Tag: maths

Questions Related to maths

 Round off $690$ to nearest $100$.

maths mental arithmetic estimation and rounding off estimations, bounds and rounding off rounding off numbers rounding of decimals

  1. $780$

  2. $690$

  3. $700$

  4. $800$

Show answer
Correct Option: C
Explanation:

$\Rightarrow$  The given number is $690$

$\Rightarrow$  Its tens digit are $9$, which is greater than $5$. 
$\Rightarrow$  So, we increase the digits at hundreds place by $1$ and replace each one of the digits at tens and ones place by $0$ to round off the given number to nearest $100$.
$\therefore$   Round off $690$ to nearest $100$ is $700.$

Round off to the nearest $10, 100,$ and $1000$:
$1289$

maths mental arithmetic estimation and rounding off estimations, bounds and rounding off rounding off numbers rounding of decimals

  1. $1290,1389,1009$

  2. $1290,1300,1000$

  3. $1280, 1300, 1200$

  4. $1280,1200,1000$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  Round off to 10 for 1289 is 1280 or 1290 but nearest to 1289 is 1290.

$\Rightarrow$  So, round off to the nearest $10$ is $1290$.
$\Rightarrow$  Round off to 100 for 1289 is 1200 or 1300 but nearest to 1289 is 1300.
$\Rightarrow$  So, round off to the nearest $100$ is $1300$.

$\Rightarrow$  Round off to 1000 for 1289 is 1000 or 2000 but nearest to 1289 is 1000.
$\Rightarrow$  So, round off to the nearest $1000$ is $1000$.

$\therefore$  The answer is $1290,\,1300,\,1000.$

Round-off $13$ to nearest $10$.

maths mental arithmetic estimation and rounding off estimations, bounds and rounding off rounding off numbers rounding of decimals

  1. $10$

  2. $20$

  3. $13$

  4. $1$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  The given number is $13$

$\Rightarrow$   Its unit digit is $3$ which is less than 5.
$\Rightarrow$   So, we replace unit digit by $0$ keep the other digits as they are to round off the given number to nearest $10$
$\therefore$   Round-off 13 to nearest 10 is $10$.

$(7268 - 2427)$ estimated to the nearest hundred is

maths mental arithmetic estimation and rounding off estimations, bounds and rounding off rounding off numbers rounding of decimals

  1. $4800$

  2. $4900$

  3. $4841$

  4. $5000$

Show answer
Correct Option: A
Explanation:

The given value is $7268-2427=4841$

the tens value is $4$ which is less than $5$, so
the estimate value is $4800$.

Rohit's teacher asked him to tell the correct answer after performing the steps given.
$\bullet$ Round off $43, 98$ and $250$ to nearest tens.
$\bullet$ After rounding off, add them.
$\bullet$ Round off the sum to the nearest hundreds.
Then, the correct answer is __________.

maths mental arithmetic estimation and rounding off estimations, bounds and rounding off rounding off numbers rounding of decimals

  1. $300$

  2. $400$

  3. $350$

  4. $480$

Show answer
Correct Option: B
Explanation:

Rounding off to nearest tens:

If unit digit is $0,1,2,3,4$ then round down to nearest tens.
If unit digit is $5,6,7,8,9$ then, round up to nearest tens.
$43$ $\rightarrow$ $40$
$98$ $\rightarrow$ $100$
$250$ $\rightarrow$ $250$
$40+100+250$ $\rightarrow$ $390$
$390$  $\rightarrow$  $400$
Hence, Option B is correct.

Differentiate $\tan^{-1} \sqrt{\dfrac{1+\cos x}{1- \cos x}}$

maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

  1. $\dfrac{-1}{2}$

  2. $\dfrac{1}{4}$

  3. $\dfrac{1}{8}$

  4. None of these

Show answer
Correct Option: A
Explanation:

Let $y = {\tan ^{ - 1}}\sqrt {\dfrac{{1 + \cos x}}{{1 - \cos x}}} $

$ = {\tan ^{ - 1}}\sqrt {\dfrac{{2{{\cos }^2}\dfrac{x}{2}}}{{2{{\sin }^2}\dfrac{x}{2}}}} $
$ = {\tan ^{ - 1}}\cot \left( {\dfrac{x}{2}} \right)$
$ = {\tan ^{ - 1}}\left[ {\tan \left( {\dfrac{\pi }{2} - \dfrac{x}{2}} \right)} \right]$
$ = \dfrac{\pi }{2} - \dfrac{x}{2}$
$ \Rightarrow \dfrac{{dy}}{{dx}} =  - \dfrac{1}{2}$

If $y=\dfrac{1+x^2+x^4}{1+x+x^2}$ and $\dfrac{dy}{dx}=ax+b$, then the values of $a$ and $b$ are,

maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

  1. $a=2,b=1$

  2. $a=-2,b=1$

  3. $a=2,b=-1$

  4. $a=-2,b=-1$

Show answer
Correct Option: C
Explanation:
$\displaystyle y=\frac{1+x^{2}+x^{4}}{1+x+x^{2}}$

$\displaystyle \frac{dy}{dx}= \frac{(2x+4x^{3})(1+x+x^{2})-(1+2x)(1+x^{2}+x^{4})}{(1+x^{2}+x^{4}+2x+2x^{2}+2x^{3})}$

$\displaystyle \frac{dy}{dx}=\frac{2x+2x^{2}+2x^{3}+4x^{3}+4x^{4}+4x^{5}-1-x^{2}-x^{4}-2x-2x^{3}-2x^{5}}{(x^{4}+2x^{3}+3x^{2}+2x+1)}$

$\displaystyle \frac{dy}{dx}=\frac{2x^{5}+3x^{4}+4x^{3}+x^{2}-1}{(x^{4}+2x^{3}+3x^{2}+2x+1)}$

$\displaystyle (\frac{dy}{dx})=\frac{(2x^{5}+4x^{4}+6x^{3}+4x^{2}+2x)-(x^{4}+2x^{3}+3x^{2}+2x+1)}{(x^{4}+2x^{3}+3x^{2}+2x+1)}$

$\displaystyle (\frac{dy}{dx})=\frac{2x[x^{4}+2x^{3}+3x^{2}+2x+1]}{(x^{4}+2x^{3}+3x^{2}+2x+1)}-1$

$\displaystyle (\frac{dy}{dx})=2x-1= ax+b$

$\Rightarrow a=2$ & $b=-1$

If $x=1(\theta+sin\,\theta), y=a(1-cos \theta)$, then at $\theta=\dfrac{\pi}{2},y'=\dfrac{2}{a}$.

maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

  1. True

  2. False

Show answer
Correct Option: B

If $f\left( x \right) =|x-2|,g\left( x \right) =f\left( f\left( x \right) \right) $, then for $x>4$, $g'(x)=$

maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

  1. 0

  2. 1

  3. -1

  4. 2

Show answer
Correct Option: A

Differential coefficient of $\log\ \sin x$ is :

maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

  1. $\cos x$

  2. $\tan x$

  3. $\text{cosec} \,x$

  4. $\cot x$

Show answer
Correct Option: D
Explanation:

We have,

$y=\log \sin x$

On differentiating w.r.t $x$, we get
$\dfrac{dy}{dx}=\dfrac{d(\log \sin x)}{dx}$
$\dfrac{dy}{dx}=\dfrac{1}{\sin x}\times \cos x$
$\dfrac{dy}{dx}=\dfrac{\cos x}{\sin x}$
$\dfrac{dy}{dx}=\cot x$

Hence, this is the answer.