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 powers and exponents scientific notation use of exponents power of 10

If $9^n = 27^{n+1}$, then calculate the value of $2^n $.

  1. $-\dfrac{10}{3}$
  2. $-\dfrac{8}{3}$
  3. $-\dfrac{3}{8}$
  4. $\dfrac{1}{8}$
  5. $\dfrac{3}{8}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given ${ 9 }^{ n }={ 27 }^{ n+1 }$, which implies
$ { 3 }^{ 2n }={ 3 }^{ 3n+3 }$
Now compare powers, we get
$2n = 3n+3$ , which implies $n = -3$
Therefore ${ 2 }^{ n }={ 2 }^{ -3 }=\dfrac { 1 }{ 8 } $

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

If $3^{n} = n^{6}$, find the value of $ n^{18} $

  1. $3^{n} n^{3}$
  2. $3^{n} n^{12}$
  3. $9^{n}$
  4. $3^{12n}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given, $3^n=n^6$
We need to find the value of $n^{18}$
$\therefore {n}^{18}= n^6. n^{12}=3^n.n^{12}$
$\therefore n^{6+12}=3^n. n^{12}$
$\therefore n^{18}= {3}^{n}{n}^{12}$
Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $5^{k^2}(25^{2k})(625) = 25\sqrt{5}$ and $k < -1$, find the value of $k$.

  1. $-3.581$
  2. $-3.162$
  3. $-2.613$
  4. $-1.581$
  5. $-0.419$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, ${ 5 }^{ { k }^{ 2 } }({ 25 }^{ 2k })(625)=25\sqrt { 5 } $
${ 5 }^{ { k }^{ 2 }+4k+4 }={ 5 }^{ \tfrac 52 }$ 

By comparing powers, we get
${ k }^{ 2 }+4k+4=\cfrac 52$ which implies ${ 2k }^{ 2 }+8k+3=0$
$\Rightarrow k = -3.581$           ...(given $k<-1$)

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $A = \left[ {{a _{ij}}} \right]$ and ${a _{ij}} = i\left( {i + j} \right)$ then trace of $A=$

  1. $\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
  2. $\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{3}$
  3. $\frac{{n\left( {n + 1} \right)}}{2}$
  4. $\frac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $A=[a _{i,j}]$ and  $a _{i,j}=i(i+j)$ 

let the order of $A$ = $n\times n$

trace of $A$ =sum of all diagonal elements i.e.,$\sum[a _{i,j}]$ where $i=j$

$a _{1,1}=1(1+1)=2$
$a _{2,2}=2(2+2)=8$
$a _{3,3}=3(3+3)=18$
              $.$
              $.$
              $.$
$a _{n,n}=n(n+n)=2n^2$


$Trace$ $of$ $ A=$ $a _{1,1}+a _{2,2}+a _{3,3}+...........+a _{n,n}$ 
              $A=$  $2+8+18+.................+2n^2$
              $A=$  $2[1+4+9+................n^2]$

              $A=$  $2 \times [\frac{n(n+1)(2n+1)}{6}]$

              $A=$  $\frac{n(n+1)(2n+1)}{3}$

               $\therefore Opt$ $is$ $[B]$

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

 $P=\left[ \begin{matrix} { 5a }^{ 2 }+2bc & 6 & 8 \ 13 & { 8b }^{ 2 }-10ac & -9 \ -7 & 5 & { 25c }^{ 2 } \end{matrix} \right]$ and $Q=\left[ \begin{matrix} { a }^{ 2 }+6bc & 3 & 5 \ 12 & { -b }^{ 2 } & 6 \ 1 & 4 & { 17bc }^{ 2 } \end{matrix} \right] a,b$ & $c \epsilon N$, if trace $\left(P\right)=trac\left(Q\right)$, and $a,b$ & $C$ are sides of $\Delta ABC$ with $BC=a,CA=b$ & $AB=C$ then $\cos A$ is:

  1. $\dfrac{-79}{120}$
  2. $\dfrac{-89}{120}$
  3. $\dfrac{-33}{40}$
  4. $\dfrac{-31}{40}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Equating the traces of P and Q gives an equation involving a, b, and c. Using the properties of triangle sides and the Law of Cosines, one can solve for cos A.

Multiple choice terms related to matrices matrices and determinants matrices algebra maths

If $\left( \begin{array} { l l } { 3 } & { 2 } \ { 7 } & { 5 } \end{array} \right) A \left( \begin{array} { c c } { - 1 } & { 1 } \ { - 2 } & { 1 } \end{array} \right) = \left( \begin{array} { c c } { 2 } & { - 1 } \ { 0 } & { 4 } \end{array} \right)$  then trace of  $A$  is equal to

  1. $-25$
  2. $-21$
  3. $-15$
  4. $-11$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let M1 * A * M2 = M3. Then A = M1^-1 * M3 * M2^-1. Calculate the inverse of the matrices and perform the multiplication to find A, then sum the diagonal elements.

Multiple choice maths place value, ordering and rounding order operations and algebra using brackets in algebraic expressions order of operations

$9+\cfrac { 3 }{ 4 } +7+\cfrac { 2 }{ 17 } -\left( 9+\cfrac { 1 }{ 15 }  \right) =$?

