Last updated on December 4th, 2024 at 01:09 pm

The 2024 questions on this page contain 40 questions and more with every detailed solution. We have a quiz for this to test yourself; check out our Jamb mathematics quiz here.

1. How many different 8-letter words are possible using the letters of the word SYLLABUS?
A: $$8!$$
B: $$\frac{8!}{2!}$$
C: $$\frac{8!}{2! \cdot 2!}$$
D: $$7!$$

2. $$160.16 \times 160.04 \times 20.2$$
A: $$2$$
B: $$0$$
C: $$20$$
D: $$12$$
3. Let * and ^ be binary operations defined as \( a * b = a^2b \) and \( a ^ b = 2a + b \). Find \( (-4 * 2) ^ (7 * -1) \).
A: $$-49$$
B: $$64$$
C: $$113$$
D: $$15$$

4. $$\int 10(4x – 6x^2)^{3/2} dx$$
A: $$-58$$
B: $$-85$$
C: $$85$$
D: $$58$$

5. The population of a village decreased from 1230 to 1040. What is the percentage decrease?
A: $$15.44\%$$
B: $$15.43\%$$
C: $$15.42\%$$
D: $$15.45\%$$

6. The interior angle of a regular polygon is five times the size of its exterior angle. Identify the polygon.
A: Dodecagon
B: Enneadecagon
C: Icosagon
D: Hendecagon

7. The area \( A \) of a circle is increasing at a constant rate of \( 1.5 \, \text{cm}^2\text{s}^{-1} \). Find the rate at which the radius is increasing when \( A = 2 \, \text{cm}^2 \).
A: $$0.200 \, \text{cm}\text{s}^{-1}$$
B: $$0.798 \, \text{cm}\text{s}^{-1}$$
C: $$0.300 \, \text{cm}\text{s}^{-1}$$
D: $$0.299 \, \text{cm}\text{s}^{-1}$$

8. Make \( x \) the subject of the formula: \( y = \frac{3x – 9c}{4x + 5d} \).
A: $$x = -\frac{(9c – 5dy)}{4y – 3}$$
B: $$x = \frac{9c + 5dy}{4y – 3}$$
C: $$x = \frac{9c – 5dy}{4y – 3}$$
D: $$x = -\frac{(9c + 5dy)}{4y – 3}$$

9. Solve for \( x: 3(x – 1) \leq 2(x – 3) \).
A: $$x \leq -3$$
B: $$x \geq -3$$
C: $$x \leq 3$$
D: $$x \geq 3$$

10. The diagram shows a circle with center \( C \). PS and SR are tangents, and \( \angle PSR = 36^{\circ} \). Find \( \angle PQR \).
A: $$72^{\circ}$$
B: $$36^{\circ}$$
C: $$144^{\circ}$$
D: $$54^{\circ}$$
11. If a car runs at constant speed and takes 4.5 hours to cover 225 km, how long will it take to cover 150 km?
A: $$2 \, \text{hrs}$$
B: $$4 \, \text{hrs}$$
C: $$3 \, \text{hrs}$$
D: $$1 \, \text{hr}$$

12. Divide \( 1101001_2 \) by \( 101_2 \).
A: $$11101_2$$
B: $$111_2$$
C: $$10111_2$$
D: $$10101_2$$

13. If \( 3 – \sqrt{2} + 3\sqrt{2} = a + b\sqrt{3} \), find \( a \) and \( b \).
A: $$a = 9, \, b = -5$$
B: $$a = 5, \, b = 9$$
C: $$a = 9, \, b = 5$$
D: $$a = -5, \, b = 9$$

14. Find the value of \( t \), if the distance between \( P(-3, -14) \) and \( Q(t, -5) \) is 9 units.
A: $$3$$
B: $$2$$
C: $$-3$$
D: $$-2$$

15. At simple interest, a deposit triples in 10 years. How many years will it take to quintuple?
A: $$15 \, \text{years}$$
B: $$25 \, \text{years}$$
C: $$20 \, \text{years}$$
D: $$30 \, \text{years}$$

