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If x= √(a+b) - √(a-b) / √(a+b) + √(a-b) , then what is bx^{2} -2ax + b equals to?

A) 0

B) 1

C) ab

D) 2ab

Correct Answer :0

Explanation : x = √(a+b) - √(a-b) / √(a+b) + √(a-b)
x = (√(a+b) - √(a-b))*(√(a+b) - √(a-b)) / (√(a+b) + √(a-b)) * (√(a+b) - √(a-b))
x = (√(a+b) - √(a-b))^{2} / (a+b) - (a-b)
x = (a+b) + (a-b) -2*√(a+b)*√(a-b) / 2b
x = 2*a -2*√(a^{2} - b^{2} ) / 2b
x = a - √(a^{2} - b^{2} ) / b
a - b*x= √(a^{2} - b^{2} )
After squaring both the sides
(a - bx )^{2} = a^{2} - b^{2}
a^{2} + b^{2} x^{2} - 2abx = a^{2} - b^{2}
b^{2} x^{2} - 2abx + b^{2} = 0
bx^{2} - 2ax + b = 0

If x = t ^{1 / t-1} and y = t ^{t / t-1} , t > 0, t ≠ 1 then what is the relation between x and y ?

A) y^{x} = x^{1/y}

B) x^{1/y} = y^{1/x}

C) x^{y} =y^{x}

D) x^{y} = y^{1/x}

Correct Answer :x^{y} =y^{x}

Explanation : Now, y = t ^{t / t-1} = ( t ^{1 / t-1} )^{t } = x^{t} …..(i)
y / x = t ^{t / t-1 } / t ^{1 / t-1}
=^{ } t ^{t / t-1 -} ^{1 / t-1}
=^{ } t ^{t / t-1 -} ^{1 / t-1}
=^{ } t ^{t -} ^{1 / t - 1}
y / x = ^{ } t …..(ii)
From Eqs. (i) and (ii), we get y = x ^{y/x}
=> y^{x} = x^{y}

If x=2 + 2^{1/3} + 2^{2/3} , then the value of x^{3} - 6x^{2 } + 6x is?

A) 3

B) 2

C) 1

D) 0

Correct Answer :2

Explanation : x = 2 + 2^{1/3} + 2^{2/3 } …..(i)
x - 2 = 2^{1/3 * } (2^{1/3 } + 1)
After cubing both the sides
(x - 2)^{3} = 2* (2^{1/3 } + 1)^{3}
x^{3} - 3*x^{2 } *2 + 3*x*2^{2} - 2^{3} = 2*( (2^{1/3} )^{3} + 3*(2^{1/3} )^{2} *1 + 3*2^{1/3} *1^{2} + 1^{3} )
x^{3} - 6x^{2} + 12x - 8 = 2*(2 + 3*2^{2/3} + 3*2^{1/3} + 1)
x^{3} - 6x^{2} + 12x - 8 = 2*(3 + 3*2^{2/3} + 3*2^{1/3} )
x^{3} - 6x^{2} + 12x - 8 = 6*(1+ 2^{2/3} + 2^{1/3} )
From Eqs (i)
x^{3} - 6x^{2} + 12x - 8 = 6*(1+ x - 2)
x^{3} - 6x^{2} + 12x - 8 = 6(x - 1)
x^{3} - 6x^{2} + 12x - 8 = 6x - 6
x^{3} - 6x^{2} + 6x = 2

If √(x/y) = 24/5 + √(y/x) and x + y = 26, then what is the value of xy?

A) 5

B) 15

C) 25

D) 30

Correct Answer :25

Explanation : Let z=√(x/y) , then, z = 24/5 + 1/z
z = 24z + 5 / 5z
5z2 - 24z - 5 = 0
5z2 - 25z + z - 5 = 0
5z( z - 5) + 1( z - 5) = 0
( z - 5)(5z + 1) = 0
z = 5 or -1/5 that is √(x/y)= 5 or -1/5
Let consider
√(x/y)= 5
x/y= 25
x = 25y ....(i)

As is given x + y = 26
then 25y + y = 26 From Eqs (i)
26y = 26
y = 1
Hence x = 25
and xy = 25 * 1 = 25

If α and β are the roots of the equation x^{2} + px + q = 0, then what is α^{2} + β^{2} equal to?

