

Understand the problem Let $$g : \mathbb{N} \to \mathbb{N}$$ with $$g(n)$$ being the product of digits of $$n$$. (a) Prove that $$g(n)\le n$$ for all $$n \in \mathbb{N}$$ . (b) Find all $$n \in \mathbb{N}$$ , for which $$n^2-12n+36=g(n)$$....

## Three Primes, ISI Entrance 2017, Subjective solution to Problem 6.

Understand the problemLet $$p_1,p_2,p_3$$ be primes with $$p_2\neq p_3$$, such that $$4+p_1p_2$$ and $$4+p_1p_3$$ are perfect squares. Find all possible values of $$p_1,p_2,p_3$$.  Start with hintsDo you really need a hint? Try it first!Let $$4+p_1p_2=m^2$$ and...

## System of n equations, ISI Entrance 2008, Solution to Subjective Problem No. 9.

Understand the problem For $$n\ge3$$, determine all real solutions of the system of $$n$$ equations : $$x_1+x_2+\cdots+x_{n-1}=\frac{1}{x_n}$$ ...

## C.M.I. 2019 Entrance – Answer Key, Sequential Hints

CMI (Chennai Mathematical Institute) Entrance 2019, Sequential hints, answer key, solutions.

## A Trigonometric Substitution, ISI Entrance 2019, Subjective Solution to Problem – 6 .

Understand the problem For all natural numbers$$n$$, let $$A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}$$ ($$n$$ many radicals) (a) Show that for $$n\ge 2, A_n=2\sin \frac{π}{2^{n+1}}$$. (b) Hence, or otherwise, evaluate the limit ...

## Two Similar Triangles, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 2

Understand the problemSuppose that $$PQ$$ and $$RS$$ are two chords of a circle intersecting at a point $$O$$ , It is given that $$PO=3$$ cm and $$SO=4$$ cm . Moreover, the area of the triangle $$POR$$ is $$7 cm^2$$ . Find the are of the triangle $$QOS$$ . I.S.I....

## Pythagorean Triple, ISI B.Math/B.Stat Entrance 2018, Subjective Solution of Problem No. 7

Understand the problem Let $$a,b,c \in \mathbb{N}$$ be such that $$a^2+b^2=c^2$$ and $$c-b=1$$.Prove that (i) $$a$$ is odd,(ii) $$b$$ is divisible by 4,(iii) $$a^b+b^a$$ is divisible by $$c$$.   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

## Powers of 2 – I.S.I. B.Math/B.Stat Entrance 2019 Subjective Solution Problem 1

Understand the problem Prove that the positive integers $$n$$ that cannot be written as a sum of $$r$$ consecutive positive integers, with $$r>1$$ ,are of the form $$n=2^l$$ for some $$l\ge 0$$.   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance...

## Clocky Rotato Arithmetic

Do you know that CLOCKS add numbers in a different way than we do? Do you know that ROTATIONS can also behave as numbers and they have their own arithmetic? Well, this post is about how clocks add numbers and rotations behave like numbers. Consider the clock on earth....

## Shortest distance between curves – I.S.I. Entrance 2019 Subjective Solution Problem 8

Shortest path between two smooth curves is along the common normal. We use this fact to solve 8th problem of I.S.I. Entrance 2019 (UG Subjective)