Inequalities and applications logarithm dpp pdf download

JEE Maths DPP

Inequalities and Applications (Free PDF)

DPP: LOG-INEQ-2026-004

Part A: Multiple Choice Questions (Q1–Q13)

  1. The domain of the function $f(x) = \sqrt{\log_{1/2}(x – 1)}$ is:
    1. $(1, 2]$
    2. $[1, 2]$
    3. $(1, \infty)$
    4. $(2, \infty)$
  2. The solution set of the inequality $\log_3(x + 2) > \log_3(2x – 1)$ is:
    1. $(1/2, 3)$
    2. $(-\infty, 3)$
    3. $(1/2, \infty)$
    4. $(3, \infty)$
  3. The domain of $f(x) = \log_{10} \left( \frac{x-3}{x+3} \right)$ is:
    1. $(-\infty, -3) \cup (3, \infty)$
    2. $(-3, 3)$
    3. $(3, \infty)$
    4. $(-\infty, -3)$
  4. If $\log_{0.3}(x – 1) < \log_{0.09}(x - 1)$, then $x$ lies in the interval:
    1. $(2, \infty)$
    2. $(1, 2)$
    3. $(-2, -1)$
    4. None of these
  5. The range of the function $f(x) = \log_2(x^2 + 2)$ is:
    1. $[1, \infty)$
    2. $(0, \infty)$
    3. $[2, \infty)$
    4. $(-\infty, \infty)$
  6. The number of integral solutions of $\log_{1/2}(x – 3) > -2$ is:
    1. 4
    2. 3
    3. 2
    4. 5
  7. The domain of $f(x) = \log_x 2$ is:
    1. $(0, \infty)$
    2. $(0, 1) \cup (1, \infty)$
    3. $(1, \infty)$
    4. $[0, \infty)$
  8. Solution of $\log_2(x^2 – 3x + 2) < 1$ is:
    1. $(0, 1) \cup (2, 3)$
    2. $(0, 3)$
    3. $(1, 2)$
    4. $(-\infty, 1) \cup (2, \infty)$
  9. If $\log_{1/3} \log_4(x^2 – 5) > 0$, then $x$ belongs to:
    1. $(-3, -\sqrt{6}) \cup (\sqrt{6}, 3)$
    2. $(-\sqrt{6}, \sqrt{6})$
    3. $(-\infty, -3) \cup (3, \infty)$
    4. None of these
  10. The domain of $f(x) = \sqrt{\log_{10} \frac{5x-x^2}{4}}$ is:
    1. $[1, 4]$
    2. $(0, 5)$
    3. $[1, 5]$
    4. $(1, 4)$
  11. If $\log_{x^2}(x+6) < 1/2$, the solution set is:
    1. $(-6, -2) \cup (3, \infty)$
    2. $(-2, 3)$
    3. $(-6, -2) \cup (-1, 0) \cup (0, 1) \cup (3, \infty)$
    4. None of these
  12. The range of $f(x) = \log_{1/2}(x^2 – 2x + 3)$ is:
    1. $(-\infty, -1]$
    2. $[-1, \infty)$
    3. $(-\infty, 1]$
    4. $(0, \infty)$
  13. The solution of $\log_{x+3}(x^2 – x) < 1$ is:
    1. $(-3, -2) \cup (-1, 0) \cup (1, 3)$
    2. $(-1, 0) \cup (1, 3)$
    3. $(-3, -2) \cup (1, 3)$
    4. $(1, 3)$

Part B: Subjective, Assertion-Reason & Integer Type Questions (Q14–Q22)

Subjective Type Questions

  1. Solve for $x$: $$ \log_{2x+3} x^2 < \log_{2x+3} (2x+3) $$
  2. Find the domain of the function: $$ f(x) = \log_{x-4}(x^2 – 11x + 24) $$

Assertion–Reason Type Questions

Choose the correct option:

  • (A) Both Assertion and Reason are true and Reason is the correct explanation of Assertion
  • (B) Both Assertion and Reason are true but Reason is not the correct explanation
  • (C) Assertion is true but Reason is false
  • (D) Assertion is false but Reason is true
  1. Assertion: The function $f(x) = \log_a x$ is increasing if $a > 1$.
    Reason: The derivative of $\log_a x$ is $$ \frac{1}{x \ln a} $$ which is positive for $x > 0$ when $a > 1$.
  2. Assertion: The inequality $$ \log_{1/2} x > \log_{1/2} y $$ implies $x < y$.
    Reason: $\log_a x$ is a decreasing function when $0 < a < 1$.

Integer Type Questions

  1. Find the number of integers in the domain of $$ f(x) = \sqrt{\log_{0.5}(x-2) + 1} $$
  2. If the range of $$ f(x) = \log_3(x^2 + 2x + 10) $$ is $[k, \infty)$, find the value of $k$.
  3. Find the number of integral solutions to $$ \log_2(3x – 2) < 4 $$
  4. Find the minimum value of $$ f(x) = 2^{\log_{10} x} + 2^{\log_{10} (100/x)} $$
  5. If $$ \log_{1/2}(x^2 – 1) \ge -1 $$ find the sum of absolute values of integral solutions.

Answer Key

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13
A A A A A B B A D A D C A
Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22
$(-2, -1) \cup (0, 3)$ $(8, \infty)$ A A 2 2 5 4 0