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# Limits and Derivatives

#### Multiple Choice Questions

1.

The solution of the differential equation    satisfying the condition y (1) = 1 is

• y = ln x + x

• y = x ln x + x2

•  y = xe(x−1)

•  y = xe(x−1)

D.

y = xe(x−1)

y = vx

Since, y (1) = 1, we have y = x log x + x

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2.

If  then the values of a and b, are

D.

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3.

is equal to

• ${\mathrm{log}}_{\mathrm{e}}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)$

• ${\mathrm{log}}_{\mathrm{e}}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

• ${\mathrm{log}}_{\mathrm{e}}\left(\mathrm{ab}\right)$

A.

${\mathrm{log}}_{\mathrm{e}}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)$

# 4.If the normal to the curve y = f(x) at (3, 4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with the positive x-axis, then f'(3) is equal to :- 1 $\frac{3}{4}$ 1 - $\frac{3}{4}$

A.

- 1

5.

• ${\mathrm{log}}_{\mathrm{e}}\left(\frac{\mathrm{\pi }}{2}\right)$

• ${\mathrm{log}}_{\mathrm{e}}\left(2\right)$

• ${\mathrm{log}}_{\mathrm{e}}\left(\mathrm{a}\right)$

• a

C.

${\mathrm{log}}_{\mathrm{e}}\left(\mathrm{a}\right)$

6.

The value of  is

• $\frac{1}{2}$

• 1

• 2

• None of these

A.

$\frac{1}{2}$

7.

equals

• $\frac{10}{3}$

• $\frac{3}{10}$

• $\frac{6}{5}$

• $\frac{5}{6}$

A.

$\frac{10}{3}$

8.

$\frac{{d}^{\mathrm{n}}}{d{\mathrm{x}}^{\mathrm{n}}}\left(\mathrm{log}\left(\mathrm{x}\right)\right)$ is equal to

D.

Let y = log(x)

On differentiating w.r.t. x from 1 to n times, we get

9.

is equal to

• e12

• e- 12

• e4

• e3

B.

e- 12

10.

The value of  is

• $\mathrm{log}\left(2\right)$

• $\mathrm{log}\left(6\right)$

• 1

• $\mathrm{log}\left(3\right)$

B.

$\mathrm{log}\left(6\right)$

We know that,