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1.

If  $f(x)=\begin{cases}\frac{\sqrt{1+ax}-\sqrt{1-ax}}{x} ,& -1\leq x <0\\ \frac{x^{2}+2}{x-2}, & 0\leq x \leq 1\end{cases} is$

continuous  on [-1,1] , than a =

A) -1

B) -2

C) 1

D) 2

2.

Consider the following  differential equations.

$D_{1}:y=4\frac{dy}{dx}+3x\frac{dx}{dy};D_{2}:\frac{d^{2}y}{dx^{2}}=\left(3+\left(\frac{dy}{dx}\right)^{2}\right)^{4/3}$

$D_{3}:\left[1+\left(\frac{dy}{dx}\right)\right]^{2}=\left(\frac{dy}{dx}\right)^{2}$

The ratio of the sum of the orders of $D_{1},D_{2}$ and $D_{3}$  to the sum of their degrees is

A) 1:2

B) 1:1

C) 2:3

D) 3:2

3.

If  $\int\frac{\sqrt{x}}{\sqrt{x}-\sqrt[3]{x}}dx=x+Ex^{5/6}+Dx^{2/3}+Cx^{1/2}+Bx^{1/3}+Ax^{1/6}+\log (\sqrt[6]{x}-1)^{6}+K,$ then A+B+C+D+E=

A) $\frac{137}{10}$

B) $\frac{129}{10}$

C) $\frac{119}{10}$

D) $\frac{117}{10}$

4.

If the curves $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $\frac{x^{2}}{16}-\frac{y^{2}}{k}=1$ cut each other orthogonally , then k=

A) 144

B) -9

C) 25

D) -21

5.

The locus  of the mid-point of the line segment joining the focus to a moving point on the parabola, $y^{2}=4ax$  is a conic . The equation of the directrix of that conic is

A) y=a

B) x=a

C) y=0

D) x=0

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