Class 10 Practice – Level 3 – Set 1 – Polynomials

Total questions: 15

For the polynomial $ \displaystyle 2{{x}^{2}}-5x+3$, if $ \displaystyle \alpha \text{ and }\beta $ are the zeros, what is the value of $ \displaystyle {{\alpha }^{2}}+{{\beta }^{2}}?$

Topic: Advanced Relationship between Zeros and Coefficients

$ \displaystyle \frac{{25}}{4}-\frac{{12}}{4}$

$ \displaystyle \frac{{13}}{4}$

$ \displaystyle \frac{5}{2}+3$

$ \displaystyle \frac{{25}}{4}$

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

None

Determine the nature of the roots of the polynomial $ \displaystyle 3{{x}^{2}}-2x+1$ using the discriminant.

Topic: Nature of Roots Using Discriminant

Real and distinct

Real and equal

Complex and distinct

Rational and equal

None

Find the roots of the equation $ \displaystyle {{x}^{4}}-16{{x}^{2}}+48=0.$

Topic: Polynomial Equations

$ \displaystyle \pm 2\sqrt{3},\pm 2$

$ \displaystyle \pm 4,\pm \sqrt{3}$

$ \displaystyle \pm 3,\pm 4$

$ \displaystyle \pm 2\sqrt{3},\pm 4$

None

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

Topic: Factoring Higher Degree Polynomials

$ \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 sum of the squares of the roots of the polynomial $ \displaystyle {{x}^{3}}-3{{x}^{2}}+3x-1.$

Topic: Roots of Cubic Polynomials

None

Using the Rational Root Theorem, find the possible rational roots of the polynomial $ \displaystyle 2{{x}^{3}}-3{{x}^{2}}-8x+12.$

Topic: Rational Root Theorem

$ \displaystyle \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12$

$ \displaystyle \pm 1,\pm 2,\pm 3,\pm 6$

$ \displaystyle \pm 1,\pm 2,\pm 3,\pm 4,\pm 6$

$ \displaystyle \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12$

None

If $ \displaystyle f(x)={{x}^{3}}-4{{x}^{2}}+5x-2,\text{ find }f(-2).$

Topic: Polynomial Substitution

-36

-10

None

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

Topic: Polynomial Identity

x<3

x>2 and x<3

x>2 or x<3

x>3

None

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

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

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

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

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

None

11.

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

Topic: Complex roots

$ \displaystyle 1\pm 2i$

$ \displaystyle -1\pm 2i$

$ \displaystyle 2\pm i$

$ \displaystyle -2\pm i$

None

12.

If the roots of the polynomial $ \displaystyle {{x}^{2}}+7x+12=0$ are $ \displaystyle \alpha \text{ and }\beta $, find the value of $ \displaystyle \frac{1}{\alpha }+\frac{1}{\beta }.$

Topic: Vieta's Formulas

$ \displaystyle -\frac{7}{{12}}$

$ \displaystyle \frac{{12}}{7}$

$ \displaystyle \frac{7}{{12}}$

$ \displaystyle -\frac{{12}}{7}$

None

13.

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

None

14.

If $ \displaystyle p(x)={{x}^{4}}-4{{x}^{3}}+6{{x}^{2}}-4x+1,\text{ find }p(-1)$

Topic: Polynomial Substitution

-11

None

15.

Solve the inequality $ \displaystyle {{x}^{2}}+4x+4>0.$

Topic: Polynomial Inequalities

$ \displaystyle x<-2$

$ \displaystyle x>-2$

$ \displaystyle x>2$

$ \displaystyle x\ne -2$

None

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Class 10 Practice – Level 3 – Set 1 – Polynomials

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