  1. $7+\cfrac { 719 }{ 1020 } $
  2. $9+\cfrac { 817 }{ 1020 } $
  3. $9+\cfrac { 719 }{ 1020 } $
  4. $7+\cfrac { 817 }{ 1020 } $

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

Given the sum $=9+\cfrac { 3 }{ 4 } +7+\cfrac { 2 }{ 17 } -\left( 9+\cfrac { 1 }{ 15 }  \right) $
$=(9+7-9)+\left( \cfrac { 3 }{ 4 } +\cfrac { 2 }{ 17 } -\cfrac { 1 }{ 15 }  \right) $
$=7+\cfrac { 765+120-68 }{ 1020 } $
$=7+\cfrac { 817 }{ 1020 } $

Multiple choice maths how much does it weigh? define weight and units of weight conversion of length measurement (length) basic operations with same units operations involving units of length

${\rm{1}}\,{\rm{g/c}}{{\rm{m}}^{\rm{3}}}$ is equal to____________.

  1. ${\rm{1}}\,{\rm{kg/c}}{{\rm{m}}^{\rm{3}}}$
  2. ${\rm{1}}{{\rm{0}}^3}\,{\rm{kg/}}{{\rm{m}}^{\rm{3}}}$
  3. ${\rm{1}}{{\rm{0}}^{ - 2}}\,{\rm{kg/c}}{{\rm{m}}^{\rm{3}}}$
  4. ${\rm{1}}{{\rm{0}}^{ - 3}}\,{\rm{kg/c}}{{\rm{m}}^{\rm{3}}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We have,

$1\ g/cm^3$

Since,
$1000\ g=1\ kg$
$1\ g=10^{-3} \ kg$

So,

$1\ g/cm^3 =10^{-3}\ kg/cm^3$

Hence, this is the answer.

Multiple choice maths calculating and mental strategies 3 finding percentage of a number how many in all? problems on percentage

Estimated value of square root of $650$ is

  1. $25.495$
  2. $24.495$
  3. $2.5495$

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

$\Rightarrow$   Here we have to find square root of $650$

$\Rightarrow$  $\sqrt{650}=\sqrt{25\times 26}$
$\Rightarrow$  $\sqrt{650}=5\sqrt{26}$
$\Rightarrow$  $\sqrt{650}=5\times  5.09$
$\Rightarrow$  $\sqrt{650}=25.495$

Multiple choice maths binomial theorem, sequence and series series introduction to series introduction to sequences and series

$ \left{ a _ { n } \right} $ and $ \left{ b _ { n } \right} $ are two sequences given by $ a _ { n } = ( x ) ^ { 1 / 2 ^ { \circ } } + ( y ) ^ { 1 / 2 ^ { \circ } } $ and $ b _ { n } = ( x ) ^ { 1 / 2 ^ { 2 } } - ( y ) ^ { 1 / 2 ^ { \circ } } $ for all $ \mathrm { n } \in \mathrm { N } . $ The value of $ \mathrm { a } _ { 1 } \mathrm { a } _ { 2 } \mathrm { a } _ { 3 } \dots \ldots \ldots \mathrm { a } _ { \mathrm { n } } $ is equal to

  1. x-y
  2. $

    \frac { x + y } { b _ { n } }

    $
  3. $

    \frac { x - y } { b _ { n } }

    $
  4. $

    \frac { x y } { b _ { n } }

    $
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice reciprocal equations theory of equations maths

$\cfrac { \left( 2x-1 \right) { \left( x-1 \right)  }^{ 4 }{ \left( x-2 \right)  }^{ 4 } }{ (x-2){ \left( x-4 \right)  }^{ 4 } } \le 0$

  1. $(\dfrac{1}{2},2)$
  2. $R$
  3. $\phi$
  4. $(1/3,2)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{(2{x}-1)(x-1)^{4}(x-2)^{4}}{(x-2)(x-4)^{4}} \le 0\implies x\neq 2$


$(2{x}-1)(x-1)^{4}(x-2)^{3}\le 0$


$(x-\dfrac{1}{2})(x-2)^{3}\le 0$

$x\in \bigg(\dfrac{1}{2},2\bigg)$

Multiple choice maths part number dividing fractions division of a fractions division of a fraction

Solve $\left[\dfrac{170}{3} +\dfrac{6}{7}\right] \div \left[\dfrac{2}{7} \times \dfrac{11}{2}\right]$

  1. $\dfrac{1208}{3\times 11}$
  2. $\dfrac{1208}{11}$
  3. $\dfrac{1208}{3}$
  4. $\dfrac{1208}{9\times 11}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\left[\dfrac{170}{3} +\dfrac{6}{7}\right] \div \left[\dfrac{2}{7} \times \dfrac{11}{2}\right]$


$=\left[ \dfrac{1190+18}{21}\right] \div \left[ \dfrac{11}{7}\right]$


$=\left[ \dfrac{1190+18}{21}\right] \times \left[ \dfrac{7}{11}\right]$

$=\left[ \dfrac{1208}{11\times 3}\right]$