OCR A Level Further Mathematics (MEI) 2025 Y540/01 Pure Core 1 – Actual Exam Paper

Oxford Cambridge and RSA

GCE


Further Mathematics A

Y540/01: Pure Core 1
*1920517432*




A Level




INSTRUCTIONS
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given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
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needed use g= 9.8 unless a different value is specified in the question.
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INFORMATION
• The total mark for this paper is 75.
• The marks for each question are shown in brackets [ ].
• This document has 8 pages.

ADVICE
• Read each question carefully before you start your answer.
© OCR 2025 [A/508/5505] DC (PQ/TC) 359052/5
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1 (a) A matrix M is given by M =JKKL05 05NOOP.


Describe the transformation represented by M.

, 2
(b) Write down the 2#2 matrix that represents a rotation of 90° anticlockwise about the
origin.




[1]
(c) Write down the 3#3 matrix that represents a reflection in the x–z plane. [1]




d
2 (a) Given that y=cosh-1b l13x , find yx in terms of x. [1]

d

(b) Determine an equation of the normal to the curve y=cosh-1b l13x at the point where
5.

Give your answer in the form ax+ = +by c lnd, where
and d are integers. [4]




© OCR 2025 Y540/01 Jun25

, 3 80
3 The region R1 is bounded by the curve y 25 - x 2 =, the line x = k and the
coordinate axes, as shown in Fig. 1.
Fig. 1

80
y= 2
y 25 - x




R1

O x




x =k

(a) In this question you must show detailed reasoning.

Given that the area of R1 is r, determine the value of k.

80
The region R2 is bounded by the curve y25
=, the2 line y = 20 and the y-axis as shown in
- x
Fig. 2.
Fig. 2

80
y= 2
y 25 - x




y =20
R2


O x




(b) Find, in an exact form, the volume of the solid formed when R2 is rotated by
2r radians
about the y-axis.
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© OCR 2025 Y540/01 Jun25

, 4
4 The equation 5x - + =4x 10
3
0 has roots a, b and c.
2


(a) Write down the values of a b c+ + , ab+ +bc ca and abc. [2]

(b) By expanding (ab+ +bc ca)2, determine the value of a b2 2+b c2 2+c a2 2. [3]

(c) By expanding a suitable expression, find the value of a b c2+ +2 2
. [2]
(d) Hence find a cubic equation with integer coefficients that has roots a2, b2, and c2. [2]



JK1NO JK 3NO JK1NO

P m n
5 A vector equation of the plane 1 is r = +KKL43OOP KK 11POO+ KLK10OOP.

L-
(a) Verify that a cartesian equation of P1 is x- +y 2z = 3. [1]
P P
For some real constant a, cartesian equations of planes 2 and 3 are

P 2| x - =3z 1

P3| ax - - =y z 4

(b) By considering a suitable matrix, show that P1, P2 and P3 intersect at a single
point for all values of a except a = 2. [3]

(c) Use the matrix from part (b) to find the coordinates of the point of
P P
intersection of 1, 2


and
P
in the case where a = 3. [2]
3



(d) In the case where a = 2, determine the geometrical arrangement of 1,
P P P
and 3. [2]
2




6 In this question you must show detailed reasoning.



JL1 # 1 NOOP+KJKL151 # 251 NOOP+ +g JKKL10n1-5 # 10n1+5NOOP for n!Z+.
The series Sn is given by Sn =KK5 15
S 1 [5]
(a) Use the method of differences to show that, for all n!Z+, n 501 .


Let S3= nlim"3Sn.

(b) Given that, for some value of k, S3= 24501 +Sk, find the value of k. [3]

© OCR 2025 Y540/01 Jun25