If `x=varphi(t), y=psi(t),t h e n(d^2 y)/(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^('')`

If x=t^2,y=t^3,t h e n(d^2y)/(dx^2) is (a) 3/2 (b) 3/(4t) (c) 3/(2t) (d) (3t)/2

If x = 3 tant and y = 3 sec t, then the value of (d^2y)/(dx^2)" at" t=pi/4 is

If g(x)=int_(sinx)^("sin"(2x))sin^(-1)(t)dt ,t h e n : (a) g^(prime)(pi/2)=-2pi (b) g^(prime)(-pi/2)=-2pi (c) g^(prime)(-pi/2)=2pi (d) g^(prime)(pi/2)=2pi

Ifx=int_0^y(dt)/(sqrt(1+9t^2))a n d(d^2y)/(dx^2)=a y ,t h e nfin da

If x^2+y^2=1, then (a) y y^('')-2(y^(prime))^2+1=0 (b) yy^('')+(y^(prime))^2+1=0 (c) y y^('')+(y^(prime))^(-2)-1=0 (d) y y^('')+2(y^(prime))^2+1=0

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to (a) n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

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

Suppose f and g are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all x and f^(prime) and g' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x)) is equal (a) (-2f^(prime)(x))/f (b) (-2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))