Prove that: `t a n 70^0=t a n 20^0+2t a n 50^0dot`

Prove that the function f(x) = x^(2n) + 1 is continuous at x = n , n > 0

Let A=[[0,1],[0,0]] , prove that: (aI+bA)^n = a^(n-1) bA , where I is the unit matrix of order 2 and n is a positive integer.

If t_1 a n d t_2 are roots of eth equation t^2+lambdat+1=0, where lambda is an arbitrary constant. Then prove that the line joining the points (a t1^2,2a t_1)a d n(a t2^2,2a t_2) always passes through a fixed point. Also, find the point.

Prove that: sin^2(pi/6)+cos^2(pi/3)-t a n^2pi/4=-1/2

Show that the sequence t n defined by t n = 3 n + 1 is an AP.

Prove that t_(75) = 2 t_(50) for 1^(st) order reaction.

A particle executes the motion described by x(t) = x_0 (1-e^(-gammat) , t le 0, x_0 > 0 . Find maximum and minimum vlaues of x(t), v(t), a (t). Show that x (t) and a (t) increase with time and v (t) decreases with time.

For an positive integer n, prove that : i^(n) + i^(n+1) + i^(n+2) + i^(n+3) + i^(n+4) + i^(n + 5) + i^(n+6) + i^(n+7) = 0 .

If t a nbeta=(n sinalphacosalpha)/(1-n sin^2alpha) , show that tan(alpha-beta)=(1-n)t a nalphadot

A sample of a radioactive isotope contains N nuclei at time t.At time (t+triangle t) ,it contains (N-triangleN) nuclei of the isotope. For the period trianglet ,state in terms of N,triangle N and trianglet , the probability of decay of a nucleus.