Algebra - Aptitude Questions and Answers

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The system of equations 2x + 4y = 6 and 4x + 8y = 8 is ?

A)

consistent with a unique solution


B)

consistent with infinitely many solutions


C)

inconsistent


D)

None of the above



Correct Answer :

inconsistent


Explanation :

We have, 2x + 4y = 6 and 4x + 8y = 8

a1 = 2, b1 = 4, c1 = -6
and
a2 = 4, b2 = 8, c2 = -8
Now,
a1/a2 =2/4 =1/2
b1/b2 =4/8 =1/2
c1/c2 =-6/-8 =3/4

When there is no solution, the equations are called inconsistent. This happens, when the lines are parallel.

Here, a1/a2 = b1/b2 = 1/2 ≠ c1/c2

Hence, system of equation is inconsistent.

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If x= √(a+b) - √(a-b) / √(a+b) + √(a-b) , then what is bx2 -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*√(a2 - b2) / 2b
x = a - √(a2 - b2) / b
a - b*x= √(a2 - b2)
After squaring both the sides
(a - bx )2 = a2 - b2
a2 + b2x2 - 2abx = a2 - b2
b2x2 - 2abx + b2 = 0
bx2 - 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)

yx = x1/y


B)

x1/y = y1/x


C)

xy =yx


D)

xy = y1/x



Correct Answer :

xy =yx


Explanation :

Now, y = t t / t-1 = ( t 1 / t-1)t = xt …..(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
=> yx = xy

If x=2 + 21/3 + 22/3, then the value of x3- 6x2 + 6x is?

A)

3


B)

2


C)

1


D)

0



Correct Answer :

2


Explanation :

x = 2 + 21/3 + 22/3 …..(i)
x - 2 = 21/3 * (21/3 + 1)
After cubing both the sides
(x - 2)3 = 2* (21/3 + 1)3
x3 - 3*x2 *2 + 3*x*22 - 23 = 2*( (21/3)3 + 3*(21/3)2*1 + 3*21/3*12 + 13)
x3 - 6x2 + 12x - 8 = 2*(2 + 3*22/3 + 3*21/3 + 1)
x3 - 6x2 + 12x - 8 = 2*(3 + 3*22/3 + 3*21/3)
x3 - 6x2 + 12x - 8 = 6*(1+ 22/3 + 21/3)
From Eqs (i)
x3 - 6x2 + 12x - 8 = 6*(1+ x - 2)
x3 - 6x2 + 12x - 8 = 6(x - 1)
x3 - 6x2 + 12x - 8 = 6x - 6
x3 - 6x2 + 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 x2 + px + q = 0, then what is α2 + β2 equal to?

A)

p2 - 2q


B)

q2 - 2p


C)

p2 + 2q


D)

q2 - q



Correct Answer :

p2 - 2q


Explanation :

As α αnd β αre the roots of the equqtion x2 + px + q = 0
therefore α + β = - p αnd αβ = q
Now,
α2 + β2 = (α + β)2 - 2αβ
= (- p)2 - 2q
= p2 - 2q

If a3 = 335 + b3 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, a3 = 335 + b3 and a = 5 + b , thus
a3 - b3 = 335 …(i)
a - b = 5 …(ii)

As, (a - b)3 = a3 - b3 - 3ab(a - b)
From Eqs (i) and (ii)
53 = 335 - 3ab(5)
125 = 335 - 15ab
ab = 14
Also, (a + b)2 = (a - b)2 + 4ab
= 52 + 4*14
= 25 + 56
= 81

Hence a + b = 9

If 9x 3y = 2187 and 23x 22y - 4xy = 0 , then what can be the value of (x + y )?

A)

1


B)

3


C)

5


D)

7



Correct Answer :

5


Explanation :

9x * 3y = 2187
(32)x . 3y = 2187
32x + y = 37
2x + y = 7 …...(i)

Again,
23x 22y - 4xy = 0
23x * 22y = 4xy
23x +2y = (22)xy
3x + 2y = 2xy …...(ii)
From Eqs. (i) and (ii)
3x + 2(7 - 2x ) = 2x(7 - 2x )
3x + 14 - 4x = 14x -4x2
4x2 - 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 51 + x +51 - x = 26 are?

A)

18 days


B)

20 days


C)

24 days


D)

25 days



Correct Answer :

20 days


Explanation :

51 + x + 51 - x = 26
(5*5x) + (5*5- x) = 26
(5*5x) + (5/5x) = 26
5(5x + 1/5x) = 26

Let 5x = y , then
5y2 - 26y +5 = 0
5y2 - 25y - y + 5 = 0
5y( y - 5) - 1( y - 5) = 0
( y - 5)(5y - 1) = 0
y = 5 , 1/5
that is , 5x = 5 or 5-1
therefore x = 1 , -1

If a + b = 5 and ab = 6, then what is the value of a3 + b3 ?

A)

35


B)

40


C)

90


D)

125



Correct Answer :

35


Explanation :

a3 + b3 = (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 6x2 - 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

The distance between the points (4,-8) and (k,0) is 10. Find k?

A)

k = 6 or - 2


B)

k = 10 or - 2


C)

k = 10 or - 4


D)

k = 6 or - 4



Correct Answer :

k = 10 or - 2


Explanation :

the distance between the points, c2 = (xA − xB)2 + (yA − yB)2

(K - 4)2 + (0 + 8)2 = (10)2
K2 + 16 - 8K + 64 = 100
K2 - 8K - 20 = 0
K2 - 10K + 2K - 20 = 0
K(K - 10) + 2(K - 10) = 0
(K + 2)(K - 10) = 0
K = - 2, K = 10

What is the equation of the line which passes through the points (2, 3) and (- 4, 1)?

A)

x - 3y = - 7


B)

x + 3y = 7


C)

x - 3y = 7


D)

x + 3y = - 7



Correct Answer :

x - 3y = - 7


Explanation :

Let (2, 3) is (x1, y1) and (-4, 1) is (x2, y2)

The equation of a line passing through two points (x1, y1) and (x2, y2) is given by
y - y1 = m(x - x1) , m is the slope

there, m = (y2 - y1)/(x2 - x1) = (1-3)/(-4-2) = -2/-6 = 1/3

Then the equation is:
y - 3 = (1/3)*(x-2)
3y - 9 = x - 2
3y - x = 7
x - 3y = - 7

Aman and Alok attempted to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in roots (4, 3). Alok made a mistake in writing down the coefficient of x to get roots (3, 2). The correct roots of the equation are?

A)

-4, -3


B)

6, 1


C)

4, 3


D)

-6, -1



Correct Answer :

6, 1


Explanation :

Let quadratic equation be
ax2 + bx + c = 0
If a and b are roots, then
α + β = -b/a
and αβ = c/a

Since Aman made a mistake in writing down the constant term.
α + β = 4 + 3 = 7
and Alok made a mistake in writing down the coefficient of x.
αβ = 3 * 2 = 6
So, equation will be
x2 - (α + β)x + αβ = 0
x2 - 7x + 6 = 0
(x - 6)(x - 1) = 0
x = 6, 1

If 3x + 2y = 12 and xy = 6, then 9x2 + 4y2 = ______.

A)

36


B)

144


C)

72


D)

52



Correct Answer :

72


Explanation :

Given, 3x+2y=12 , xy=6

(3x+2y)2 =122
(3x)2 + (2y)2 + 2 (3x) (2y)=144
9x2 +4y2 +12xy=144
9x2 +4y2 +(12×6)=144
9x2 +4y2 +72=144
9x2 +4y2 =144 - 72
9x2 +4y2 =72

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