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1. (132)7 × (132)? = (132)11.5.

A. 3

B. 3.5

C. 4

D. 4.5

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%a^m.a^n = a^{m+n}$MF#%
(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

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%a^{n} = \dfrac{1}{a^{-n}}$MF#%

$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

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 = 82 = 4



4. (0.04)-2.5 = ?

A. 125

B. 25

C. 3125

D. 625

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%a^{-n} = \dfrac{1}{a^n}$MF#%

$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

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#%

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#%

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%(a - b)^2 = a^2 - 2ab + b^2$MF#%

$MF#%a^2 - b^2 = (a - b)(a + b)$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

Here is the answer and explanation

Answer : Option D

Explanation :

-------------------------------------------------------------------------
Solution 1
--------------------------------------------------------------------------

$MF#%\dfrac{a^m}{a^n} = a^{m-n}$MF#%
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 

6
46656
6
7776
6
1296
6
216
6
36
6

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

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%\dfrac{a^m}{a^n} = a^{m-n}$MF#%
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

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

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

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%a^{n} = \dfrac{1}{a^{-n}}$MF#%

$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

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m $MF#%
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)

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

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

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%a^m.a^n = a^{m+n}$MF#%

$MF#%\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m $MF#%
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

Here is the answer and explanation

Answer : Option B

Explanation :

$MF#%\left({a^m}\right)^n = a^{mn} = \left({a^n}\right)^m $MF#%
(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#%

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%a^{p/q} = \sqrt[q]{a^p}$MF#%

$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

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%a^m.a^n = a^{m+n}$MF#%

$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

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\left(ab\right)^n = a^nb^n$MF#%

$MF#%\dfrac{a^m}{a^n} = a^{m-n}$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

Here is the answer and explanation

Answer : Option C

Explanation :

  3
6561
3
2187
3
729
3
243
3
81
3
27
3
9
3

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

Here is the answer and explanation

Answer : Option A

Explanation :

  3
6561
3
2187
3
729
3
243
3
81
3
27
3
9
3

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

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

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

Here is the answer and explanation

Answer : Option D

Explanation :

$MF#%\left(ab\right)^n = a^nb^n$MF#%

$MF#%a^m.a^n = a^{m+n}$MF#%
(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

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

Here is the answer and explanation

Answer : Option A

Explanation :

$MF#%a^{-n} = \dfrac{1}{a^n}$MF#%

$MF#%a^{n} = \dfrac{1}{a^{-n}}$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

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

Here is the answer and explanation

Answer : Option C

Explanation :

$MF#%a^{p/q} = \sqrt[q]{a^p}$MF#%

$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#%

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#%





 
 
 
 
Comments(4) Sign in (optional)
showing 1-4 of 4 comments,   sorted newest to the oldest
Harpreet
2015-02-14 09:59:31 
Good One !! More chapters like Ratio & Proportions , Sequence & Series and some statistics topics like Central tendency, Dispersion , Correlation & regression should be added.
(0) (0) Reply
pooja
2014-03-10 18:35:52 
it's very nice method each que.each farmula are used 
so that's very good simplification thank you very much..........
(0) (0) Reply
S.Sathiyendraan
2013-11-25 07:04:03 
It is a very useful to my PGD Entrance Exam
(0) (0) Reply
Prabha
2013-07-25 11:35:55 
This page is very useful for my MBA entrance exam :)
(1) (0) Reply
 
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