Q1. If x = 2 then the value of x+√(x+√(x+√x) ) ….. is –
(a) 1 or 2
(b) 2
(c) 1 or 4
(d) 4
Sol.
let x+√(x+√(x+√(x……)) ) = y
2+√(2+√(2+√(2+√(2……)) ) ) = y
2+√y=y
(√y)^2-(y-2)^2
y^2-4y+4=y
y^2-5y+4=0
(y – 4) (y – 1) = 0
y = 1, 4
Q2. If x + y = 5 and xy = 6, then the value of 1/x^2 +1/y^2 will be –
(a) 13/24
(b) 13/30
(c) 13/32
(d) 13/36
Sol.
1/x^2 +1/y^2 =(y^2 + x^2)/(x^2 y^2 )
=((x + y)^2 – 2xy)/(xy)^2
=(25 – 12)/36=13/36
Q3. If a^x=b^3,b^y=c^3,c^z=a^3, then the value of xyz will be –
(a) 9
(b) 0
(c) 27
(d) None of these
Sol.
a^x=b^3⇒(a^x )^y⇒(b^3 )^y=a^xy=b^3y
a^xyz=((c^3 )^3 )^z
a^xyz=(c^z )^9=a^27
⇒ xyz = 27
Q4. If 2^(x+y) and 2^(x-2y)=1/8, then value of x will be –
(a) 3
(b) 1
(c) 1/3
(d) 0
Sol.
2^(x+y)=(2)^2,2^(x-2y)=(2)^(-3)
x + y = 2
x – 2y = –3
Put y = 2 – x
x – 2 (2 – x) = –3
⇒ 3x = 1
⇒ x=1/3
Q5. The difference of 1/(x^2 + y^2 ) and 1/(x^2 – y^2 ) is –
(a) (-2x^2)/(x^4 -〖 y〗^4 )
(b) (2x^2)/(x^4 – y^4 )
(c) (2y^2)/(x^4 – y^4 )
(d) (-2y^2)/(x^(4 )- y^4 )
Sol.
1/(x^2 + y^2 )-1/(x^2 – y^2 )
⇒((x^(2 )- y^2 )-(x^2 +〖 y〗^2 ))/((x^2 +〖 y〗^2)(x^2 – y^2))
⇒ (-2y^2)/(x^4 – y^4 )
Q6. If 7x-1/(x^2 – 4)=21-1/(x^2 – 4) then x will be –
(a) 3
(b) 4
(c) 5
(d) None of these
Sol.
7x-1/(x^2 – 4)=21-1/(x^2 – 4)
⇒ 7x = 21
⇒ x = 3
Q7. If 4^(x+2y)=2^(4x+10y), then y is equal to –
(a) x/3
(b) (-1)/3 x
(c) 1/2 x
(d) – 3x
Sol.
4^(x+2y)=2^(4x+10y)=4^(2x+5y )
x + 2y = 2x + 5y
⇒ x = – 3y
y=-x/3
Q8. Factors of 2y^2-9y+4 are –
(a) (4y – 1) (y – 2)
(b) (y – 2) (2y – 1)
(c) (y – 4) (2y – 1)
(d) (y – 2) (y – 4)
Sol.
2y^2-9x+4
=2y^2-8y-y+4
=2y(y-4)-1(y-4)
= (2y – 1) (y – 4)
Q9. If 2^(x+y)=32 and 2^(x-y)=4^2, then x^2+y^2 is –
(a) 41/2
(b) 21/2
(c) 41/4
(d) 41/5
Sol.
2^(x+y)=2^5,2^(x-y)=2^4
x + y = 5
x – y = 4
⇒ 2x = 9
⇒ x=9/2,y=1/2
So (x^2+y^2 )=81/4+1/4=41/2
Q10. If y+1/y=12, then y^3+1/y^3 is equal to –
(a) 1680
(b) 1686
(c) 1692
(d) None of these
Sol.
(y+1/y)^3=y^3+1/y^3 +3(y+1/y)
⇒ y^3+1/y^3 =(12)^3-3(12)=1692
