Tag: maths

Questions Related to maths

If $3sin\alpha =5sin\beta ,\quad then\quad \frac { \tan { \frac { \alpha +\beta }{ 2 } } }{ \tan { \frac { \alpha -\beta }{ 2 } } } $ is equal to

trigonometric equations trigonometric functions trigonometry maths

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: A
Explanation:

Given $3\sin \alpha=5\sin \beta\implies \dfrac{\sin \alpha}{\sin \beta}=\dfrac{5}{3}$


Applying componendo and dividendo rule


$\implies \dfrac{\sin \alpha+\sin \beta}{\sin \alpha-\sin \beta}=\dfrac{5+3}{5-2}$

$\implies \dfrac{2\sin \dfrac{\alpha+\beta}{2}\cos \dfrac{\alpha-\beta}{2}}{2\sin \dfrac{\alpha-\beta}{2}\cos \dfrac{\alpha+\beta}{2}}=\dfrac{8}{2}$

$\implies \dfrac{\tan \dfrac{\alpha+\beta}{2}}{\tan \dfrac{\alpha-\beta}{2}}=4$

If $y\tan (A+B+C)=x\tan (A+B-C)=\lambda$, then $\tan 2C=?$

trigonometric equations trigonometric functions trigonometry maths

  1. $\dfrac{\lambda(x+y)}{\lambda^2-xy}$

  2. $\dfrac{\lambda(x+y)}{\lambda^2+xy}$

  3. $\dfrac{\lambda(x-y)}{xy-\lambda^2}$

  4. $\dfrac{\lambda (x-y)}{xy+\lambda^2}$

Show answer
Correct Option: B
Explanation:

Given $y\tan (A+B+C)=x\tan (A+B-C)=\lambda$

$\implies \tan (A+B+C)=\dfrac{\lambda}{y},\tan (A+B-C)=\dfrac{\lambda}{x}$

$\tan 2 C=\tan ((A+B+C)-(A+B-C))=\dfrac{\tan (A+B+C)-\tan (A+B-C)}{1+\tan (A+B+C)\tan (A+B-C)}$

                                                                                $=\dfrac{\frac{\lambda}{y}-\frac{\lambda}{x}}{1+\frac{\lambda^2}{x y}}$

                                                                                $=\dfrac{\lambda(x-y)}{x y+\lambda^2}$

If $4^{2\, sin^2x}.16^{tan^2x}.2^{4\, cos^2x} = 256 $ such that $0 < x < \dfrac{\pi}{2}$ then $x$ is equal to ___________.

trigonometric equations trigonometric functions trigonometry maths

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

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

  3. $\dfrac{\pi}{12}$

  4. $\dfrac{\pi}{24}$

Show answer
Correct Option: B
Explanation:
${4}^{2{\sin}^{2}{x}}.{16}^{{\tan}^{2}{x}}.{2}^{4{\cos}^{2}{x}}=256$

$\Rightarrow\,{2}^{4{\sin}^{2}{x}}.{2}^{4{\tan}^{2}{x}}.{2}^{4{\cos}^{2}{x}}={2}^{8}$

$\Rightarrow\,{2}^{4{\sin}^{2}{x}+4{\tan}^{2}{x}+4{\cos}^{2}{x}}={2}^{8}$

$\Rightarrow\,4{\sin}^{2}{x}+4{\tan}^{2}{x}+4{\cos}^{2}{x}=8$

$\Rightarrow\,{\sin}^{2}{x}+{\tan}^{2}{x}+{\cos}^{2}{x}=2$

$\Rightarrow\,\left({\sin}^{2}{x}+{\cos}^{2}{x}\right)+{\tan}^{2}{x}=2$

$\Rightarrow\,1+{\tan}^{2}{x}=2$ since $\left({\sin}^{2}{x}+{\cos}^{2}{x}=1\right)$

$\Rightarrow\,{\sec}^{2}{x}=2$ since $1+{\tan}^{2}{x}={\sec}^{2}{x}$

$\Rightarrow\,{\cos}^{2}{x}=\dfrac{1}{2}$

$\Rightarrow\,\cos{x}=\pm\dfrac{1}{\sqrt{2}}$

$\Rightarrow\,\cos{x}=\dfrac{1}{\sqrt{2}}$ since $0<x<\dfrac{\pi}{2}$

$\Rightarrow\,x=\dfrac{\pi}{4}$

The number of zeroes that comes after $1$ for $10$ millions are

maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. $8$

  2. $7$

  3. $6$

  4. $9$

Show answer
Correct Option: B
Explanation:

Million $=1,000,000$ ($6$ zeros)
$10$   million $=10,000,000$
Hence the number of zeros after $1$ in $10$ million is $7.$

Match the following.

