Solution of Cambridge International AS and A Level Mathematics 9709-01. Paper 1 Pure Mathematics 1

Cambridge International AS and A Level

9709-01

For examination from 2020

shahbaz ahmed


August 2024




MATHEMATICS

Paper 1 Pure Mathematics 1

SPECIMEN PAPER


Introduction


It is not merely solutions of problems but verification along

with graph to create comprehensive learning and problem

solving strategy.




1

, Q1

The following points

A(0, 1), B(1, 6), C(1.5, 7.75), D(1.9, 8.79) and E(2, 9)

lie on the curve y = f(x). The table below shows the gradients

of the chords AE and BE.




Chord AE BE CE DE

Gradient 4 3

of chord




(a) Complete the table to show the gradients of CE and DE.

(b) State what the values in the table indicate about the value of

f ′ (2).

Solution

(a) Complete the table to show the gradients of CE and DE.

Here

C(1.5,7.75), E(2,9)

2

, 9−7.75 1.25 5
Gradient of CE= 2−1.5 = 0.5 = 2 = 2.5

D(1.9,8.79), E(2,9)

Gradient of DE= 9−8.79
2−1.9 =
21
10 = 2.1

Hence




Chord AE BE CE DE

Gradient 4 3 2.5 2.1

of chord




(b) State what the values in the table indicate about the value of

f ′ (2).

Since

Gradient of DE=2.1

f ′ (x) ≈ 2.1

=⇒ x = 2

Verification



3

, Let

f (x) = ax2 + bx + c

Putting A(0,1)=A(x,f(x))

1=c

Or

c=1

Hence equation becomes

f (x) = ax2 + bx + 1

Putting B(1,6)=B(x,f(x))

6=a+b+1

a+b=5

b=5−a

Again putting E(2,9)=E(x,f(x)) in the equation

f (x) = ax2 + bx + 1

=⇒

9 = 4a + 2b + 1

4a + 2b = 8

Putting b=5-a

4a + 2(5 − a) = 8

4