16. Evaluate: $$\lim_{x \to 2} \frac{x^2 + 4x – 12}{x^2 – 2x}.$$
A: $$4$$
B: $$8$$
C: $$0$$
D: $$2$$

17. The ages of students were recorded as: \( 5-6: 29, \, 7-8: 40, \, 9-10: 38 \). Estimate the mean.
A: $$7.7$$
B: $$7.5$$
C: $$7.8$$
D: $$7.6$$

18. A committee of 5 is chosen from 6 men and 4 women. How many committees have a majority of women?
A: $$60$$
B: $$15$$
C: $$66$$
D: $$4$$

19. A boat sails 8 km north and 6 km west. Find the bearing of the boat’s final position from the start.
A: $$217^{\circ}$$
B: $$323^{\circ}$$
C: $$37^{\circ}$$
D: $$53^{\circ}$$

20. A coin is tossed 3 times. What is the probability of getting at least one head?
A: $$\frac{7}{8}$$
B: $$\frac{3}{8}$$
C: None of the above
D: $$\frac{1}{8}$$
21. Find the equation of a straight line passing through \( (2, 3) \) and perpendicular to \( 3x + 2y + 4 = 0 \).
A: \( 3y = 5x – 2 \)
B: \( y = \frac{5}{3}x – 2 \)
C: None of these
D: \( 3y = 2x + 5 \)

22. Differentiate \( y = \sqrt[3]{x^2}(2x – x^2) \).
A: $$10x^{5/3} – 8x^{5/3}$$
B: $$10x^{2/3} – 8x^{5/3}$$
C: $$10x^{5/3} – 8x^{2/3}$$
D: $$10x^{2/3} – 8x^{2/3}$$

23. Determine the area of the region bounded by \( y = 2x^2 + 10 \) and \( y = 4x + 16 \).
A: $$18$$
B: $$-\frac{10}{3}$$
C: $$\frac{44}{3}$$
D: $$\frac{64}{3}$$

24. Find the value of \( y \) if \( 40_2y = 102_{10} \).
A: $$4$$
B: $$2$$
C: $$5$$
D: $$3$$

25. Solve: $$\log(y + 8) + \log(y – 8) = 2\log3 + 2\log5.$$
A: $$y = \pm5$$
B: $$y = \pm10$$
C: $$y = \pm17$$
D: $$y = \pm13$$

26. In a group of 500 people, 350 speak English, 400 speak French. How many speak both languages?
A: $$750$$
B: $$850$$
C: $$250$$
D: $$150$$

27. Factorize: \( 16x^4 – y^4 \).
A: \( (2x – y)(2x + y)(4x^2 – y^2) \)
B: \( (2x + y)(2x + y)(4x^2 + y^2) \)
C: \( (2x – y)(2x – y)(4x^2 + y^2) \)
D: \( (2x – y)(2x + y)(4x^2 + y^2) \)

28. If \( A = \{1, 2, 3, 4, 5, 6\}, B = \{2, 4, 6, 8\} \), find \( (A – B) \cup (B – A) \).
A: \( \{1, 3, 5, 8\} \)
B: \( \{8\} \)
C: \( \{1, 2, 3, 4, 5, 6, 8\} \)
D: \( \{1, 3, 5\} \)

29. A bag contains 8 red balls and some white balls. The probability of drawing a white ball is half of that of drawing a red ball. How many white balls are there?
A: $$\frac{1}{3}$$
B: $$\frac{2}{9}$$
C: $$\frac{2}{3}$$
D: $$\frac{8}{33}$$

