Tag: maths

Questions Related to maths

Face value of '$3$' in $31005660$ is:

maths numbers and place value face value of digit large numbers general form of number

  1. $3$ crores

  2. $30$ lakhs

  3. $3$

  4. $0$

Show answer
Correct Option: C
Explanation:

Face value of a digit is the value of the digit in the number. 

So, the face value of $3$ in $31005660$ is $3$.
Hence, the answer is $3$.

How many times does the digit $1$ appear in numbers from $1$ to $100$?

maths numbers and place value face value of digit large numbers general form of number

  1. $18$

  2. $19$

  3. $20$

  4. $21$

Show answer
Correct Option: D
Explanation:

$\Rightarrow$  1–10 = 2 times

$\Rightarrow$  11–20 = 10 times
$\Rightarrow$  21–30 = 1 time
$\Rightarrow$  31–40 = 1 time
$\Rightarrow$  41–50 = 1 time
$\Rightarrow$  51–60 = 1 time
$\Rightarrow$  61–70 = 1 time
$\Rightarrow$  71–80 = 1 time
$\Rightarrow$  81–90 = 1 time
$\Rightarrow$  91–100 = 2 times
$\Rightarrow$  $Total = 2+10+1+1+1+1+1+1+1+2=21$
$\therefore$   The digit 1 appear in number from 1 to 100 is $21$.

What is the sum of all integers between $50$ and $350$ which have $1$ as the units digit?

maths numbers and place value face value of digit large numbers general form of number

  1. $5880$

  2. $5985$

  3. $6230$

  4. $6800$

Show answer
Correct Option: A
Explanation:

The sequence $51+61+71...+341$ is an arithmetic progression.

$\Rightarrow$  To find the sum of n terms of an AP we use the formula.
$\Rightarrow$  Here, $n=30,\,a=51$ and $d=10$.
$\therefore$   $S _n=\dfrac{n}{2}[2a+(n-1)d]$

$\therefore$   $S _n=\dfrac{30}{2}[2\times 51+(30-1)10]$

$\therefore$   $S _n=15[102+290]$
$\therefore$   $S _n=15\times 392$
$\therefore$   $S _n=5880$

A number consists of two digits whose sum is $11$. If $27$ is added to the number, then the digits change their places. What is the number?

maths numbers and place value face value of digit large numbers general form of number

  1. $47$

  2. $65$

  3. $83$

  4. $92$

Show answer
Correct Option: A
Explanation:

Let the ten's digit be $x$. Then, unit's digit $= \left(11 - x\right)$.
So, number $= 10x + \left(11 - x\right) = 9x + 11$.
Therefore $\left(9x + 11\right) + 27 = 10 \left(11 - x\right) + x  \Leftrightarrow  9x + 38 = 110 - 9x \Leftrightarrow   18x = 72 \Leftrightarrow   x = 4$.
Thus, ten's digit $= 4$ and unit's digit $= 7$.
Hence, required number $= 47$.

For $Z _1=\displaystyle \sqrt[6]{\frac{1-i}{1+i\sqrt{3}}}; Z _2=\sqrt[6]{\frac{1-i}{\sqrt{3}+i}}; Z _3=\sqrt[6]{\frac{1+i}{\sqrt{3}-i}}$ which of the following holds good?

maths numbers and place value face value of digit large numbers general form of number

