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11th Maths Weekly Test - 3 ( Differential Calculus - Differentiability and Methods of Differentiatio

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test - 3 ( Differential Calculus - Differentiability and Methods of Differentiatio-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

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

1
\(\pi \)
\(\frac { \pi }{ 2 } \)
0

\({d\over dx}(e^{x+5log \ x})\)is

e^x.x⁴(x+5)
e^x.x(x+5)
e^x\(+{5\over x}\)
e^x\(-{5\over x}\)

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

2
-2
1
-1

If f(x)=x²-3x, then the points at which f(x)=f'(x) are

both positive integers
both negative integers
both irrational
one rational and another irrational

If y=mx+c and f(0)=\(f'(0)=1\),then f(2) is

1
2
3
-3

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

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

If the derivative of (ax - 5)e^3x at x=0 is -13, then the value of a is

8
-2
5
2

If \(y={1\over a-z}\) ,then \({dz\over dy}\) is

\((a-z)^2\)
-(z-a)²
(z+a)²
-(z+a)²

If f(x)=x+2, then f'(f(x)) at x= 4 is

8
1
4
5

Choose the correct statement

Derivative of odd functionis odd
Derivative of even function is even
Inverse of odd function is even
Inverse function of sin x is sin^-1 x

Choose the correct or the most suitable answer from the given four alternatives.
If \(y=\log _{ a }{ x } \) then \(\frac { dy }{ dx } \) is

\(\frac { 1 }{ x } \)
\(\frac { 1 }{ x\log _{ e }{ a } } \)
\(\log { _{ e }^{ a } } \)
\(\frac { 1 }{ \log _{ a }{ x } } \)

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

\(\frac { \sin { x } }{ 2y-1 } \)
\(\frac { \sin { x } }{ 1-2y } \)
\(\frac { \cos { x } }{ 1-2y } \)
\(\frac { \cos { x } }{ 2y-1 } \)

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)

The differential coefficient of log₁₀ x with respect to log_x10 is

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

If \(y={(1-x)^2\over x^2}\) ,then \({dy \over dx}\) is

8
1
4
5

Match List - I with List II

List I List II

i \(\begin{bmatrix} a & b \\ b & a \end{bmatrix}\) a Identity

ii \(\begin{bmatrix} 0 & b \\- b & 0 \end{bmatrix}\) b Singular matrix

iii \(\begin{bmatrix} a & a \\ b & b \end{bmatrix}\) c Skew-Symmetric

iv \(\begin{bmatrix} 1 &0 \\0 & 1 \end{bmatrix}\) d Symmetric

The Correct match is

i ii iii iv

d c b a
i ii iii iv

c d b a
i ii iii iv

b a d c
i ii iii iv

b d a c

Choose the correct or the most suitable answer from the given four alternatives.
If f(x) is an odd function, then f^'(x) is an ______ function

even
odd
even or odd
neither even nor odd

Choose the correct or the most suitable answer from the given four alternatives.
If f(x) is an even functions, thenf^'(x) is an ______function

even
odd
even and odd
even or odd

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

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

If y=\({1\over4}u^4,u={2\over 3}x^3+5,\) then \({dy\over dx}\) is

\({1\over 27}x^2(2 x^3+15)^3\)
\({2\over 27}x(2 x^3+5)^3\)
\({2\over 27}x^2(2 x^3+15)^3\)
\(-{2\over 27}x(2 x^3+5)^3\)

If x=a sin \(\theta\) and y= b cos \(\theta\),then \({d^2y\over dx^2}\)is

\({a \over b^2}sec^2 \theta\)
\(-{b \over a}sec^2 \theta\)
\(-{b \over a^2}sec^3 \theta\)
\(-{b^2\over a^2}sec^3 \theta\)

If pv=81,then \({dp\over dv}\) at v=9 is

1
-1
2
-2

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

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

Choose the correct or the most suitable answer from the given four alternatives.
For the curve \(\sqrt { x } +\sqrt { y } =1,\quad \frac { dy }{ dx } at\left( \frac { 1 }{ 4 } ,\frac { 1 }{ 4 } \right) is\)

\(\frac { 1 }{ 2 } \)
1
-1
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.
\(Iff\left( x \right) =4{ x }^{ 8 },\quad then\)

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

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

6
-6
does not exist
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

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

\(x={1-t^2\over 1+t^2},y={2t\over 1+t^2}\) then \({dy\over dx}\)is

\(-{y\over x}\)
\({y\over x}\)
\(-{x\over y}\)
\({x\over y}\)

Choose the correct or the most suitable answer from the given four alternatives.
The derivative of \(\cos ^{ -1 }{ (2{ x }^{ 2 } } -1)\) with respect to \(\cos ^{ -1 }{ x } \) is

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

If g(x)=(x²+2x+3) f(x) and f(0)=5 and \(\underset{x \rightarrow 0}{lim}{f(x)-5\over x}=4\),then g'(0) is

20
14
18
12

The number of points in R in which the function \(f(x)=|x-1|+|x-3|+sin \ x\) is not differentiable, is

3
2
1
4

Choose the incorrect pair

1. 330^o \(\frac{11\pi}{6}\)radians

2. \(\frac{7\pi^c}{3}\) 200^o

3. 0^o 0^o

4. \(2\pi^c\) 360^o

330^o - \(\frac{11\pi}{6}\)radians
\(\frac{7\pi^c}{3}\) - 200^o
0^o - 0^c
2\(\pi\)^c - 360^o

If y= f(x²+2) and f'(3)=5,then \({dy\over dx}\) at x=1 is

5
25
15
10

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