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11th Maths Weekly Test - 3 ( Differential Calculus - Limits and Continuity )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test - 3 ( Differential Calculus - Limits and Continuity )-Aug 2020

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

The value of \(\underset{x \rightarrow 0}{lim}{sin x\over \sqrt{x^2}}\) is

-1
1
0
limit does not exist

Choose the incorrect statement

log 1 to any base is zero
\(\frac{d}{dx}\) (e^x)=e^x
Inverse function of log x is \(\frac{1}{x}\)
|x| is not differentiable at x =0

\(\underset{x\rightarrow\infty}{lim}{sin \ x \over x} \)

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

\(\underset{\theta\rightarrow0}{lim}{Sin\sqrt{\theta}\over \sqrt{sin \theta}} \)

1
-1
0
2

If f(x)=x(-1)^{\(\left\lfloor 1\over x \right\rfloor \)},\(x\le0\),then the value of \(\underset{x\rightarrow 0}{lim}f(x)\) is equal to

-1
0
2
4

\(\underset{x \rightarrow 0}{lim}{e^{sin \ x}-1\over x}=\)

1
e
\({1\over e}\)
0

Let f be a continuous function on [2, 5]. If f takes only rational values for all x and f(3) =12, then (4.5) f is equal to

\({f(3)+f(4.5)\over 7.5}\)
12
17.5
\(f(4.5)-f(3)\over 1.5\)

Find the odd one out of the following

sin x=2
e^x=-2
tan x=7
|x|=1

At x\(={3\over 2}\) the function \(f(x)={|2x-3|\over 2x-3}\) is

continuous
discontinuous
differentiable
non-zero

For what values of x is the rate of increase of x³ - 2x² + 3x + 8 is twice the rate of increase of x?

\(\left( -\frac { 1 }{ 3 } ,-3 \right) \)
\(\left( \frac { 1 }{ 3 } ,3 \right) \)
\(\left( -\frac { 1 }{ 3 } ,3 \right) \)
\(\left( \frac { 1 }{ 3 } ,1 \right) \)

\(\underset{ x\rightarrow \infty }{lim}{ \left( \frac { 1 }{ x } +2 \right) } \)is equal to

\(\infty \)
0
1
2

If y= 6x -x³ and x increases at the ratio of 5 units per second, the rate of change of slope when x = 3 is ______ units/sec.

-90
90
180
-180

\(\underset{ x\rightarrow 1 }{lim}{ \frac { { x }^{ m }-1 }{ { x }^{ n }-1 } } is\)

mn
m+n
m-n
\(\frac { m }{ n } \)

Choose the incorrect pair

\(\underset { x\rightarrow \infty }{ lim } \frac { sin\quad x }{ x } =0\)
\(\underset { x\rightarrow \frac { \pi }{ 2 } }{ lim } \frac { 2x-\pi }{ cos\quad x } =-2\)
\(\underset { x\rightarrow \infty }{ lim } \frac { { a }^{ x }-{ b }^{ x } }{ x } =log\left( \frac { a }{ b } \right) \)
\(\frac { { a }^{ x }-1 }{ x } =log\ a>0\)

\(\underset{x \rightarrow 3}{lim}\left\lfloor x \right\rfloor =\)

2
3
does not exist
0

Find the odd one out of the following

|x|
sin x
cos x
\(\frac{1}{x}\)

\(\underset{x\rightarrow 0}{lim}{xe^x-sin \ x\over x}\) is

1
2
3
0

The function \(f\left( x \right) =\tan { x } \) is discontinuous on the set

\(\left\{ n\pi :\quad n\in z \right\} \)
\(\left\{ 2n\pi :\quad n\in z \right\} \)
\(\left\{ (2n+1)\frac { \pi }{ 2 } ,\quad n\in z \right\} \)
\(\left\{ n\frac { \pi }{ 2 } ,\quad n\in z \right\} \)

\(\underset{\alpha \rightarrow {\pi/4}}{lim}{sin \alpha -cos \alpha \over \alpha -{\pi\over 4}}\) is

\(\sqrt{2}\)
\(1\over \sqrt{2}\)
1
2

\(\underset{n \rightarrow \infty}{lim}({1\over n^2}+{2\over n^2}+{3\over n^2}+..+{n\over n^2})\) is

\(1\over 2\)
0
1
\(\infty\)

\(\underset{x \rightarrow 0}{lim}{e^{tan \ x}-e^x\over tan x-x}=\)

1
e
\({1\over2}\)
0

\(\underset{x\rightarrow0}{lim}{\sqrt{1-cos 2x}\over x} \)

0
1
\(\sqrt{2}\)
does not exist

The rate of change of area A of a circle of radius r is

\(2\pi r\)
\(2\pi r\frac { dr }{ dt } \)
\(\pi { r }^{ 2 }\frac { dr }{ dt } \)
\(\pi \frac { dr }{ dt } \)

The slope of the graph of \(f\left( x \right) =\frac { \left| x \right| }{ x } ,x>0\quad is\)

1
0
-1
undefined

\(\underset{ x\rightarrow 2 }{lim}{ \frac { 2{ x }^{ 2 }+x+1 }{ x+2 } } \) is equal to

\(\frac { 1 }{ 2 } \)
2
\(\frac { 11 }{ 4 } \)
0

\(\lim _{ x\rightarrow \infty }{ \frac { 1+2+3+....+n }{ { 2n }^{ 2 }+6 } } \)

2
6
\(\frac { 1 }{ 4 } \)
\(\frac { 1 }{ 2 } \)

Find the odd one of the following

x²
x⁴
cos x
sin x

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

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

The value of \(\underset{x\rightarrow k^-}{lim}x-\left\lfloor x \right\rfloor \) , where k is an integer is

-1
1
0
2

Find the odd one of out of the following

tan x
\(\frac{1}{x}\)
\(\\ \\ \\ \\ \\ \\ \\ \frac { { x }^{ 2 }+5x+4 }{ { x }^{ 2 }+4x+4 } \)
cos x

\(\underset{x\rightarrow o}{lim}{8^x-4^x-2^x+1^x\over x^2}=\)

2 log 2
2( log2)²
log 2
3 log 2

Let a function f be defined by \(f(x)={x-|x|\over x}\) for x \(\neq\)0 and f(0)=2. Then f is

continuous nowhere
continuous everywhere
continuous for all x except x = 1
continuous for all x except x = 0

If f : \(R \rightarrow R\) is defined by f(x)=\(\left\lfloor x-3 \right\rfloor +|x-4|\) for \(x \in R\), then \(\underset{x\rightarrow 3^-}{lim}f(x)\) is equal to

-2
-1
0
1

The slop of the curve \(y={ x }^{ 3 }-7{ x }^{ 2 }+5x+10\quad \) at x=1 is

6
-6
5
15

\(\underset{x \rightarrow \infty}{lim}{\sqrt{x^2-1}\over 2x-1}=\)

1
0
-1
\(1\over 2\)

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