A) p^{2} - 2q

B) q^{2} - 2p

C) p^{2} + 2q

D) q^{2} - q

Correct Answer :p^{2} - 2q

Explanation : As α αnd β αre the roots of the equqtion x2 + px + q = 0
therefore α + β = - p αnd αβ = q
Now,
α^{2} + β^{2} = (α + β)^{2} - 2αβ
= (- p)^{2} - 2q
= p^{2} - 2q

If a^{3} = 335 + b^{3} and a = 5 + b, then what is the value of a + b (given that a > 0 and b > 0)?

A) 7

B) 9

C) 16

D) 49

Correct Answer :9

Explanation : As is given, a^{3} = 335 + b^{3 } and a = 5 + b , thus
a^{3} - b^{3} = 335 …(i)
a - b = 5 …(ii)

As, (a - b)^{3} = a^{3} - b^{3} - 3ab(a - b)
From Eqs (i) and (ii)
5^{3} = 335 - 3ab(5)
125 = 335 - 15ab
ab = 14
Also, (a + b)^{2} = (a - b)^{2} + 4ab
= 5^{2} + 4*14
= 25 + 56
= 81

Hence a + b = 9

If 9^{x } 3^{y} = 2187 and 2^{3x } 2^{2y} - 4^{xy } = 0 , then what can be the value of (x + y )?

A) 1

B) 3

C) 5

D) 7

Correct Answer :5

Explanation : 9^{x} * 3^{y} = 2187
(3^{2} )^{x} . 3^{y} = 2187
3^{2x + y} = 3^{7}
2x + y = 7 …...(i)
Again,
2^{3x } 2^{2y} - 4^{xy} = 0
2^{3x} * 2^{2y} = 4^{xy}
2^{3x +2y} = (2^{2} )^{xy}
3x + 2y = 2xy …...(ii)
From Eqs. (i) and (ii)
3x + 2(7 - 2x ) = 2x(7 - 2x )
3x + 14 - 4x = 14x -4x^{2}
4x^{2} - 15x + 14 = 0
(x - 2)(4x - 7) = 0
Thus x = 2 or 7/4
y = 3 or 7/2

x + y = 5 or 21/4

The pair of linear equations kx + 3y +1 = 0 and 2x + y + 3 = 0 intersect each other, if?

A) k = 6

B) K ≠ 6

C) k = 0

D) k ≠ 0

Correct Answer :K ≠ 6

Explanation : linear equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0
Has unique solution if a1/a2 ≠ b1/b2

for the euqation,
kx + 3y + 1 = 0
2x + y + 3 = 0
the unique solution,
k/2 ≠ 3/1
k ≠ 6

The values of x which satisfy the equation 5^{1 + x} +5^{1 - x} = 26 are?

A) 18 days

B) 20 days

C) 24 days

D) 25 days

Correct Answer :20 days

Explanation : 5^{1 + x } + 5^{1 - x } = 26
(5*5^{x} ) + (5*5^{- x} ) = 26
(5*5^{x} ) + (5/5^{x} ) = 26
5(5^{x} + 1/5^{x} ) = 26

Let 5^{x} = y , then
5y^{2} - 26y +5 = 0
5y^{2} - 25y - y + 5 = 0
5y( y - 5) - 1( y - 5) = 0
( y - 5)(5y - 1) = 0
y = 5 , 1/5
that is , 5^{x} = 5 or 5^{-1}
therefore x = 1 , -1

If a + b = 5 and ab = 6, then what is the value of a^{3} + b^{3} ?

A) 35

B) 40

C) 90

D) 125

Correct Answer :35

Explanation : a^{3} + b^{3} = (a + b)^{3} - 3ab(a + b)
= (5)^{3} - 3 * 6 * 5 = 125 - 90 = 35

If (7 - 12x) - (3x - 7) = 14, then the value of x is ?

A) -4

B) 0

C) 5

D) 2

Correct Answer :0

Explanation : (7 - 12x) - (3x - 7) = 14
7 - 12x - 3x + 7 = 14
- 15x + 14 = 14
- 15x = 0
x = 0

Find the roots of the quadratic equation 6x^{2} - 11x - 35 = 0

A) 5/3, - 7/2

B) - 5/3, 7/2

C) - 3/5, 2/7

D) 3/5, - 2/7

Correct Answer :- 5/3, 7/2

Explanation : 6x2 - 11x - 35 = 0
6x2 - 21x + 10x - 35 = 0
3x(2x - 7) + 5(2x - 7) = 0
(3x + 5)(2x - 7) = 0
x = -5/3, x = 7/2