# Algebra - Aptitude Questions and Answers

 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 Post/View Answer Post comment Cancel Thanks for your comment.! Write a comment(Click here) ...
 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 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.