P1P2 Jan ’26 Batch

P1P2 Revision Worksheet

Month: 06 (June 2025)
Syllabus: P1 Ch 1, 2, 3, 4, 5, 6, 7, 8
P2 Chapter 2
8𝑥
1. Solve the equation 8 + 𝑥 √12 = . Give your answer in the form 𝑎√𝑏, where 𝑎 and 𝑏 are integers. [4√3]
√3

2. Do not use your calculator for this question.

A rectangle has a length of (1 + √3)cm and area of √12cm2.Calculate the width of the rectangle in cm.
Express your answer in the form 𝑎 + 𝑏√3, where 𝑎 and 𝑏 are integers to be found. [3 − √3 cm]

5
4𝑥 3 +𝑥2
3. Given that can be written in the form 4𝑥 𝑎 + 𝑥 𝑏 , write down the value of 𝑎 and the value of 𝑏.
√𝑥
5
[a = , 𝑏 = 2]
2

4. 𝑥 2 − 14𝑥 + 1 = (𝑥 + 𝑝) 2 + 𝑞, where 𝑝 and 𝑞 are constants.

a. Find the values of 𝑝 and 𝑞. [p = -7, q = -48]
b. Using your answer to part a, or otherwise, show that the solutions to the equation 𝑥 2 − 14𝑥 + 1 = 0 can be
written in the form 𝑟 ± 𝑠√3, where 𝑟 and 𝑠 are constants to be found. [r = 7, s = 4]

5. The function f is defined as 𝑓(𝑥) = 32𝑥 − 28(3𝑥 ) + 27, 𝑥 ∈ ℝ.

a. Write 𝑓(𝑥) in the form (3𝑥 − 𝑎)(3𝑥 − 𝑏), where 𝑎 and 𝑏 are real constants. [ 𝑎 = 27, 𝑏 = 1]
b. Hence find the two roots of 𝑓(𝑥). [0 and 3]

6. Find the set of values of 𝑥 for which:

1
a. 6𝑥 − 7 < 2𝑥 + 3 [𝑥 < 2 ]
2
1
b. 2𝑥 2 − 11𝑥 + 5 < 0 [ < 𝑥 < 5]
2
20
c. 5 < [0 < 𝑥 < 4]
𝑥

1 21
d. Both 6𝑥 − 7 < 2𝑥 + 3 and 2𝑥 2 − 11𝑥 + 5 < 0. [ <𝑥< ]
2 2

7. a) Sketch the graphs of 𝑦 = 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 15 and 𝑔(𝑥) = 6 − 2𝑥 on the same axes.

b) Find the coordinates of any points of intersection. [(−7, 20), (3,0)]

c) Write down the set of values of 𝑥 for which 𝑓(𝑥) > 𝑔(𝑥). [𝑥 < −7, 𝑥 > 3]

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8. a) On a coordinate grid, shade the region that satisfies the inequalities 𝑦 + 𝑥 < 6, 𝑦 < 2𝑥 + 9, 𝑦 > 3 and 𝑥 > 0
9
b) Work out the area of the shaded region. [ ]
2

9. The functions f and g are defined as f(x) = 9 − 𝑥 2 and g(x) = 14 − 6𝑥, x ε R.

a) On the same set of axes, sketch the graphs of y = f(x) and y = g(x). Indicate clearly the coordinates of any points where
the graphs intersect with each other or the coordinate axes.
b) On your sketch, shade the region that satisfies the inequalities y> 0 and f(x) > g(x).

10. a) Sketch the graph of 𝑦 = 𝑥 3 − 5𝑥 2 + 6𝑥, marking clearly the points of intersection with the axes.
b) Hence sketch 𝑦 = (𝑥 − 2)3 − 5(𝑥 − 2)2 + 6(𝑥 − 2).

11. f(x) = (𝑥 − 1)(𝑥 − 2)(𝑥 + 1)

a) State the coordinates of the point at which the graph y = f(x) intersects the y-axis. [ (0, 2) ]
b) The graph of y = af(x) intersects the y-axis at (0, -4). Find the value of a. [-2]
c) The graph of y = f(x + b) passes through the origin. Find the three possible values of b. [-1, 1, 2]

12. The point (6, -8) lies on the graph of y = f(x). State the coordinates of the point to which P is transformed on the graph
with equation:
a) 𝑦 = −𝑓(𝑥) [ (6, 8) ]
b) 𝑦 = 𝑓(𝑥 – 3) [ (9, −8) ]
c) 2𝑦 = 𝑓(𝑥) [ (6, −4) ]

13. The distance between the points (−1, 13) and (𝑥, 9) is √65. Find the two possible values of 𝑥. [−8, 6]

1
14. The straight line 𝑙1 passes through the point (−4, 14) and has gradient - .
4
a) Find an equation for 𝑙1 in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0, where a, b and c are integers. [𝑥 + 4𝑦 − 52 = 0]
b) Write down the coordinates of A, the point where the straight line 𝑙1 crosses the y-axis. [ 𝐴(0, 13) ]
The straight line 𝑙2 passes through the origin and has gradient 3.
The lines 𝑙1 and 𝑙2 intersect at the point B.
c) Calculate the coordinates of B. [ 𝐵(4, 12) ]
d) Calculate the exact area of OAB. [26]

