Quantitative Aptitude

Algebra

3,715 Questions
All Polynomial Roots and Coefficients 2451 Zero-Product Root Finding 667 Linear System Manipulation 162 Polynomial Theorems and Reduction 85 Algebraic Identity Simplification 82 Surds and Indices 70 Modulus and Absolute Value 54 Algebraic Expression Substitution 46 Algebraic Fraction Simplification 33 Logarithmic Equations 25 Repeated Series Sum 12 Linear Equation Word Problems 9

Algebra Questions

Multiple choice

Series I: 64, 192, 48, 144, 36, 108, 27, 81, 20.25, 60.75 Series II: X, 216, Y, Z, 40.5, W, 30.375, 91.125, P, Q If the roots of a quadratic equation are Z Y and ( X − Y 9 ) , then find the quadratic equation whose variable is r.

  1. r2 + 5r + 6 = 0

  2. r2 + 5r - 6 = 0

  3. r2 - 5r + 6 = 0

  4. r2 - 5r - 6 = 0

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Series II follows multiplication by 3 and division by 4 alternately, giving X = 72, Y = 54 and Z = 162. Interpreting the roots as Z/Y and (X - Y)/9 gives roots 3 and 2, so the equation is r^2 - 5r + 6 = 0.

Multiple choice

Directions: In the question given below, equations I and II are given. You have to solve both the equations and give answer. I. 16p2 - 22p + 7 = 0 II. 14q2 + q - 4 = 0

  1. p ≥ q

  2. p < q

  3. p > q

  4. p ≤ q

  5. p = q or relation cannot be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Solving the first quadratic equation gives the roots p = 1/2 and p = 7/8. Solving the second quadratic equation gives the roots q = 1/2 and q = -4/7. Comparing the values, we find that p is always greater than or equal to q.

Multiple choice

Solve the given quadratic equations and answer the following question. Equation A: (x - 4)2 = (-2x)2 + 45 - 22x - R - 9 Equation B: (6t2 + 7t + 5 6 × ( 12 5 t − 18 ) ) = 0 One root of equation A is 8. The largest root of equation A, when multiplied by the largest prime number less than 60, is equal to

  1. 472

  2. 236

  3. 708

  4. 826

  5. Cannot be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Expanding equation A gives 3x^2 - 14x + 20 - R = 0. Since 8 is a root, substitution gives R = 100, and the other root is -10/3. The largest root is therefore 8, and the largest prime below 60 is 59, giving 8 x 59 = 472.

Multiple choice

The roots of the equation 7x2 - 6x + 1 = 0 are tanα and tanβ, where 2α and 2β are the angles of a triangle. Which one of the following is correct?

  1. The triangle is equilateral.

  2. The triangle is isosceles but not right-angled.

  3. The triangle is right-angled.

  4. The triangle is right-angled isosceles.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

tan(alpha) + tan(beta) = 6/7, tan(alpha)*tan(beta) = 1/7. tan(alpha+beta) = (tan(alpha)+tan(beta)) / (1 - tan(alpha)*tan(beta)) = (6/7) / (1 - 1/7) = (6/7) / (6/7) = 1. Thus alpha+beta = 45 degrees. Since 2alpha and 2beta are angles of a triangle, 2alpha + 2beta = 90 degrees. The third angle is 180 - 90 = 90 degrees, so it is right-angled.

Multiple choice

The roots of quadratic equation 2x2 + x + 4 = 0 are:

  1. Positive and negative

  2. Both Positive

  3. Both Negative

  4. No real roots
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The discriminant D = b^2 - 4ac = (1)^2 - 4(2)(4) = 1 - 32 = -31. Since the discriminant is negative, the equation has no real roots.

Multiple choice

Y = a(x + 3)(x – 5) In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy- plane has the vertex (h, k). If the value of k is -8, what is the value of a?

  1. ¾

  2. ½

  3. 5/8

  4. 4/6
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The vertex x-coordinate h is the midpoint of the roots -3 and 5, so h = (-3+5)/2 = 1. The vertex is (1, k). Given k = -8, substitute into the equation: -8 = a(1+3)(1-5) = a(4)(-4) = -16a. So a = -8 / -16 = 1/2.

Multiple choice

Y = 2(x-a)(x-b) In the quadratic equation above, a and b are positive constants, with a > b. If the graph of the equation in the xy- plane is a parabola with vertex (6, -18), what is the value of a?

  1. 3

  2. 6

  3. 9

  4. 12
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Vertex form is y = 2(x - 6)^2 - 18. Expanding 2(x - a)(x - b) = 2(x^2 - (a+b)x + ab). Comparing coefficients, a+b = 12 and ab = 6 + 9 = 15. Solving x^2 - 12x + 27 = 0 gives roots 9 and 3. Since a > b, a = 9.

Multiple choice

g(x) = bx2 – 6x + d In the given quadratic function, b and d are constants. The graph of y = g(x) in the xy-plane is a parabola that opens upward and has a vertex at the point (m, n) where m and n are constants. If n > 0 and g(-4) = g(10), which of the following must be true? I. d > 0 II. b > 0

  1. I only

  2. II only

  3. I and II

  4. Neither I nor II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Parabola opens upward, so b > 0. Vertex x-coordinate m = -(-6)/(2b) = 3/b. Since g(-4) = g(10), the axis of symmetry is x = (-4+10)/2 = 3. Thus 3/b = 3, so b = 1. g(x) = x^2 - 6x + d. Vertex y-coordinate n = g(3) = 9 - 18 + d = d - 9. Given n > 0, d - 9 > 0, so d > 9. Both I (d > 0) and II (b > 0) are true.