Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Continuity and Differentiability

Multiple Choice Questions

671.

Let y = $(\frac{3^{x} - 1}{3^{x} + 1}) \sin (x) + \log_{e} (1 + x), x > - 1$ . Then, at x = 0, $\frac{dy}{dx}$ equals

672.

If f is a real-valued differentiable function such that f(x)f' (x) < 0 for all real x, then

f(x) must be an increasing function
f(x) must be a decreasing function
$|f (x)|$ must be an increasing function
$|f (x)|$ must be a decreasing function

673.

Rolle's theorem is applicable in the interval [- 2, 2] for the function

f(x) = x³
f(x) = 4x⁴
f(x) = 2x³ + 3
f(x) = $π |x|$

674.
The solution of $25 \frac{d^{2} y}{d x^{2}} - 10 \frac{d y}{d x} + y = 0$ , y(0) = 1, y(1) = $2 e^{\frac{1}{5}}$ is

$y = e^{5 x} + e^{- 5 x}$

$y = (1 + x) e^{5 x}$

$y = (1 + x) e^{\frac{x}{5}}$

$y = (1 + x) e^{\frac{- x}{5}}$

$y = (1 + x) e^{\frac{x}{5}}$

Let y = e^mx be the solution of given differential equation,

$\Rightarrow \frac{d y}{d x} = {me}^{mx} \Rightarrow \frac{d^{2} y}{{dx}^{2}} = m^{2} e^{mx} ∴ 25 \frac{d^{2} y}{{dx}^{2}} - 10 \frac{d y}{d x} + y = 0 25 m^{2} e^{mx} - 10 {me}^{mx} + e^{mx} = 0 \Rightarrow e^{mx} (25 m^{2} - 10 m + 1) = 0 \Rightarrow Auxiliary equation \Rightarrow 25 m^{2} - 10 m + 1 = 0 e^{mx} \neq 0 \Rightarrow {(5 m)}^{2} - 2 (5 m) \times 1 + 1 = 0 \Rightarrow {(5 m - 1)}^{2} = 0 \Rightarrow m = \frac{1}{5}, \frac{1}{5}$

Since, roots are real and euqal.

$∴ General solution is y = (c_{1} + c_{2} x) e^{\frac{x}{5}} . . . (i) y (0) = 1 \Rightarrow c_{1} = 1 y (1) = 2 e^{\frac{1}{5}} \Rightarrow 2 e^{\frac{e}{5}} = (c_{1} + c_{2}) e^{\frac{1}{5}} \Rightarrow c_{1} + c_{2} = 2 \Rightarrow c_{1} = 1$