Practice | Hashtable

1. (17)^3.5 × (17)^? = 17⁸

A. 2.29

B. 2.75

C. 4.25

D. 4.5

Discuss

Solution:

Let (17)^3.5 × (17)^x = 17⁸
Then, (17)^{3.5 + x} = 17⁸
∴ 3.5 + x = 8
⇒ x = (8 - 3.5)
⇒ x = 4.5

2. $If {(\frac{a}{b})}^{x - 1} = {(\frac{b}{a})}^{x - 3},$ then the value of x is

A. $\frac{1}{2}$

B. 1

C. 2

D. $\frac{7}{2}$

Discuss

Solution:

\begin{aligned} Given {(\frac{a}{b})}^{x - 1} = {(\frac{b}{a})}^{x - 3} \\ \Rightarrow {(\frac{a}{b})}^{x - 1} = {(\frac{a}{b})}^{- (x - 3)} = {(\frac{a}{b})}^{(3 - x)} \\ \Rightarrow x - 1 = 3 - x \\ \Rightarrow 2 x = 4 \\ \Rightarrow x = 2 \end{aligned}

3. Given that 10^0.48 = x, 10^0.70 = y and x^z = y², then the value of z is close to:

A. 1.45

B. 1.88

C. 2.9

D. 3.7

Discuss

Solution:

\begin{aligned} x^{z} = y^{2} \Leftrightarrow 10^{(0.48 z)} = 10^{2 \times 0.70} = 10^{1.40} \\ \Rightarrow 0.48 z = 1.40 \\ \Rightarrow z = \frac{140}{48} = \frac{35}{12} = 2.9 (approx) \end{aligned}

4. If 5^a = 3125, then the value of 5^{(a - 3)} is:

A. 25

B. 125

C. 625

D. 1625

Discuss

Solution:

5^a = 3125 ⇔ 5^a = 5⁵
⇒ a = 5.
∴ 5^{(a - 3)} = 5^{(5 - 3)} = 5² = 25

5. If 3^{(x - y)} = 27 and 3^{(x + y)} = 243, then x is equal to:

A. 0

B. 2

C. 4

D. 6

Discuss

Solution:

3^{x - y} = 27 = 3³ ⇔ x - y = 3 ....(i)
3^{x + y} = 243 = 3⁵ ⇔ x + y = 5 ....(ii)
On solving (i) and (ii), we get x = 4.

6. (256)^0.16 × (256)^0.09 = ?

A. 4

B. 16

C. 64

D. 256.25

Discuss

Solution:

(256)^0.16 × (256)^0.09 = (256)^{(0.16 + 0.09)}

\begin{aligned} = {(256)}^{0.25} \\ = {(256)}^{\frac{25}{100}} \\ = {(256)}^{\frac{1}{4}} \\ = {(4^{4})}^{\frac{1}{4}} \\ = 4^{1} \\ = 4 \end{aligned}

7. The value of [(10)¹⁵⁰ ÷ (10)¹⁴⁶]

A. 1000

B. 10000

C. 100000

D. 10⁶

Discuss

Solution:

\begin{aligned} {(10)}^{150} \div {(10)}^{146} = \frac{10^{150}}{10^{146}} \\ = 10^{150 - 146} \\ = 10^{4} \\ = 10000 \end{aligned}

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

A. 0

B. 1

C. x^{a - b - c}

D. None of these

Discuss

Solution:

Given exp. =

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

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

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

= \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})}

\begin{aligned} = \frac{x^{a} + x^{b} + x^{c}}{x^{a} + x^{b} + x^{c}} \\ = 1 \end{aligned}

9. (25)^7.5 × (5)^2.5 ÷ (125)^1.5 = 5^?

A. 8.5

B. 13

C. 16

D. 17.5

Discuss

Solution:

\begin{aligned} Let {(25)}^{7.5} \times {(5)}^{2.5} \div {(125)}^{1.5} = 5^{x} \\ Then, \frac{{(5^{2})}^{7.5} \times {(5)}^{2.5}}{{(5^{3})}^{1.5}} = 5^{x} \\ \Rightarrow \frac{5^{(2 \times 7.5)} \times 5^{2.5}}{5^{(3 \times 1.5)}} = 5^{x} \\ \Rightarrow \frac{5^{15} \times 5^{2.5}}{5^{4.5}} = 5^{x} \\ \Rightarrow 5^{x} = 5^{(15 + 2.5 - 4.5)} \\ \Rightarrow 5^{x} = 5^{13} \\ ∴ x = 13 \end{aligned}

10. (0.04)^-1.5 = ?

A. 25

B. 125

C. 250

D. 625

Discuss

Solution:

\begin{aligned} {(0.04)}^{- 1.5} = {(\frac{4}{100})}^{- 1.5} \\ = {(\frac{1}{25})}^{- (3 / 2)} \\ = {(25)}^{(3 / 2)} \\ = {(5^{2})}^{(3 / 2)} \\ = {(5)}^{2 \times (3 / 2)} \\ = 5^{3} \\ = 125 \end{aligned}