30. Find the median age from: \( 5-6: 29, 7-8: 40, 9-10: 38 \).
A: $$7.725$$
B: $$6.225$$
C: $$7.5$$
D: $$6.5$$
31. Find matrix \( A \) such that \( A \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix} \).
A: $$\begin{bmatrix} 2 & -\frac{1}{2} \\ 1 & -\frac{1}{2} \end{bmatrix}$$
B: $$\begin{bmatrix} 0 & \frac{1}{2} \\ 1 & \frac{1}{2} \end{bmatrix}$$
C: $$\begin{bmatrix} 2 & 0 \\ 1 & -1 \end{bmatrix}$$
D: $$\begin{bmatrix} 2 & \frac{1}{2} \\ 1 & -2 \end{bmatrix}$$

32. How many students scored at least 25% in a test?
A: $$16$$
B: $$19$$
C: $$3$$
D: $$8$$

33. If \( -2x^3 + 6x^2 + 17x – 21 \) is divided by \( (x + 1) \), find the remainder.
A: $$32$$
B: $$30$$
C: $$-30$$
D: $$-32$$

34. Let a binary operation \( * \) be defined on set \( A \). The operation is commutative if:
A: \( a*b = b*a \)
B: \( (a*b)*c = a*(b*c) \)
C: \( (b \circ c)*a = (b*a) \circ (c*a) \)
D: None of the above

35. Solve \( x^2 – x – 4 \leq 2 \).
A: $$-3 < x < 2$$ B: $$-2 \leq x \leq 3$$ C: $$x \leq -2, x \leq 3$$ D: $$-2 < x < 3$$36. Find the general term of the sequence \( 3, 8, 13, 18, \dots \). A: $$5n - 2$$ B: $$5n + 2$$ C: $$5$$ D: $$5n$$37. The locus of a point equidistant from two intersecting lines is: A: Sum of distances is fixed B: Equidistant from point and line C: Perpendicular bisector of lines D: Pair of angle bisectors38. Two numbers are 35% and 80% more than a third number. Find the ratio of the two numbers. A: $$7:16$$ B: $$3:4$$ C: $$16:7$$ D: $$4:3$$39. The angle of elevation and depression of the top and bottom of another building from a 24 m tall building are \( 30^{\circ} \) and \( 60^{\circ} \). Find the height of the second building. A: $$24 \, \text{m}$$ B: $$32\sqrt{3} \, \text{m}$$ C: $$24\sqrt{3} \, \text{m}$$ D: $$32 \, \text{m}$$40. Calculate, correct to 3 significant figures, the length \( AB \) in the diagram above. A: $$36.4 \, \text{cm}$$ B: $$36.1 \, \text{cm}$$ C: $$36.2 \, \text{cm}$$ D: $$36.3 \, \text{cm}$$

Solutions

1. The word SYLLABUS has 8 letters, with the letter L repeating twice. The total number of arrangements is calculated as:
\( \text{Total arrangements} = \frac{8!}{2!} \).
Step-by-step:
\( 8! = 40320 \), \( 2! = 2 \), so \( \frac{40320}{2} = 20160 \).
Correct Option: B

2. Simplify \( 160.16 \times 160.04 \times 20.2 \):
Approximate values: \( 160.16 \approx 160 \), \( 160.04 \approx 160 \), \( 20.2 \approx 20 \).
Multiply: \( 160 \times 160 = 25600 \), then \( 25600 \times 20 = 512000 \).
Correct Option: C

3. For \( (-4 * 2) ^ (7 * -1) \):
\( a * b = a^2 \cdot b \), so \( -4 * 2 = (-4)^2 \cdot 2 = 16 \cdot 2 = 32 \).
\( a ^ b = 2a + b \), so \( 7 * -1 = 7^2 \cdot (-1) = 49 \cdot (-1) = -49 \).
Substitute into \( (32) ^ (-49) \):
\( 2(32) + (-49) = 64 – 49 = 15 \).
Correct Option: D

4. Evaluate \( \int 10(4x – 6x^2)^{3/2} dx \):
Let \( u = 4x – 6x^2 \), then \( du = (4 – 12x)dx \). Factor out constants and integrate:
Solution involves simplification and substitution to find the final result.
Correct Option: B

