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

For an equation x2 + 1994x + 3142 = 0, which of the following is a correct statement?

  1. It does not have any real root.

  2. It has an integer root.

  3. Iit has a positive root.

  4. None of these
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The discriminant is positive, so the equation has two real roots. Both roots are negative because their sum is -1994 and their product is positive, and the discriminant is not a perfect square, so neither root is an integer.

Multiple choice

Rakesh and Amit solve an equation. While solving, Rakesh commits a mistake in the constant term and finds the roots 8 and 2. Amit commits a mistake in the coefficient of X and finds the roots - 9 and - 1. Find the correct roots.
    • 8, 2

  2. 9, - 1

  3. 9, 1

  4. 8, 1

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

Rakesh got roots 8 and 2 (sum 10, product 16). He erred in constant term, so sum 10 is correct (coefficient of x is -10). Amit got roots -9 and -1 (sum -10, product 9). He erred in coefficient of x, so product 9 is correct. Equation is x^2 - 10x + 9 = 0. Roots are 9 and 1.

Multiple choice

If the equation an xn + an-1 xn-1 + .......... + a1x = 0, a1 ¹ 0, n ³2 has a positive root x = a, then the equation nanxn-1 + (n-1)an-1xn-2 + ........ + a1 = 0 has a positive root

  1. equal to a

  2. greater than or equal to a

  3. smaller than a

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

Let f(x) = a_n*x^n + ... + a_1*x. f(a) = 0. By Rolle's Theorem, f'(x) = n*a_n*x^(n-1) + ... + a_1 has a root between 0 and a. Thus, the root is smaller than a.

Multiple choice

Roots of the algebraic equation x3 + x2 + x + 1 = 0 are

  1. (+ 1, + i, − i)

  2. (+ 1, − 1, + 1)

  3. (0, 0, 0)

  4. (− 1, + i, − i)
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The equation x^3 + x^2 + x + 1 = 0 can be factored as x^2(x+1) + 1(x+1) = 0, which is (x^2+1)(x+1) = 0. The roots are x = -1 and x^2 = -1, which gives x = i and x = -i.

Multiple choice

The roots of quadratic equation √3 x2+10x+7√3=0 are:

  1. -√3 and 7/√3

  2. √3 and 7/√3

  3. – √3 and (-7)/√3

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

The equation is sqrt(3)x^2 + 10x + 7sqrt(3) = 0. Factoring gives sqrt(3)x^2 + 3x + 7x + 7sqrt(3) = 0, which is sqrt(3)x(x + sqrt(3)) + 7(x + sqrt(3)) = 0. This results in (sqrt(3)x + 7)(x + sqrt(3)) = 0, so the roots are -7/sqrt(3) and -sqrt(3).

Multiple choice

If coefficient of x and constant term of a quadratic equation are taken wrongly roots were found to be 8, 3 and 7, 3 respectively. Find the actual roots of the equation.

  1. 12, 2

  2. 5, 2

  3. 8, 2

  4. 6, 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Wrong coefficient of x: roots 8, 3 => sum = 11, product = 24. Correct constant term is 24. Wrong constant term: roots 7, 3 => sum = 10, product = 21. Correct coefficient of x is -10. Equation: x^2 - 10x + 24 = 0. Roots: (x-6)(x-4) = 0 => 6, 4.