The sum of series `1/2!+1/4!+16!+……….` is (A) `(e^2-1)/2` (B) `(e^2-2)/e` (C) `(e^2-1)/(2e)` (D) `)(e-1)^2)/(2e)`

The sum of the series 1+1/4.2!1/16.4!+1/64.6!+………to oo is (A) (e+1)/(2sqrt(e)) (B) (e-1)/sqrt(e) (C) (e-1)/(2sqrt(e)) (D) (e+1)/2sqrt(e)

The sum of the series 1+1/4.2!1/16.4!+1/64.6!+………to oo is (A) (e+1)/2sqrt(e) (B) (e-1)/sqrt(e) (C) (e-1)/+2sqrt(e) (D) (e-1)/sqrt(e)

The sum of the series 1/(2!)-1/(3!)+1/(4!)-... upto infinity is (1) e^(-2) (2) e^(-1) (3) e^(-1//2) (4) e^(1//2)

The value of int_0^(pi/4) (e^(1/(cos^2x)).sinx)/(cos^3 x)dx equals (A) (e^2-e)/2 (B) (e^4-1)/4 (C) (e^2+e)/4 (D) (e^4-1)/2

The minimum value of f(x)=int_0^4e^(|x-t|)dt where x in [0,3] is : (A) 2e^2-1 (B) e^4-1 (C) 2(e^2-1) (D) e^2-1

The maximum value of x^4e^ (-x^2) is (A) e^2 (B) e^(-2) (C) 12 e^(-2) (D) 4e^(-2)

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x)a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

If xlog_e(log_ex)-x^2+y^2=4 then ((dy)/(dx))_(atx=e) is equal to (A) (2e+1)/ sqrt(4+e^2) (B) e/(2sqrt(4+e^2)) (C) (2e+1)/(2(4+e^2)) (D) (2e-1)/(2sqrt(4+e^2))

The value of int_1^e((x/e)^(2x)+(e/x)^x)log_ex dx is equal to (A) e-1/(2e^2)-1/2 (B) e-1/(2e^2)+1/2 (C) e^3-1/(2e^2)-1/2 (D) none of these

A variable point P on the ellipse of eccentricity e is joined to the foci S and S' . The eccentricity of the locus of incentre of the triangle PSS' is (A) sqrt((2e)/(1+e)) (B) sqrt(e/(1+e)) (C) sqrt((1-e)/(1+e)) (D) e/(2(1+e))