Mathematics · Quantitative Aptitude

Surds and Indices

408 Questions

Surds and indices questions focus on evaluating square roots, cube roots, and fractional exponents. These topics are a core part of the quantitative aptitude section in many competitive exams. Practicing these problems builds speed and accuracy for solving numerical equations.

Square roots evaluationCube roots calculationExponents and powersFractional exponentsSurds multiplication

Surds and Indices Questions

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the square root of each of the following correct to three places of decimal.
$17$
$1.7$
$2.5$
$\displaystyle\frac{7}{8}$

  1. $4.153\;;\;1.304\;;\;1.581\;;\;0.935$
  2. $4.123\;;\;1.304\;;\;1.581\;;\;0.995$
  3. $4.123\;;\;1.304\;;\;1.581\;;\;0.935$
  4. $4.123\;;\;1.304\;;\;1.501\;;\;0.935$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$17$

$4.123$
$4$$+4$ $17$$16$
$822$$+2$ $100$$81$
$8243$ $1900$$1644$
$25600$$24729$                 $871$

$\therefore$ The square root of $17=4.123$

$1.7$

$0$ $1.303$
$1$$+1$ $1.7$$1$
$23$$+3$ $70$$69$
$2603$ $10000$$7809$
$1191$

$\therefore$ The square root of $1.7=1.303$

$2.5$

$0$ $1.581$
$1$$+1$ $2.5$$1$
$25$$+5$ $150$$125$
$308$$+8$ $2500$$2464$
$3161$ $3600$$3161$                 $439$

$\therefore$ The square root of $2.5=1.581$

$\frac{7}{8}$

$8$ $70$$64$ $0.875$
$60$$56$
$40$$40$             $0$
$0.935$
$0$ $0.875$$0$
$9$$9$ $87$$81$
$183$$+3$ $650$$549$
$1865$ $10100$$9325$
$925$

$\therefore$ The square root of $\frac{7}{8}=0.935$

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Estiamate the square root of $850$ 

  1. $29.15$
  2. $30.21$
  3. $98.23$
  4. $23.11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The square root of $850$ is $\sqrt {850}=\sqrt {25 \times 34}=5\sqrt {34}$

Square root of $34$ lie between $5$ and $6$.
 square of $5.5$ is $30.25$ 
Now, we can say that square root of $34$ lie between $5.5$ and $6$.
Now, square of $5.75$ is $33.06$
So, square root of $34$ lie between $5.75$ and $6$.
Now, we have to choose the number $5.85$
$(5.85)^2=34.225$ which is greater than $34$ and close to $34$
So, assume a number $5.84$.
$(5.84)^2=34.1056$
$(5.83)^2=33.9889$
Hence, we can say that square root of 34 lie between $5.83$ and $5.84$.
So, $5\times 5.83=29.15$.

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

The number of common roots of the 15th and  of 25th roots of unity are

  1. 1

  2. 5

  3. 6

  4. 10
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$15th$ roots of unity are $e^{\dfrac{{i}2{k}\pi}{15}}$ where $k=1,2,\cdots,15$

$25th$ roots of unity are $e^{\dfrac{{i}2{m}\pi}{25}}$ where $m=1,2,3,\cdots,25$
The common roots are $e^{i\dfrac{2{n}}{5}}$ where $n=1,2,3,4,5$
Number of common roots will be $5$

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

If $1, \alpha _1, \alpha _2, \alpha _3, \alpha _4, \alpha _5, \alpha _6$, are seven, $7^{th}$ root of unity them $|(3-\alpha _1)(3-\alpha _3)(3-\alpha _5)|$ is?

  1. $\sqrt{2186}$
  2. $\sqrt{1093}$
  3. $\sqrt{1023}$
  4. $\sqrt{511}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$x^7 -1 = 0$
$(x-1)(x-\alpha _1).....(x-\alpha _6) = x^7-1$
Putting $x=3$, we get
$2.(x-\alpha _1).(x-\alpha _2)...(x-\alpha _6) = 2186$
$|(x-\alpha _1)|^2 |(x- \alpha _3)|^2 |(x-\alpha)|^2 = 1093$
$|(x-\alpha _1)(x- \alpha _3)(x-\alpha _5)| = \sqrt{1093}$

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

If $1,\omega ,\omega ^{2},....\omega ^{n-1}$ are $n,n^{th}$ roots ofunity then the value of $\left ( 13-\omega  \right )\left ( 13-\omega ^{n-1} \right )$ equals

  1. $ \displaystyle \frac{13^{n}+1}{3}$
  2. $ \displaystyle\frac{13^{n}-1}{3}$
  3. $ \displaystyle 13^{n}-1$

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$(13-w)(13-w^{n-1})$
$=(13-w)(13-\dfrac{w^{n}}{w})$
$=(13-w)(13-\overline{w})$
$=169-13(w+\overline{w})+w\overline{w}$
$=169-13(Re(w))+1$
$=170-13(Re(w))$
Hence answer is none of these.

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

The value of ${ \left( 16 \right)  }^{ 1/4 }$ are

  1. $\pm 2,\pm 2i$
  2. $\pm 4,\pm 4i$
  3. $\pm 1,\pm i$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $x={ \left( 16 \right)  }^{ { 1 }/{ 4 } }$

${ x }^{ 4 }=16$

${ x }^{ 4 }-16=0$

${ x }^{ 4 }-{ 2 }^{ 4 }=0$

$\left( { x }^{ 2 }-{ 2 }^{ 2 } \right) \left( { x }^{ 2 }+{ 2 }^{ 2 } \right) =0$

$\left( x-2 \right) \left( x+2 \right) \left( { x }^{ 2 }+4 \right) =0$

$x-2=0,x+2=0,{ x }^{ 2 }+4=0$

$x=2,-2$ or ${ x }^{ 2 }=-4\Longrightarrow x=\pm \sqrt { -4 } =\pm 2i$

$\therefore x=\pm 2,\pm 2i$

Multiple choice business maths applications of matrices and determinants elementary transformations of a matrix multiplicative inverse of a matrix inverse of a matrix

If $A = \left[ {\begin{array}{*{20}{c}}1&2\3&4\end{array}} \right]$, then $8A^{-4}$ is equal to

  1. $145A^{-1}+27I$
  2. $145A^{-1}-27I$
  3. $27I - 145A^{-1}$
  4. $29A^{-1} +9I$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the Cayley-Hamilton theorem for matrix A = [[1, 2], [3, 4]], the characteristic equation is A^2 - 5A - 2I = 0. By manipulating this equation, one can express higher powers of A in terms of A and I, eventually leading to the expression 27I - 145A^-1.