Mathematics Paper 1: Form 4 QA Model – Document ID 20250302001

Section I (Answer all 16 questions – 50 Marks)

1. Evaluate the following expression using square roots and reciprocals tables, rounded to 3 decimal places:
   1644+0.62510\frac{164}{4} + \frac{0.625}{10}4164​+100.625​ (3 marks)
2. By letting $P=4-y$ in the equation 4−2y+1−3×4−y−10=04 – 2y + 1 – 3x^{4-y} – 10 = 04−2y+1−3x4−y−10=0:
   • a) Write the above equation in terms of $P$. (1 mark)
   • b) Hence, find the possible values of $y$. (3 marks)
3. The heights of two similar pails are 12 cm and 8 cm. The larger pail can hold 2 litres. What is the capacity of the smaller pail? (3 marks)
4. A tourist arrived in Kenya with £4680, spent Ksh. 52,352, and converted the rest into U.S dollars. Calculate the amount received in U.S dollars using the given exchange rates. (3 marks)
5. Find the equation of the perpendicular bisector of line AB where A is (3, 9) and B is (7, 5) in the form $ax+by+c=0$. (3 marks)
6. Solve the following inequality and show your solution on a number line:
   4x−3≤12(x+8)<x+54x – 3 \leq \frac{1}{2}(x + 8) < x + 54x−3≤21​(x+8)<x+5 (3 marks)
7. From a viewing tower 30 metres above the ground, the angle of depression of an object on the ground is 30° and the angle of elevation of an aircraft vertically above the object is 42°. Calculate the height of the aircraft above the ground. (3 marks)
8. Estimate the area bounded by the curve y=52×2+1y = \frac{5}{2}x^2 + 1y=25​x2+1, the x-axis, the line $x=1$, and $x=5$ using the trapezium rule with 4 trapezia. (3 marks)
9. Given that $CD=4$ cm, $DT=8$ cm, and $AB=6$ cm, calculate the length $BT$ when chords AT and CT meet at point T. (3 marks)
10. Find the standard deviation of the data below:
    $35,37,37,40,30,34,33,38$ (3 marks)
11. On the line segment AB:
    • i. Construct the locus of P such that $\angle PAB=90°$.
    • ii. Measure the length $AB$ and calculate the area enclosed by the locus P and the line segment AB. (4 marks)
12. A carpenter constructed a closed wooden box with internal measurements 1.5 metres long, 0.8 metres wide, and 0.4 metres high. If the wood used has a density of 0.6 g/cm³, determine the mass of the wood used in constructing the box (to 1 decimal place). (3 marks)
13. The sum of interior angles of a regular polygon is 1440°. Find the number of sides of the polygon. (3 marks)
14. Simplify the expression:
    x2+2x+5×2+3x+2\frac{x^2 + 2x + 5}{x^2 + 3x + 2}x2+3x+2x2+2x+5​ (3 marks)
15. Given that cos⁡A=513\cos A = \frac{5}{13}cosA=135​ and angle A is acute, find the value of $2\tan A+3\sin A$. (2 marks)
16. Given that $a=0.45$ and $b=1.5$, find the percentage error in evaluating ab\frac{a}{b}ba​, given the values used are $a=0.45$ and $b=1.5$ (to 2 decimal places). (4 marks)

Section II (Answer any five questions – 50 Marks)

17. A surveyor recorded measurements of a field with XY=400m as the baseline. Use a scale of 1 cm to represent 50 m to draw the map of the field and find the area in hectares. (10 marks)
18. (a) Use graphical methods to solve the simultaneous equations:
    3x + 2y = 7 \\ 2x + y = 12 \] (5 marks)

(b) Use matrix method to solve for the values of x and y. (5 marks)

19. a) Train A leaves a station 45 minutes before train B. Both trains travel in the same direction at speeds of 36 km/h and 48 km/h respectively.

• i) How long will it take train B to catch up with train A? (3 marks)
• ii) How far from the start were the two trains when they met? (2 marks)
  b) A car accelerated from rest to a velocity of 10 m/s in 10 seconds. Calculate the initial acceleration, distance traveled, and average velocity. (5 marks)

20. Given a circle with center O and diameters QOT, calculate various angles based on the provided angles. (10 marks)
21. In a parallelogram OABC, express certain vectors in terms of vectors a and c, and find the scalars in the expressions. (10 marks)
22. Four towns P, R, T, and S are located based on given bearings and distances. Make a scale drawing to show their positions and find specific distances and bearings. (10 marks)
23. Prepare a frequency distribution table for the masses of 25 students. Estimate the median mass and draw a histogram for the data. (10 marks)
24. A solution of 80 litres is made up of 40% water and 60% alcohol. When x litres of water is added, the percentage of alcohol drops to 40%. Calculate x, and find the percentage of alcohol after adding 30 litres of water to the new solution. (10 marks)