JEE Maths DPP – Exponential and Logarithmic Equations
Keywords: Exponential Equations, Logarithmic Equations, Base Transformation, Variable Substitution, JEE Advanced Algebra
DPP Reference Key: QE-EXP-LOG-22-015
Part I: Multiple Choice Questions (Q1–Q13)
Question 1: The number of real solutions of the equation $4^x – 3 \cdot 2^x + 2 = 0$ is:
Answer:
- (a) 1
- (b) 2
- (c) 0
- (d) 3
Question 2: If $\log_2(x^2 – 1) = 3$, then the value of $x$ is:
Answer:
- (a) 3
- (b) $\pm 3$
- (c) 9
- (d) $\pm \sqrt{7}$
Question 3: The solution set of the equation $9^x – 4 \cdot 3^x + 3 = 0$ is:
Answer:
- (a) $\{0, 1\}$
- (b) $\{1, 3\}$
- (c) $\{0, 3\}$
- (d) $\{1, 2\}$
Question 4: If $\log_{10}x + \log_{10}(x-3) = 1$, then $x$ equals:
Answer:
- (a) 5
- (b) -2
- (c) 5 and -2
- (d) 2
Question 5: The number of real roots of $e^{2x} – e^x – 6 = 0$ is:
Answer:
- (a) 2
- (b) 1
- (c) 0
- (d) 4
Question 6: If $2^{2x+1} – 5 \cdot 2^x + 2 = 0$, then $x$ is:
Answer:
- (a) 1
- (b) -1
- (c) 1 or -1
- (d) 0
Question 7: The solution of $\log_x 2 + \log_2 x = 2.5$ is:
Answer:
- (a) 4
- (b) $\sqrt{2}$
- (c) 4 or $\sqrt{2}$
- (d) 2
Question 8: The value of $x$ satisfying $5^{x+1} + 5^{1-x} = 26$ is:
Answer:
- (a) 1
- (b) -1
- (c) $\pm 1$
- (d) 0
Question 9: If $\log_3(5 + 4\log_3(x-1)) = 2$, then $x$ is:
Answer:
- (a) 2
- (b) 4
- (c) 3
- (d) 10
Question 10: The product of roots of the equation $x^{\log_{10}x} = 100x$ is:
Answer:
- (a) 10
- (b) 100
- (c) 1
- (d) 0.1
Question 11: The number of real solutions of $2^x + 3^x + 4^x = 9$ is:
Answer:
- (a) 1
- (b) 2
- (c) 0
- (d) 3
Question 12: If $7^{\log_x 2} = 2^{\log_x 7}$, the number of solutions for $x$ is:
Answer:
- (a) 0
- (b) 1
- (c) Infinite
- (d) 2
Question 13: The sum of the roots of the equation $\log_2 x \cdot \log_2(x/4) = 3$ is:
Answer:
- (a) 8.5
- (b) 8.25
- (c) 7.5
- (d) 9
Part II: Subjective Questions (Q14–Q15)
Question 14: Solve for real $x$: $3^{2x^2 – 7x + 7} = 9$.
Answer:
[Enter solution here]
Question 15: Solve the equation $\log_2(9 – 2^x) = 3 – x$.
Answer:
[Enter solution here]
Part III: Integer Answer Type (Q16–Q20)
Question 16: Find the number of real roots of $(x-3)^{x^2-5x+6} = 1$.
Answer:
Question 17: If $\log_{10}(x^2-12x+36) = 2$, find the sum of all possible values of $x$.
Answer:
Question 18: Solve $4^x – 2^{x+3} + 12 = 0$. Find the sum of the roots. (Note: $\log_2 a + \log_2 b = \log_2(ab)$).
Answer:
Question 19: Find the number of real solutions of $\log_e x = \sin x$.
Answer:
Question 20: If $2^x \cdot 5^x = 0.1 \cdot (10^{x-1})^5$, find $x$.
Answer:
Part IV: Assertion-Reason (Q21–Q22)
Question 21:
Assertion (A): The equation $\log_2 x = -x$ has exactly one real solution.
Reason (R): $f(x) = \log_2 x$ is an increasing function and $g(x) = -x$ is a decreasing function.
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): The equation $a^x = b$ ($a, b > 0, a \neq 1$) always has a unique real solution.
Reason (R): The function $f(x) = a^x$ is a bijection from $\mathbb{R}$ to $(0, \infty)$.
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 | B |
| Q3 | A | Q4 | A |
| Q5 | B | Q6 | C |
| Q7 | C | Q8 | C |
| Q9 | B | Q10 | A |
| Q11 | A | Q12 | C |
| Q13 | B | Q14 | 1, 2.5 |
| Q15 | 0, 3 | Q16 | 4 |
| Q17 | 12 | Q18 | 2.58 |
| Q19 | 1 | Q20 | 1.5 |
| Q21 | A | Q22 | A |