1. (132)7 × (132)? = (132)11.5. | |
A. 3 | B. 3.5 |
C. 4 | D. 4.5 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
(132)7 × (132)x = (132)11.5
=> 7 + x = 11.5
=> x = 11.5 - 7 = 4.5
2. $MF#%\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}.\text{ What is the value of x ?}$MF#% | |
A. 3 | B. 3.5 |
C. 4 | D. 4.5 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
$MF#%\begin{align}&\left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{b}{a}\right)^{x-7}\\\\
&\Rightarrow \left(\dfrac{a}{b}\right)^{x-2} = \left(\dfrac{a}{b}\right)^{-(x-7)}\\\\
&\Rightarrow x - 2 = -(x - 7)\\\\
&\Rightarrow x - 2 = -x + 7\\\\
&\Rightarrow x-2 = -x + 7\\\\
&\Rightarrow 2x = 9\\\\
&\Rightarrow x = \dfrac{9}{2} = 4.5
\end{align} $MF#%
3. If 7(x - y) = 343 and 7(x + y) = 16807, what is the value of x? | |
A. 4 | B. 3 |
C. 2 | D. 1 |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
7(x - y) = 343 = 73
=> x - y = 3 ---------------------------(Equation 1)
7(x + y) = 16807 = 75
=> x + y = 5 ---------------------------(Equation 2)
(Equation 1)+ (Equation 2) => 2x = 3 + 5 = 8
=> x = 8⁄2 = 4
4. (0.04)-2.5 = ? | |
A. 125 | B. 25 |
C. 3125 | D. 625 |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
$MF#%\begin{align}&(0.04)^{-2.5} = \left(\dfrac{1}{.04}\right)^{2.5} = \left(\dfrac{100}{4}\right)^{2.5}
= \left(25\right)^{2.5} = \left(5^2\right)^{2.5}\\\\
&= \left(5^2\right)^{\left(\dfrac{5}{2}\right)}= 5^5 = 3125\end{align}$MF#%
5. (6)6.5 × (36)4.5 ÷ (216)4.5 = (6)? | |
A. 1 | B. 2 |
C. 4 | D. 6 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
(6)6.5 × (36)4.5 ÷ (216)4.5
= (6)6.5 × [(6)2]4.5 ÷ [(6)3]4.5
= (6)6.5 × (6)9 ÷ (6)13.5
= (6)(6.5 + 9 - 13.5)
= (6)2
6. $MF#%\dfrac{1}{1 + P^{(n - m)}} + \dfrac{1}{1 + P^{(m - n)}} = ?$MF#% | |
A. $MF#%2$MF#% | B. $MF#%\dfrac{1}{1 + P}$MF#% |
C. $MF#%1$MF#% | D. $MF#%\dfrac{1}{P}$MF#% |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
$MF#%\begin{align}&\dfrac{1}{1 + P^{(n - m)}} + \dfrac{1}{1 + P^{(m - n)}}\\\\
&= \dfrac{1}{1 + \dfrac{P^n}{P^m}} + \dfrac{1}{1 + \dfrac{P^m}{P^n}}\\\\\\\\
&= \dfrac{P^m}{P^m + P^n} + \dfrac{P^n}{P^n + P^m}\\\\\\\\
&= \dfrac{P^m + P^n}{P^m + P^n} \\\\\\\\
&= 1\end{align} $MF#%
7. $MF#%\text{If }x = \left(8 + 3\sqrt{7}\right),\text{ what is the value of }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)?$MF#% | |
A. $MF#%\sqrt{12}$MF#% | B. $MF#%4$MF#% |
C. $MF#%2$MF#% | D. $MF#%\sqrt{14}$MF#% |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
$MF#%a^2 - b^2 = (a - b)(a + b)$MF#%
$MF#%\begin{align}&\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2\\\\
&= x - 2 + \dfrac{1}{x}\\\\
