Tag: maths

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Questions Related to maths

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

 Round off $690$ to nearest $100$.

  1. $780$
  2. $690$
  3. $700$
  4. $800$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $700.$

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

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

  1. $1290,1389,1009$
  2. $1290,1300,1000$
  3. $1280, 1300, 1200$
  4. $1280,1200,1000$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $1290,\,1300,\,1000.$

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

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

  1. $10$
  2. $20$
  3. $13$
  4. $1$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $10$.

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

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 __________.

  1. $300$
  2. $400$
  3. $350$
  4. $480$
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

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

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

  1. $\dfrac{-1}{2}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{8}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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}$

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

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,

  1. $a=2,b=1$
  2. $a=-2,b=1$
  3. $a=2,b=-1$
  4. $a=-2,b=-1$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$\displaystyle y=\frac{1+x^{2}+x^{4}}{1+x+x^{2}}$

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

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

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

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

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

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

$\Rightarrow a=2$ & $b=-1$
Multiple choice maths differencial calculus - differenciability and methods of differnciation differentiation by substitution methods of differentiation derivative of a function

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

  1. $\cos x$
  2. $\tan x$
  3. $\text{cosec} \,x$
  4. $\cot x$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.