Column I Column II 
(i). 100 crores (P) 1 thousand
(ii) 10 lakhs (Q) 1 lakh
(iii) 100 thousands (R) 1 billion
(iv) 100 tens (S) 1 million
maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. (i)>(P), (ii) > (Q), (iii) > (S), (iv) > (R)

  2. (i)>(R), (ii) > (S), (iii) > (P), (iv) > (Q)

  3. (i)>(R), (ii) > (S), (iii) > (Q), (iv) > (P)

  4. (i)>(P), (ii) > (S), (iii) > (Q), (iv) > (R)

Show answer
Correct Option: C
Explanation:

(i) $100$ crores $= 1,000,000,000 = 1$ billion (R)

(ii) $10$ lakhs $= 1,000,000 =  1$ million (S)
(iii) $100$ thousands $= 1,00,000 = 1$ lakh (Q)
(iv) $100$ tens $= 1,000 = 1$ thousand (P)

A train overtakes two persons who are walking in the same direction in which the train is going at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is:

maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. 20 m

  2. 30 m

  3. 40 m

  4. 50 m

Show answer
Correct Option: D
Explanation:
Let the length of the train be x metres and speed  be y $m/s$

Speed of the first person$=2kmph$

                                          $=2\times\dfrac{5}{18}$

                                            $=\dfrac{5}{9}m/s$

Speed of the second train $=4kmph$

                                              $=4\times\dfrac{5}{18}$

                                               $=\dfrac{10}{9}m/s$

$\dfrac{x}{y-\dfrac{5}{9}}=9$

$9y-5=x$   
                                       
$90y-50=10x$...........................................(1)

                                            
$\dfrac{x}{y-\dfrac{10}{9}}=10$

$90y-100=9x$...........................................(2)

Substracting $1$ and $2$

$90y-50-90y+100=10x-9x$

$x=50$

So, the length of the train$=50m$



Express in Indian system of numeration.
Three million four hundred thirty five thousand two hundred twenty five.

maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. $345,225$

  2. $34,352,25$

  3. $3,43,52,25$

  4. $34,35,225$

Show answer
Correct Option: D
Explanation:

Let us convert the given number in words into numerals as follows:


Three million$=30,00,000$
Four hundred thirty five thousand$=435,000$
Two hunderd$=200$
Twenty$=20$
Five$=5$

Now, add the numerals as shown below:

$3000000+435000+200+20+5=34,35,225$

Therefore, the required number is $34,35,225$.

Hence, the Indian system of numeration is $34,35,225$.

Express in Indian number system:
$35,987,123$

maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. $35,987,123$

  2. $3,59,87,123$

  3. $3,987,123$

  4. $3,59,8,71,23$

Show answer
Correct Option: B
Explanation:
Let us convert the given numerals into words as follows:

$3,00,00,000=$Three crores
$59,00,000=$Fifty nine lakh
$87,000=$Eighty seven thousand
$100=$One hundred
$20=$Twenty
$3=$Three

Therefore, the number can be written in words as Three crores fifty nine lakh eighty seven thousand one hundred and twenty three.

Now, add the above numerals as shown below:

$3,00,00,000+59,00,000+87,000+100+20+3=3,59,87,123$ 
where $3$ is at one's place, $2$ is at tens place, $1$ is at hundredth place, $7$ is at thousandth place and so on. 

Hence, the Indian number system is $3,59,87,123$.

Express in Indian number system.
$4,456,765$

maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

  1. Four crore forty five lakh seven hundred sixty five,

  2. Forty four lakh fifty six thousand seven hundred sixty five,

  3. Forty four crore forty five lakh seven hundred sixty five,

  4. Forty five lakh seven hundred sixty five,

Show answer
Correct Option: B
Explanation:
Let us convert the given numerals into words as follows:

$44,00,000=$Forty four lakh
$56,000=$Fifty six thousand
$700=$Seven hundred
$60=$Sixty
$5=$Five

Now, combine the numbers in words as shown below:

Fourty four lakh fifty six thousand seven hundred sixty five.

Hence, the Indian number system is fourty four lakh fifty six thousand seven hundred sixty five.

Number name for 83,07,80,120 is

maths number work indian place value chart place value of digit largest and smallest numbers

  1. eighty three crore seven thousand eighty lakh one hundred twenty

  2. eighty three crore seven lakh eighty thousand one hundred twenty

  3. eighty three crore crore seventy thousand eighty lakh one twenty

  4. none

Show answer
Correct Option: B
Explanation:

$83,07,80,120 = 83x1,00,00,000 + 07x1,00,000 +80x1,000 + 120$

So this will equal to Eighty three crore seven lakh eighty thousand one hundred twenty
So correct answer will be option B