Questions covered in Mock 3

Review the question prompts below before starting the timed test. Try each problem independently, then use the interactive test to record your answers and review the worked solutions.

1. If $A$ and $B$ are two sets, then $A \cap (A \cup B)'$ is equal to
2. The domain of the function $f(x) = \sqrt{1 - \log_{10} x}$ is
3. The value of $\sin 20^\circ \sin 40^\circ \sin 60^\circ \sin 80^\circ$ is
4. If $\left(\frac{1+i}{1-i}\right)^m = 1$, then the least positive integer $m$ is
5. The solution of the inequality $3(2-x) \ge 2(1-x)$ is
6. The rank of the word 'MOTHER' in dictionary order is
7. The coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$ is
8. The sum of the infinite series $1 + \frac{2}{3} + \frac{4}{9} + \dots$ is
9. The distance between the parallel lines $3x + 4y = 5$ and $6x + 8y = 15$ is
10. The eccentricity of the hyperbola $9x^2 - 16y^2 = 144$ is
11. The distance of the point $P(1, -2, 3)$ from the origin is
12. $\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{\sin x}$ is
13. The derivative of $e^{\sin x}$ with respect to $x$ is
14. If the variance of a dataset is 25, and each observation is multiplied by 2, the new variance is
15. A card is drawn from a pack of 52 cards. The probability that it is a King or a Heart is
16. The number of non-empty subsets of a set having 4 elements is
17. The minimum value of $3 \cos x + 4 \sin x + 5$ is
18. The smallest positive integer $n$ for which $(1+i)^{2n} = (1-i)^{2n}$ is
19. The solution of $4x + 3 < 6x + 7$ is
20. The number of diagonals of a decagon (10 sides) is
21. Prove that $\tan 3x \tan 2x \tan x = \tan 3x - \tan 2x - \tan x$.
22. Solve the equation $\sqrt{3}x^2 - \sqrt{2}x + 3\sqrt{3} = 0$.
23. Find $r$ if $^5P_r = 2 \cdot ^6P_{r-1}$.
24. Find the derivative of $\frac{\sin x + \cos x}{\sin x - \cos x}$.
25. Evaluate $\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}$.
26. Find the domain and range of the real function $f(x) = \frac{x^2}{1+x^2}$.
27. Prove that $\cos^2 x + \cos^2(x + \frac{\pi}{3}) + \cos^2(x - \frac{\pi}{3}) = \frac{3}{2}$.
28. Find the coefficient of $a^4$ in the product $(1+2a)^4 (2-a)^5$.
29. Find the equation of the circle passing through the points $(4, 1)$ and $(6, 5)$ and whose center is on the line $4x + y = 16$.
30. Find the derivative of $x \sin x \cos x$ with respect to $x$.
31. Solve the system of inequalities graphically: $x + 2y \le 10, x + y \ge 1, x - y \le 0, x \ge 0, y \ge 0$.
32. Insert 5 numbers between 8 and 26 such that the resulting sequence is an A.P.
33. Calculate the Mean Deviation about Median for the following data:
    Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60
    Frequency: 6, 7, 15, 16, 4, 2
34. Find the image of the point $(3, 8)$ with respect to the line $x + 3y = 7$ assuming the line to be a plane mirror.
35. How many numbers greater than 1,000,000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
36. Case Study (Sets): In a town of 10,000 families, 40% buy newspaper A, 20% buy B, 10% buy C. 5% buy A and B, 3% buy B and C, 4% buy A and C. 2% buy all three.
    (i) How many families buy none of the newspapers?
    (ii) How many buy only newspaper A?
37. Case Study (Derivatives): A particle moves along a straight line so that its distance $s$ in meters from a fixed point at time $t$ seconds is given by $s = t^3 - 6t^2 + 9t + 5$.
    (i) Find the velocity at time $t$.
    (ii) When is the particle at rest?
38. Case Study (Conics): An arch is in the form of a semi-ellipse. It is 8m wide and 2m high at the center.
    (i) Find the equation of the ellipse.
    (ii) Find the height of the arch at a point 1.5m from one end.