&= x + \dfrac{1}{x} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{1}{\left(8 + 3\sqrt{7}\right)} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{\left(8 + 3\sqrt{7}\right)\left(8 - 3\sqrt{7}\right)} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{8^2 - \left(3\sqrt{7}\right)^2} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{64 - 63} - 2 \\\\
&= \left(8 + 3\sqrt{7}\right) + \dfrac{\left(8 - 3\sqrt{7}\right)}{1} - 2 \\\\
&= 8 + 3\sqrt{7} + 8 - 3\sqrt{7} - 2 \\\\
&= 14\end{align} $MF#%
$MF#%\begin{align}&\text{We got that }\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right)^2 = 14\\\\
&\text{Hence ,}\left(\sqrt{x} - \dfrac{1}{\sqrt{x}}\right) = \sqrt{14}\end{align} $MF#%
8. if 6m = 46656, What is the value of 6m-2 | |
A. 36 | B. 7776 |
C. 216 | D. 1296 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
-------------------------------------------------------------------------
Solution 1
--------------------------------------------------------------------------
Given that 6m = 46656
$MF#%6^{m-2} = \dfrac{6^m}{6^2} = \dfrac{46656}{6^2} = \dfrac{46656}{36} = 1296$MF#%
------------------------------------------------------------------------- Solution 2 -------------------------------------------------------------------------- 6m = 46656
646656677766129662166366
Hence, 46656 = 66
Thus, we got that 6m = 66
=> m = 6
6m-2 = 6(6-2) = 64 = 1296
9. 10222 ÷ 10220 = ? | |
A. 10 | B. 100 |
C. 1000 | D. 10000 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
10222 ÷ 10220 = 10222 - 220 = 102 = 100
10. If m and n are whole numbers and mn = 196, what is the value of (m - 3)(n+1) ? | |
A. 2744 | B. 1 |
C. 121 | D. 1331 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
mn = 196
We know that 142 = 196
Hence we can take m = 14 and n = 2
(m - 3)(n+1) = (14 - 3)(2+1) = 113 = 1331
11. $MF#%\dfrac{(1024)^{n/5} \times 4^{2n + 1}}{16^n \times 4^{n-1}}\text{= ?}$MF#% | |
A. 256 | B. 64 |
C. 16 | D. 9 |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
$MF#%\begin{align}&\dfrac{(1024)^{n/5} \times 4^{2n + 1}}{16^n \times 4^{n-1}}\\\\
&= \dfrac{(4^5)^{n/5} \times 4^{2n + 1}}{(4^2)^n \times 4^{n-1}}\\\\
&= \dfrac{4^n \times 4^{2n + 1}}{4^{2n} \times 4^{n-1}}\\\\
&= 4^{(n + 2n + 1) - (2n)-(n-1)}\\\\
&= 4^2\\\\
&= 16
\end{align} $MF#%
12. $MF#%\text{If }\left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{b}{a}\right)^{x-8}\text{, what is the value of x?}$MF#% | |
A. 4 | B. 6 |
C. 8 | D. 10 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
$MF#%\begin{align}&\left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{b}{a}\right)^{x-8}\\\\
