Aptitude - Surds and Indices

Exercise :: Surds and Indices - General Questions

Surds and Indices - Important Formulas
Surds and Indices - General Questions

11.

(243)^n/5 x 3^{2n + 1} = ?

9ⁿ x 3^{n - 1}

A.	1
B.	2
C.	9
D.	3ⁿ

Answer: Option C

Explanation:

Given Expression

= (243)^(n/5) x 3^{2n + 1}

9ⁿ x 3^{n - 1}

= (3⁵)^(n/5) x 3^{2n + 1}

(3²)ⁿ x 3^{n - 1}

= (3^{5 x (n/5)} x 3^{2n + 1})

(3²ⁿ x 3^{n - 1})

= 3ⁿ x 3^{2n + 1}

3²ⁿ x 3^{n - 1}

= 3^{(n + 2n + 1)}

3^{(2n + n - 1})

=
3^{3n + 1}

3^{3n - 1}

= 3^{(3n + 1 - 3n + 1)} = 3² = 9.

12.

1 + 1 = ?

1 + a^{(n - m)} 1 + a^{(m - n)}

0

1

2

1

a^{m + n}

Answer: Option C

Explanation:

1 + 1 =

1 + 1

1 + aⁿ

a^m

1 + a^m

aⁿ

1 + a^{(n - m)} 1 + a^{(m - n)}

= a^m + aⁿ

(a^m + aⁿ) (a^m + aⁿ)

= (a^m + aⁿ)

(a^m + aⁿ)

= 1.

13.

If m and n are whole numbers such that mⁿ = 121, the value of (m - 1)^{n + 1} is:

A.	1
B.	10
C.	121
D.	1000

Answer: Option D

Explanation:

We know that 11² = 121.

Putting m = 11 and n = 2, we get:

(m - 1)^{n + 1} = (11 - 1)^{(2 + 1)} = 10³ = 1000.

14.

x^b (b + c - a) . x^c (c + a - b) . x^a ^{(a + b - c)} = ?

x^c x^a x^b

A.	x^abc
B.	1
C.	x^{ab + bc + ca}
D.	x^{a + b + c}

Answer: Option B

Explanation:

Given Exp.

= x^{(b - c)(b + c - a)} . x^{(c - a)(c + a - b)} . x^{(a - b)(a + b - c)}

= x^{(b - c)(b + c) - a(b - c)} . x^{(c - a)(c + a) - b(c - a)}
. x^{(a - b)(a + b) - c(a - b)}

= x^{(b² - c² + c² - a² + a² - b²)} . x^{-a(b - c) - b(c - a) - c(a - b)}

= (x⁰ x x⁰)

= (1 x 1) = 1.

15.

If x = 3 + 22, then the value of x - 1 is:

x

A.	1
B.	2
C.	22
D.	33

Answer: Option B

Explanation:

x - 1 2 = x + 1 - 2

x x

= (3 + 22) + 1 - 2

(3 + 22)

= (3 + 22) + 1 x (3 - 22) - 2

(3 + 22) (3 - 22)

= (3 + 22) + (3 - 22) - 2

= 4.

x - 1 = 2.

x

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