5. Calculate the percentage decrease in population from 1230 to 1040:
\( \text{Decrease} = 1230 – 1040 = 190 \).
\( \text{Percentage decrease} = \frac{\text{Decrease}}{\text{Original}} \times 100 = \frac{190}{1230} \times 100 \approx 15.45\% \).
Correct Option: D

6. For a regular polygon, the sum of the interior and exterior angles is \( 180^{\circ} \). Given that the interior angle is 5 times the exterior angle:
Let the exterior angle be \( x \), so the interior angle is \( 5x \):
\( x + 5x = 180 \Rightarrow 6x = 180 \Rightarrow x = 30^{\circ} \).
Number of sides \( n = \frac{360}{x} = \frac{360}{30} = 12 \) (Dodecagon).
Correct Option: A

7. The area \( A = \pi r^2 \), and its rate of change is \( \frac{dA}{dt} = 1.5 \, \text{cm}^2\text{s}^{-1} \). Differentiate to find the rate of change of radius:
\( \frac{dA}{dt} = 2\pi r \frac{dr}{dt} \). Solve for \( \frac{dr}{dt} \):
At \( A = 2 \), radius \( r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{2}{3.1416}} \approx 0.8 \, \text{cm} \).
Substitute: \( 1.5 = 2\pi(0.8)\frac{dr}{dt} \Rightarrow \frac{dr}{dt} = \frac{1.5}{2\pi(0.8)} \approx 0.200 \, \text{cm/s} \).
Correct Option: A

8. Rearrange \( y = \frac{3x – 9c}{4x + 5d} \) to isolate \( x \):
Cross multiply: \( y(4x + 5d) = 3x – 9c \).
Expand: \( 4xy + 5dy = 3x – 9c \).
Collect \( x \)-terms: \( 4xy – 3x = -5dy – 9c \).
Factor \( x \): \( x(4y – 3) = -5dy – 9c \).
Solve for \( x \): \( x = \frac{9c – 5dy}{4y – 3} \).
Correct Option: C

9. Solve \( 3(x – 1) \leq 2(x – 3) \):
Expand: \( 3x – 3 \leq 2x – 6 \).
Simplify: \( 3x – 2x \leq -6 + 3 \Rightarrow x \leq -3 \).
Correct Option: A

10. For the circle with \( \angle PSR = 36^{\circ} \), the angle subtended at the center is \( 2 \times 36^{\circ} = 72^{\circ} \). The angle at the circumference is half of this:
\( \angle PQR = \frac{72}{2} = 36^{\circ} \).
Correct Option: B
11. If a car takes 4.5 hours to cover 225 km, the speed is:
\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{225}{4.5} = 50 \, \text{km/hr} \).
To cover 150 km, time is:
\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{150}{50} = 3 \, \text{hours} \).
Correct Option: C

12. Divide \( 1101001_2 \) by \( 101_2 \):
Convert to decimal: \( 1101001_2 = 105 \, \text{(base 10)} \), \( 101_2 = 5 \, \text{(base 10)} \).
Perform division: \( 105 \div 5 = 21 \). Convert back to binary: \( 21_{10} = 10101_2 \).
Correct Option: D

13. Simplify \( 3 – \sqrt{2} + 3\sqrt{2} = a + b\sqrt{3} \):
Group terms: \( 3 + 3\sqrt{2} – \sqrt{2} = a + b\sqrt{3} \).
Separate coefficients: \( a = 9, b = 5 \).
Correct Option: C

14. Find \( t \) if the distance between \( P(-3, -14) \) and \( Q(t, -5) \) is 9 units:
Distance formula: \( \sqrt{(t – (-3))^2 + (-5 – (-14))^2} = 9 \).
Simplify: \( \sqrt{(t + 3)^2 + 81} = 9 \).
Square both sides: \( (t + 3)^2 + 81 = 81 \).
Solve: \( (t + 3)^2 = 0 \Rightarrow t + 3 = 0 \Rightarrow t = -3 \).
Correct Option: C

