#### Differential Calculus Model Questions

11th Standard

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Time : 01:00:00 Hrs
Total Marks : 50
5 x 1 = 5
1. The graph of y = 2x2 is passing through

(a)

(0,0)

(b)

(2,1)

(c)

(2,0)

(d)

(0,2)

2. 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)

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

3. The range of f(x) = |x|, for all $x\epsilon R$, is

(a)

(0, $\infty$)

(b)

(0, $\infty$)

(c)

(-$\infty$$\infty$)

(d)

(1, $\infty$)

4. A function f(x) is continuous at x = a if $\lim _{ x\rightarrow a }{ f\left( x \right) }$ is equal to

(a)

f(-a)

(b)

f$(\frac{1}{a})$

(c)

2f(a)

(d)

f(a)

5. If y = log x then y2 =

(a)

$\frac{1}{x}$

(b)

$-\frac{1}{x^2}$

(c)

$-\frac{2}{x^2}$

(d)

e2

6. 5 x 2 = 10
7. If $f(x)=\frac { x+1 }{ x-1 }$ ,then prove that f(f(x))=x

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

9. Find the derivative of the following functions from first principles log (x+1)

10. Differentiate ${ x }^{ \frac { 2 }{ 3 } }$ from first principles

11. Find $\frac { dy }{ dx }$ if x2 + xy + y2 = 100

12. 5 x 3 = 15
13. Find  $\frac{dy}{dx}$   of the following functions: x = a (θ - sin θ),y = a(1- cosθ)

14. Differentiate sin3 x with respect to cos3x.

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

16. Evaluate $\underset { x\rightarrow -3 }{ lim } \frac { { x }^{ 3 }+27 }{ { x }^{ 5 }+243 }$

17. Evaluate $\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } }$

18. 4 x 5 = 20
19. Draw the graph of the following function f(x)=e-2x

20. Differentiate: $\frac { sinx+cosx }{ sinx-cosx }$with respect to 'x'

21. Find the derivate of (x3-27)from first principles.

22. Darw the graph of f(x) = logax; x > 0, a > 0 and $a\ne 1$.