Questions covered in Mock 2

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 set $A$ has $n$ elements, then the total number of subsets of $A$ is
2. The domain of the function $f(x) = \frac{1}{\sqrt{x+|x|}}$ is
3. The value of $\sin(\frac{\pi}{3} - x)\cos(\frac{\pi}{6} + x) + \cos(\frac{\pi}{3} - x)\sin(\frac{\pi}{6} + x)$ is
4. The conjugate of the complex number $z = \frac{1}{3+4i}$ is
5. The solution of $24x < 100$, when $x$ is a natural number, is
6. How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 6, 7 if the digits can be repeated?
7. The coefficient of $x^5$ in the expansion of $(x+3)^8$ is
8. If the third term of a G.P. is 4, then the product of its first 5 terms is
9. The slope of a line which makes an angle of $30^\circ$ with the positive direction of y-axis measured anticlockwise is
10. The eccentricity of the circle $x^2 + y^2 = a^2$ is
11. The distance of the point $(1, -2, 3)$ from the origin is
12. $\lim_{x \to 0} \frac{\sin 4x}{\tan 2x}$ is
13. The derivative of $\cot x$ with respect to $x$ is
14. The mean of the first $n$ natural numbers is
15. If A and B are mutually exclusive events with $P(A)=0.3$ and $P(B)=0.5$, then $P(A \text{ or } B)$ is
16. If $A = \{x : x \text{ is a letter in BETTER}\}$ and $B = \{x : x \text{ is a letter in BEAT}\}$, then $A - B$ is
17. The radian measure corresponding to $40^\circ 20'$ is
18. The value of $i^9 + i^{19}$ is
19. The region represented by $x > 2$ and $y > 3$ lies in which quadrant?
20. If $^nC_8 = ^nC_2$, then $^nC_2$ is equal to
21. If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $A = \{2, 4, 6, 8\}$ and $B = \{2, 3, 5, 7\}$, verify that $(A \cup B)' = A' \cap B'$.
22. Prove that $\cos(\frac{\pi}{4} + x) + \cos(\frac{\pi}{4} - x) = \sqrt{2} \cos x$.
23. Solve the quadratic equation $x^2 + 3x + 9 = 0$.
24. How many chords can be drawn through 21 points on a circle?
25. Evaluate $\lim_{x \to 2} \frac{x^3 - 4x^2 + 4x}{x^2 - 4}$.
26. Let $f(x) = x^2$ and $g(x) = 2x + 1$ be two real functions. Find $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$ and $(f/g)(x)$.
27. Find the values of $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$ if $\tan x = -\frac{4}{3}$ and $x$ is in quadrant II.
28. Solve the following system of inequalities graphically: $2x + y \ge 4, x + y \le 3, 2x - 3y \le 6$.
29. Find the term independent of $x$ in the expansion of $(\frac{3}{2}x^2 - \frac{1}{3x})^6$.
30. Find the equation of the ellipse whose major axis is on the x-axis and passes through the points $(4, 3)$ and $(6, 2)$.
31. Find the derivative of $(x + \sec x)(x - \tan x)$.
32. Find the sum of the series: $5 + 55 + 555 + \dots$ to $n$ terms.
33. Find the mean and variance of the first $n$ natural numbers.
34. Find the equation of the line passing through the intersection of the lines $4x + 7y - 3 = 0$ and $2x - 3y + 1 = 0$ that has equal intercepts on the axes.
35. A committee of 7 is to be formed from 9 boys and 4 girls. In how many ways can this be done if the committee consists of: (i) exactly 3 girls? (ii) at least 3 girls? (iii) at most 3 girls?
36. Case Study (Sets): In a survey of 200 students, 120 were found to be taking Apple juice, 150 taking Orange juice and 80 taking both.
    (i) Find how many students were taking neither Apple nor Orange juice?
    (ii) How many were taking Apple juice only?
37. Case Study (Trigonometry): The height of a point on a Ferris wheel is given by $h(t) = 20 + 15 \sin(\frac{\pi t}{10})$, where $t$ is in seconds.
    (i) What is the maximum height reached?
    (ii) What is the time period of rotation?
38. Case Study (GP): The number of bacteria in a culture doubles every hour. If there were 30 bacteria originally.
    (i) How many bacteria will be there after 2nd hour?
    (ii) How many after $n$th hour?