Tag: maths

Quizzes (518) Questions (9324)

Questions Related to maths

Multiple choice the nth roots of unity complex numbers maths

Simplify the expressions of the sums

$\displaystyle cot^2 \frac{\pi}{2n + 1} + cot^2 \frac{2\pi}{2n + 1} + cot^2 \frac{3\pi}{2n + 1} + ...... + cot^2 \frac{n\pi}{2n + 1}=$

  1. $\displaystyle \frac{n (2n +1)}{3}$
  2. $\displaystyle \frac{n (2n + 1)}{6}$
  3. $\displaystyle \frac{n (2n - 1)}{3}$
  4. $\displaystyle \frac{n (2n - 1)}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ (cosx+i sinx) }^{ n }=cos(nx)+i sin(nx)$
Comparing imaginary parts: $sin(nx)= _{ 1 }^{ n }{ C }sinx{ .cos }^{ n-1 }x- _{ 3 }^{ n }{ C }sin^{ 3 }x{ .cos }^{ n-3 }x+.....$
Divide both sides by $sin^{ n }x$
$\dfrac { sin(nx) }{ sin^{ n }x } = _{ 1 }^{ n }{ C }cot^{ n-1 }x- _{ 3 }^{ n }{ C }cot^{ n-3 }x+...$
Replace $n$ with $2n+1:$ $\dfrac { sin((2n+1)x) }{ sin^{ 2n+1 }x } = _{ 1 }^{ 2n+1 }{ C }cot^{ 2n }x- _{ 3 }^{ 2n+1 }{ C }cot^{ 2n-2 }x+...$
$ = _{ 1 }^{ 2n+1 }{ C }(cot^{ 2 }x)^{ n }- _{ 3 }^{ 2n+1 }{ C }(cot^{ 2 }x)^{ n-1 }+.... ----1)$
So $cot^{ 2 }x _{ i }$ are the roots of the polynomial for LHS=0 where $cot^{ 2 }x _{ i }=cot^{ 2 }(\dfrac { \pi i }{ 2n+1 } )$ where i=1,2,3,...n.
Sum of the roots is the ratio of the first two coefficients on the RHS.
Thus $\sum _{ i=1 }^{ n }{ cot^{ 2 }(\dfrac { \pi i }{ 2n+1 } )= } -\dfrac { - _{ 3 }^{ 2n+1 }{ C } }{ _{ 1 }^{ 2n+1 }{ C } } =\dfrac { (2n+1)2n(2n-1) }{ 6 } .\dfrac { 1 }{ 2n+1 } =\dfrac { n(2n-1) }{ 3 } $
Hence, (c) is correct.

Multiple choice maths square and square root scientific notation use of exponents power of 10

Evaluate the product of $2.3\times 10^4$ and $3\times 10^3$.

  1. $6.9\times 10^4$
  2. $6.9\times 10^7$
  3. $6.9\times 10^8$
  4. $6.9\times 10^6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$(2.3\times 10^4) \times (3 \times 10^3)$
$\Rightarrow (2.3 \times 3) \times(10^{3+4})$

$\therefore 6.9 \times 10^7$
Ans- Option $B$.

Multiple choice maths square and square root scientific notation use of exponents power of 10

Solve:

$\cfrac{2.3^{n+1} + 7.3^{n-1}}{3^{n+2}-2 \left ( \cfrac{1}{3} \right )^{1-n}} $

  1. $0$
  2. $1$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have,

$\cfrac{2.3^{n+1} + 7.3^{n-1}}{3^{n+2}-2 \left ( \cfrac{1}{3} \right )^{1-n}}  $

$\Rightarrow \cfrac{6.3^n+ \dfrac{7}{3}.3^n}{9.3^n-\dfrac{2}{3}. 3^n}  $

$\Rightarrow \cfrac{18.3^n+ 7.3^n}{27.3^n-2. 3^n}  $

$\Rightarrow \cfrac{25.3^n}{25.3^n}  $

$\Rightarrow 1  $

Hence, this is the answer.

Multiple choice maths square and square root scientific notation use of exponents power of 10

The average distance between the Sun and a certain planet is approximately $\displaystyle 2.3\times { 10 }^{ 14 }$ inches. Which of the following is closest to the average distance between the Sun and the planet, in kilometers? (1 kilometer is approximately $\displaystyle 3.9\times { 10 }^{ 4 }$ inches )

  1. $\displaystyle 7.1\times { 10 }^{ 8 }$
  2. $\displaystyle 5.9\times { 10 }^{ 9 }$
  3. $\displaystyle 1.6\times { 10 }^{ 10 }$
  4. $\displaystyle 1.6\times { 10 }^{ 111 }$
  5. $\displaystyle 5.9\times { 10 }^{ 11 }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The average distance in KM will be $\frac { 2.3\times { 10 }^{ 14 } }{ 3.9\times { 10 }^{ 4 } } =0.59\times { 10 }^{ 10 }=5.9\times { 10 }^{ 9 }\quad km$

So correct answer will be option B

Multiple choice maths square and square root scientific notation use of exponents power of 10

Express $(3000)^2\times (20)^3$ in scientific notation:

  1. $7.2\times 10^{10}$
  2. $3.6\times 10^{8}$
  3. $6\times 10^9$
  4. $4.8\times 10^{12}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

${ 3000 }^{ 2 }\times { 20 }^{ 3 }=(9\times { 10 }^{ 6 })\times (8\times { 10 }^{ 3 })\ =72\times { 10 }^{ 9 }=7.2\times { 10 }^{ 10 }$

So correct answer will be option A

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

Write the number of significant digits in:
$0.97 \times 10^{-2}$

  1. $1$
  2. $5$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$0.97\times 10^{-2}$
There are $2$ significant figures $9$ and $7$. When a number is  written in scientific notation, only significant figures are placed into the numerical portion.

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

Which of the following is greater than 1000.01 ?

  1. 0.00010001 x 10$\displaystyle ^{7}$
  2. 0.00001 x 10$\displaystyle ^{8}$
  3. 1.1 x 10$\displaystyle ^{2}$
  4. 1.00001 x 10$\displaystyle ^{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

A $\Rightarrow \displaystyle 0.00010001\times 10^{7}=1000.1 $
B $\Rightarrow \displaystyle 0.00001\times 10^{8}=1000 $
C $\Rightarrow \displaystyle 1.1\times 10^{2}=110 $
D $\Rightarrow \displaystyle 1.00001\times 10^{3}=1000.01 $

We observe that $\displaystyle 1000.1> 1000.01 $
i.e., $\displaystyle 0.00010001\times 10^{7}> 1000.01 $