Class 10 Practice – Level 3 – Set 2 – Polynomials

Total questions: 15

For the polynomial $ \displaystyle {{x}^{2}}-2px+q=0$ ,if the zeros are in the ratio 3:4 , find the relationship between p and q:

Topic: Relationship between zeros and Coefficients

$ \displaystyle {{p}^{2}}=12q$

$ \displaystyle {{p}^{2}}=13q$

$ \displaystyle {{p}^{2}}=7q$

$ \displaystyle {{p}^{2}}=6q$

None

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

Topic: Nature of Roots

Real and distinct

Real and equal

Complex and distinct

Rational and equal

None

If $ \displaystyle f\left( x \right)={{x}^{4}}-6{{x}^{3}}+11{{x}^{2}}-6x+1$ , find f (2).

Topic: Polynomial Substitution

None

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

Topic: Symmetric polynomial

-1

None

Factorize the polynomial $ \displaystyle {{x}^{4}}+5{{x}^{2}}+4$ .

Topic: Factoring Higher Degree Polynomials

$ \displaystyle \left( {{{x}^{2}}-1} \right)\left( {{{x}^{2}}-4} \right)$

$ \displaystyle \left( {x-1} \right)\left( {{{x}^{3}}+4x+1} \right)$

$ \displaystyle \left( {{{x}^{2}}+1} \right)\left( {{{x}^{2}}-4} \right)$

$ \displaystyle \left( {{{x}^{2}}-1} \right)\left( {{{x}^{2}}-4} \right)$

None

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

1, 3

1, 4

3, 5

2, 3

None

Using the rational root theorem , find the possible rational roots of the polynomial $ \displaystyle 4{{x}^{3}}-3{{x}^{2}}-2x+1$ .

Topic: Rational Root Theorem

$ \displaystyle \pm 1,\pm \frac{1}{2},\pm \frac{1}{4}$

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

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

$ \displaystyle \pm 1,\pm 2,\pm \frac{1}{2},\pm \frac{1}{4}$

None

Is the given expression true or not $ \displaystyle {{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz=\left( {x+y+z} \right)\left( {{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx} \right)$:

Topic: Polynomial Identity

True

False

Cannot be predicted

Incomplete information

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

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

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

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

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

None

10.

For the polynomial $ \displaystyle 4{{x}^{2}}-12x+9=0$ , calculate the discriminant and determine the nature of the roots.

Topic: Discriminant Roots

0, Real and equal

9, Real and distinct

-9, Complex and distinct

16, Rational and equal

None

11.

Find the roots of the equation $ \displaystyle {{x}^{2}}+4x+8=0$ .

Topic: Complex roots

$ \displaystyle -2\pm 2i$

$ \displaystyle -4\pm 2i$

$ \displaystyle -2\pm i$

$ \displaystyle -4\pm i$

None

12.

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

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

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

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

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

None

13.

If $ \displaystyle p\left( x \right)={{x}^{4}}-8{{x}^{3}}+18{{x}^{2}}-16x+5$ ,find p (3).

Topic: Polynomial Substitution

None

14.

Is the given expression true or not $ \displaystyle {{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=\left( {a+b+c} \right)\left( {{{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca} \right)when\text{ a+b+c=0}$

Topic: Polynomial Identity

True

False

Cannot be determined

Incomplete information

None

15.

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

Topic: Polynomial Inequalities

$\displaystyle x\le -1$

$ \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

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Class 10 Practice – Level 3 – Set 2 – Polynomials

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