JEE Maths DPP – Irrational Equations
SEO Keywords: Irrational Equations, Radical Equations, Extraneous Roots, Squaring Method, JEE Advanced Algebra, Square Root Equations
DPP Reference Key: QE-IRR-22-014
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: The number of real roots of the equation $\sqrt{x+3} = x-3$ is:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) 4
Question 2: When solving an irrational equation by squaring both sides, the roots of the squared equation that do not satisfy the original equation are called:
Answer:
- (a) Imaginary roots
- (b) Real roots
- (c) Extraneous roots
- (d) Repeated roots
Question 3: The solution set of $\sqrt{2x+9} + x = 13$ is:
Answer:
- (a) $\{8, 20\}$
- (b) $\{8\}$
- (c) $\{20\}$
- (d) $\emptyset$
Question 4: The number of real solutions of $\sqrt{x^2 – x + 1} + \sqrt{x^2 + x + 1} = 1$ is:
Answer:
- (a) 0
- (b) 1
- (c) 2
- (d) Infinite
Question 5: If $\sqrt{x-1} – \sqrt{x-4} = 1$, then the value of $x$ is:
Answer:
- (a) 4
- (b) 5
- (c) 6.25
- (d) 1
Question 6: For the equation $\sqrt{3x+1} – \sqrt{x-1} = 2$, the value of $x$ is:
Answer:
- (a) 1
- (b) 5
- (c) 1 and 5
- (d) 0
Question 7: The equation $\sqrt{x+1} = -2$ has:
Answer:
- (a) One real root
- (b) Two real roots
- (c) No real roots
- (d) Infinitely many roots
Question 8: To solve $\sqrt[3]{x+1} + \sqrt[3]{x-1} = \sqrt[3]{2x}$, the appropriate substitution or method is:
Answer:
- (a) Squaring both sides
- (b) Cubing both sides
- (c) $x = \cos \theta$
- (d) $x = 0$ is the only root
Question 9: The number of real roots of $x^2 – 3x + \sqrt{x^2 – 3x + 11} = 1$ is:
Answer:
- (a) 1
- (b) 2
- (c) 3
- (d) 4
Question 10: If $x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots \infty}}}$, then $x$ equals:
Answer:
- (a) 2
- (b) 3
- (c) -2
- (d) 6
Question 11: The equation $\sqrt{x+5} + \sqrt{x+21} = \sqrt{6x+40}$ has:
Answer:
- (a) $x=4$ as a root
- (b) $x=0$ as a root
- (c) $x=4$ and $x=-4$ as roots
- (d) No real roots
Question 12: The sum of real roots of the equation $\sqrt{x^2-4x+3} + \sqrt{x^2-9} = \sqrt{x^2-x-6}$ is:
Answer:
- (a) 3
- (b) 5
- (c) 0
- (d) 7
Question 13: If $\sqrt{x+3-4\sqrt{x-1}} + \sqrt{x+8-6\sqrt{x-1}} = 1$, then $x$ lies in the interval:
Answer:
- (a) $[2, 5]$
- (b) $[5, 10]$
- (c) $[1, 2]$
- (d) $[10, \infty)$
Part II: Subjective Questions (Q14–Q15)
Question 14: Solve for real $x$: $\sqrt{x^2 – 5x + 6} + \sqrt{x^2 – 7x + 10} = x – 2$.
Answer:
[Enter solution here]
Question 15: Solve the equation $\sqrt{3x^2 – 7x – 30} – \sqrt{2x^2 – 7x – 5} = x – 5$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of real roots of $\sqrt{x+1} – \sqrt{x-1} = \sqrt{4x-1}$.
Answer:
Question 17: If the equation $\sqrt{x-k} = x-2$ has exactly one real solution, find the number of integer values of $k$ in the range $[0, 5]$.
Answer:
Question 18: Solve $\sqrt{2x+3} + \sqrt{x+1} = 1$. How many real solutions exist?
Answer:
Question 19: Find the value of $x$ satisfying $x + \sqrt{x-1} = 3$.
Answer:
Question 20: Find the sum of all values of $x$ satisfying $\sqrt{5x^2 – 6x + 8} = 4x – 8$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): Squaring the equation $\sqrt{x} = -2$ gives $x=4$, which is a root of the original equation.
Reason (R): Squaring can introduce extraneous roots that do not satisfy the original radical expression.
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 $\sqrt{x^2+1} = x-1$ has no real solution.
Reason (R): For $x < 1$, the RHS is negative while LHS is positive, and for $x \ge 1$, $(x^2+1) > (x-1)^2$ always.
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 | B | Q4 | A |
| Q5 | C | Q6 | C |
| Q7 | C | Q8 | B |
| Q9 | B | Q10 | B |
| Q11 | A | Q12 | A |
| Q13 | B | Q14 | See Sol |
| Q15 | See Sol | Q16 | 0 |
| Q17 | 4 | Q18 | 0 |
| Q19 | 2 | Q20 | 4 |
| Q21 | C | Q22 | A |