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 the roots of the equation $a-bx-x^2=0$; $(a > 0; b > 0)$ which statement is true-

  1. positive and same sign

  2. negative and same sign

  3. greater root in magnitude, negative and opposite in signs

  4. greater root in magnitude, positive and opposite in signs
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation is x^2 + bx - a = 0. The product of roots is -a. Since a > 0, the product is negative, meaning the roots have opposite signs. The sum of roots is -b. Since b > 0, the sum is negative, meaning the root with the larger absolute value must be negative.

Multiple choice

Let $f(x) = x^2 + ax + b$ be a quadratic polynomial in which a and b are integers. If for a given integer $n$, $f(n) f(n + 1) = f(m)$ for some integer $m$, then the value of m is
  1. $n(a + b) + ab$
  2. $n^2 + an + b$
  3. $n(n + 1) + an + b$
  4. $n^2 + n + a + b$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

f(n) = n^2 + an + b. f(n+1) = (n+1)^2 + a(n+1) + b = n^2 + 2n + 1 + an + a + b. f(n)f(n+1) = (n^2 + an + b)(n^2 + 2n + 1 + an + a + b). This looks like it should match f(m) = m^2 + am + b. By testing values or algebraic expansion, m = n(n+1) + an + b.

Multiple choice

The least value of $|a|$ for which $\tan q$ and $\cot q$ are the roots of the equation $x^2 + ax + b = 0$ is
  1. $2$
  2. $1$
  3. $1/2$
  4. $0$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Roots are tan(q) and cot(q). Their product is tan(q) * cot(q) = 1. From the equation x^2 + ax + b = 0, the product of roots is b = 1. The sum of roots is tan(q) + cot(q) = -a. Since tan(q) + cot(q) = 2/sin(2q), its range is (-inf, -2] U [2, inf). Thus |a| >= 2. The least value is 2.

Multiple choice

The set of values of $\displaystyle p$ for which the roots of the equation $\displaystyle 3x^{2} + 2x + p\left ( p-1 \right ) = 0$ have opposite signs is
  1. $\displaystyle \left ( -1, -\infty \right )$
  2. $\displaystyle \left ( -\infty, \infty \right )$
  3. $\displaystyle \left ( 2, 6 \right )$
  4. $\displaystyle \left ( 0,  1 \right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For roots to have opposite signs, the product of the roots (c/a) must be negative. Here, p(p-1)/3 < 0, which implies 0 < p < 1.

Multiple choice

One root of the equation $\displaystyle x^{2} - ix - \left ( 1 + i \right )=0$ is $\displaystyle 1 + i$, where $i=\sqrt{-1}$, then other root is
  1. $\displaystyle 1 - i$
  2. $\displaystyle -1 - i$
  3. $\displaystyle -1 + i$
  4. $\displaystyle -1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a quadratic equation with complex coefficients, the roots do not necessarily come in conjugate pairs. Using the sum of roots: (1+i) + r2 = -(-i) = i. Therefore, r2 = i - (1+i) = -1.

Multiple choice

If $\displaystyle \alpha , \beta $ are roots of the equation $\displaystyle 3x^{2}-11x+19=0 $, then it is possible to find the value of $\displaystyle \alpha^{2}+\beta^{2}$.

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

If $\displaystyle \sin \alpha, : \cos \alpha$ are the roots of the equation $\displaystyle cx^{2} + bx + a = 0$, then $\displaystyle a, : b, : c$ satisfies
  1. $\displaystyle c^{2} + 2ac + b^{2} = 0$
  2. $\displaystyle b^{2} - 2ac - c^{2} = 0$
  3. $\displaystyle 2ac - b^{2} - c^{2} = 0$
  4. $\displaystyle b^{2} + 2ac - c^{2} = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Roots are sin(a) and cos(a). Sum = sin(a) + cos(a) = -b/c. Product = sin(a)cos(a) = a/c. Squaring the sum: sin^2(a) + cos^2(a) + 2sin(a)cos(a) = b^2/c^2. 1 + 2(a/c) = b^2/c^2. Multiplying by c^2: c^2 + 2ac = b^2, or b^2 - 2ac - c^2 = 0.

Multiple choice

Let $\displaystyle \alpha, \beta $ are roots of $\displaystyle f\left ( x \right )=3x^{2}-4x+5=0. $ The equation whose roots are $\displaystyle 2\alpha, 2\beta $ is given by $\displaystyle 3x^{2}+8x-20=0$.

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is incorrect but Reason is correct

  4. Both Assertion and Reason are incorrect
Reveal answer Fill a bubble to check yourself
C Correct answer
Multiple choice

A quadratic equation with rational coefficients one of whose roots is $\displaystyle \tan \left| \frac{\pi}{12}\right|$ is
  1. $x^2 - 2x + 1 = 0$
  2. $x^2- 2x + 4 = 0$
  3. $x^2 - 4x + 1 = 0$
  4. $x^2 -4x - 1 = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The root is tan(pi/12) = tan(15 degrees). Using the half-angle formula, tan(15) = 2 - sqrt(3). For a quadratic with rational coefficients, the other root must be the conjugate, 2 + sqrt(3). The sum of roots is 4 and the product is (2-sqrt(3))(2+sqrt(3)) = 4 - 3 = 1. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 4x + 1 = 0.