JEE Maths DPP – Reciprocal Equations
SEO Keywords: Reciprocal Equations, Symmetric Equations, Polynomial Transformation, JEE Advanced Algebra, Solving Higher Degree Equations
DPP Reference Key: QE-REC-22-009
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: A polynomial $P(x)$ is called a reciprocal polynomial if $P(x) = x^n P(1/x)$. If $x = \alpha$ is a root of such an equation, then another root is necessarily:
Answer:
- (a) $-\alpha$
- (b) $1/\alpha$
- (c) $\bar{\alpha}$
- (d) $1 – \alpha$
Question 2: The number of real roots of the equation $x^4 – 10x^3 + 26x^2 – 10x + 1 = 0$ is:
Answer:
- (a) 0
- (b) 2
- (c) 4
- (d) 1
Question 3: To solve the reciprocal equation $ax^4 + bx^3 + cx^2 + bx + a = 0$, the substitution $y = x + 1/x$ reduces the equation to a quadratic in $y$ of the form:
Answer:
- (a) $ay^2 + by + (c-2a) = 0$
- (b) $ay^2 + by + c = 0$
- (c) $ay^2 + by + (c-a) = 0$
- (d) $ay^2 + by + 2c = 0$
Question 4: One root of the equation $x^5 – 1 = 0$ is 1. The remaining four roots are the roots of the reciprocal equation:
Answer:
- (a) $x^4 + x^3 + x^2 + x + 1 = 0$
- (b) $x^4 – x^3 + x^2 – x + 1 = 0$
- (c) $x^4 + 1 = 0$
- (d) $x^4 – 1 = 0$
Question 5: The value of $x + 1/x$ if $x^2 – 3x + 1 = 0$ is:
Answer:
- (a) 1
- (b) 2
- (c) 3
- (d) 0
Question 6: If the equation $x^4 + mx^3 + nx^2 + mx + 1 = 0$ has two roots equal to 1, then the value of $2m + n$ is:
Answer:
- (a) -2
- (b) 0
- (c) 2
- (d) 4
Question 7: In a reciprocal equation of odd degree and class I (coefficients equidistant from ends are equal), one root is always:
Answer:
- (a) 0
- (b) 1
- (c) -1
- (d) 2
Question 8: If $x + 1/x = 2 \cos \theta$, then $x^n + 1/x^n$ is equal to:
Answer:
- (a) $2 \cos(n\theta)$
- (b) $2^n \cos \theta$
- (c) $2 \sin(n\theta)$
- (d) $\cos(n\theta)$
Question 9: The product of all real roots of $x^4 – 4x^3 + 5x^2 – 4x + 1 = 0$ is:
Answer:
- (a) 1
- (b) -1
- (c) 2
- (d) 5
Question 10: For the equation $6x^4 – 35x^3 + 62x^2 – 35x + 6 = 0$, the values of $x + 1/x$ are:
Answer:
- (a) $3, 2/3$
- (b) $13/6, 6/13$
- (c) $13/6, 3$
- (d) $5/2, 10/3$
Question 11: If $\alpha$ is a root of $x^4 + ax^3 + bx^2 + ax + 1 = 0$, then $\alpha^2 + 1/\alpha^2$ can be expressed as:
Answer:
- (a) $(x+1/x)^2 – 2$
- (b) $a^2 – 2b$
- (c) $y^2 – 2$ where $y$ is a root of the auxiliary quadratic
- (d) None of these
Question 12: The roots of $x^4 + 1 = 0$ are $\alpha, 1/\alpha, \beta, 1/\beta$. The value of $\alpha + 1/\alpha$ is:
Answer:
- (a) $\pm \sqrt{2}$
- (b) $\pm i\sqrt{2}$
- (c) $\pm 1$
- (d) $\pm 2$
Question 13: Which of the following is NOT a property of reciprocal equations?
Answer:
- (a) Roots occur in reciprocal pairs.
- (b) If the degree is odd, at least one root is 1 or -1.
- (c) The leading coefficient must be 1.
- (d) They can be reduced in degree by half if the degree is even.
Part II: Subjective Questions (Q14–Q15)
Question 14: Solve the reciprocal equation $2x^4 + x^3 – 6x^2 + x + 2 = 0$.
Answer:
[Enter solution here]
Question 15: Find the condition on $a, b$ such that the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has four real and distinct roots.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of real roots of $x^4 + x^3 + x^2 + x + 1 = 0$.
Answer:
Question 17: If one root of $x^3 – 3x^2 + 3x – 1 = 0$ is 1, find the sum of the other two roots.
Answer:
Question 18: If $y = x + 1/x$ transforms $x^4 – 2x^3 + x^2 – 2x + 1 = 0$ into $y^2 + Ay + B = 0$, find $A + B$.
Answer:
Question 19: How many real roots does the equation $x^6 – 1 = 0$ have?
Answer:
Question 20: If $x^4 – 10x^2 + 1 = 0$, find the value of $x^2 + 1/x^2$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): Every reciprocal equation of odd degree has a root $x = -1$ if it is of Class II (coefficients equidistant from ends are equal in magnitude but opposite in sign).
Reason (R): For $P(x)$ of Class II, $P(x) = -x^n P(1/x)$. Putting $x=1$ gives $P(1) = -P(1) \implies P(1)=0$.
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 false but R is true.
- (d) A is true but R is false.
Question 22:
Assertion (A): The equation $x^4 + 1 = 0$ is a reciprocal equation.
Reason (R): The coefficients of $x^k$ and $x^{n-k}$ are equal for $n=4$ in $x^4 + 0x^3 + 0x^2 + 0x + 1 = 0$.
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 | B | Q2 | C |
| Q3 | A | Q4 | A |
| Q5 | C | Q6 | A |
| Q7 | C | Q8 | A |
| Q9 | A | Q10 | D |
| Q11 | C | Q12 | B |
| Q13 | C | Q14 | See Sol |
| Q15 | See Sol | Q16 | 0 |
| Q17 | 2 | Q18 | -3 |
| Q19 | 2 | Q20 | 10 |
| Q21 | C | Q22 | A |