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

Given that Tan A, Tan B are the roots of the equation ${x^2} - bx -c = 0$ then the value of ${\sin ^2}\left( {A + B} \right)$ is
  1. $\dfrac{b}{{{{\left( {b + c} \right)}^2}}}$
  2. $\dfrac{{{b^2}}}{{{b^2} + {c^2}}}$
  3. $\dfrac{{{b^2}}}{{{c^2} + {{\left( {1 - b} \right)}^2}}}$
  4. $\dfrac{{{b^2}}}{{{b^2} + {{\left( {1 - c} \right)}^2}}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

tan A + tan B = b, tan A * tan B = -c. tan(A+B) = (tan A + tan B) / (1 - tan A tan B) = b / (1 + c). sin^2(A+B) = tan^2(A+B) / (1 + tan^2(A+B)) = [b^2 / (1+c)^2] / [1 + b^2 / (1+c)^2] = b^2 / ((1+c)^2 + b^2). The provided option D uses (1-c)^2, which implies a sign difference in the original equation or identity.

Multiple choice

If $\cos \frac { 2 \pi } { 7 } ; \cos \frac { 4 \pi } { 7 } \text { and } \cos \frac { 6 \pi } { 7 }$ are the roots of the equation $8 x ^ { 3 } + 4 x ^ { 2 } - 4 x - 1 = 0$ then the value of $\sin ^ { 2 } \frac { \pi } { 7 } \sin ^ { 2 } \frac { 2 \pi } { 7 } + \sin ^ { 2 } \frac { 2 \pi } { 7 } \sin ^ { 2 } \frac { 3 \pi } { 7 } + \sin ^ { 2 } \frac { \pi } { 7 } \sin ^ { 2 } \frac { 3 \pi } { 7 } =$
  1. $\frac { \sqrt { 7 } } { 4 }$
  2. $\frac { \sqrt { 7 } } { 2 }$
  3. $\frac { \sqrt { 7 } } { 8 }$
  4. $\frac { 7 } { 4 }$
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice

If $\cos^4\theta+\alpha$ and $\sin^4\theta+\alpha$ are the roots of the equation $x^2+4x+2=0$, then values of $\alpha$ are
  1. $2$
  2. $-1$
  3. $-2$

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

The roots are cos^4(theta)+alpha and sin^4(theta)+alpha. Their sum is 1+2alpha = -4 (so alpha = -2.5) and their product is (cos^4(theta)+alpha)(sin^4(theta)+alpha) = 2. Substituting alpha = -2.5 leads to a contradiction, as the values of sin and cos are bounded. Thus, no real alpha satisfies the equation.

Multiple choice

If ABCD is a cyclic quadrilateral such that $\displaystyle 12\tan A-5$ and $5\cos B+3=0$ then the quadratic equation whose roots are $\displaystyle \cos C,\tan D$ is
  1. $\displaystyle 39x^{2}-16x-48=0$
  2. $\displaystyle 39x^{2}+88x-48=0$
  3. $\displaystyle 39x^{2}-88x+48=0$

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

Interpreting the first condition as 12 tan A - 5 = 0 gives tan A = 5/12, so cos A = 12/13. The second condition gives cos B = -3/5 and tan D = 4/3 because A + C = 180 degrees and B + D = 180 degrees. The roots are -12/13 and 4/3, producing 39x^2 - 16x - 48 = 0.

Multiple choice

In the $\displaystyle \triangle ABC$, $BC$ is produced to D and $\displaystyle \angle ACD= \frac{3\pi }{4}$and $\tan A, \tan B$ are roots of the equation $\displaystyle x^{2}-\lambda x+\mu =0.$ Then
  1. $\displaystyle \lambda ^{2}-\mu ^{2}=1+6\mu$
  2. $\displaystyle \lambda ^{2}-\mu ^{2}=1$
  3. $\displaystyle \lambda =\mu -1$

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

Angle C = 180 - 135 = 45 degrees. A + B = 135 degrees. tan(A+B) = (tanA + tanB) / (1 - tanA*tanB) = tan(135) = -1. Given roots are lambda and mu, so (lambda) / (1 - mu) = -1. lambda = -1 + mu, or lambda = mu - 1.

