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Quantitative Aptitude

SURDS AND INDICES MCQs

Surds & Indices, Indices And Surds, Power

Total Questions : 753 | Page 4 of 76 pages
Question 31.

If  \(\left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3}\)  , then the value of x is:

  1.    \(\frac{1}{2}\)
  2.    1
  3.    2
  4.    \(\frac{7}{2}\)
 Discuss Question
Answer: Option C. -> 2

Givan \(\left(\frac{a}{b}\right)^{x-1} = \left(\frac{b}{a}\right)^{x-3}\)


\(\left(\frac{a}{b}\right)^{x-1} = \left(\frac{a}{b}\right)^{-(x-3)}= \left(\frac{a}{b}\right)^{(3-x)}\)


 x - 1 = 3 - x


 2x = 4


 x = 2.


 

Question 32.

Given that 100.48 = x, 100.70 = y and xz = y2, then the value of z is close to:

  1.    1.45
  2.    1.88
  3.    2.9
  4.    3.7
 Discuss Question
Answer: Option C. -> 2.9

xz = y2     \(\Leftrightarrow \)   10(0.48z) = 10(2 x 0.70) = 101.40


 0.48z = 1.40


 z = \(\frac{140}{48}\)\(\frac{35}{12}\) =  2.9 (approx.)

Question 33.

If 5a = 3125, then the value of 5(a - 3) is:

  1.    25
  2.    125
  3.    625
  4.    1625
 Discuss Question
Answer: Option A. -> 25

5a = 3125    \(\Leftrightarrow \)    5a = 55


 a = 5.


So, 5(a - 3) = 5(5 - 3) = 52 = 25.

Question 34.

If 3(x - y) = 27 and 3(x + y) = 243, then x is equal to:

  1.    0
  2.    2
  3.    4
  4.    6
 Discuss Question
Answer: Option C. -> 4

3x - y = 27 = 33     \(\Leftrightarrow \)   x - y = 3 ....(i)


3x + y = 243 = 35   \(\Leftrightarrow \)     x + y = 5 ....(ii)


On solving (i) and (ii), we get x = 4.

Question 35.

(256)0.16 x (256)0.09 = ?

  1.    4
  2.    16
  3.    64
  4.    256.25
 Discuss Question
Answer: Option A. -> 4

(256)0.16 x (256)0.09 = (256)(0.16 + 0.09)


   = (256)0.25


= \((256)^{\left(\frac{25}{100}\right)}\)


= \((256)^{\left(\frac{1}{4}\right)}\)


= \((4^{4})^{\left(\frac{1}{4}\right)}\)


= \(4^{4\left(\frac{1}{4}\right)}\)


= 41


   = 4


 

Question 36.

The value of [(10)150 ÷ (10)146]

  1.    1000
  2.    10000
  3.    100000
  4.    106
 Discuss Question
Answer: Option B. -> 10000

(10)150 ÷ (10)146 = \(\frac{10^{150}}{10^{146}}\)


= 10150 - 146


   = 104


   = 10000.

Question 37.

\(\frac{1}{1+x^{(b-a)}+x^{(c-a)}} +\frac{1}{1+x^{(a-b)}+x^{(c-b)}} + \frac{1}{1+x^{(b-c)}+x^{(a-c)}} = ?\)

  1.    0
  2.    1
  3.    xa - b - c
  4.    None of these
 Discuss Question
Answer: Option B. -> 1

Given Exp. =  \(\frac{1}{\left(1+\frac{x^{b}}{x^{a}}+\frac{x^{c}}{x^{a}}\right)} +\frac{1}{\left(1+\frac{x^{a}}{x^{b}}+\frac{x^{c}}{x^{b}}\right)}+\frac{1}{\left(1+\frac{x^{b}}{x^{c}}+\frac{x^{a}}{x^{c}}\right)}\)


= \(\frac{x^{a}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{b}}{(x^{a}+x^{b}+x^{c})}+\frac{x^{c}}{(x^{a}+x^{b}+x^{c})}\)


= \(\frac{(x^{a}+x^{b}+x^{c})}{(x^{a}+x^{b}+x^{c})}\)


= 1

Question 38.

