JEE Maths DPP – Biquadratic Equations
SEO Keywords: Biquadratic Equations, Quartic Equations, Reduction to Quadratic, Roots of Polynomials, JEE Advanced Algebra, Symmetric Biquadratic
DPP Reference Key: QE-BIQ-22-012
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: The number of real roots of the biquadratic equation $x^4 + 5x^2 + 6 = 0$ is:
Answer:
- (a) 0
- (b) 2
- (c) 4
- (d) 1
Question 2: If the equation $x^4 – 10x^2 + k = 0$ has four real and distinct roots, then the range of $k$ is:
Answer:
- (a) $(0, 25)$
- (b) $(-\infty, 25)$
- (c) $(0, 10)$
- (d) $(25, \infty)$
Question 3: To solve the equation $(x^2 – 5x + 7)^2 – (x-2)(x-3) = 1$, the best substitution is:
Answer:
- (a) $y = x^2 – 5x$
- (b) $y = x^2 + 5x$
- (c) $y = x – 5$
- (d) $y = x^2 – 5x + 6$
Question 4: If $\alpha, \beta, \gamma, \delta$ are the roots of $x^4 + px^2 + q = 0$, then the value of $\alpha + \beta + \gamma + \delta$ is:
Answer:
- (a) $p$
- (b) $-p$
- (c) $0$
- (d) $\sqrt{q}$
Question 5: The product of all roots of the equation $2x^4 – 8x^2 + 3 = 0$ is:
Answer:
- (a) $3/2$
- (b) $-3/2$
- (c) $4$
- (d) $3$
Question 6: If the roots of $x^4 – 4x^2 + 4 = 0$ are $\alpha, \beta, \gamma, \delta$, then the value of $\alpha^2 + \beta^2 + \gamma^2 + \delta^2$ is:
Answer:
- (a) 4
- (b) 8
- (c) 0
- (d) 16
Question 7: The substitution $y = x + \frac{1}{x}$ is used to solve a biquadratic of the form $ax^4 + bx^3 + cx^2 + bx + a = 0$. After dividing by $x^2$, the term $(x^2 + \frac{1}{x^2})$ becomes:
Answer:
- (a) $y^2$
- (b) $y^2 – 2$
- (c) $y^2 + 2$
- (d) $2y^2 – 1$
Question 8: If $x^4 – 13x^2 + 36 = 0$, the roots are:
Answer:
- (a) $\pm 2, \pm 3$
- (b) $\pm 1, \pm 6$
- (c) $2, 3, 4, 9$
- (d) $\pm \sqrt{2}, \pm \sqrt{3}$
Question 9: The biquadratic equation whose roots are $\pm \sqrt{2}$ and $\pm \sqrt{5}$ is:
Answer:
- (a) $x^4 – 7x^2 + 10 = 0$
- (b) $x^4 + 7x^2 + 10 = 0$
- (c) $x^4 – 10x^2 + 7 = 0$
- (d) $x^4 – 3x^2 + 10 = 0$
Question 10: If $x^4 – ax^2 + 1 = 0$ has four real roots, then the minimum value of $a$ is:
Answer:
- (a) 1
- (b) 2
- (c) 0
- (d) -2
Question 11: If the sum of squares of the roots of $x^4 – kx^2 + 25 = 0$ is 20, then $k$ is:
Answer:
- (a) 10
- (b) 20
- (c) 5
- (d) 15
Question 12: The number of real roots of $(x-1)(x-2)(x-3)(x-4) = 120$ is:
Answer:
- (a) 0
- (b) 2
- (c) 4
- (d) 1
Question 13: If $x = \sqrt{3 + \sqrt{3 + \dots}}$, then $x$ is a root of:
Answer:
- (a) $x^2 – x – 3 = 0$
- (b) $x^4 – x^2 – 3 = 0$
- (c) $x^4 – 6x^2 + x + 9 = 0$
- (d) $x^2 + x + 3 = 0$
Part II: Subjective Questions (Q14–Q15)
Question 14: Solve the equation $x^4 – 10x^3 + 26x^2 – 10x + 1 = 0$ by reducing it to a quadratic equation.
Answer:
[Enter solution here]
Question 15: Find all real roots of the equation $(x+1)(x+2)(x+3)(x+4) = 24$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of distinct real roots of the equation $x^4 – 4x^3 + 4x^2 – 1 = 0$.
Answer:
Question 17: If the roots of $x^4 – 8x^2 + m = 0$ are in A.P., find the value of $9m/4$.
Answer:
Question 18: If $x^4 + px^2 + q = 0$ has roots $\alpha, -\alpha, \beta, -\beta$, find the value of $\alpha^2 + \beta^2$ if $p=-13$.
Answer:
Question 19: Find the sum of all real roots of the equation $x^4 – 5x^2 + 4 = 0$.
Answer:
Question 20: If $f(x) = x^4 – 4x^2 + 3$, find the number of points where $f(x)$ crosses the x-axis.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): The equation $x^4 + x^2 + 1 = 0$ has no real roots.
Reason (R): For $ax^4 + bx^2 + c = 0$, if $a, b, c > 0$, the sum of positive terms cannot be zero for any real $x$.
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): Every biquadratic equation can be solved by reducing it to two quadratic equations.
Reason (R): Ferrari’s method provides a general way to factorize a quartic into two quadratics.
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 | A |
| Q3 | D | Q4 | C |
| Q5 | A | Q6 | B |
| Q7 | B | Q8 | A |
| Q9 | A | Q10 | B |
| Q11 | A | Q12 | B |
| Q13 | A | Q14 | — |
| Q15 | — | Q16 | 2 |
| Q17 | 9 | Q18 | 13 |
| Q19 | 0 | Q20 | 4 |
| Q21 | A | Q22 | A |