JEE Maths DPP – Cubic Equations
SEO Keywords: Cubic Equation Roots, Vieta’s Relations, Transformation of Roots, Cubic Polynomials, JEE Advanced Maths, Cardan’s Method basics
DPP Reference Key: QE-CUB-22-011
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If $\alpha, \beta, \gamma$ are the roots of $x^3 – 6x^2 + 11x – 6 = 0$, then the value of $\alpha^2 + \beta^2 + \gamma^2$ is:
Answer:
- (a) 14
- (b) 12
- (c) 10
- (d) 16
Question 2: If the roots of $x^3 – 12x^2 + 39x – 28 = 0$ are in A.P., then the common difference is:
Answer:
- (a) $\pm 3$
- (b) $\pm 2$
- (c) $\pm 1$
- (d) $\pm 4$
Question 3: If the roots of $x^3 – px^2 + qx – r = 0$ are in G.P., then:
Answer:
- (a) $p^3r = q^3$
- (b) $q^3r = p^3$
- (c) $pr^3 = q^3$
- (d) $r p^3 = q^3$
Question 4: The condition that the roots of $x^3 + px^2 + qx + r = 0$ are in H.P. is:
Answer:
- (a) $2q^3 = 27r^2 + 9pqr$
- (b) $2q^3 = 27pr^2 + 9qr$
- (c) $27r^2 + 9pqr + 2q^3 = 0$
- (d) $2q^3 = r(9pq – 27r)$
Question 5: If $\alpha, \beta, \gamma$ are roots of $x^3 + qx + r = 0$, then the value of $\sum \frac{1}{\alpha + \beta}$ is:
Answer:
- (a) $q/r$
- (b) $-q/r$
- (c) $1/r$
- (d) $q/r^2$
Question 6: To remove the second term ($x^2$ term) from $x^3 + 6x^2 + 12x + 10 = 0$, the roots must be diminished by:
Answer:
- (a) 2
- (b) -2
- (c) 1
- (d) -1
Question 7: If $\alpha, \beta, \gamma$ are roots of $x^3 – 2x + 1 = 0$, then the equation whose roots are $\alpha^2, \beta^2, \gamma^2$ is:
Answer:
- (a) $y^3 – 4y^2 + 4y – 1 = 0$
- (b) $y^3 + 4y^2 + 4y + 1 = 0$
- (c) $y^3 – 4y^2 – 4y + 1 = 0$
- (d) $y^3 – 2y^2 + 4y – 1 = 0$
Question 8: If $a, b, c$ are roots of $x^3 – x^2 + 1 = 0$, then the value of $a^{-2} + b^{-2} + c^{-2}$ is:
Answer:
- (a) 0
- (b) 1
- (c) -1
- (d) 2
Question 9: If the product of two roots of $x^3 – 5x^2 – 2x + 24 = 0$ is 12, then the roots are:
Answer:
- (a) 3, 4, -2
- (b) 2, 6, -2
- (c) 1, 12, -8
- (d) 3, 4, 2
Question 10: If $x^3 + 3px^2 + 3qx + r = 0$ has two equal roots, then:
Answer:
- (a) $(pq-r)^2 = 4(p^2-q)(q^2-pr)$
- (b) $(pq-r) = 2(p^2-q)$
- (c) $(p^2-q)^2 = (q^2-pr)$
- (d) $p^2 = q$
Question 11: If $\alpha, \beta, \gamma$ are roots of $x^3 – 7x + 6 = 0$, then the value of $\sum \alpha^3$ is:
Answer:
- (a) -18
- (b) 18
- (c) 0
- (d) 21
Question 12: If $\alpha, \beta, \gamma$ are the roots of $x^3 – px^2 + qx – r = 0$, the value of $(\beta+\gamma)(\gamma+\alpha)(\alpha+\beta)$ is:
Answer:
- (a) $pq – r$
- (b) $pq + r$
- (c) $r – pq$
- (d) $p+q+r$
Question 13: If one root of $x^3 – 2x^2 – x + 2 = 0$ is 1, the other two roots are:
Answer:
- (a) 2, -1
- (b) -2, 1
- (c) 2, 1
- (d) -2, -1
Part II: Subjective Questions (Q14–Q15)
Question 14: Let $\alpha, \beta, \gamma$ be the roots of $x^3 + x + 1 = 0$. Find the value of $\alpha^5 + \beta^5 + \gamma^5$.
Answer:
[Enter solution here]
Question 15: Find the values of $a$ and $b$ for which $x^3 – 6x^2 + ax + b = 0$ has roots in A.P. and the sum of their squares is 20.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: If $\alpha, \beta, \gamma$ are roots of $x^3 – 3x^2 + 1 = 0$, find the value of $\sum \alpha^4$.
Answer:
Question 17: If the roots of $x^3 – 9x^2 + 23x – 15 = 0$ are $\alpha, \beta, \gamma$, find the value of $|\alpha – \beta| + |\beta – \gamma| + |\gamma – \alpha|$.
Answer:
Question 18: Find the value of $k$ if one root of $x^3 – 7x + k = 0$ is double the other.
Answer:
Question 19: If the roots of $x^3 + ax^2 + bx + c = 0$ are $\alpha, \beta, \gamma$, find the value of $\sum \frac{1}{\alpha}$ if $b=6$ and $c=-2$.
Answer:
Question 20: Find the sum of the roots of the equation formed by diminishing the roots of $x^3 – 3x^2 + 4x – 5 = 0$ by 1.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): Every cubic equation with real coefficients must have at least one real root.
Reason (R): Imaginary roots of a polynomial with real coefficients always occur in conjugate pairs.
Answer:
- (a) Both A and R are true and R is the correct explanation of A.
- (b) Both A and R are true but R is NOT the correct explanation of A.
- (c) A is true but R is false.
- (d) A is false but R is true.
Question 22:
Assertion (A): If $\sum \alpha = 0$ for a cubic, then $\sum \alpha^3 = 3\alpha\beta\gamma$.
Reason (R): For any three numbers $a, b, c$, if $a+b+c=0$, then $a^3+b^3+c^3 = 3abc$.
Answer:
- (a) Both A and R are true and R is the correct explanation of A.
- (b) Both A and R are true but R is NOT the correct explanation of A.
- (c) A is true but R is false.
- (d) A is false but R is true.
Answer Key
| Question | Answer | Question | Answer |
|---|---|---|---|
| Q1 | A | Q2 | C |
| Q3 | A | Q4 | D |
| Q5 | A | Q6 | B |
| Q7 | A | Q8 | A |
| Q9 | A | Q10 | A |
| Q11 | A | Q12 | A |
| Q13 | A | Q14 | — |
| Q15 | — | Q16 | 57 |
| Q17 | 4 | Q18 | 6 |
| Q19 | 3 | Q20 | 0 |
| Q21 | A | Q22 | A |