&\Rightarrow \left(\dfrac{a}{b}\right)^{x-4} = \left(\dfrac{a}{b}\right)^{-(x-8)}\\\\
&\Rightarrow x-4 = -(x-8)\\\\
&\Rightarrow x-4 = -x + 8\\\\
&\Rightarrow 2x = 12\\\\
&\Rightarrow x = \dfrac{12}{2} = 6\end{align} $MF#%
13. If 1000.20 = x, 100.60 = y and xz = y2, what is the value of z? | |
A. 3 | B. 6 |
C. 4.2 | D. 2.2 |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
x = (100)0.2
y = (10)0.6
xz = y2
=> (1000.2)z = (100.6)2
=> 1000.2z = 10(0.6 × 2)
=> (102)0.2z = 10(0.6 × 2)
=> 10(2 × 0.2z)= 10(0.6 × 2)
=> 2 × 0.2z = 0.6 × 2(∵ powers are equal as base values are equal in both side)
=> 0.2z = 0.6
$MF#%\Rightarrow z = \dfrac{.6}{.2} = \dfrac{6}{2} = 3$MF#%
14. $MF#%\left(\dfrac{x^q}{x^r}\right)^{(q + r - p)} . \left(\dfrac{x^r}{x^p}\right)^{(r + p - q)} . \left(\dfrac{x^p}{x^q}\right)^{(p + q - r)}\text{ = ?}$MF#% | |
A. x(a - b - c) | B. .5 |
C. 1 | D. x(a + b + c) |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
$MF#%\begin{align}&\left(\dfrac{x^q}{x^r}\right)^{(q + r - p)} . \left(\dfrac{x^r}{x^p}\right)^{(r + p - q)}
. \left(\dfrac{x^p}{x^q}\right)^{(p + q - r)} \\\\
&= \left[x^{(q-r)}\right]^{(q + r - p)} . \left[x^{(r-p)}\right]^{(r + p - q)}
. \left[x^{(p-q)}\right]^{(p + q - r)} \end{align} $MF#%
= x[(q - r)(q + r - p) + (r - p)(r + p - q) + (p - q)(p + q - r)]
= x[(q - r)(q + r)- p(q - r) + (r - p)(r + p) - q(r - p) + (p - q)(p + q) - r(p - q)]
= x[(q2 - r2) - p(q - r) + (r2 - p2) - q(r - p) + (p2 - q2) - r(p - q)]
= x[(q2 - r2 + r2 - p2 + p2 - q2) - p(q - r) - q(r - p) - r(p - q)]
= x[0 - p(q - r) - q(r - p) - r(p - q)]
= x[- p(q - r) - q(r - p) - r(p - q)]
= x[-pq + pr -qr + pq - rp + rq ]
= x[0]
= 1
15. $MF#%\dfrac{1}{1 + x^{(q - p)} + x^{(r - p)}} + \dfrac{1}{1 + x^{(p - q)} + x^{(r - q)}} + \dfrac{1}{1 + x^{(q - r)} + x^{(p - r)}}\text{= ?}$MF#% | |
A. 0 | B. 1 |
C. .5 | D. 2 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
$MF#%\begin{align}&\dfrac{1}{1 + x^{(q - p)} + x^{(r - p)}} + \dfrac{1}{1 + x^{(p - q)} + x^{(r - q)}}
+ \dfrac{1}{1 + x^{(q - r)} + x^{(p - r)}}\\\\\\\\
&= \dfrac{1}{1 + \dfrac{x^q}{x^p} + \dfrac{x^r}{x^p}} + \dfrac{1}{1 + \dfrac{x^p}{x^q}
+ \dfrac{x^r}{x^q}} + \dfrac{1}{1 + \dfrac{x^q}{x^r} + \dfrac{x^p}{x^r}}\\\\\\\\
&= \dfrac{x^p}{x^p + x^q + x^r} + \dfrac{x^q}{x^q + x^p + x^r} + \dfrac{x^r}{x^r + x^q + x^p}\\\\\\\\
&= \dfrac{x^p + x^q + x^r}{x^p + x^q + x^r}\\\\
&= 1 \end{align} $MF#%
16. 65610.14 × 65610.11 = ? | |
A. 16 | B. 9 |
C. 4 | D. 1 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
65610.14 × 65610.11 = 6561(0.14 + 0.11)= 6561(0.25)
= 6561(1/4) = (38)(1/4) = 38/4 = 32 = 9
17. What is the value of (2 × 4 × 5)5n | |
A. 25n + 45n + 55n | B. (405)n |
