Show that `sin^(2)x=p+1/p` is impossible if x is real.

Show that the statement . p : 'If x is a real number such that x^(3)+ 4x=0, then x=0' is true by Direct method

Show that the statement . p : 'If x is a real number such that x^(3)+ 4x=0, then x=0' is true by Method of contradiction

Show that the statement . p : 'If x is a real number such that x^(3)+ 4x=0, then x=0' is true by Method of contrapositive

A quadratic trinomial P(x)=a x^2+b x+c is such that the equation P(x)=x has no real roots. Prove that in this case equation P(P(x))=x has no real roots either.

A quadratic trinomial P(x)=a x^2+b x+c is such that the equation P(x)=x has o real roots. Prove that in this case equation P(P(x))=x has no real roots either.

(a) If r^(2) = pq , show that p : q is the duplicate ratio of (p + r) : (q + r) . (b) If (p - x) : (q - x) be the duplicate ratio of p : q then show that : (1)/(p) + (1)/(q) = (1)/(r) .

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) .

The random variable X can the values 0, 1, 2, 3, Give P(X = 0) = P(X = 1)= p and P(X = 2) = P(X = 3) such that sum p_i x_i^2=2sum p_i x_i then find the value of p

Let P(x) be a polynomial with real coefficients such that P(sin^2x) = P(cos^2x) , for all xϵ[0,π/2] . Consider the following statements: I. P(x) is an even function . II. P(x) can be expressed as a polynomial in (2x−1) 2 III. P(x) is a polynomial of even degree . Then

The equation (cosp-1)x^2+(cos p)x+sin p=0 in the variable x has real roots. The p can take any value in the interval (a) (0,2pi) (b) (-pi,0) (c) (-pi/2,pi/2) (d) (0,pi)