Show that `|1 1+p1+p+q2 3+2p1+3p+2q3 6+3p 106 p+3q|=1.`

Using properties of determinants in Exercise 11 to 15 prove that |{:(1,1+p,1+p+q),(2,3+2p,4+3p+2q),(3,6+3p,10+6p+3q):}|=1

Show that the points P(-2, 3, 5), Q(1, 2, 3) and R(7, 0, -1) are collinear.

if the roots of the equation 1/ (x+p) + 1/ (x+q) = 1/r are equal in magnitude but opposite in sign, show that p+q = 2r & that the product of roots is equal to (-1/2)(p^2+q^2) .

Suppose E_1, E_2 and E_3 be three mutually exclusive events such that P(E_i)=p_i" for "i=1, 2, 3 . If p_1, p_2 and p_3 are the roots of 27x^3-27x^2+ax-1=0 the value of a is

Find equation of plane passing through the points P(1, 1, 1), Q(3, -1, 2) and R(-3, 5, -4) .

If the sum of p terms of an AP is q and the sum of q terms is p, then show that the sum of p+q terms is -(p+ q) . Alos, find the sum of first p-q terms (where, p gt q )

If p + r = 2q and 1/q + 1/s = 2/r , then prove that p : q = r : s .

If the p th and qth terms of a GP are q and p respectively, then show that its (p+ q) th term is ((q^(p))/(p^(q)))^((1)/(p-q))

Show that 1.272727 …..= 1. bar(2.7) can be expressed in the form (p)/(q) where p and q are integer and q ne 0

Q. Let p and q real number such that p!= 0 , p^2!=q and p^2!=-q . if alpha and beta are non-zero complex number satisfying alpha+beta=-p and alpha^3+beta^3=q , then a quadratic equation having alpha/beta and beta/alpha as its roots is