Multiple choice

Consider the cubic equation ${ x }^{ 3 }-\left( 1+\cos { \theta } +\sin { \theta } \right) { x }^{ 2 }+\left( \cos { \theta } \sin { \theta } +\cos { \theta } +\sin { \theta } \right) x-\sin { \theta } \cos { \theta } =0$ whose roots are ${ x }{ 1 },{ x }{ 2 },{ x }_{ 3 }$ The greatest possible difference between two of the roots if $\displaystyle \theta \epsilon \left [ 0,2\pi \right ]$ is
  1. $2$
  2. $1$
  3. $\displaystyle\sqrt{2}$
  4. $\displaystyle 2\sqrt{2} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The cubic equation is (x - 1)(x - cos(theta))(x - sin(theta)) = 0. The roots are 1, cos(theta), and sin(theta). We want to maximize the difference between any two roots. The range of sin and cos is [-1, 1]. The maximum difference occurs between 1 and -1, which is 2.

Multiple choice

Consider the cubic equation ${ x }^{ 3 }-\left( 1+\cos { \theta } +\sin { \theta } \right) { x }^{ 2 }+\left( \cos { \theta } \sin { \theta } +\cos { \theta } +\sin { \theta } \right) x-\sin { \theta } \cos { \theta } =0$ whose roots are ${ x }{ 1 },{ x }{ 2 },{ x }_{ 3 }$ The number of values of $\displaystyle \theta $ in $\displaystyle \left [ 0,2\pi \right ]$ for which at least two roots are equal
  1. $3$
  2. $4$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The cubic equation is (x-1)(x-sin theta)(x-cos theta) = 0. Roots are 1, sin theta, cos theta. At least two roots are equal if: 1 = sin theta (theta = pi/2), 1 = cos theta (theta = 0, 2pi), or sin theta = cos theta (theta = pi/4, 5pi/4). Total values: pi/2, 0, 2pi, pi/4, 5pi/4. That is 5 values.

Multiple choice

If $\tan \theta_{1},\tan \theta _{2},\tan \theta _{3},\tan \theta _{4}$ are the roots of the equation $x^{4}-x^{3}\sin 2\beta + x^{2}\cos 2\beta - x \cos \beta -\sin \beta =0$ then $\tan (\theta _1+\theta _{2}+\theta _{3}+\theta _{4})=$
  1. $\sin \beta$
  2. $\cos \beta$
  3. $ \tan \beta$
  4. $ \cot \beta $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Using Vieta's formulas for the equation x^4 - x^3*sin(2B) + x^2*cos(2B) - x*cos(B) - sin(B) = 0. The sum of roots tan(t1)+tan(t2)+tan(t3)+tan(t4) = sin(2B). The product of roots taken three at a time is cos(B). The product of all four is -sin(B). The formula for tan(t1+t2+t3+t4) involves these symmetric sums, which simplifies to cot(B).

Multiple choice

$\cos \theta \cos 2 \theta \cos 3 \theta = \dfrac{1}{4} (0 \leq \theta \leq \pi)$ Sum of the roots of this equation is
  1. $\pi$
  2. $2 \pi$
  3. $3 \pi$
  4. $4\pi$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation cos(theta)cos(2theta)cos(3theta) = 1/4 can be solved by multiplying both sides by 8sin(theta) to use the identity 2sin(A)cos(A) = sin(2A). This simplifies to sin(4theta)cos(2theta) = sin(theta), leading to roots in the interval (0, pi). The sum of these roots is 3pi.

