Class 10 Practice – Level 3 – Set 2 – Polynomials Leave a Comment / By Gulam Hamza / August 19, 2024 Class 10 Practice – Level 3 – Set 2 – Polynomials Total questions: 15 1. 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 2. 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 3. If $ \displaystyle f\left( x \right)={{x}^{4}}-6{{x}^{3}}+11{{x}^{2}}-6x+1$ , find f (2). Topic: Polynomial Substitution 1 2 3 4 None 4. 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 0 1 2 None 5. 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 6. 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 7. 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 8. 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 9. 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 16 14 10 13 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 1 out of 15 Time's up
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