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# Limits and Derivatives

#### Multiple Choice Questions

11.

equals

• e

• e2

• e3

• e5

C.

e3

12.

The differential equation which represents the family of curves y=c1ec2xe, where c1 and c2 are arbitrary constants, is

• y' =y2

•  y″ = y′ y

• yy″ = y′

• yy″ = y′

D.

yy″ = y′

y c1ec2x = …..(i)
y' = c1c2ec2x
y' = c2y.....(from (i) ....(ii)
y" = c2y' ....... (iii)
from (ii) & (iii)

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13.

The value of  is

• 10$\sqrt{2}$

• $\infty$

• $\infty$

• does not exist

D.

does not exist

Thus, limit does not exist.

14.

If y = , then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = $\frac{\mathrm{\pi }}{6}$ is

• $\frac{1}{2}$

• $\frac{1}{2}$

• 1

• - 1

A.

$\frac{1}{2}$

15.

The value of

• $\frac{3}{2}$

• 3

• - 3

• - 1

C.

- 3

16.

The value of the limit

• 0

• e

• $\frac{1}{\mathrm{e}}$

• 1

D.

1

17.

The value of  is

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{log}\left(2\right)$

• 0

• 1

A.

$\frac{\mathrm{\pi }}{4}$

18.

Let f(x) = , then the value of  is

• 0

• does not exist

• $\frac{1}{2}$

B.

does not exist

But  is not defined on left hand side of - 3.

Hence, function is not defined.

19.

The equation of the tangent to the curve y = x +4/x2, that is parallel to the x-axis, is

• y= 0

• y= 1

• y= 2

• y= 2

D.

y= 2

We have,

On differentiating w.r.t x, we get

since the tangent is parallel to X- axis, therefore
dy/dx = 0
⇒ x3 = 8

⇒ x = 2 abd y =3

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20.

If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y) + g(xy) - 2 for all real x and y and g(2) = 5, then $\underset{\mathrm{x}\to 3}{\mathrm{lim}}$ g(x) is

• 9

• 10

• 25

• 20

B.

10

Since, g(x) g(y) = g(x) + g(y) + g(xy) - 2

Now, at x = 0, y = 2, we get

g(0) g(2) = g(0) + g(2) + g(0) - 2

g(x) is given in a polynomial and by the relation given g(x) cannot be linear.

Let g(x) = x2 + k