Tag: maths

Questions Related to maths

Simplify the expressions of the sums

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

the nth roots of unity complex numbers maths

  1. $\displaystyle \frac{n (2n +1)}{3}$

  2. $\displaystyle \frac{n (2n + 1)}{6}$

  3. $\displaystyle \frac{n (2n - 1)}{3}$

  4. $\displaystyle \frac{n (2n - 1)}{6}$

Show answer
Correct Option: C
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.

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

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

  1. $6.9\times 10^4$

  2. $6.9\times 10^7$

  3. $6.9\times 10^8$

  4. $6.9\times 10^6$

Show answer
Correct Option: B
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 $B$.

Evaluate $\dfrac{6.3\times 10^3}{3\times 10^7}$

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

  1. $2\times 10^{-3}$

  2. $2.1\times 10^{-3}$

  3. $2.1\times 10^{-4}$

  4. $2\times 10^{-5}$

Show answer
Correct Option: C
Explanation:

$\dfrac{6.3\times 10^{3}}{3\times 10^7}$


$\Rightarrow \dfrac{6.3}{3} \times 10^{3-7}$


$\therefore 2.1\times 10^{-4}$

Ans-Option $C$.

Solve:

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

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

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: B
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.

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 )

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

  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. $\displaystyle 5.9\times { 10 }^{ 11 }$

Show answer
Correct Option: B
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

Simplify: $\dfrac{1.8\times 10^{11}}{2\times 10^4}$

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

  1. $9\times 10^7$

  2. $0.9\times 10^6$

  3. $0.9\times 10^5$

  4. $9\times 10^6$

Show answer
Correct Option: D
Explanation:

$\dfrac{1.8\times 10^{11}}{2\times 10^4}$


$= \dfrac{1.8}{2} \times 10^{11-4}$


$=0.9 \times 10^7$

$= 9\times 10^6$


Ans-Option $D$.

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

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

  1. $7.2\times 10^{10}$

  2. $3.6\times 10^{8}$

  3. $6\times 10^9$

  4. $4.8\times 10^{12}$

Show answer
Correct Option: A
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

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

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

  1. $1$

  2. $5$

  3. $2$

  4. $3$

Show answer
Correct Option: C
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.

$225$ can be expressed as

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

  1. $5\times 3^2$

  2. $5^2\times 3$

  3. $5^2\times 3^2$

  4. $5^3\times 3$

Show answer
Correct Option: C
Explanation:

$ 225 = 25 \times 9$
$ 225 = (5\times 5)  \times (3 \times 3)$

$ 225 = 5^2 \times 3^2$
Option C is correct

Which of the following is greater than 1000.01 ?

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

  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$\displaystyle ^{3}$

Show answer
Correct Option: A
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 $