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 $\alpha, \beta$ are zeros of quadratic polynomial $kx^2 +4x + 4$, find the value of k such that $(\alpha + \beta)^2 -2\alpha \beta= 24.$
  1. $k = 1$ or $k = \displaystyle \frac{11}{3}$
  2. $k = - 1$ or $k = \displaystyle \frac{2}{3}$
  3. $k = 1$ or $k = \displaystyle \frac{1}{3}$
  4. $k = - 1$ or $k = \displaystyle \frac{10}{3}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For kx^2 + 4x + 4, alpha+beta = -4/k and alpha*beta = 4/k. (alpha+beta)^2 - 2*alpha*beta = 16/k^2 - 8/k = 24. 16 - 8k = 24k^2. 3k^2 + k - 2 = 0. (3k-2)(k+1) = 0. k = 2/3 or k = -1.

Multiple choice

The quadratic equations $\displaystyle x^{2}-5x+3=0$ has

  1. no real roots

  2. two distinct real roots

  3. two equal real roots

  4. more than two real roots
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Discriminant D = b^2 - 4ac = (-5)^2 - 4(1)(3) = 25 - 12 = 13. Since D > 0, the equation has two distinct real roots.

Multiple choice

Which of the following equation has $(-1)$ as a root ?
  1. $\displaystyle x^{2}+3x-10=0$
  2. $\displaystyle x^{2}-x-12=0$
  3. $\displaystyle 3x^{2}-2x-5=0$
  4. $\displaystyle 9x^{2}+24x+16=0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Plug x = -1 into each equation. A: 1-3-10 != 0. B: 1+1-12 != 0. C: 3(1)-2(-1)-5 = 3+2-5 = 0. D: 9-24+16 != 0.

Multiple choice

The roots of the equation are $lx^2 + nx+n=0$ are in the ratio $p:q$. Then
  1. $\displaystyle\sqrt{ \frac {p} {q}} +\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =0$
  2. $\displaystyle\sqrt{ \frac {p} {q}} -\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =0$
  3. $\displaystyle \sqrt{\frac {p} {q}} -\sqrt{\frac {q} {p}} -\sqrt{\frac {n} {l}} =0$
  4. $\displaystyle \sqrt{\frac {p} {q}} +\sqrt{\frac {q} {p}} +\sqrt{\frac {n} {l}} =1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

If one root of a quadratic equation with rational coefficient is $\begin{pmatrix}\displaystyle\frac{3\sqrt{5}}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}\end{pmatrix}$ then quadratic equation is
  1. $x^2+2x-1=0$
  2. $x^2-2x-1=0$
  3. $x^2-2\sqrt{2}+1=0$
  4. $x^2+2x-3=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Simplify the root: 3*sqrt(5) / (sqrt(10) + 2*sqrt(5) + 2*sqrt(10) - sqrt(5) - 2*sqrt(5)*sqrt(2)) = 3*sqrt(5) / (3*sqrt(10) - sqrt(5)). Rationalizing this leads to 1 - sqrt(2). If roots are rational coefficients, the other root is 1 + sqrt(2). Sum = 2, Product = 1 - 2 = -1. Equation: x^2 - 2x - 1 = 0.

Multiple choice

The roots of the equation ${x}^{2}-\cfrac{2}{3}x=32$ are
  1. $6,\cfrac{-15}{4}$
  2. $-6,\cfrac{-16}{3}$
  3. $3,16$
  4. $\cfrac{-16}{3},6$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

x^2 - (2/3)x - 32 = 0. Multiply by 3: 3x^2 - 2x - 96 = 0. Using quadratic formula: x = [2 +/- sqrt(4 - 4*3*(-96))] / 6 = [2 +/- sqrt(4 + 1152)] / 6 = [2 +/- sqrt(1156)] / 6 = [2 +/- 34] / 6. Roots are 36/6 = 6 and -32/6 = -16/3.

Multiple choice

The roots of the equation $2{x}^{2}+x-4=0$ are
  1. $1,-4$
  2. $-3,\cfrac {1}{\sqrt {3}}$
  3. $\cfrac{\sqrt {33}-1}{4},\cfrac{-\sqrt {33}-1}{4}$

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

Using the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a for 2x^2 + x - 4 = 0, where a=2, b=1, c=-4. The discriminant is 1^2 - 4(2)(-4) = 1 + 32 = 33. The roots are (-1 +/- sqrt(33)) / 4.