ISI Entrance Paper 2018 – from Indian Statistical Institute’s B.Stat Entrance

1. Find all pairs ( (x,y) ) with (x,y) real, satisfying the equations $$sinbigg(frac{x+y}{2}bigg)=0~,~vert xvert+vert yvert=1$$

2. Suppose that (PQ) and (RS) are two chords of a circle intersecting at a point (O). It is given that (PO=3 text{cm}) and (SO=4 text{cm}). Moreover, the area of the triangle (POR) is (7 text{cm}^2). Find the area of the triangle (QOS).

3. Let (f:mathbb{R}tomathbb{R}) be a continuous function such that for all (xinmathbb{R}) and for all (tgeq 0), $$f(x)=f(e^tx)$$Show that (f) is a constant function.

4. Let (f:(0,infty)tomathbb{R}) be a continuous function such that for all (xin(0,infty)), $$f(2x)=f(x)$$Show that the function (g) defined by the equation $$g(x)=int_{x}^{2x} f(t)frac{dt}{t}~~text{for}~x>0$$is a constant function.

5. Let (f:mathbb{R}tomathbb{R}) be a differentiable function such that its derivative (f’) is a continuous function. Moreover, assume that for all (xinmathbb{R}), $$0leq vert f'(x)vertleq frac{1}{2}$$Define a sequence of real numbers ( {a_n}_{ninmathbb{N}}) by :$$a_1=1~~text{and}~~a_{n+1}=f(a_n)~text{for all}~ninmathbb{N}$$Prove that there exists a positive real number (M) such that for all (ninmathbb{N}), $$vert a_nvert leq M$$

# Hint:

The sequence is Cauchy

6. Let, (ageq bgeq c >0) be real numbers such that for all natural number (n), there exist triangles of side lengths (a^{n} , b^{n} ,c^{n}). Prove that the triangles are isosceles.

7. Let (a, b, c) are natural numbers 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

8. Let (ngeq 3). Let (A=((a_{ij}))_{1leq i,jleq n}) be an (ntimes n) matrix such that (a_{ij}in{-1,1}) for all (1leq i,jleq n). Suppose that $$a_{k1}=1~~text{for all}~1leq kleq n$$and (~~sum_{k=1}^n a_{ki}a_{kj}=0~~text{for all}~ineq j). Show that n is a multiple of 4.

# Hint: 