If `P(1)=0a n d(d P(x))/(dx),sinx+2xgeq(3x(x+1))/pi` . Explain the identity, if any, used in the proof.

Prove that sin x+2x>=(3x(x+1))/(pi),AA x in[0,(pi)/(2)] (Justify the inequality,if any used).

int_(0)^(pi)(x)/(1+sinx)dx .

If polynomials P and Q satisfy int[(3x-1)cosx+(1-2x)sinx ]dx=P cosx+Qsinx (ignore the constant of integration) then :

If |(x^2+x, x-1, x+1), (x, 2x, 3x-1), (4x+1, x-2, x+2)|= px^4 +qx^3+rx^2+sx+t be n identity in x and omega be an imaginary cube root of unity, (a+bomega+comega^2)/(c+aomega+bomega^2)+(a+bomega+comega^2)/(b+comega+aomega^2)= (A) p (B) 2p (C) -2p (D) -p

Let f(x)=(2x-pi)^(3)+2x-cos x .If the value of (d)/(dx)(f^(-1) (x)) at x=pi can be expressed in the form of (p)/(q) ( where p and q are natural numbers in their lowest form), then the value of (p+q) is

The equation (cos p-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) ( 0,2 pi)( b) (-pi,0) (c) (-(pi)/(2),(pi)/(2))( d )(0,pi)

P(x) is a least degree polynomial such that (P(x)-1) in divisible by (x-1)^(2) and (P(x)-3) is divisible by ((x+1)^2) ,then (A) graph of y=P(x) ,is symmetric about origin (B) y=P(x) ,has two points of extrema (C) int_(-1)^(2)P(x)dx=0 for exactly one value of lambda . (D) int_(-1)^(2)P(x)dx=0 for exactly two values of lambda

Given P(x)""=""x^4+""a x^3+""c x""+""d such that x""=""0 is the only real root of P '(x)""=""0.""If""P(1)""<""P(1) , then in the interval [1,""1] . (1) P(1) is the minimum and P(1) is the maximum of P (2) P(1) is not minimum but P(1) is the maximum of P (3) P(1) is the minimum but P(1) is not the maximum of P (4) neither P(1) is the minimum nor P(1) is the maximum of P