#### Differential Calculus Model Question Paper

11th Standard

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Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. Let $f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}$, then f(5) is

(a)

-1

(b)

2

(c)

5

(d)

7

2. The graph of y = ex intersect the y-axis at

(a)

(0,0)

(b)

(1,0)

(c)

(0,1)

(d)

(1,1)

3. Which one of the following functions has the property f (x) = $f\left( \frac { 1 }{ x } \right)$

(a)

$f\left( x \right) =\frac { { x }^{ 2 }-1 }{ x }$

(b)

$f\left( x \right) =\frac { 1-{ x }^{ 2 } }{ x }$

(c)

f(x) = x

(d)

$f\left( x \right) =\frac { { x }^{ 2 }+1 }{ x }$

4. The graph of f(x) = ex is identical to that of

(a)

f(x) = ax, a > 1

(b)

f(x) = ax, a < 1

(c)

f(x) = ax, 0 < a < 1

(d)

y = ax +b, a $\ne$ 0

5. $\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =$

(a)

e

(b)

nx(n-1)

(c)

1

(d)

0

6. 5 x 2 = 10
7. Determine whether the following functions are odd or even?
f(x)=x+x2

8. Let f be defined by f(x) =x3-kx2+2x,x ∈ R,Find k,if 'f' is an odd function.

9. For $f(x)=\frac { x-1 }{ 3x+1 }$ ,write the expression of $f\left( \frac { 1 }{ x } \right)$ and $\frac { 1 }{ f(x) }$

10. Evaluate $\underset { h\rightarrow 0 }{ lim } \frac { \sqrt { x+h } -\sqrt { x } }{ h }$

11. Differentiate (sec x -1) (sec x +1)

12. 5 x 3 = 15
13. Differentiate sin2 x with respect to x2.

14. Evaluate the following $\lim _{ x\rightarrow \infty }{ \frac { 2x+5 }{ { x }^{ 2 }+3x+9 } }$

15. Find y2 of the following function x = a cos $\theta$, y = a sin $\theta$

16. Find $\frac{dy}{dx}$ if x = 15(t - sin t); y = 18(1 - cos t).

17. If ey (x + 1) = 1, show that $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } ={ \left( \frac { dy }{ dx } \right) }^{ 2 }$

18. 4 x 5 = 20
19. Draw the graph of the following function f(x)=e2x

20. Draw the graph of the following function f(x)=e-2x

21. If $\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 4 }-1 }{ x-1 } \end{matrix}=\begin{matrix} \underset { x\rightarrow k }{ lim } & \frac { { x }^{ 3 }-{ k }^{ 3 } }{ { x }^{ 2 }-{ k }^{ 2 } } \end{matrix}$,then find the value of K

22. If $x=a\left( t+\frac { 1 }{ t } \right) ;y=a\left( t-\frac { 1 }{ t } \right)$ show that $\frac { dy }{ dx } =\frac { x }{ y }$