1) The order and the degree of the differential equation
$$ \frac{d^2y}{dx^2} + (\frac{dy}{dx})^4 + 6y^7 = e^x $$ is:
(A) 3rd order and 1st degree
(B) 2nd order and 4th degree
(C) 1st order and 3rd degree
(D) 4th order and 2nd degree
(02) The differential equation $$ \frac{dy}{dx} - y^3 x = \cos x $$ is:
(A) Linear (B) Non-Linear (C) Quasilinear (D) None of them
(03) Which of the following differential equations are separable?
(i) $$ \frac{dy}{dx} = xy $$
(ii) $$ \frac{dy}{dx} = x + y $$
(iii) $$ \frac{dy}{dx} = xy + y $$
(A) All three are separable,
(B) Equation (i) only,
(C) Equations (i) and (iii) only,
(D) Equation (ii) only.
(04) Integrating factor of $$ \frac{dy}{dx} = 500x^2 - x^{-2} $$ is:
(A) x (B) 2x (C) 3x (D) 4x
(05) If the roots of an auxiliary equation is ($100 \pm \sqrt{500}i$), what is the
complimentary function?
(A) $e^{100x} (A \cos \sqrt{500} x + B \sin \sqrt{500} x)$,
(B) $e^{100x} (A \cos 500 x + B \sin 500 x)$,
(C) $e^{500x} (A \cos 100 x + B \sin 100 x)$,
(D) $e^{\sqrt{500}x} (A \cos 100 x + B \sin 100 x)$.
(06) If $$ \frac{dy}{dx} = x^2 $$, what is the equation of y in terms of x if the curve passes through
the point (1,1).
(A) $x^2 - 3y + 3 = 0$, (B) $x^3 - 3y + 2 = 0$.
(C) $x^2 + 3y^2 + 2 = 0$, (D) $x^3 + 2y + 2 = 0$.
(07) First order partial derivative of $f(x, y) = e^{3x} \cos y$ with respect to x is:
(A) $$ \frac{\partial f}{\partial x} = 3 e^{3x} \cos y + e^{3x} \sin y $$
(B) $$ \frac{\partial f}{\partial x} = 3 e^{3x} \cos y + e^{3x} \cos y $$,
(C) $$ \frac{\partial f}{\partial x} = -e^{3x} \sin y $$
(D) $$ \frac{\partial f}{\partial x} = 3 e^{3x} \cos y $$.
(08) The double integral of $$ \int_{x=0}^{1} \int_{y=0}^{1} 4x^2 y \ dy \ dx $$ is:
(A) $$ \frac{1}{3} $$ (B) $$ \frac{15}{15} $$ (C) $$ \frac{16}{3} $$ (D) 4.
(09) The equation of the circle $$(x - \frac{1}{2})^2 + y^2 = \frac{1}{4}$$ in polar coordinates is:
(A) $r = \sin \theta$ (B) $r = \cos \theta$ (C) $r = 4 \sin \theta$ (D) $r = 4 \cos \theta$.
