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

Determine the number of distinct integral roots of the equation (x2 + 16)2 + 80x2 = 18x(x2 + 16).

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let y = x^2 + 16. The equation is y^2 + 80x^2 = 18xy. Substitute y = x^2 + 16: (x^2 + 16)^2 + 80x^2 = 18x(x^2 + 16). x^4 + 32x^2 + 256 + 80x^2 = 18x^3 + 288x. x^4 - 18x^3 + 112x^2 - 288x + 256 = 0. Testing roots: x=4 gives 256 - 1152 + 1792 - 1152 + 256 = 0. Dividing by (x-4)^2 gives (x-4)^2(x^2 - 10x + 16) = 0, which is (x-4)^2(x-8)(x-2) = 0. The distinct roots are 2, 4, 8.

Multiple choice

If a, b and c are the sides of a triangle ABC and equations ax2 + bx + c = 0 and 5x2 + 12x + 13 = 0 have both the roots common, then the given triangle is

  1. acute

  2. obtuse

  3. right

  4. either 1 or 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If ax^2 + bx + c = 0 and 5x^2 + 12x + 13 = 0 have both roots common, then a/5 = b/12 = c/13 = k. Thus a=5k, b=12k, c=13k. Since a, b, c are sides of a triangle, they must satisfy the triangle inequality. However, 5+12=17 > 13, so it is a valid triangle. Checking for a right triangle: 5^2 + 12^2 = 25 + 144 = 169 = 13^2. Since a^2 + b^2 = c^2, it is a right triangle.

Multiple choice

If the roots of the equation x2 - bx + c = 0 are α and β, then the equation whose roots are -α and -β will be

  1. x2 - bx - c = 0

  2. x2 + bx + c = 0

  3. x2 + bx - c = 0

  4. More than one of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If roots are alpha and beta, x^2 - bx + c = 0. Sum = alpha+beta = b, Product = alpha*beta = c. New roots are -alpha and -beta. New sum = -(alpha+beta) = -b. New product = (-alpha)*(-beta) = alpha*beta = c. Equation: x^2 - (sum)x + (product) = 0 -> x^2 - (-b)x + c = 0 -> x^2 + bx + c = 0.

Multiple choice

If the roots of the quadratic equation 2x2 – (7m – 2)x + 18 = 0 are equal and m > 0, then find the quadratic equation if one of the roots is 'm' and the other is -3.

  1. x2 – 9 = 0

  2. x2 + x + 6 = 0

  3. x2 + x - 6 = 0

  4. More than one of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For equal roots, the discriminant D = b^2 - 4ac must be 0. Solving (7m - 2)^2 - 4(2)(18) = 0 gives (7m - 2)^2 = 144, so 7m - 2 = 12 (since m > 0), resulting in m = 2. With roots 2 and -3, the quadratic equation is (x - 2)(x + 3) = x^2 + x - 6 = 0.

Multiple choice

The roots of the equation ax2 + x + b = 0 are equal if

  1. b2 = 4a

  2. b2 < 4a

  3. b2 > 4a

  4. ab = 1/4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For the quadratic equation ax^2 + x + b = 0 to have equal roots, the discriminant D = b^2 - 4ac must be zero. Here, a=a, b=1, c=b. So 1^2 - 4(a)(b) = 0, which means 1 = 4ab, or ab = 1/4.

Multiple choice

If the roots of the equation (a + b)x2 + (a + 2b + c)x + (b + c) = 0 are equal, which of the following statements is always true?

  1. a = c

  2. a + b + c = 0

  3. a = b = c

  4. b = a + c
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For the quadratic equation to have equal roots, the discriminant D = b'^2 - 4a'c' must be 0. Here a' = a+b, b' = a+2b+c, c' = b+c. (a+2b+c)^2 - 4(a+b)(b+c) = 0. Expanding this: (a+2b+c)^2 - 4(ab+ac+b^2+bc) = 0. a^2 + 4b^2 + c^2 + 4ab + 2ac + 4bc - 4ab - 4ac - 4b^2 - 4bc = 0. Simplifying: a^2 - 2ac + c^2 = 0, which is (a-c)^2 = 0. Therefore, a = c.

Multiple choice

The roots of the equation 3x2 - 12x + 10 = 0 are

  1. irrational and unequal

  2. rational and equal

  3. rational and unequal

  4. complex
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Discriminant D = b^2 - 4ac = (-12)^2 - 4(3)(10) = 144 - 120 = 24. Since 24 is positive but not a perfect square, the roots are irrational and unequal.

Multiple choice

Solve the given equation 12x + 6 = 3x - 12

  1. x = - 5

  2. x = - 2

  3. x = 4

  4. x = 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Subtract 3x from both sides to get 9x + 6 = -12. Then, subtract 6 from both sides to get 9x = -18, which simplifies to x = -2.

Multiple choice

Ganesh and Sarath were given a quadratic equation in x to solve. Ganesh made a mistake in copying the constant term of the equation and got the root as 12. Sarath made a mistakes in copying the coefficient of x, as well as the constant term and got the root as 2. Later, they realised that the mistake they committed was only in copying the signs. The difference between the roots of the original equation is

  1. 2

  2. 10

  3. 4

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