Class 10 Quiz Level 3 (Difficult) – Polynomials

Total questions: 10

If $ \displaystyle \alpha \text{ and }\beta $ are the roots of the polynomial $ \displaystyle {{x}^{2}}+px+q=0,\,\text{express }{{\alpha }^{3}}\text{+}{{\beta }^{3}}$ in terms of $ \displaystyle p\text{ and }q$.

Topic: Symmetric polynomial

Deselect Answer

$ \displaystyle -p({{p}^{2}}+3q)$

$ \displaystyle -p({{p}^{2}}-3q)$

$ \displaystyle {{p}^{3}}+3pq$

$ \displaystyle {{p}^{3}}-3pq$

None

If the roots of the cubic polynomial $ \displaystyle {{x}^{3}}-6{{x}^{2}}+11x-6=0$ are in arithmetic progression, find the common difference.

Topic: Cubic polynomial

Deselect Answer

None

Solve the inequality $ \displaystyle {{x}^{2}}-5x+6>0$

Topic: Polynomial Identity

Deselect Answer

x<3

x>2 and x<3

x>2 or x<3

x>3

None

If $ \displaystyle \alpha ,\beta ,\text{ and }\gamma $ are the roots of the polynomial $ \displaystyle {{x}^{3}}-4{{x}^{2}}+5x-2=0,$ find the value of $ \displaystyle \alpha +\beta +\gamma $.

Topic: Polynomial Roots

Deselect Answer

None

Factorize the polynomial $ \displaystyle {{x}^{4}}-4{{x}^{3}}+6{{x}^{2}}-4x+1.$

Topic: Factoring Higher Degree Polynomials

Deselect Answer

$ \displaystyle {{(x-1)}^{4}}$

$ \displaystyle \left( {x-1} \right)\left( {{{x}^{3}}-3{{x}^{2}}+3x-1} \right)$

$ \displaystyle {{\left( {x-1} \right)}^{3}}\left( {x+1} \right)$

$ \displaystyle {{\left( {x-1} \right)}^{4}}+1$

None

Find the nature of the roots of the equation $ \displaystyle 2{{x}^{2}}+3x+5=0$ .

Topic: Nature of Roots

Deselect Answer

Real and distinct

Real and equal

Complex and distinct

Rational and equal

None

Solve the inequality $ \displaystyle {{x}^{2}}+3x+2\ge 0$

Topic: Polynomial Inequalities

Deselect Answer

$\displaystyle x\le -1$

$ \displaystyle x\le -1\text{ or }x\ge -2$

$ \displaystyle x\le -2\text{ or }x\ge -1$

$ \displaystyle x\ge -2\text{ or }x\ge -1$

None

If $ \displaystyle \alpha \text{ and }\beta $ are the roots of the polynomial $ \displaystyle {{x}^{2}}-px+q=0$ , express $ \displaystyle {{\alpha }^{3}}+{{\beta }^{3}}$ in terms of p and q.

Topic: Symmetric polynomial

Deselect Answer

$ \displaystyle {{p}^{3}}-pq$

$ \displaystyle -p\left( {{{p}^{2}}-3q} \right)$

$ \displaystyle p\left( {{{p}^{2}}-3q} \right)$

$ \displaystyle {{p}^{3}}+3pq$

None

If the roots of the polynomial $ \displaystyle 2{{x}^{2}}+3x+1=0$ are $ \displaystyle \alpha \text{ and }\beta $ , find the value of $ \displaystyle \frac{1}{\alpha }+\frac{1}{\beta }$ .

Topic: Vieta's Formulas

Deselect Answer

$ \displaystyle \frac{3}{2}$

$ \displaystyle \frac{2}{3}$

$ \displaystyle -\frac{3}{1}$

$ \displaystyle \frac{2}{1}$

None

10.

If one root of the cubic polynomial $ \displaystyle {{x}^{3}}-6{{x}^{2}}+11x-6=0$ is 2 , find the other roots.

Topic: Cubic polynomial

Deselect Answer

1, 3

1, 4

3, 5

2, 3

None

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Class 10 Quiz Level 3 (Difficult) – Polynomials

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