JEE Advanced 2016 Paper 1 Offline | Differential Equations Question 10 | Mathematics

JEE Advanced 2016 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Let $$f:(0,\infty ) \to R$$ be a differentiable function such that $$f'(x) = 2 - {{f(x)} \over x}$$ for all $$x \in (0,\infty )$$ and $$f(1) \ne 1$$. Then

$$\mathop {\lim }\limits_{x \to {0^ + }} f'\left( {{1 \over x}} \right) = 1$$

$$\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = 2$$

$$\mathop {\lim }\limits_{x \to {0^ + }} {x^2}f'(x) = 0$$

$$\left| {f(x)} \right| \le 2$$ for all $$x \in (0,2)$$

JEE Advanced 2015 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Let $$y(x)$$ be a solution of the differential equation
$$\left( {1 + {e^x}} \right)y' + y{e^x} = 1.$$
If $$y(0)=2$$, then which of the following statement is (are) true?

$$y(-4)=0$$

$$y(-2)=0$$

$$y(x)$$ has a critical point in the interval $$(-1, 0)$$

$$y(x)$$ has no critical point in the interval $$(-1,0)$$

JEE Advanced 2015 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Consider the family of all circles whose centres lie on the straight line $$y=x,$$ If this family of circle is represented by the differential equation $$Py'' + Qy' + 1 = 0,$$ where $$P, Q$$ are functions of $$x,y$$ and $$y'$$ $$\left( {here\,\,\,y' = {{dy} \over {dx}},y'' = {{{d^2}y} \over {d{x^2}}}} \right)$$ then which of the following statements is (are) true?

$$P = y + x$$

$$\,P = y - x$$

$$\,P + Q = 1 - x + y + y' + {\left( {y'} \right)^2}$$

$$\,P - Q = 1 - x + y - y' - {\left( {y'} \right)^2}$$

IIT-JEE 2012 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

If $$y(x)$$ satisfies the differential equation $$y' - y\,tan\,x = 2x\,secx$$ and $$y(0)=0,$$ then

$$y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {8\sqrt 2 }}$$

$$y'\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {18}}$$

$$y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over 9}$$

$$y'\left( {{\pi \over 3}} \right) = {{4\pi } \over 3} + {{2{\pi ^2}} \over {3\sqrt 3 }}$$

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