C. (40)5n | D. (40n)5 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
(2 × 4 × 5)5n = (40)5n = (405)n = (40n)5
18. $MF#%\text{If }\left(\sqrt{3}\right)^n = 6561\text{, }(n)^{3/2}\text{ = ?}$MF#% | |
A. $MF#%64$MF#% | B. $MF#%64\sqrt{3}$MF#% |
C. $MF#%16\sqrt{3}$MF#% | D. $MF#%16$MF#% |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
$MF#%\begin{align}&\left(\sqrt{3}\right)^n = 6561\\\\
&\Rightarrow \left(\sqrt{3}\right)^n = \left(\sqrt{3}\right)^{16}\\\\
&\Rightarrow n = 16\\\\\\\\
&(n)^{3/2} = (16)^{3/2} = \sqrt{16^3} = 16 \sqrt{16} = 16 \times 4 = 64\end{align} $MF#%
19. 5x × 23 = 36. 5(x+1) = ? | |
A. 22 | B. 21 |
C. 20.5 | D. 22.5 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
$MF#%\begin{align}&5^x \times 2^3 = 36\\\\
&\Rightarrow 5^x = \dfrac{36}{2^3}\\\\
&5^{(x+1)} = 5^x × 5 = \dfrac{36}{2^3} × 5 = \dfrac{36 \times 5}{2 \times 2 \times 2}
= \dfrac{9 \times 5}{2} = 22.5\end{align} $MF#%
20. $MF#%36^{120} = (36 \times x)^{40}\text{. What is the value of x?}$MF#% | |
A. 44 | B. 44 |
C. 62 | D. 64 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
$MF#%\dfrac{a^m}{a^n} = a^{m-n}$MF#%
$MF#%\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m $MF#%
$MF#%\begin{align}&36^{120} = (36 \times x )^{40}\\\\
&\Rightarrow 36^{120} = 36^{40} \times x^{40}\\\\
&\Rightarrow x^{40} = \dfrac{36^{120}}{36^{40}} = 36^{(120 - 40)} = 36^{80}\\\\
&\Rightarrow (\sqrt{x}^2)^{40} = 36^{80}\\\\
&\Rightarrow (\sqrt{x})^{80} = 36^{80}\\\\
&\Rightarrow \sqrt{x} = 36\\\\
&\Rightarrow x = 36^2 = 6^4\end{align} $MF#%
21. (6561)(1/2) + (6561)(1/4) + (6561)(1/8) = ? | |
A. 98 | B. 86 |
C. 93 | D. 81 |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
365613218737293243381327393
Hence, 6561 = 38
(6561)(1/2) + (6561)(1/4) + (6561)(1/8)
= (38)(1/2) + (38)(1/4) + (38)(1/8)
= (3)(8/2) + (3)(8/4) + (3)(8/8)
= (3)4 + (3)2 + (3)1
= 81 + 9 + 3
= 93
22. $MF#%\sqrt{3^n} = 6561. \quad 3^{\sqrt{n}}\text{ = ?}$MF#% | |
A. 81 | B. 9 |
C. 16 | D. 25 |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
365613218737293243381327393
Hence, 6561 = 38
$MF#%\begin{align}&\text{Given that }\sqrt{3^n} = 6561 \\\\
&\sqrt{3^n} = 3^{8}\\\\
&3^n = 3^{16}\\\\
&\Rightarrow n = 16\\\\
&3^{\sqrt{n}} = 3^{\sqrt{16}} = 3^{4} = 81\end{align} $MF#%
23. If 5(a + b) = 5 × 25 × 125 , what is (a + b)2 | |
A. 12 | B. 16 |
C. 34 | D. 36 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
5(a + b) = 5 × 25 × 125 = 51 × 52 × 53 = 5(1 + 2 + 3) = 56
=> (a + b) = 6
=> (a + b)2 = 62 = 36
24. (7-1 - 11-1) + (7-1 + 11-1) = ? | |
A. 2 × 7-1 | B. 2 × 11-1 |
C. 14 | D. 22 |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