(25)7.5 x (5)2.5 ÷ (125)1.5 = 5?

  1.    8.5
  2.    13
  3.    16
  4.    17.5
  5.    None of these
 Discuss Question
Answer: Option B. -> 13

Let (25)7.5 x (5)2.5 ÷ (125)1.5 = 5x.


Then, \(\frac{(5^{2})^{7.5}\times (5)^{2.5}}{(5^{3})^{1.5}}\)  =5x


       \(\frac{5^{(2\times7.5)}\times 5^{2.5}}{5^{(3\times1.5)}}\) = 5x


      \(\frac{5^{15}\times5^{2.5}}{5^{4.5}}\)  =  5x


 5x = 5(15 + 2.5 - 4.5)


 5x = 513


Therefore x = 13.


 

Question 39.

(0.04)-1.5 = ?

  1.    25
  2.    125
  3.    250
  4.    625
 Discuss Question
Answer: Option B. -> 125

(0.04)-1.5 =  \(\left(\frac{4}{100}\right)^{-1.5}\)


\(\left(\frac{1}{25}\right)^{-(\frac{3}{2})}\)


= \(\left(25\right)^{\left(\frac{3}{2}\right)}\)


= \(\left(5^{2}\right)^{\left(\frac{3}{2}\right)}\)


= \(\left(5\right)^{2\times\left(\frac{3}{2}\right)}\)


= 53


= 125.

The given expression is (0.04)^-1.5. We can simplify this expression using the following formula:

a^(-n) = 1/(a^n)

where a is a non-zero real number and n is a positive integer.

Using this formula, we get:

(0.04)^-1.5 = 1/(0.04)^1.5

Now, we can simplify the expression inside the parentheses using the following formula:

a^n = (a^m)^(n/m)

where a is a non-zero real number and m and n are integers.

Using this formula with a=0.04, n=3, and m=2, we get:

(0.04)^1.5 = (0.04^2)^(3/2) = 0.0016^(3/2) = 0.000064

Therefore,

(0.04)^-1.5 = 1/(0.04)^1.5 = 1/0.000064 = 15625/1 = 125

Hence, the correct answer is option B, 125.

In summary, we used the formula a^(-n) = 1/(a^n) to convert the negative exponent to a positive exponent, and then used the formula a^n = (a^m)^(n/m) to simplify the expression inside the parentheses. Finally, we obtained the answer by taking the reciprocal of the simplified expression.

If you think the solution is wrong then please provide your own solution below in the comments section .

Question 40.

\(\frac{\left(243\right)^{\frac{n}{5}}\times3^{2n+1}}{9^{n}\times3^{n-1}}= ?\)

  1.    1
  2.    2
  3.    9
  4.    3n
 Discuss Question
Answer: Option C. -> 9

Given Expression = \(\frac{\left(243\right)^{\frac{n}{5}}\times3^{2n+1}}{9^{n}\times3^{n-1}}\)


= \(\frac{\left(3^{5}\right)^{\left(\frac{n}{5}\right)} \times3^{2n+1}}{\left(3^{2}\right)^{n}\times3^{n-1}}\)


= \(\frac{\left(3^{5\times(\frac{n}{5})}\times3^{2n-1}\right)}{\left(3^{2n}\times3^{n-1}\right)}\)


= \(\frac{3^{n}\times3^{2n-+1}}{3^{2n}\times3^{n-1}}\)


= \(\frac{3^{(n+2n+1)}}{3^{(2n+n-1)}}\)


= \(\frac{3^{3n+1}}{3^{3n-1}}\)


= 3(3n + 1 - 3n + 1)   


= 32   = 9.


 

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