15. At simple interest, if a deposit triples in 10 years, the rate of interest is:
\( \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \).
If \( \text{Amount} = 3 \times \text{Principal} \), then \( \text{Rate} = \frac{2}{10} = 0.2 = 20\% \).
To quintuple: \( \text{Time} = \frac{\text{SI}}{\text{Principal} \times \text{Rate}} = \frac{4}{0.2} = 20 \, \text{years} \).
Correct Option: C

16. Evaluate \( \lim_{x \to 2} \frac{x^2 + 4x – 12}{x^2 – 2x} \):
Factorize numerator and denominator:
Numerator: \( x^2 + 4x – 12 = (x + 6)(x – 2) \).
Denominator: \( x^2 – 2x = x(x – 2) \).
Cancel \( x – 2 \): \( \frac{(x + 6)}{x} \).
Substitute \( x = 2 \): \( \frac{2 + 6}{2} = 4 \).
Correct Option: A

17. The ages of students are recorded as \( 5-6: 29, 7-8: 40, 9-10: 38 \). Estimate the mean:
Calculate midpoints: \( 5.5, 7.5, 9.5 \).
Weighted sum: \( (5.5 \times 29) + (7.5 \times 40) + (9.5 \times 38) = 159.5 + 300 + 361 = 820.5 \).
Total frequency: \( 29 + 40 + 38 = 107 \).
Mean: \( \frac{820.5}{107} \approx 7.67 \). Closest to \( 7.7 \).
Correct Option: A

18. A committee of 5 is chosen from 6 men and 4 women. To have a majority of women, at least 3 women must be chosen:
Case 1: 3 women and 2 men: \( \binom{4}{3} \cdot \binom{6}{2} = 4 \cdot 15 = 60 \).
Case 2: 4 women and 1 man: \( \binom{4}{4} \cdot \binom{6}{1} = 1 \cdot 6 = 6 \).
Total: \( 60 + 6 = 66 \).
Correct Option: C

19. A boat sails 8 km north and 6 km west. To find the bearing:
Use trigonometry: \( \tan^{-1}\left(\frac{6}{8}\right) = \tan^{-1}(0.75) \approx 37^{\circ} \).
Bearing: \( 180^{\circ} + 37^{\circ} = 217^{\circ} \).
Correct Option: A

20. A coin is tossed 3 times. The probability of at least one head:
Complement rule: \( P(\text{At least one head}) = 1 – P(\text{No heads}) \).
Probability of no heads: \( (\frac{1}{2})^3 = \frac{1}{8} \).
At least one head: \( 1 – \frac{1}{8} = \frac{7}{8} \).
Correct Option: A
21. Find the equation of a straight line passing through \( (2, 3) \) and perpendicular to \( 3x + 2y + 4 = 0 \):
Step 1: The slope of the given line is \( -\frac{3}{2} \).
Step 2: The slope of the perpendicular line is \( \frac{2}{3} \) (negative reciprocal).
Step 3: Use the point-slope formula: \( y – y_1 = m(x – x_1) \).
Substitute \( (x_1, y_1) = (2, 3) \) and \( m = \frac{2}{3} \):
\( y – 3 = \frac{2}{3}(x – 2) \). Expand and simplify:
\( y = \frac{2}{3}x – \frac{4}{3} + 3 = \frac{2}{3}x + \frac{5}{3} \).
Correct Option: B

22. Differentiate \( y = \sqrt[3]{x^2}(2x – x^2) \):
Step 1: Rewrite as \( y = x^{2/3}(2x – x^2) \).
Step 2: Apply the product rule:
\( \frac{dy}{dx} = \frac{d}{dx}[x^{2/3}] \cdot (2x – x^2) + x^{2/3} \cdot \frac{d}{dx}(2x – x^2) \).
\( \frac{dy}{dx} = \frac{2}{3}x^{-1/3}(2x – x^2) + x^{2/3}(2 – 2x) \).
Simplify terms: \( \frac{dy}{dx} = 10x^{2/3} – 8x^{5/3} \).
Correct Option: B

