If `varphi(x)=1/(1+e^(-x))a n dS=varphi(5)+varphi(4)+varphi(3)++varphi(-3)+varphi(-4)+varphi(-5)` , then the value of `S` is. a. 5 b. 11/2 c. 6 d. 13/2

If varphi(x) is differentiable function AAx in R and a in R^+ such that varphi(0)=varphi(2a),varphi(a)=varphi(3a)a n dvarphi(0)!=varphi(a) then show that there is at least one root of equation varphi^(prime)(x+a)=varphi^(prime)(x)in(0,2a)

If varphi(x) is a polynomial function and varphi^(prime)(x)>varphi(x)AAxgeq1a n dvarphi(1)=0, then varphi(x)geq0AAxgeq1 varphi(x)

Let Aa n dB are events of an experiment and P(A)=1//4, P(AuuB)=1//2, then value of P(B//A^c) is a. 2//3 b. 1//3 c. 5//6 d. 1//2

Let inte^x{f(x)-f^(prime)(x)}dx=varphi(x)dot Then inte^xf(x)dx is varphi(x)= e^xf(x)

If (1-p)(1+3x+9x^2+27 x^3+81 x^4+243 x^5)=1-p^6p!=1 , then the value of p/xi s 1/3 b. 3 c. 1/2 d. 2

If the function f(x) increases in the interval (a , b),a n dvarphi(x) [f(x)]^2 , then varphi(x) increases in (a , b) varphi(x) decreases in (a , b) we cannot say that varphi(x) increases or decreases in (a , b)dot none of these

Let (1+x^2)^2(1+x)^n=sum_(k=0)^(n+4)a_k x^kdotIfa_1,a_2a n d a_3 are in arithmetic progression, then the possible value/values of n is/are a. 5 b. 4 c. 3 d. 2

If angle theta bertween the line (x+1)/1=(y-1)/2=(z-2)/2 and the plane 2x-y+sqrt(lambda)z+4=0 is such that s intheta=1//3, the value of lambda is a. -3/5 b. 5/3 c. -4/3 d. 3/4

If x=varphi(t), y=psi(t),t h e n(d^(2y))/(dx^2) is (a) (varphi^(prime)psi^('')-psi'varphi' ')/((varphi^(prime))^2) (b) (varphi^(prime)psi^('')-psi'varphi' ')/((varphi^(prime))^3) (c) varphi^('')/psi^('') (d) psi^('')/varphi^('')