15. In ABC, 𝐴𝐵 = 𝑥 cm, 𝐵𝐶 = 5 cm, 𝐴𝐶 = (10 – 𝑥) cm.

4𝑥−15
a) Show that cos ABC = .
2𝑥
1
b) Given that cos ABC = − , work out the value of 𝑥. [3.5]
7

16. a) Sketch on the same set of axes, in the interval 0 ≤ 𝑥 ≤ 180°, the graphs of 𝑦 = tan(𝑥 − 45°) and 𝑦 = −2 cos 𝑥,
showing the coordinates of points of intersection with the axes.
b) Deduce the number of solutions of the equation tan(𝑥 − 45°) + 2 cos 𝑥 = 0, in the interval 0 ≤ 𝑥 ≤ 180°.
[No solutions]

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17. Two points, 𝐴 and 𝐵 are on level ground. A tower at point 𝐶 has an angle of elevation from 𝐴 of 15° and an angle of
elevation from 𝐵 of 32°. 𝐴 and 𝐵 are both on the same side of 𝐶, and 𝐴, 𝐵 and 𝐶 lie on the same straight line. The distance
𝐴𝐵 = 75 m. Find the height of the tower. [35.2 m]

18. Two radar stations A and B are 16 km apart and A is due north of B. A ship is known to be on a bearing of 150° from A
and 10 km from B. Show that this information gives two positions for the ship, and calculate the distance between these
two positions. [12 km]

19. Sketch, on separate sets of axes, the graphs of the following, in the interval −180° ≤ 𝑥 ≤ 180°. Give the coordinates
of the points of intersection with the axes, and of maximum and minimum points where appropriate.

a) 𝑦 = −2𝑠𝑖𝑛𝑥
b) 𝑦 = tan(−𝑥)
c) 𝑦 = cos (𝑥 − 90°)

20. In the diagram 𝑂𝐴𝐵 is a sector of the circle, with centre 𝑂 and radius 𝑅 cm,
and ∠𝐴𝑂𝐵 = 2𝜃 radians. A circle centre 𝐶 and radius 𝑟 cm, touches the arc
𝐴𝐵 at 𝑇, and touches 𝑂𝐴 and 𝑂𝐵 at 𝐷 and 𝐸 respectively, as shown.

(a) Write down, in terms of 𝑅 and 𝑟, the length of 𝑂𝐶. [(R – r)]
(b) Using Δ𝑂𝐶𝐸, show that 𝑅𝑠𝑖𝑛 𝜃 = 𝑟(1 + sin 𝜃).
3
(c) Given that 𝑠𝑖𝑛𝜃 = and that the perimeter of the sector 𝑂𝐴𝐵 is 21 cm,
4
find 𝑟,giving your answer to 3 significant figures. [2.43 cm]

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21. The diagram shows a triangular garden, 𝑃𝑄𝑅, with 𝑃𝑄 = 12 𝑚, 𝑃𝑅 = 7𝑚 and

∠𝑄𝑃𝑅 = 0.5 radians.. The curve 𝑆𝑅 is a small path separating the shaded patio
area and the lawn, and is an arc of a circle with centre at 𝑃 and radius 7 m.
Find:
(a) The length of the path 𝑆𝑅. [3.5 m]
(b) The perimeter of the shaded patio , giving your answer to 3 significant
figures.[15.3 m]

22. In the diagram, AB and AC are tangents to a circle, centre O and radius 3.6 cm.
2𝜋
Calculate the area of the shaded region, given that ∠𝐵𝑂𝐶 = radians. [8.88 𝑐𝑚 2]
3

23. The diagram shows two intersecting sectors:

𝐴𝐵𝐷, with radius 5 cm and angle 1.2 radians,
and 𝐶𝐵𝐷 with radius 12cm.
Find the area of the overlapping section. [4.62 𝑐𝑚 2]

❖ P2 Chapter 2
24. A circle C has equation (𝑥 − 5)2 + (𝑦 + 3)2 = 10. The line l is a tangent to the circle and has gradient -3. Find two
possible equations for l, giving your answers in the form 𝑦 = 𝑚𝑥 + 𝑐. [𝑦 = −3𝑥 + 22 or 𝑦 = −4]

25. The points 𝐴(−7, 7), 𝐵(1, 9), 𝐶(3, 1) and 𝐷(−7, 1) lie on a circle.
(a) Find the equation of the perpendicular bisector of: (i) 𝐴𝐵 (ii) 𝐶𝐷
(b) Find the equation of the circle. [(𝑥 + 2)2 + (𝑦 − 4)2 = 34]

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26. The line with equation 2𝑥 + 𝑦 − 5 = 0 is a tangent to the circle with equation (𝑥 − 3)2 + (𝑦 − 𝑝)2 = 5.
(a) Find the two possible values of 𝑝. [4, −6]
(b) Write down the coordinates of the centre of the circle in each case. [(3, 4) & (3, −6)]

27. The line segment 𝑄𝑅 is a diameter of the circle centre 𝐶, where 𝑄 and 𝑅 have coordinates (11, −12) and (−5, 0)
respectively. The point 𝑃 has coordinates [(13, 6)]
(a) Find the coordinates of 𝐶. [(3, 6)]
(b) Find the radius of the circle. [10]
2 2
(c) Write down the equation of the circle. [(𝑥 − 3) + (𝑦 − 6) = 100]
(d) Show that P lies on the circle.

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