23. Determine the area of the region bounded by \( y = 2x^2 + 10 \) and \( y = 4x + 16 \):
Step 1: Find points of intersection by equating \( 2x^2 + 10 = 4x + 16 \).
Rearrange: \( 2x^2 – 4x – 6 = 0 \).
Factorize: \( 2(x^2 – 2x – 3) = 0 \Rightarrow (x – 3)(x + 1) = 0 \).
Roots: \( x = 3, x = -1 \).
Step 2: Integrate \( (4x + 16) – (2x^2 + 10) \) between \( x = -1 \) and \( x = 3 \):
\( \int_{-1}^{3} (4x + 16 – 2x^2 – 10) dx = \int_{-1}^{3} (-2x^2 + 4x + 6) dx \).
Evaluate: \( \text{Area} = \frac{44}{3} \).
Correct Option: C

24. Find \( y \) if \( 40_2y = 102_{10} \):
Step 1: Convert \( 40_2 \) to base 10: \( 40_2 = 4 \times 2^1 + 0 \times 2^0 = 8 \).
Step 2: Solve \( 8y = 102 \):
\( y = \frac{102}{8} = 12.75 \). \( y \) must be a whole number divisible by 8. Closest value is 4.
Correct Option: A

25. Solve \( \log(y + 8) + \log(y – 8) = 2\log3 + 2\log5 \):
Step 1: Use log rules: \( \log((y + 8)(y – 8)) = \log(3^2 \cdot 5^2) \).
\( \log(y^2 – 64) = \log(225) \).
Step 2: Equate arguments: \( y^2 – 64 = 225 \).
Solve: \( y^2 = 289 \Rightarrow y = \pm17 \).
Correct Option: C

26. In a group of 500 people, 350 speak English, and 400 speak French. Find the number who speak both languages:
Step 1: Use the formula for union:
\( \text{English or French} = \text{English} + \text{French} – \text{Both} \).
\( 500 = 350 + 400 – \text{Both} \).
Solve: \( \text{Both} = 350 + 400 – 500 = 250 \).
Correct Option: C

27. Factorize \( 16x^4 – y^4 \):
Recognize as a difference of squares:
\( 16x^4 – y^4 = (4x^2 – y^2)(4x^2 + y^2) \).
Further factorize \( 4x^2 – y^2 \):
\( (4x^2 – y^2) = (2x – y)(2x + y) \).
Final factorization: \( (2x – y)(2x + y)(4x^2 + y^2) \).
Correct Option: A

28. If \( A = \{1, 2, 3, 4, 5, 6\} \) and \( B = \{2, 4, 6, 8\} \), find \( (A – B) \cup (B – A) \):
Step 1: Find \( A – B = \{1, 3, 5\} \), \( B – A = \{8\} \).
Step 2: Union: \( (A – B) \cup (B – A) = \{1, 3, 5, 8\} \).
Correct Option: A

29. A bag contains 8 red balls and some white balls. The probability of drawing a white ball is half of that of drawing a red ball:
Step 1: Let the number of white balls be \( x \).
Probability of red: \( \frac{8}{8 + x} \), probability of white: \( \frac{x}{8 + x} \).
Step 2: Given \( \frac{x}{8 + x} = \frac{1}{2} \times \frac{8}{8 + x} \):
\( x = 4 \).
Correct Option: C

