Tag: maths

Quizzes (518) Questions (9324)

Questions Related to maths

Multiple choice trigonometric equations trigonometric functions trigonometry maths

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

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

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


Applying componendo and dividendo rule


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

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

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

Multiple choice trigonometric equations trigonometric functions trigonometry maths

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

  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. $\dfrac{\lambda (x-y)}{xy+\lambda^2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

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

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

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

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

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

Multiple choice trigonometric equations trigonometric functions trigonometry maths

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 ___________.

  1. $\dfrac{\pi}{3}$
  2. $\dfrac{\pi}{4}$
  3. $\dfrac{\pi}{12}$
  4. $\dfrac{\pi}{24}$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $0<x<\dfrac{\pi}{2}$

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

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

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

  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)
Reveal answer Fill a bubble to check yourself
C Correct answer
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)

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

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:

  1. 20 m

  2. 30 m

  3. 40 m

  4. 50 m
Reveal answer Fill a bubble to check yourself
D Correct answer
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$=50m$



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

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

  1. $345,225$
  2. $34,352,25$
  3. $3,43,52,25$
  4. $34,35,225$
Reveal answer Fill a bubble to check yourself
D Correct answer
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 $34,35,225$.

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

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

  1. $35,987,123$
  2. $3,59,87,123$
  3. $3,987,123$
  4. $3,59,8,71,23$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $3,59,87,123$.
Multiple choice maths number work indian place value chart place value of digit largest and smallest numbers writing and expanding numbers

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

  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,
Reveal answer Fill a bubble to check yourself
B Correct answer
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.
Multiple choice maths number work indian place value chart place value of digit largest and smallest numbers

Number name for 83,07,80,120 is

  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
Reveal answer Fill a bubble to check yourself
B Correct answer
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