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11th Business Maths Monthly Test - 1 ( Differential Calculus )

MABS Institution

11th Business Maths Monthly Test - 1 ( Differential Calculus )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

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

(0, \(\infty \))
(0, \(\infty \))
(-\(\infty \), \(\infty \))
(1, \(\infty \))

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

e
nx^(n-1)
1
0

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

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

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

an identity function
modulus function
exponential function
constant function

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

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

Which of the following function is neither even nor odd?

f(x) = x³ + 5
f(x) = x⁵
f(x) = x¹⁰
f(x) = x²

If y = log x then y₂ =

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

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

f(-a)
f\((\frac{1}{a})\)
2f(a)
f(a)

If f(x) = \(\frac{1-x}{1+x}\) then f(-x) is equal to

-f(x)
\(\frac{1}{f(x)}\)
\(\frac{-1}{f(x)}\)
f(x)

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

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

Part - B

Generic 2 - Answer All Question

Differentiate the following functions with respect to x, 7e^x

If f(x) = x and and g(x) = |x|, then find (f + g)(x)

Evaluate \(\underset { h\rightarrow 0 }{ lim } \frac { \sqrt { x+h } -\sqrt { x } }{ h } \)
If ax² + 5hxy + by² + 2gx + 2fy + c = 0 then find \(\frac{dy}{dx}\).

Find the second order derivative of the following functions with respect to x, x sin x

Find \(\frac{dy}{dx}\) if x = at², y = 2at

If f(x) = x and and g(x) = |x|, then find (f - g)(x)

Differentiate \({ x }^{ \frac { 2 }{ 3 } }\) from first principles

Differentiate the following functions with respect to x, x² sin x

If f(x) = \(\frac{x-1}{x+1}\), 0\(-\frac{1}{x}\)

Part - C

Generic 3 - Answer All Question

If \(f\left( x \right) =\begin{cases} \frac { x-\left| x \right| }{ x } \quad ifx\neq 0 \\ \quad 2\quad \quad ifx=0 \end{cases}\) , then show that \(\underset { x\rightarrow 0 }{ lim } f\left( x \right) \) does not exist.

If 4x+3y=log(4x-3y), then find \(\frac { dy }{ dx } \).

Evaluate the following \(\lim _{ x\rightarrow \infty }{ \frac { 2x+5 }{ { x }^{ 2 }+3x+9 } } \)

Draw the graph y = 9 - x²

If x = \(a\ \theta\) and \(y=\frac{a}{\theta}\), then prove that \(\frac{dy}{dx}+\frac{y}{x}=0\)
Differentiate the following with respect to x. \(\frac { { x }^{ 2 }+x+1 }{ { x }^{ 2 }-x+1 } \)

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

Evaluate the following \(\lim _{ x\rightarrow a }{ \frac { { x }^{ \frac { 5 }{ 8 } }-{ a }^{ \frac { 5 }{ 8 } } }{ { x }^{ \frac { 2 }{ 3 } }-a^{ \frac { 2 }{ 3 } } } } \)

Show thatf(x) =|x| is continuous at x = 0.

Evaluate the following \(\lim _{ x\rightarrow 2 }{ \frac { { x }^{ 3 }+2 }{ x+1 } } \)

Part - D

Generic 5 - Answer All Question

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

Draw the graph of the following function f(x)=|x-2|

if y = 2 sin x + 3 cos x, then show that y₂ + y = 0

A group of students wish to charter a bus for an educational tour which can accommodate atmost 50 students. The bus company requires at least 35 students for that trip. The bus company charges Rs 200 per student up to the strength of 45. For more than 45 students, company charges each student Rs 200 less \(\frac{1}{5}\) times the number more than 45. Consider the number of students who participates the tour as a function, find the total cost and its domain.

Examine the following functions for continuity at indicated points
\(f(x)=\begin{cases} \frac { { x }^{ 2 }-4 }{ x-2 } ,\quad if\quad x\neq 2 \\ \quad \quad \quad 0,\quad \quad if\quad x=2 \end{cases}at\quad x=2\)

Draw the graph of f(x) = a^x, \(a\ne 1\) and a > 0

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

If the function \(f\left( x \right) =\begin{cases} 6ax+3b\quad if\quad x>1 \\ ax-2b\quad if\quad x<1\quad is\quad continuous\quad at\quad x=1 \\ 15\quad if\quad x=1 \end{cases}\) Find the value of a and b.
Differentiate: \(\frac { sinx+cosx }{ sinx-cosx } \)with respect to 'x'

Examine the following functions for continuity at indicated points
\(f(x)=\begin{cases} \frac { { x }^{ 2 }-9 }{ x-3 } ,if\quad x\neq 3 \\ \quad \quad \quad \quad 6,\quad \quad \quad if\quad x=3 \end{cases}\) at x =3

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