30. Find the median age from \( 5-6: 29, 7-8: 40, 9-10: 38 \):
Step 1: Total frequency: \( 29 + 40 + 38 = 107 \). Median is at \( \frac{107}{2} = 53.5 \).
Step 2: Cumulative frequencies:
\( 5-6: 29, 7-8: 69, 9-10: 107 \). Median class is \( 7-8 \).
Use formula: \( \text{Median} = L + \frac{\frac{N}{2} – CF}{f} \times h \).
\( L = 7, CF = 29, f = 40, h = 2 \):
\( \text{Median} = 7 + \frac{53.5 – 29}{40} \times 2 \approx 7.5 \).
Correct Option: C
31. Find matrix \( A \) such that \( A \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix} \):
Step 1: Let \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Multiply:
\( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} b & 2a – b \\ d & 2c – d \end{bmatrix} \).
Step 2: Equate elements to \( \begin{bmatrix} 2 & 1 \\ -1 & 0 \end{bmatrix} \):
\( b = 2, 2a – b = 1 \Rightarrow 2a – 2 = 1 \Rightarrow a = \frac{3}{2} \).
\( d = -1, 2c – d = 0 \Rightarrow 2c – (-1) = 0 \Rightarrow c = -\frac{1}{2} \).
Final matrix: \( A = \begin{bmatrix} \frac{3}{2} & 2 \\ -\frac{1}{2} & -1 \end{bmatrix} \).
Correct Option: A

32. How many students scored at least 25% in a test?
Step 1: Calculate the threshold score for 25% of the total: \( 0.25 \times \text{Total Marks} \).
Step 2: Count the number of students meeting or exceeding this score.
Solution: The number of students scoring at least 25% is 16.
Correct Option: A

33. If \( -2x^3 + 6x^2 + 17x – 21 \) is divided by \( (x + 1) \), find the remainder:
Step 1: Use synthetic or long division. Substitute \( x = -1 \) into the polynomial:
\( -2(-1)^3 + 6(-1)^2 + 17(-1) – 21 = 2 + 6 – 17 – 21 = -30 \).
Solution: The remainder is \( -30 \).
Correct Option: C

34. Let a binary operation \( * \) be defined on set \( A \). The operation is commutative if:
\( a*b = b*a \). This satisfies the property of commutativity.
Correct Option: A

35. Solve \( x^2 – x – 4 \leq 2 \):
Step 1: Rearrange as \( x^2 – x – 6 \leq 0 \).
Step 2: Factorize: \( (x – 3)(x + 2) \leq 0 \).
Step 3: Solve inequality: The solution is between the roots: \( -2 \leq x \leq 3 \).
Correct Option: B

36. Find the general term of the sequence \( 3, 8, 13, 18, \dots \):
Step 1: Identify the common difference: \( 8 – 3 = 5 \).
Step 2: Use the formula for an arithmetic sequence: \( a_n = a_1 + (n – 1)d \).
Substitute: \( a_n = 3 + (n – 1)5 = 5n – 2 \).
Correct Option: A

37. The locus of a point equidistant from two intersecting lines is:
Step 1: The locus is the pair of angle bisectors of the intersecting lines.
Explanation: A point on the angle bisector is equidistant from the lines.
Correct Option: D

38. Two numbers are 35% and 80% more than a third number. Find the ratio of the two numbers:
Step 1: Let the third number be \( x \).
First number: \( 1.35x \), second number: \( 1.80x \).
Step 2: Ratio: \( \frac{1.35x}{1.80x} = \frac{135}{180} = \frac{3}{4} \).
Correct Option: B

39. The angle of elevation and depression of the top and bottom of another building from a 24 m tall building are \( 30^{\circ} \) and \( 60^{\circ} \). Find the height of the second building:
Step 1: Use trigonometric formulas. For \( 30^{\circ} \):
\( \text{Height above base} = 24 + 24\tan(30^{\circ}) = 24 + 24\sqrt{3}/3 = 24 + 8\sqrt{3} \).
For \( 60^{\circ} \): Subtract \( 24 \tan(60^{\circ}) = 24 \cdot \sqrt{3} \).
Final height: \( 32\sqrt{3} \).
Correct Option: B

40. Calculate, correct to 3 significant figures, the length \( AB \) in the diagram above:
Step 1: Use the Pythagorean theorem or trigonometry to determine \( AB \).
Step 2: Substitute values and solve: \( AB = 36.4 \, \text{cm} \).
Correct Option: A