Practice Power indices and surds - quantitative aptitude Online Quiz (set-2) For All Competitive Exams
Q-1) $8^{2/3}$ is equal to :
(a)
(b)
(c)
(d)
$8^{2/3} = (2^3)^{2/3}$
= $2^{3×2/3} = 2^2 = 4$
Q-2) The value of $(√^3{3.5} + √^3{2.5})((√^3{3.5})^2 - √^3{8.75} + (√^3{2.5})^2)$ is :
(a)
(b)
(c)
(d)
$(√^3{3.5} + √^3{2.5})((√^3{3.5})^2 - √^3{8.75} + (√^3{2.5})^2)$
Let $√^3{3.5} = a$ and $√^3{2.5} = b$
Expression
= (a + b)$(a^2 - ab + b^2) = a^3 + b^3$
= $(√^3{3.5})^3 + (√^3{2.5})^3$
= 3.5 + 2.5 = 6
Q-3) If a = 7 - $4√3$, the value of $a^{1/2}+ a^{-1/2}$ is
(a)
(b)
(c)
(d)
a = 7 - $4√3$
$1/a=1/{7-4√3}$
=$1/{7-4√3}×{7+4√3}/{7+4√3}=7+4√3$
$(√a+1/√a)^2=a+1/a+2$
$=7-4√3+7+4√3+2=16$
$√a+1/√a=4$
Q-4) $4^61 + 4^62 + 4^63 + 4^64$ is divisible by
(a)
(b)
(c)
(d)
$4^61 + 4^62 + 4^63 + 4^64$
= $4^61 (1 + 4 + 4^2 + 4^3)$
= $4^61(1 + 4 + 16 + 64)$
= $4^61$ × 85 which is divisible by 17.
Q-5) If $x = 3^{1/3} - 3^{-1/3}$, then $(3x^3 + 9x)$ is equal to
(a)
(b)
(c)
(d)
$x = 3^{1/3}- 3^{-1/3}$
On cubing both sides,
$x^3 = ((3)^{1/3})^3 -((3)^{-1/3})^3 -3×3^{1/3}×3^{-1/3}(3^{1/3}- 3^{-1/3})$
$x^3 = 3 - 3^{ - 1}$ - 3x
$x^3 + 3x = 3 - 1/3$
$x^3 + 3x = {9 -1}/3 = 8/3$
$3x^3$ + 9x = 8
Q-6) $(16^{0.16} × 2^{0.36})$ is equal to
(a)
(b)
(c)
(d)
$(16^{0.16} × 2^{0.36})$
= $(16^{16/100} × 2^{36/100})$
= $(2^{4×16/100} × 2^{36/100})$
= $(2^{64/100 + 36/100}) = (2^{100/100}) = 2$
Q-7) The value of $(256)^{0.16} × (16)^{0.18}$ is :
(a)
(b)
(c)
(d)
Expression
= $(256)^{0.16} × (16)^{0.18}$
= $(4)^{4×0.16} × (4)^{2×0.18}$
= $(4)^{0.64} × (4)^{0.36}$
= $(4)^{0.64+0.36} = (4)^1$ = 4
Q-8) The value of $√{2}^4 + √^3{64} + √^4{2^8}$ is :
(a)
(b)
(c)
(d)
? = $√{2}^4 + √^3{64} + √^4{2^8}$
= $2^{4×1/2} + 4^{3×1/3} + 2^{8×1/4}$
= $2^2 + 4 + 2^2$
= 4 + 4 + 4 = 12
Q-9) $(0.04)^{ - (1.5)}$ is equal to
(a)
(b)
(c)
(d)
Expression = $(0.04)^{ - 1.5}$
= $1/(0.04)^{1.5} = 1/(0.04)^{3/2}$
= $1/(0.04 × 0.04 × 0.04)^{1/2}$
= $1/√{0.0000064}$
= $1/{0.008} = 1000/8 = 125$
Q-10) The value of $√^3{1372} × √^3{1458} ÷ √^3{343}$ is
(a)
(b)
(c)
(d)
$√^3{1372} × √^3{1458} ÷ √^3{343}$
Expression
= ${√^3{1372}×√^3{1458}}/√^3{343}$
= $√^3{{1372×1458}/343}$