Multiple choice

If $1$ lies between the roots of the equation $\displaystyle 3x^{2}-3\sin \alpha x-2\cos ^{2}\alpha = 0$, then $\alpha $ lies in the interval
  1. $\displaystyle \left ( 0,\frac{\pi }{2} \right )$
  2. $\displaystyle \left ( \frac{\pi }{12},\frac{\pi }{2} \right )$
  3. $\displaystyle \left ( \frac{\pi }{6},\frac{5\pi }{6} \right )$
  4. $\displaystyle \left ( \frac{\pi }{6},\frac{\pi }{2} \right )\cup \left ( \frac{\pi }{2},\frac{5\pi }{6} \right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let f(x) = 3x^2 - 3sin(a)x - 2cos^2(a). If 1 lies between the roots, then f(1) < 0. So, 3 - 3sin(a) - 2cos^2(a) < 0. Substituting cos^2(a) = 1 - sin^2(a), we get 3 - 3sin(a) - 2 + 2sin^2(a) < 0, which is 2sin^2(a) - 3sin(a) + 1 < 0. Factoring gives (2sin(a) - 1)(sin(a) - 1) < 0. This holds when 1/2 < sin(a) < 1. This corresponds to a in (pi/6, pi/2) U (pi/2, 5pi/6).

Multiple choice

$\tan\alpha, \tan\beta, \tan\gamma$ are the roots of the equation $au^{3}+(20-x)u+y=0$ for fixed $\mathrm{x}$ and $\mathrm{y}$, and $\tan\alpha+\tan\beta=h$, then $ah^{3}+(20-x)h=$
  1. $y$
  2. $-y$
  3. $2a - x$
  4. $a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By Vieta's formulas for au^3 + 0u^2 + (20-x)u + y = 0, the sum of roots is 0. Thus, tan(alpha) + tan(beta) + tan(gamma) = 0. Given tan(alpha) + tan(beta) = h, then tan(gamma) = -h. Since tan(gamma) is a root, a(-h)^3 + (20-x)(-h) + y = 0, which simplifies to -ah^3 - (20-x)h + y = 0, or ah^3 + (20-x)h = y.

Multiple choice

In $\triangle PQR, \,\angle R = \dfrac { \pi }{ 2 } $. If $\tan \left(\dfrac{P}{2}\right)$ and $\tan \left(\dfrac{Q}{2}\right)$ are the roots of the equation $ax^2 + bx + c= 0$, then which one of the following is correct?
  1. $a = b + c$
  2. $b = c + a $
  3. $c= a+ b$
  4. $ b = c$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a right triangle, P + Q = 90 degrees, so P/2 + Q/2 = 45 degrees. Thus, tan(P/2 + Q/2) = tan(45) = 1. Using the tangent sum formula, (tan(P/2) + tan(Q/2)) / (1 - tan(P/2)tan(Q/2)) = 1. Let roots be x1, x2. Then (x1+x2) / (1-x1x2) = 1. From the quadratic, x1+x2 = -b/a and x1x2 = c/a. Substituting gives (-b/a) / (1 - c/a) = 1, which simplifies to -b / (a-c) = 1, so -b = a - c, or c = a + b.

Multiple choice

The roots of the equation ${ \left( z+\alpha \beta \right) }^{ 3 }={ \alpha }^{ 3 }$ represent the vertices of a triangle, one of whose sides is of length
  1. $\sqrt { 3 } \left| \alpha \beta \right|$
  2. $\sqrt { 3 } \left| \alpha \right|$
  3. $\sqrt { 3 } \left| \beta \right|$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation (z + alpha*beta)^3 = alpha^3 implies z + alpha*beta = alpha * omega^k, where omega is a cube root of unity. The roots are z = alpha(omega^k - beta). The distance between two roots is |alpha(omega^k1 - omega^k2)|. For k1=0, k2=1, the distance is |alpha(1 - omega)| = |alpha| * sqrt(3).

Multiple choice

If $z_1,z_2,z_3,$ and $z_4$ are the complex roots of the equation $z^4+3z^2+1=0$, then the value of $\prod { i=1 }^{ 4 }{ 4+\left( z{ i }^{ 2 } \right) } $ is
  1. $9$
  2. $12$
  3. $25$
  4. $27$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The roots of z^4 + 3z^2 + 1 = 0 satisfy z^2 = (-3 +/- sqrt(9-4))/2 = (-3 +/- sqrt(5))/2. Let the roots be a, b, c, d. The product is (4+a^2)(4+b^2)(4+c^2)(4+d^2). Using the polynomial P(z) = z^4 + 3z^2 + 1, we evaluate P(2i) or similar substitutions.