1

(a)  

(4, −4)

Transformation: (x,y) → (x−2, y)

1

1

(b)

(−4, 6)

Inverse: swap x, y

1

1

(c)

(6, 5)

(x,y) → (x, 2

2

2

(a)(i)

Centre: (−3, 4)

From (x−h)²+(y−k)²=r²

1

2

(a)(ii)

Radius: 2√6

• √24

2

2

(b) in

No, origin is outside. Distance to centre = 5 > 2√6 ≈ 4.899

Calculate √[(−3)²+4²] and compare to radius

2

3

(a)

k = 5/2

Arithmetic sequence: 10−6k = 2k−10

2

3

(b)

S₅₀ = −5375

Sₙ = n/2[2a+(n−1)d], a=15, d=−5

3

4

(a)

f(x) = (x+4)(2x²−5x+4)

Factor theorem at x=−4, then divide

3

4

(b)

Discriminant = 25−32 = −7 ⇒ no real roots ⇒ only x=−4

Check b²−4ac

2

5

(a)

∫₀.₄₄².⁸₉ (2/√x) dx

Definite integral from limit-of-sum

1

5

(b)

k = 2

Direct evaluation with limits 1.44 and 2.89 for ∫(2/√x)

2

6

(a)

h = 20 − 0.3 t¹·⁵

Solve 17.6 = A−B·4¹·⁵ and 11.9 = A−B·9¹·⁵ ⇒ A=20, B=0.3

4

6

(b)

20 m

h(0)=A

1

6

(c)

T ≤ (200/3)²ᐟ³ ≈ 16.44

20−0.3 t¹·⁵ ≥ 0

2

7

(a)

{x: x≤ −1} ∪ {x: 2 ≤ x ≤ 5}

f′(x)≥0 on those intervals

2

7

(b)

f(x)=−3(x+1)²(x−5)²

Roots at x=−1, 5 double; use y-int to find leading coefficient

3

7

(c)

0 < k < 243

f(2)=−243 so shift up by k to get four real roots

2

8

(a)

Correct (5k+2)²=25k²+20k+4 ⇒ 5(5k²+4k+1)−1

Student had 10k term error

1

8

(b)

m=5k+3 → form 5n−1; m=5k+4 → form 5n+1

OR

m=5k-1

m=5k-2

Square and factor each to show they fit 5n±1

4

9

a)

13.5

15t - te^0.2t

t(15-e^0.2t), solve 15 - e^0.2t = 0

Value when y axis (think it was V?) was 0

2

9

b)

dv/dt then show that it looks like 5ln(75/t+5

4

9

c) i)

t3 =

3

9

c) ii)

8.554 seconds

Repeated integration to find the time in seconds when the car reaches max velocity

10

(a)

PR = 8i + 8j + 4k

PR = PQ + QR

2

10

(b)

∥PQ∥=∥QR∥=6√2, ∥PR∥=12 ⇒ isosceles & right at Q

Check magnitudes & dot(PQ,QR)=0 (FM) but u can also do pythag or cosine

4

11

(a)

V=1500+18500 e^(−kt), k≈0.227

Use V(0)=20000 → A=18500; V(2.5)=12000 → solve for k

4

11

(b)

dV/dt=−kAe^(−kt) ⇒ −k(V-500)

Differentiate & substitute A e^(−kt)=V−1500

3

11

(c)

V→1500 as t→∞

e^(−kt)→0

1

12

(a)

y=15x/[(2x+3)(x−3)] at x=6 → y=2; line: y=2x−10 at x=6 → y=2

Shows C and ℓ meet at same y

2

12

(b)

4x³−26x²−3x+90=0

Equate and rearrange

2

12

(c)

x=(1−√61)/4

Divide cubic by (x−6) then solve quadratic

4

13

—

2/25

Likely ∫₀² [x/(2x+1)³] dx by parts (typo in original). Substitution u=2x+1 worked too

5

14

(a)

2 cos x

Expand sin(x+30)+√3 cos(x+30) via compound-angle

3

14

(b)

θ=19.5°, 90°, 160.5°

Solve 2 cosθ(1−3 sinθ)=0 for 0≤θ<180

4

15

(a)

P=2r+480/r

θ=480/r²−1/5; P=2r+r/5+rθ

4

15

(b)

r=4√15

dP/dr=0 ⇒ r²=240

3

15

(c)

d²P/dr²=960/r³>0 at r=4√15 ⇒ minimum (0.258)

2

16

(a)

∫_{π/6}^{π/3} (1/4)cosec²u du

x=2 sin u ⇒ dx and √(4−x²)=2 cos u; change of variables

4

16

(b)

√3/6

(1/4)(−cot u)

3

1 TB SSD

33

32 GB RAM

67

Ryzen 7 7800X3D

225

Case (pre-owned)

0 (60 when bought 9 years prior)

Subtotal

325

Thermal paste

5

AliExpress, PTM7950 40×80mm

CPU cooler

32

Phantom spirit 120SE

PSU

60

CCL - £55 for a 750W MSI A750GN Gold, some users report loud fans with AGL version.

Motherboard

Under 100

This part is difficult, Mobos are more important now, than before.

GPU

500

I want to do AI stuff +gaming Rumoured 5070TI super looks really juicy with 24 GB but most likely over 1K

Subtotal

around   700