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11th Maths Monthly Test - 1 ( Differential Calculus - Differentiability and Methods of Differentiat

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 1 ( Differential Calculus - Differentiability and Methods of Differentiat-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

It is given that f '(a) exists, then \(\underset{x\rightarrow a}{lim}{xf(a)-af(x)\over x-a}\) is

f(a)-af'(a)
f '(a)
-f '(a)
f(a)+af '(a)

If f(x)=x tan^-1 x, then f'(1) is

\(1+{\pi\over 4}\)
\({1\over 2}+{\pi\over 4}\)
\({1\over 2}-{\pi\over 4}\)
2

\({d\over dx}({2\over \pi}sin \ x^o)\)is

\({\pi\over 180}cos \ x^o\)
\({1\over 90} cos \ x^o\)
\({\pi\over 90}cos \ x^o\)
\({2\over \pi}cos \ x^o\)

Choose the correct or the most suitable answer from the given four alternatives.
\(If\quad y=\sin ^{ -1 }{ \left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) } then\quad \frac { dy }{ dx } is\)

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

Choose the correct or the most suitable answer from the given four alternatives.
\(If\quad y=\log { \left( \frac { 1-{ x }^{ 2 } }{ 1+{ x }^{ 2 } } \right) } then\quad \frac { dy }{ dx } \quad is\)

\(\frac { 4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)
\(-\frac { 4x }{ 1-{ x }^{ 4 } } \)
\(\frac { 1 }{ 4-{ x }^{ 4 } } \)
\(\frac { -4{ x }^{ 3 } }{ 1-{ x }^{ 4 } } \)

Choose the correct or the most suitable answer from the given four alternatives.
\(If\quad x=a(\theta +\sin { \theta ),y=a(1+\cos { \theta } ) } then\quad \frac { dy }{ dx } is\)

\(\tan { \frac { \theta }{ 2 } } \)
\(-\tan { \frac { \theta }{ 2 } } \)
\(\cot { \frac { \theta }{ 2 } } \)
\(-\cot { \frac { \theta }{ 2 } } \)

The differential coefficient of log₁₀ x with respect to log_x10 is

1
-(log₁₀x)²
(log_x10)²
\(x^2\over100\)

If y=cos (sin x²),then \({dy\over dx}\) at x= \(\sqrt{\pi\over 2}\) is

-2
2
\(-2\sqrt{\pi\over 2}\)
0

Assertion (A):f (x) =\(\begin{cases} \begin{matrix} x+1, & x<2 \end{matrix} \\ \begin{matrix} 2x-1, & x\ge 2 \end{matrix} \end{cases}\) then f'(2) does not exist.
Reason (R) :f(x) is not continuous at 2.

Both A and it are true and R is the correct explanation of A
Both A and R are true but R is not the correct explantion of A
A is true R is false
A is false R is true

The derivative of f(x)=x|x| at x =−3 is

6
-6
does not exist
0

Part - B

Generic 2 - Answer All Question

Differentiate the following with respect to x :\(y={log x \ x \over e^x}\)
Find \(\frac { dy }{ dx } if\quad x=a\log { t,\quad y=b\sin { t. } } \)

Find the slopes of the tangent lines to the graph of x²+y²=4 at the points corresponding to x = 1.

Differentiate x² (x + 1)³ (x + 2)⁴ with respect to' 'x'.

Differentiate the following: \(h(t)=(t-{1\over t})^{3\over2}\)

Differentiate \(\sqrt { { e }^{ \sqrt { x } } } ,x>0.\)

Find y''' if y=\({1\over x}\)

Find the derivatives of the following functions with respect to corresponding independent variables:f(x) = x - 3 sinx

Differentiate the following:\(y={3\sqrt{1+x^3}}\)

Find the derivation
(3x² + 1)²

Part - C

Generic 3 - Answer All Question

Determine whether the following function is differentiable at the indicated values.f(x) = x | x | at x = 0

Find the derivatives of the following \(x={1-t^2\over 1+t^2},y=\frac{2t^{2}}{1+t^{2}}\)

Find the derivative of tan^-1(1+x²) with respect to x²+x+1.

Show that the following functions are not differentiable at the indicated value of x.
f(x) = \(\begin{cases} 3x, & x<0 \\ -4x & x\ge 0 \end{cases};\) x=0

Determine whether the following function is differentiable at the indicated values. f(x) = |x| + |x - 1| at x = 0, 1
Differentiate the following:y=(2x-5)⁴(8x²-5)^-3

If f(x)=2x²+3x-5, then prove that f' (0) + 3 f' (-1) = 0

Find the derivatives of the following functions with respect to corresponding independent variables:\(y={ \ x\over sin \ x+ cos \ x}\)

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

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

Part - D

Generic 5 - Answer All Question

Differentiate \({ \left( \sin { x } \right) }^{ { \cos { ^{ -1x } } } }\) with respect to 'x'.

Differentiate the following: \(y=\sqrt{1+2 \ tan \ x}\)

If y = sin^-1x then find y''.

Differentiate the following: \(y=\sqrt{x+\sqrt{x}}\)

Find the derivatives of the following functions with respect to corresponding independent variables:y=(x²+5)log(1+x)e^-3x
Find the derivation
(3 see x - 4 cosec x) (2 sin x + 5 cos x)

Differentiate the following: y=e^xcosx

Examine the differentiability of functions in R by drawing the diagrams |sin x|

If \(\log { ({ x }^{ 2 }+{ y }^{ 2 }) } =2\tan ^{ -1 }{ \frac { y }{ x } , } \) Show that \(\frac { dy }{ dx } =\frac { x+y }{ x-y } .\)

Find the slope of the tangent line to the graph of f(x) = 7x + 5 at any point (x₀, f(x₀)).

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