6672 edexcel gce pure mathematics p2 advanced advanced subsidiary 3

1. The equation of a line with gradient 2 and passing through the point (1,3)(1, 3)(1,3) is:

A. y=2x+1y = 2x + 1y=2x+1

B. y=2x+3y = 2x + 3y=2x+3

C. y=3x+2y = 3x + 2y=3x+2

D. y=x+3y = x + 3y=x+3
Answer: B) y=2x+3y = 2x + 3y=2x+3

Rationale:

Using the point-slope form y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1=m(x−x1), with m=2m =
2m=2 and (x1,y1)=(1,3)(x_1, y_1) = (1, 3)(x1,y1)=(1,3), we get:
y−3=2(x−1)y - 3 = 2(x - 1)y−3=2(x−1), which simplifies to:

y=2x+3y = 2x + 3y=2x+3.



2. Solve for x: 2x−3=72x - 3 = 72x−3=7

A. x=5x = 5x=5

B. x=3x = 3x=3

C. x=2x = 2x=2

D. x=10x = 10x=10
Answer: A) x=5x = 5x=5
Rationale:

To solve 2x−3=72x - 3 = 72x−3=7, add 3 to both sides:

2x=102x = 102x=10, then divide by 2:

x=5x = 5x=5.



3. Solve for xxx: log⁡2(x−3)=4\log_2 (x - 3) = 4log2(x−3)=4

A. x=16x = 16x=16

,B. x=7x = 7x=7

C. x=4x = 4x=4

D. x=8x = 8x=8

Answer: B) x=7x = 7x=7

Rationale:

We solve log⁡2(x−3)=4\log_2 (x - 3) = 4log2(x−3)=4 by rewriting it in exponential form:

x−3=24=16x - 3 = 2^4 = 16x−3=24=16, so x=16+3=7x = 16 + 3 = 7x=16+3=7.


4. The derivative of f(x)=3x2−5x+2f(x) = 3x^2 - 5x + 2f(x)=3x2−5x+2 is:

A. 6x−56x - 56x−5
B. 6x+56x + 56x+5

C. 3x−53x - 53x−5

D. 6x−26x - 26x−2

Answer: A) 6x−56x - 56x−5

Rationale:

The derivative of f(x)=3x2−5x+2f(x) = 3x^2 - 5x + 2f(x)=3x2−5x+2 is found by applying basic
differentiation rules:

f′(x)=6x−5f'(x) = 6x - 5f′(x)=6x−5.


5. What is the value of ∫01(3x2) dx\int_0^1 (3x^2) \, dx∫01(3x2)dx?
A. 111

B. 13\frac{1}{3}31

C. 333

D. 12\frac{1}{2}21

Answer: B) 13\frac{1}{3}31

Rationale:

Integrating 3x23x^23x2 with respect to xxx:

, ∫3x2 dx=x3\int 3x^2 \, dx = x^3∫3x2dx=x3, so

∫013x2 dx=[x3]01=1−0=1\int_0^1 3x^2 \, dx = [x^3]_0^1 = 1 - 0 = 1∫013x2dx=[x3]01=1−0=1.



6. Find the value of sin⁡45∘\sin 45^\circsin45∘:

A. 12\frac{1}{2}21

B. 22\frac{\sqrt{2}}{2}22

C. 32\frac{\sqrt{3}}{2}23

D. 111

Answer: B) 22\frac{\sqrt{2}}{2}22

Rationale:

Using the exact value for sin⁡45∘\sin 45^\circsin45∘, we know that sin⁡45∘=22\sin 45^\circ =
\frac{\sqrt{2}}{2}sin45∘=22.



7. Find the solution to the system of equations:


A) x=4,y=2x = 4, y = 2x=4,y=2

B) x=5,y=1x = 5, y = 1x=5,y=1

C) x=3,y=3x = 3, y = 3x=3,y=3

D) x=7,y=1x = 7, y = 1x=7,y=1

Answer: A) x=4,y=2x = 4, y = 2x=4,y=2
Rationale:

By solving the system, adding both equations gives:

2x=82x = 82x=8, so x=4x = 4x=4.

Substituting x=4x = 4x=4 into x+y=6x + y = 6x+y=6, we get 4+y=64 + y = 64+y=6, so y=2y =
2y=2.



8. What is the integral of ∫x2 dx\int x^2 \, dx∫x2dx?
A. x33+C\frac{x^3}{3} + C3x3+C