  1. $\displaystyle\sum|Z _1|^2=\frac{3}{2}$

  2. $\displaystyle|Z _1|^4+|Z _2|^4=|Z _3|^{-8}$

  3. $\displaystyle\sum|Z _1|^3+|Z _2|^3=|Z _3|^{-6}$

  4. $|Z _1|^4+|Z _2|^4=|Z _3|^8$

Show answer
Correct Option: B
Explanation:
$z _1 =\sqrt {\dfrac {1-i}{1+i\sqrt 3}}, z _2=\sqrt {\dfrac {1-i}{\sqrt 3 +i}}, z _3=\sqrt {\dfrac {1+i}{\sqrt 3-1}}$
$z _1 =\sqrt [6]{\dfrac {1-i}{1+\sqrt 3}}=\sqrt [6]{\dfrac {(1-i)(1-i\sqrt 3)}{1+3}}=\sqrt [6]{\dfrac {1(1-\sqrt 3)-i(1+\sqrt 3)}{4}}$
$|z _1|^2 =z _1 \bar {z} _1 =\sqrt [6]{\dfrac {(1-\sqrt 3)}{4}}\times \sqrt [6]{\dfrac {(1-\sqrt 3)+i(1+\sqrt 3)}{4}}$
$|z _1|^2 =\sqrt [6]{\dfrac {(1-\sqrt 3)^2 +(1+\sqrt 3)^2}{16}}$
$|z _1|^2 =\sqrt [6]{\dfrac {8}{16}}=\dfrac {1}{(2) 1/6}$
$z _2 =\sqrt [6]{\dfrac {1-i}{\sqrt 3+i}}=\sqrt [6]{\dfrac {(1-i) (\sqrt 3 -i)}{(3+1)}}=\sqrt [6]{\dfrac {(\sqrt 3-1)-i (1+\sqrt 3)}{4}}$
$|z _2|^2 =z _2 \bar {z} _2=\sqrt [6]{\dfrac {(\sqrt 3-1)-i (1+\sqrt 3)+i(1+\sqrt 3)}{4}}$
$|z _2|^2 =\sqrt [6]{\dfrac {(\sqrt 3-1)^2 +(1+\sqrt 3)^2}{16}}=\sqrt {\dfrac {8}{16}}=\dfrac {1}{(2) 1/6}$
$z _3 =z _3 \bar {z} _3 =\sqrt [6]{\dfrac {(\sqrt 3-1)+(1+\sqrt 3)}{4}\times \dfrac {(\sqrt 3-1)-i (1+\sqrt 3)}{4}}$
$=\sqrt [6]{\dfrac {(\sqrt 3-1)^2 +(1+\sqrt 3)^2}{16}}=\sqrt {\dfrac {8}{16}}=\dfrac {1}{(2)1/6}$
$|z _1|^4 =\dfrac {1}{2^{2/6}}\quad |z _2|^4 =\dfrac {1}{2^{2/6}}$
$|z _3|^8 =\dfrac {1}{2^{4/6}}\ \Rightarrow \ |z _3|^{-8}=2^{4/6}$
$\Rightarrow \ |z _1|^4 +|z _2|^4 =\dfrac {1}{2^{2/6}}+\dfrac {1}{2^{2/6}}=\dfrac {2}{2^{2/6}}=2^{4/6}$
$=|z _3|^{-8}$
so, $\boxed {|z _1|^4 +|z _2|^4 =|z _3|^{-8}}$ as
so, option $(B)$ is right.

What is a curve?

maths fundamental concepts - geometry terms related to polygons curves open and closed figures

  1. A line which is not straight and does not any sharp edges.

  2. It is a polygon

  3. It is a quadrilateral

  4. A line with sharp edges.

Show answer
Correct Option: A
Explanation:

In mathematics, a straight line also is a curve with no bends.
Therefore, A is the correct answer.

Write the number of significant digits in:

$3.005$.

maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

  1. $4$

  2. $2$

  3. $1$

  4. $0$

Show answer
Correct Option: A
Explanation:

Zeroes placed between other digits are always significant.
$\therefore  3.005$ has $4$ significant digits.

Write the number of significant digits in:

$5.16 \times 10^8$.

maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

  1. $3$

  2. $1$

  3. $2$

  4. $9$

Show answer
Correct Option: A
Explanation:

$5.16\times 10^8$
There are $3$ significant figures. When a number is  written in scientific notation, only significant figures are placed into the numerical portion.


Write the number of significant digits in:

$16.000$.

maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

  1. $2$

  2. $5$

  3. $16$

  4. $1$

Show answer
Correct Option: B
Explanation:

All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are themselves significant.
$\therefore  16.000$ has $5$ significant digits

Write the number of significant digits in $23.4$

maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

  1. $2$

  2. $23.4$

  3. $3$

  4. $1$

Show answer
Correct Option: C
Explanation:

Non-zero digits are always significant.

$\therefore  23.4$ has $3$ significant digits.