1

### IIT-JEE 2001 Screening

Let $$f:\left( {0,\infty } \right) \to R$$ and $$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.}$$ If $$F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$$, then $$f(4)$$ equals
A
$$5/4$$
B
$$7$$
C
$$4$$
D
$$2$$
2

### IIT-JEE 2000

If $${x^2} + {y^2} = 1$$ then
A
$$yy'' - 2{\left( {y'} \right)^2} + 1 = 0$$
B
$$yy'' + {\left( {y'} \right)^2} + 1 = 0$$
C
$$yy'' + {\left( {y'} \right)^2} - 1 = 0$$
D
$$yy'' + 2{\left( {y'} \right)^2} + 1 = 0$$
3

### IIT-JEE 1994

If $$y = {\left( {\sin x} \right)^{\tan x}},$$ then $${{dy} \over {dx}}$$ is equal to
A
$${\left( {\sin x} \right)^{\tan x}}\left( {1 + {{\sec }^2}x\,\log \,\sin \,x} \right)$$
B
$$\tan x{\left( {\sin x} \right)^{\tan x - 1}}.\cos x$$
C
$${\left( {\sin x} \right)^{\tan x}}{\sec ^2}x\,\log \,\sin \,x$$
D
$$\tan x{\left( {\sin x} \right)^{\tan x - 1}}$$
4

### IIT-JEE 1990

Let $$f(x)$$ be a quadratic expression which is positive for all the real values of $$x$$. If $$g(x)=f(x)+f''(x)$$, then for any real $$x$$,
A
$$g(x)<0$$
B
$$g(x)>0$$
C
$$g(x)=0$$
D
$$g\left( x \right) \ge 0$$

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NEET

Class 12