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11th Business Maths Monthly Test - 2 ( Differential Calculus )

MABS Institution

11th Business Maths Monthly Test - 2 ( Differential Calculus )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

f(x) = - 5 , for all \(x\epsilon R\), is a

an identity function
modulus function
exponential function
constant function

\(\lim _{ x\rightarrow \infty }{ \frac { \tan { \theta } }{ \theta } } =\)

1
\(\infty\)
\(-\infty\)
\(\theta\)

The graph of y = e^x intersect the y-axis at

(0,0)
(1,0)
(0,1)
(1,1)

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

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

For \(f\left( x \right) =\begin{cases} { x }^{ 2 }-4x\quad ifx\ge 2 \\ x+2\quad ifx<2 \end{cases}\) then f(0) is

2
5
-1
0

\(\lim _{ x\rightarrow 0 }{ \frac { { e }^{ x }-1 }{ x } } =\)

e
nx^(n-1)
1
0

If y = log x then y₂ =

\(\frac{1}{x}\)
\(-\frac{1}{x^2}\)
\(-\frac{2}{x^2}\)
e²

The minimum value of the function f(x)= Ixl is

0
-1
+1
\(-\infty \)

\(\frac{d}{dx}(a^x)=\)

\(\frac { 1 }{ x\log { \begin{matrix} a \\ e \end{matrix} } } \)
a^a
\(x\log { \begin{matrix} a \\ e \end{matrix} } \)
\({ a }^{ x }\log { \begin{matrix} a \\ e \end{matrix} } \)

\(\frac{d}{dx}(\frac{1}{x})\) is equal to

\(-\frac{1}{x^2}\)
\(-\frac{1}{x}\)
log x
\(\frac{1}{x^2}\)

Part - B

Generic 2 - Answer All Question

If f(x) = log \(\frac{1+x}{1-x}\), show that \(f(\frac{2x}{1+x^2})=2f(x)\)

Differentiate the following functions with respect to x, \(x^{\frac{3}{2}}\)

Evaluate \(\underset { x\rightarrow \frac { 1 }{ 2 } }{ lim } \frac { { 4x }^{ 2 }-1 }{ 2x-1 } \)

Find the domain for which the functions f(x) = 2x² - 1 and g(x) = 1 - 3x are equal.

Evaluate: \(\underset { x\rightarrow a }{ lim } \frac { { x }^{ \frac { 3 }{ 5 } }-{ a }^{ \frac { 3 }{ 5 } } }{ { x }^{ \frac { 1 }{ 5 } }-{ a }^{ \frac { 1 }{ 5 } } } \)

f = {(1, 1),(2, 3)} be a function described by the formula f(x) = ax + b. Determine a and b?

If \(f(x)={ x }^{ 3 }-\frac { 1 }{ { x }^{ 3 } } \) then show that \(f(x)+f\left( \frac { 1 }{ x } \right) =0\)

Part - C

Generic 3 - Answer All Question

Differentiate the following with respect to x.
\(\frac { 5 }{ { x }^{ 4 } } -\frac { 2 }{ { x }^{ 3 } } +\frac { 5 }{ x } \)

Differentiate the following with respect to x. \(\frac { { e }^{ x } }{ 1+x } \)

If \(y=\sqrt { x } +\frac { 1 }{ \sqrt { x } } \) show that \(2x\frac { dy }{ dx } +y=2\sqrt { x } \).

Find \(\frac{d}{dx}(e^{3x})\) from first principle.

Evaluate \(\underset { x\rightarrow 0 }{ lim } \frac { 2sinx-sin2x }{ { x }^{ 3 } } \)

Differentiate the following with respect to x. \(\sqrt { x } +\frac { 1 }{ \sqrt [ 3 ]{ x } } +{ e }^{ x }\)

Find y₂ of the following functions y = e^3x+2

Part - D

Generic 5 - Answer All Question

Draw the graph of the following function f(x) =x|x|

Show that f(x) = x² differentiable at x = 1 and find f'(1).

Evaluate \(\begin{matrix} \underset { x\rightarrow 1 }{ lim } & \frac { { x }^{ 7 }-2{ x }^{ 5 }+1 }{ { x }^{ 3 }-{ 3x }^{ 2 }+2 } \end{matrix}\)

if y = 2 sin x + 3 cos x, then show that y₂ + y = 0
Draw the graph of the following function f(x)=e^-2x

Verify the continuity and differentiability of \(\\ f\left( x \right) =\begin{cases} 1-x\quad \quad \quad \quad \quad if\ x<1 \\ \left( 1-x \right) \left( 2-x \right) \quad if\quad 1\le x\le 2\ at\quad x-1\quad and\quad x=2 \\ 3-x\quad \quad \quad \quad \quad if\quad x>2 \end{cases}\)

Draw the graph of the following function \(f(x)=\frac { |x| }{ x } \)

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