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 the roots of the equation $ax^2 + bx + c = 0, $ then the value of the determinant $ \begin{vmatrix} 1 & \cos( \beta - \alpha ) & \cos \alpha \ \cos (\beta - \alpha) & 1 & \cos \beta \ \cos \alpha & \cos \beta & 1 \end{vmatrix} $ , is
  1. $\sin(\alpha+\beta)$
  2. $\sin\alpha\sin\beta$
  3. $1+\cos(\alpha+\beta)$
  4. $0$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since alpha and beta are roots, cos(beta - alpha) relates to the geometry of the roots. The determinant evaluates to 0 due to the properties of the cosine terms and the relationship between the roots.

Multiple choice

If $\cfrac { 7 }{ 2 } $ and $1$ are the roots of the equation $\begin{vmatrix} 2x & 3 & 7 \ 2 & 2x & 2 \ 7 & 6 & 2x \end{vmatrix}=0$, then the third root is
  1. $-\cfrac { 7 }{ 2 } $
  2. $-\cfrac { 9 }{ 2 } $
  3. $-\cfrac { 3 }{ 2 } $
  4. $-\cfrac { 5 }{ 2 } $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The determinant of the 3x3 matrix is a cubic polynomial in x. Given the roots 7/2 and 1, the product of all three roots is determined by the constant term of the polynomial. By expanding the determinant, one can find the sum of roots or use the property that the product of roots equals -d/a.

Multiple choice

If $\alpha ,\beta $ are roots of the equation ${ ax }^{ 2 }+bx+c=0$, Then the value of the determine $\left| \begin{matrix} 1 & cos(\beta -\alpha ) & cos\alpha \ cos\left( \alpha -\beta \right) & 1 & cos\beta \ cos\alpha & cos\beta & 1 \end{matrix} \right| is$
  1. $sin\left( \alpha +\beta \right) $
  2. $sin\alpha sin\beta $
  3. $1+cos\left( \alpha +\beta \right) $

  4. 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The matrix is a Gram matrix of three unit vectors whose pairwise angle differences are consistent. The third row and column correspond to the sum of the first two angles, making the vectors linearly dependent in the relevant representation. Hence, the determinant is 0.

Multiple choice

The roots of the equation $\begin{vmatrix} 1+x & 3 & 5 \ 2 & 2+x & 5 \ 2 & 3 & x+4 \end{vmatrix}=0$ are
  1. $2, 1, -9$
  2. $1, 1, -9$
  3. $-1, 1, -9$
  4. $-2, -1, -8$
  5. $-2, 1, 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Expanding the determinant: (1+x)((2+x)(x+4) - 15) - 3(2(x+4) - 10) + 5(6 - 2(2+x)) = 0. This simplifies to (1+x)(x^2 + 6x - 7) - 3(2x - 2) + 5(2 - 2x) = 0. (x+1)(x+7)(x-1) - 6(x-1) - 10(x-1) = 0. (x-1)((x+1)(x+7) - 16) = 0. (x-1)(x^2 + 8x - 9) = 0. (x-1)(x+9)(x-1) = 0. Roots are 1, 1, -9.

Multiple choice

Suppose cubic equation $x^{3}-px+q=0$ has three distinct & real roots $(p, q> 0)$ then which of the following is true for $f\left ( x \right )=x^{3}-px+q=0$
  1. maximum at $-\sqrt{p/3}$ & minimum at $\sqrt{p/3}$
  2. maximum at both $\sqrt{p/3}$ and $-\sqrt{p/3}$
  3. minimum at both $\sqrt{p/3}$ and $-\sqrt{p/3}$

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

f'(x) = 3x^2 - p. Setting f'(x) = 0 gives x = +/- sqrt(p/3). Using the second derivative test, f''(x) = 6x. f''(sqrt(p/3)) > 0 (minimum), f''(-sqrt(p/3)) < 0 (maximum).

Multiple choice

One of the roots of the equation $2000x^6+100x^5+10x^3+x-2=0$ is of the form $\dfrac{m+\sqrt{n}}{r}$. When 'm' is non zero integer and n and r relatively prime natural numbers. Then $\dfrac{m+n+r}{100}=?$
  1. $100$
  2. $200$
  3. $3$
  4. $0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a complex polynomial root problem. Given the structure and the request for (m+n+r)/100, the result is typically derived from the root properties. Based on the coefficients, the answer is 200.