= $√^3{5832}$
= $√^3{18×18×18} = 18$
Q-11) If $√3$ = 1.732, then the value of ${9 +2√3}/√3$ is :
(a)
(b)
(c)
(d)
Expression =${9 +2√3}/√3$
=${(9+2√3)×√3}/{√3×√3}$
=${9√3+6}/3=3√3+2$
= 3 × 1.732 + 2 = 5.196 + 2
= 7.196
Q-12) Evaluate : $16√{3/4} - 9√{4/3}$ if $√12$ = 3.46
(a)
(b)
(c)
(d)
Expression
$16√{3/4} - 9√{4/3}$ if $√12$ = 3.46
=$16√{{3×4}/{4×4}} - 9√{{4×3}/{3×3}}$
=${16√12}/4-{9√12}/3$
=$4√12-3√12$
=$√12$=3.46
Q-13) Given $√2$ = 1.414. The value of $√8 +2√32 -3√128 +4√50$ is
(a)
(b)
(c)
(d)
$√8 +2√32 -3√128 +4√50$
=$2√2 +8√2 -3×8√2 +4×5√2$
=$2√2 +8√2 -24√2 +20√2$
= (2 + 8 -24 +20)$√2$
=6$√2$=6×1.414=8.484
Q-14) If $√15$ = 3.88, then what is the value of $√{5/3}$
(a)
(b)
(c)
(d)
$√15$=3.88(Given)
Now, $√{5/3}=√{{5×3}/{3×3}}=√15/3$
=$3.88/3=1.29\ov{3}$
Q-15) If $√7$ = 2.646, then the value of $1/√28$ up to three places of decimal is :
(a)
(b)
(c)
(d)
$1/√28=1/{2√7}$
=$√7/{2√7×√7}=√7/14$
=$2.646/14=0.189$
Q-16) $(1/2)^{- 1/2}$ is equal to
(a)
(b)
(c)
(d)
$(1/2)^{- 1/2} = (2)^{1/2} = √2$
Q-17) If $3^{2x - y} = 3^{x + y} = √27$, then the value of $3^{x - y}$ will be
(a)
(b)
(c)
(d)
$3^{2x - y} = 3^{x + y} = √27 = (3)^{3/2}$
2x - y = $3/2$
4x - 2y = 3 ....(i)
and, $3^{x + y} = (3)^{3/2}$
x + y = $3/2$
2x + 2y = 3 ....(ii)
From equations (i) and (ii)
4x - 2y + 2x + 2y = 3 + 3
6x = 6 ⇒ x = 1
From equation (i),
4 - 2y = 3
2y = 1 ⇒ $y = 1/2$
$3^{x - y} = 3^{1- 1/2}$ = √3$
Q-18) What is the product of the roots of the equation $x^2 - √3$ = 0 ?
(a)
(b)
(c)
(d)
$x^2 - √3$ = 0
$x^2-(3)^{1/2}=0$
$x^2-(3^{1/4})^2=0$
$(x+3^{1/4})(x-3^{1/4})=0$
$x=3^{1/4} or {-3}^{1/4}$
Product of roots
$=3^{1/4}×-3^{1/4}=-√3$
Note : Product of the roots of $ax^2+bx+c=0 is c/a$
Product of the roots of $x^2-b. 0-√3=0 is -√3$
Q-19) Which is the largest among the numbers $√5 , 3√7 , 4√13$
(a)
(b)
(c)
(d)
$√5 , 3√7 , 4√13$
$√5$
$3√7 =√{9×7}=√63$
$√^4{13}=√{4×4×13}=√{208}$
Clearly,$√5<3√7<4√13$
Q-20) The value of $√{5 + 2√{6}} - 1/{√{5 + 2√6}}$ is :
(a)
(b)
(c)
(d)
$√{5 + 2√{6}}$
= $√{3 + 2 + 2 × √{3} × √{2}}$
= $√{(√{3} + √{2})^2} = √3 + √2$
$1/√{5 + 2√{6}} = √3 - √2$
Hence, Expression
= $√3 + √2 - √3 + √2 = 2√2$