1.
Find the gradient of the line through (1, 2) and (5, 14).
[2 marks]
- (14−2)/(5−1)
- Final answer: 3
JD SCIENCE
Pure — coordinate geometry and trigonometry
Indicative marking points. Award marks for equivalent scientific or mathematical wording. Do not issue this sheet with the student worksheet.
2.
Find the equation of the line perpendicular to y = 3x − 1 that passes through (3, 4).
[3 marks]
- m = −1/3
- Final answer: y − 4 = −(1/3)(x − 3)
3.
Find the distance between (2, −1) and (6, 2).
[2 marks]
- √(16+9)
- Final answer: 5
4.
Find the midpoint of (2, −1) and (6, 2).
[1 mark]
- Final answer: (4, 0.5)
5.
A circle has equation (x − 3)² + (y + 1)² = 25. State the centre and radius.
[2 marks]
- Final answer: Centre (3, −1), radius 5
6.
Find the equation of the circle with centre (1, 2) passing through (4, 6).
[3 marks]
- Radius 5
- Final answer: (x−1)²+(y−2)² = 25
7.
Show that the line y = 2x + 1 intersects the circle x² + y² = 5 and find the points.
[4 marks]
- Show two intersection points or a repeated root for tangent
- Final answer: Substitute; solve the quadratic
8.
State the condition for a line to be a tangent to a circle in terms of the discriminant.
[2 marks]
- Repeated root / discriminant zero
9.
Convert 30° to radians and 2π/3 to degrees.
[2 marks]
- Final answer: π/6 ; 120°
10.
Find the exact value of sin 150° and cos 210°.
[2 marks]
- CAST
- Final answer: 1/2 ; −√3/2
11.
Solve sin θ = 0.5 for 0 ≤ θ ≤ 360°.
[3 marks]
- Sine is positive in 1st and 2nd
- Final answer: 30° and 150°
12.
Solve 2 cos θ = −1 for 0 ≤ θ ≤ 2π.
[3 marks]
- Final answer: 2π/3 and 4π/3
13.
Prove sin²θ + cos²θ = 1 using a right triangle or the unit circle.
[3 marks]
- Final answer: By definition on the unit circle x²+y²=1
14.
Express 2 sin θ cos θ as a single sine.
[1 mark]
- Double angle
- Final answer: sin 2θ
15.
Use cos 2θ = 2cos²θ − 1 to find the exact value of cos 15° if needed, or find cos 2θ when cos θ = 0.6.
[3 marks]
- Final answer: 2(0.36)−1 = −0.28
16.
Find the area of a triangle with sides 7 cm and 10 cm and included angle 30°.
[3 marks]
- ½ab sin C
- Final answer: 17.5 cm²
17.
Use the cosine rule to find the third side when a = 5, b = 8, C = 60°.
[3 marks]
- c² = 25+64−80cos60 = 49
- Final answer: √49 = 7
18.
Use the sine rule: a/sin A = b/sin B. A = 40°, a = 10, B = 60°. Find b.
[3 marks]
- Leave exact or 3 s.f.
- Final answer: b = 10 sin60 / sin40
19.
Sketch y = 2 sin(x − 30°) for 0 ≤ x ≤ 360°, stating amplitude, period and shift.
[4 marks]
- Amplitude 2
- Period 360°
- Shift 30° right
20.
Find the period of y = 3 cos(2x).
[2 marks]
- 2π/2
- Final answer: π or 180°
21.
The line l1: x = 1 + 2t, y = −1 + t. Find a Cartesian equation.
[3 marks]
- Eliminate t
- Final answer: y + 1 = (1/2)(x − 1)
22.
Find the angle between the lines y = x and y = 2x.
[3 marks]
- m1=1, m2=2, tanθ = |(m2−m1)/(1+m1m2)|
- Final answer: tan⁻¹(2) − 45°
23.
State the small-angle approximations for sin θ, cos θ and tan θ when θ is in radians.
[3 marks]
- sin θ ≈ θ
- cos θ ≈ 1 − θ²/2
- tan θ ≈ θ
24.
A sector of radius 6 cm has angle 0.5 rad. Find the arc length and sector area.
[3 marks]
- s=rθ, A=½r²θ
- Final answer: s=3 cm, A=9 cm²
25.
Find the equation of the tangent to the circle x² + y² = 25 at (3, 4).
[3 marks]
- Radius is perpendicular to tangent
- Final answer: 3x + 4y = 25
26.
Solve tan 2θ = 1 for 0 ≤ θ ≤ 180°.
[3 marks]
- 2θ = 45°, 225°
- Final answer: θ = 22.5° and 112.5°
27.
Write the identities for sin(A+B) and cos(A−B).
[3 marks]
- sin A cos B + cos A sin B
- cos A cos B + sin A sin B
28.
Find R and α such that 3 sin θ + 4 cos θ = R sin(θ + α).
[4 marks]
- R=√(9+16)
- Final answer: R = 5, tan α = 4/3
29.
The points A(0,1), B(4,3), C(2,7) form a triangle. Find the area using the determinant / shoelace method.
[3 marks]
- ½|0(3−7)+4(7−1)+2(1−3)|
- Final answer: 10
30.
On OCR papers, a 5-mark trig equation needs all solutions in the interval. Write a checklist.
[3 marks]
- Find the principal value
- Use the correct quadrants
- Do not lose solutions after a double-angle substitution
31.
Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.
[3 marks]
- Complete the square
- Final answer: Centre (3, −2), radius 5
32.
A triangle has sides 6, 7, 8. Find the largest angle using the cosine rule.
[3 marks]
- Largest angle opposite 8
- Final answer: cos C = (36+49−64)/84 = 21/84 = 0.25