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

If $\sin A$ and $\cos A$ are roots of the equation $p{x^2} + qx + m = 0$, then the relation among $p,q$ and $m$ is :
  1. ${q^2} + {m^2} = {\left( {p + m} \right)^2}$
  2. ${q^2} - {m^2} = {\left( {p + m} \right)^2}$
  3. ${q^2} + {m^2} = {\left( {p - m} \right)^2}$

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

Roots are sin A and cos A. Sum = sin A + cos A = -q/p. Product = sin A cos A = m/p. Square the sum: sin^2 A + cos^2 A + 2 sin A cos A = q^2/p^2. 1 + 2(m/p) = q^2/p^2. p^2 + 2mp = q^2. Adding m^2 to both sides: p^2 + 2mp + m^2 = q^2 + m^2. (p+m)^2 = q^2 + m^2.

Multiple choice

If $min. (2x^{2} - ax + 2) > max. (b - 1 + 2x - x^{2})$ then roots of the equation $2x^{2} + ax + (2 - b) = 0$, are

  1. Positive and distinct

  2. Negative and distinct

  3. Opposite in sign

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

Min(2x^2 - ax + 2) = 2 - a^2/8. Max(b - 1 + 2x - x^2) = b - 1 + 1 = b. Condition: 2 - a^2/8 > b, so a^2/8 + b < 2. Equation 2x^2 + ax + (2 - b) = 0. Discriminant D = a^2 - 4(2)(2 - b) = a^2 - 16 + 8b = a^2 + 8b - 16. Since a^2 + 8b < 16, D < 0. Roots are imaginary.

Multiple choice

The values of the parameter $a$ for which the quadratic equations $(1 - 2a)x^{2} - 6ax - 1 = 0$ and $ax^{2} - x + 1 = 0$ have at least one root in common are
  1. $0, \dfrac{1}{2}$
  2. $\dfrac{1}{2}, \dfrac{2}{9}$
  3. $\dfrac{2}{9}$
  4. $0, \dfrac{1}{2}, \dfrac{2}{9}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For common roots, the resultant of the two equations must be zero. Alternatively, subtracting the equations or using the condition for common roots leads to a = 2/9. Checking a=2/9 satisfies the condition.

Multiple choice

If the equations $ax^{3}+3bx^{2}+3cx+d=0$ and $ax^{2}+2bx+c=0$ have a common root, then
  1. $\left(bc-ad\right)^{2}=3\left(ac-b^{2}\right)(bd+c^{2})$
  2. $\left(bc-ad\right)^{2}=2\left(ac-b^{2}\right)(bd+c^{2})$
  3. $\left(bc-ad\right)^{2}=4\left(ac+b^{2}\right)(bd+c^{2})$
  4. $\left(bc-ad\right)^{2}=4\left(ac-b^{2}\right)(bd-c^{2})$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

This is a standard condition for a cubic and its derivative to share a root, derived from the resultant of the two polynomials.

Multiple choice

If the equations $ax^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root, where a, b, c are the lengths of sides of a triangle ABC, then $(sin^2\, A+sin^2 B+sin^2C)$ is equal to

  1. 1
  2. $\dfrac{3}{2}$
  3. $\dfrac{5}{2}$

  4. 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For the equations to have a common root, the ratio of coefficients must be consistent. Comparing ax^2 + 2bx + 3c = 0 and 3x^2 + 8x + 15 = 0, we get a/3 = 2b/8 = 3c/15, which simplifies to a/3 = b/4 = c/5. Since a, b, c are sides of a triangle, they can be represented as 3k, 4k, 5k. This is a right-angled triangle (3^2 + 4^2 = 5^2), so A + B = 90 degrees and C = 90 degrees. Thus, sin^2 A + sin^2 B + sin^2 C = sin^2 A + cos^2 A + 1 = 2.