(7-1 - 11-1) + (7-1 + 11-1)
= 7-1 + 7-1 - 11-1 + 11-1
= 7-1 + 7-1
= 2 × 7-1
25. (5)1.25 × (12)0.25 × (60)0.75 = ? | |
A. 420 | B. 260 |
C. 200 | D. 300 |
| Discuss |
Here is the answer and explanation
Answer : Option D
Explanation :
(5)1.25 × (12)0.25 × (60)0.75
=(5)1.25 × (12)0.25 × (12 × 5)0.75
= (5)1.25 × (12)0.25 × (12)0.75 × (5)0.75
= (5)(1.25 + 0.75) × (12)(0.25 + 0.75)
= (5)2 × (12)1
= 25 × 12
= 300
26. 36 × 36 × 36 × 36 = 6? | |
A. 10 | B. 8 |
C. 4 | D. 6 |
| Discuss |
Here is the answer and explanation
Answer : Option B
Explanation :
36 × 36 × 36 × 36 = 62 × 62 × 62 × 62 = 6(2+2+2+2) = 68
27. What is the value of (7-14 - 7-15) ? | |
A. 6 × 7-15 | B. 6 × 7-14 |
C. 7 × 7-15 | D. 7 × 7-14 |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
$MF#%a^{n} = \dfrac{1}{a^{-n}}$MF#%
$MF#%\begin{align}&\left(7^{-14} - 7^{-15}\right) = \dfrac{1}{7^{14}} - \dfrac{1}{7^{15}} = \dfrac{7}{7^{15}} - \dfrac{1}{7^{15}}\\\\
&= \dfrac{7 - 1}{7^{15}} =\dfrac{6}{7^{15}} = 6 × 7^{-15}\end{align} $MF#%
28. If 3(n + 4) - 3(n + 2) = 8, What is the value of n? | |
A. 0 | B. -1 |
C. -2 | D. 2 |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
3(n + 4) - 3(n + 2) = 8
3(n + 2 + 2) - 3(n + 2) = 8
3(n + 2) × 3(2) - 3(n + 2) = 8
3(n + 2)[3(2) - 1] = 8
3(n + 2) × 8 = 8
3(n + 2) = 1
=> n + 2 = 0
=> n = -2
29. $MF#%\text{If }2^x = \sqrt[7]{1024}\text{, what is the value of }x \text{ ?}$MF#% | |
A. None of these | B. -7/10 |
C. 10/7 | D. 7/10 |
| Discuss |
Here is the answer and explanation
Answer : Option C
Explanation :
$MF#%\begin{align}&2^x = \sqrt[7]{1024}\\\\
&\Rightarrow 2^x = \sqrt[7]{2^{10}}\\\\
&\Rightarrow 2^x = 2^{10/7}\\\\
&\Rightarrow x = 10/7\end{align} $MF#%
30. $MF#%\left[\left(3^{-2} - 5^{-2}\right)^{17} ÷ \left(3^{-2} - 5^{-2}\right)^{18} \right]^{1/2}\text{=?}$MF#% | |
A. $MF#%\dfrac{15}{4}$MF#% | B. $MF#%\dfrac{4}{15}$MF#% |
C. $MF#%\dfrac{7}{4}$MF#% | D. $MF#%\dfrac{4}{7}$MF#% |
| Discuss |
Here is the answer and explanation
Answer : Option A
Explanation :
$MF#%\begin{align}&\left[\left(3^{-2} - 5^{-2}\right)^{17} ÷ \left(3^{-2} - 5^{-2}\right)^{18} \right]^{1/2}\\\\
&= \left[\left(3^{-2} - 5^{-2}\right)^{(17 - 18)} \right]^{1/2}\\\\
&= \left[\left(3^{-2} - 5^{-2}\right)^{(-1)} \right]^{1/2}\\\\
&= \left[\dfrac{1}{\left(3^{-2} - 5^{-2}\right)} \right]^{1/2}\\\\
&= \left[\dfrac{1}{\left(\dfrac{1}{3^2} - \dfrac{1}{5^2}\right)} \right]^{1/2}\\\\
&= \left[\dfrac{3^2 × 5^2}{\left(5^2 - 3^2\right)} \right]^{1/2}\\\\
&= \left[\dfrac{3^2 × 5^2}{16} \right]^{1/2}\\\\
&= \dfrac{3 \times 5}{4}\\\\
&= \dfrac{15}{4}\end{align} $MF#%
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