JEE Maths DPP – Higher Degree Equations
SEO Keywords: Polynomial Roots, Vieta’s Relations, Factor Theorem, Remainder Theorem, Cubic Equations, Biquadratic Equations, JEE Advanced Maths
DPP Reference Key: QE-HDE-22-008
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + px^2 + qx + r = 0$, then the value of $\sum \alpha^2$ is:
Answer:
- (a) $p^2 – 2q$
- (b) $p^2 + 2q$
- (c) $q^2 – 2pr$
- (d) $p^2 – q$
Question 2: If $x-1$ and $x+2$ are factors of $x^3 + ax^2 + bx – 4 = 0$, then the values of $a$ and $b$ are:
Answer:
- (a) $a=3, b=0$
- (b) $a=1, b=2$
- (c) $a=3, b=-4$
- (d) $a=5, b=-2$
Question 3: If the roots of the equation $x^3 – 12x^2 + 39x – 28 = 0$ are in A.P., then the common difference is:
Answer:
- (a) $\pm 1$
- (b) $\pm 2$
- (c) $\pm 3$
- (d) $\pm 4$
Question 4: The remainder when $x^{100} – 2x^{99} + 3$ is divided by $(x-1)(x-2)$ is:
Answer:
- (a) $2x + 1$
- (b) $x + 2$
- (c) $3$
- (d) $0$
Question 5: If $\alpha, \beta, \gamma$ are the roots of $x^3 – 7x + 6 = 0$, then the value of $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$ is:
Answer:
- (a) $7/6$
- (b) $-7/6$
- (c) $1/6$
- (d) $0$
Question 6: If the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has roots $1, 2, 3, 4$, then the value of $a$ is:
Answer:
- (a) $10$
- (b) $-10$
- (c) $24$
- (d) $-24$
Question 7: If $\alpha, \beta, \gamma$ are roots of $x^3 + 4x + 1 = 0$, then the value of $(\alpha+\beta)^{-1} + (\beta+\gamma)^{-1} + (\gamma+\alpha)^{-1}$ is:
Answer:
- (a) $4$
- (b) $-4$
- (c) $0$
- (d) $1$
Question 8: A cubic equation whose roots are $1, 2$ and the sum of whose roots is 6 is:
Answer:
- (a) $x^3 – 6x^2 + 11x – 6 = 0$
- (b) $x^3 – 6x^2 + 3x + 10 = 0$
- (c) $x^3 – 6x^2 + 5x + 12 = 0$
- (d) $x^3 – 6x^2 + 11x + 6 = 0$
Question 9: If two roots of $x^3 – 5x^2 – 16x + 80 = 0$ are equal in magnitude but opposite in sign, then the third root is:
Answer:
- (a) $4$
- (b) $5$
- (c) $-5$
- (d) $16$
Question 10: If the product of two roots of $x^3 – 3x^2 – 10x + k = 0$ is -2, then the value of $k$ is:
Answer:
- (a) $8$
- (b) $-8$
- (c) $12$
- (d) $-24$
Question 11: If $f(x)$ is a polynomial such that $f(1)=2$ and $f(2)=1$, then the remainder when $f(x)$ is divided by $x^2-3x+2$ is:
Answer:
- (a) $x+3$
- (b) $-x+3$
- (c) $3x-1$
- (d) $x-3$
Question 12: If $\alpha, \beta, \gamma, \delta$ are roots of $x^4 + qx^2 + rx + s = 0$, then $\sum \alpha \beta \gamma$ is:
Answer:
- (a) $r$
- (b) $-r$
- (c) $0$
- (d) $q$
Question 13: The number of real roots of $x^4 + 4x^3 + 6x^2 + 4x + 5 = 0$ is:
Answer:
- (a) $0$
- (b) $2$
- (c) $4$
- (d) $1$
Part II: Subjective Questions (Q14–Q15)
Question 14: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial with real coefficients. If $P(1) = 1, P(2) = 2,$ and $P(3) = 3,$ find the value of $P(4) – P(0)$.
Answer:
[Enter solution here]
Question 15: Find the sum of the squares of the roots of the equation $x^4 – 8x^3 + 22x^2 – 24x + 9 = 0$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: If $\alpha, \beta, \gamma$ are roots of $x^3 – 2x^2 + 3x – 4 = 0$, find the value of $(1+\alpha)(1+\beta)(1+\gamma)$.
Answer:
Question 17: Find the remainder when $x^{2026} + x^{2025} + 1$ is divided by $x+1$.
Answer:
Question 18: If the roots of $x^3 – 9x^2 + 23x – 15 = 0$ are $a-d, a, a+d$, find the value of $a$.
Answer:
Question 19: If $x^3 + 2x^2 + 3x + 1 = 0$ has roots $\alpha, \beta, \gamma$, find the value of $\alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2$.
Answer:
Question 20: Find the sum of roots of the equation $(x-1)^4 + (x-5)^4 = 82$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): A polynomial of degree $n$ has exactly $n$ roots in the complex number system.
Reason (R): Fundamental Theorem of Algebra states every non-zero single-variable polynomial with complex coefficients has at least one complex root.
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 $\alpha, \beta, \gamma$ are roots of $x^3 – px^2 + qx – r = 0$, then $\alpha\beta\gamma = r$.
Reason (R): For a cubic equation, the product of roots is always equal to the constant term.
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 | C | Q4 | B |
| Q5 | A | Q6 | B |
| Q7 | B | Q8 | A |
| Q9 | B | Q10 | D |
| Q11 | B | Q12 | B |
| Q13 | A | Q14 | 10 |
| Q15 | 20 | Q16 | 10 |
| Q17 | 1 | Q18 | 3 |
| Q19 | 1 | Q20 | 12 |
| Q21 | A | Q22 | C |