## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 8 Indices Chapter Test

Question 1.

If 2^{x} . 3^{y}. 5^{z} = 2160 find the values of x, y and z. Hence compute the value of 3^{x}. 2^{-y} 5^{-z}.

Answer:

2^{x} . 3^{y}. 5^{z} = 2160

=> 2^{x} . 3^{y}. 5^{z} = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5

=> 2^{x} . 3^{y}. 5^{z} = (2)^{4}.(3)^{3} . (5)^{1}

Comparing powers of 2, 3 and 5, on both sides of above equation we get

x = 4, y = 3, z = 1

Also 3^{x}. 2^{-y} 5^{-z} = (3)^{4} × (2)^{-3} × (5)^{-1}

Question 2.

If x = 2 and y = -3, find the values of

(i) x^{x} + y^{y}

(ii) x^{y} + y^{x}.

Answer:

(i) x^{x} + y^{y}, Given that x = 2 and y = 3

Question 3.

If p = x^{m+n} . y^{t} , q = x^{n+l}. y^{m} and r = x^{l+m} . y^{n}. Prove that p^{m-n}. q^{n-l} . r^{l-n} = 1

Answer:

Given that p = x^{m+n}. y^{t} ………(1)

q = x^{n+l}. y^{m} ….(2)

r = x^{l+m} . y^{n} ..(3)

L.H.S. = p^{m-n} . q^{n-l} . r^{l-m} ……(4)

Putting the value of a, b, c from (1), (2), (3) respectively in (4), we get

L.H.S = (x^{m+n}y^{t} )^{m-n} ,(x^{n+l} ,y^{m} )^{n-l} .(x^{l+m} y^{n})^{l-m}

= (x^{m+n})^{m-n} . y^{l(m-n)} . (x)^{(n+l)(n-l)} . y^{m(n-l)}. (x)^{(l+m)(l-m)} y^{m(l-m)}

= (x^{(m+n)(m-n)} y^{lm+ln} . (x)^{(n+l)(n-l)} . y^{mn-l} . (x)^{(l+m)(l-m)} . y^{ln+nm}

= (x)^{m2-n2} . y^{lm-ln} . (x)^{n2-l2} . y^{mn-ml} . (x) ^{l2-m2} . y ^{nl-nm}

= (x)^{m2 – n2 + n2 – l2 + l2 = m2} (y)^{lm-ln+mn-ml+nl-nm}

= (x)^{o}(y)^{o} = 1 × 1 = 1

Hence, Proved L.H.S = R.H.S.

Question 4.

If x = a^{m+n}, y = a^{n+1} and z = a^{l+m}, prove that x^{m}.y^{n}z^{l} = x^{n} y^{t} z^{m}

Answer:

x = a^{m+n},y = a^{n+l}, z = a^{l+m}

L.H.S. =x^{m} y^{n} z^{p}

= a^{m[m+n]} . y^{x[n+l]} . z^{l[l+m]}

= a^{m2+mn} . y^{n2+nl} . z^{l2+lm}

= a^{m2+mn+n2+nl+l2+lm} = a^{l2+m2+n2+lm+mn+np}

R.H.S. = x^{n} . y^{l} . z^{m}

= a^{n(m+n)} . a^{l(n+p)} . a^{m(l+m)}

= a^{mn+n2} . a^{l(n+p)} . a^{m(l+m)}

= a^{mn+n2} . a^{ln+l2} . a^{lm+m2}

= a^{mn+n2+ln+l2+lm+m2}

= a^{l2+m2+n2+lm+mn+nl}

∴ L.H.S. = R.H.S.

Question 5.

Show that

Answer:

Hence, L.H.S = R.H.S, Proved the result.

Question 6.

If x is a positive real number »nd exponents are rational numbers, then simplify the following :

Answer:

Question 7.

Show that:

Answer:

L.H.S =

Question 8.

If 3^{x} = 5^{y} = (75)^{z} show that z = \(\frac{x y}{2 x+y}\).

Answer:

Let 3^{x} = 5^{y} = (75)^{z} = k

Question 9.

Solve the following equations:

Answer: