GCSE Further Maths Paper 2
Covers Algebraic Manipulation, Surds and Indices, Simultaneous Equations and Inequalities and 9 more.
Questions
Question 1 [2 marks]
Surds and Indices
Simplify sqrt(50) - sqrt(18), giving your answer in the form k*sqrt(2).
Question 2 [2 marks]
Geometric Proof
Triangle ABC is isosceles with AB = AC.
D is the midpoint of BC.
Prove that triangle ABD is congruent to triangle ACD.
Question 3 [4 marks]
Trigonometric Identities and Equations
Show that (cos(x) - sin(x))^2 can be simplified to 1 - 2sin(x)cos(x).
Question 4 [4 marks]
Geometric Proof
Triangle PQR and triangle PSR share the side PR.
Angle PQR = angle PSR = 90 degrees, and QR = SR.
Prove that triangle PQR is congruent to triangle PSR.
Question 5 [4 marks]
Simultaneous Equations and Inequalities
Solve the simultaneous equations 4x + y = 16 and 2x - 3y = -6.
Question 6 [4 marks]
Algebraic Manipulation
Simplify fully (x^2 - 9)/(x^2 + x - 6).
Question 7 [4 marks]
Sequences
The nth term of a sequence is given by Un = 2n^2 + 3n.
Find the 6th term of the sequence, and determine whether 200 is a term of the sequence, showing your working.
Question 8 [5 marks]
Non-Right-Angled Triangle Trigonometry
Triangle XYZ has XY = 10 cm, XZ = 13 cm, and area 55 cm^2.
Find the possible size(s) of angle YXZ, giving your answer(s) to 1 decimal place.
Question 9 [5 marks]
Coordinate Geometry
A circle has equation x^2 + y^2 - 6x + 4y - 12 = 0.
Find the centre and radius of the circle by completing the square.
Question 10 [5 marks]
Applications of Differentiation
A curve has equation y = x^2 + 5x - 2.
Find the equation of the normal to the curve at the point where x = -2, giving your answer in the form y = mx + c.
Question 11 [5 marks]
Matrices and Transformations
The matrix N represents an enlargement, scale factor 2, centre the origin, followed by a reflection in the x-axis.
Write down the matrix representing the enlargement and the matrix representing the reflection, and hence find N as a single 2x2 matrix.
State which of the two matrices is applied first when N is used to transform a point.
Question 12 [5 marks]
Differentiation
A curve has equation y = x^3 - 3x^2 - 9x + 4.
Find the range of values of x for which the curve is decreasing.
Question 13 [5 marks]
Sequences
The sum of the first n terms of an arithmetic series is given by Sn = n(2n - 3).
Find the first term and the common difference of the series.
Hence find the 15th term of the series.
Question 14 [6 marks]
Functions and their Graphs
The function k is defined by k(x) = -2x^2 + 12x - 7 for all real x.
Express k(x) in the form a - b(x - c)^2, where a, b and c are integers to be found.
Hence state the maximum value of k(x) and the value of x at which it occurs.
Model solutions
| Question 1[2 marks] | |
|---|---|
| Answer or working | Marks |
| writing sqrt(50) = 5sqrt(2) and sqrt(18) = 3sqrt(2) | M1 |
| 2sqrt(2) | A1 |
| Question 2[2 marks] | |
|---|---|
| Answer or working | Marks |
| stating BD = CD (D is the midpoint of BC) and AD is common to both triangles | B1 |
| concluding triangle ABD is congruent to triangle ACD by SSS, using AB = AC given | B1 |
| Final answer: Congruent by SSS: AB = AC (given), BD = CD (D midpoint), AD common | |
| Question 3[4 marks] | |
|---|---|
| Answer or working | Marks |
| expanding the brackets to give cos^2(x) - 2sin(x)cos(x) + sin^2(x) | M1 |
| stating the identity sin^2(x) + cos^2(x) = 1 | B1 |
| substituting sin^2(x) + cos^2(x) = 1 | M1 |
| 1 - 2sin(x)cos(x) | A1 |
| Final answer: 1 - 2sin(x)cos(x), shown using sin^2(x) + cos^2(x) = 1 | |
| Question 4[4 marks] | |
|---|---|
| Answer or working | Marks |
| stating angle PQR = angle PSR = 90 degrees (given) | B1 |
| stating PR is common to both triangles, and is the hypotenuse of each | B1 |
| stating QR = SR (given) | B1 |
| concluding triangle PQR is congruent to triangle PSR by RHS | B1 |
| Final answer: Congruent by RHS: angle PQR = angle PSR = 90 degrees, PR common (hypotenuse), QR = SR (given) | |
| Question 5[4 marks] | |
|---|---|
| Answer or working | Marks |
| rearranging 4x + y = 16 to make y the subject, y = 16 - 4x | M1 |
| substituting into 2x - 3y = -6 | M1 |
| simplifying to 14x = 42 | M1 |
| x = 3 and y = 4 | A1 |
| Final answer: x = 3, y = 4 | |
| Question 6[4 marks] | |
|---|---|
| Answer or working | Marks |
| factorising the numerator as (x - 3)(x + 3) | M1 |
| factorising the denominator as (x + 3)(x - 2) | M1 |
| cancelling the common factor (x + 3) | M1 |
| (x - 3)/(x - 2) | A1 |
| Question 7[4 marks] | |
|---|---|
| Answer or working | Marks |
| substituting n = 6 into Un = 2n^2 + 3n | M1 |
| the 6th term = 90 | A1 |
| forming the equation 2n^2 + 3n - 200 = 0 and using the discriminant | M1 |
| concluding 200 is not a term since n is not a positive integer, as the discriminant 1609 is not a perfect square | A1 |
| Final answer: 6th term = 90; 200 is not a term of the sequence | |
| Question 8[5 marks] | |
|---|---|
| Answer or working | Marks |
| using Area = (1/2)(XY)(XZ)sin(X) | M1 |
| substituting to get 55 = (1/2)(10)(13)sin(X) | M1 |
| rearranging to sin(X) = 55/65 | M1 |
| X = 57.8 degrees | A1 |
| X = 122.2 degrees | A1 |
| Final answer: angle YXZ = 57.8 degrees or 122.2 degrees (1 d.p.) | |
| Question 9[5 marks] | |
|---|---|
| Answer or working | Marks |
| completing the square on the x terms: (x - 3)^2 - 9 | M1 |
| completing the square on the y terms: (y + 2)^2 - 4 | M1 |
| rearranging to (x - 3)^2 + (y + 2)^2 = 25 | M1 |
| centre (3, -2) | A1 |
| radius 5 | A1 |
| Final answer: Centre (3, -2), radius 5 | |
| Question 10[5 marks] | |
|---|---|
| Answer or working | Marks |
| finding y = -8 at x = -2 | M1 |
| differentiating to get dy/dx = 2x + 5 | M1 |
| finding the tangent gradient = 1 at x = -2 | M1 |
| using the perpendicular gradient rule to get normal gradient -1 | M1 |
| y = -x - 10 | A1 |
| Question 11[5 marks] | |
|---|---|
| Answer or working | Marks |
| the enlargement matrix, the matrix with rows (2, 0) and (0, 2) | B1 |
| the reflection matrix, the matrix with rows (1, 0) and (0, -1) | B1 |
| multiplying the reflection matrix by the enlargement matrix in that order | M1 |
| N = the matrix with rows (2, 0) and (0, -2) | A1 |
| stating the enlargement is applied first, since it is written on the right of the product | B1 |
| Final answer: N = the matrix with rows (2, 0) and (0, -2), found as (reflection matrix) x (enlargement matrix) | |
| Question 12[5 marks] | |
|---|---|
| Answer or working | Marks |
| differentiating to get dy/dx = 3x^2 - 6x - 9 | M1 |
| setting dy/dx = 0 | M1 |
| solving to find x = -1 and x = 3, e.g. factorising as 3(x - 3)(x + 1) = 0 | M1 |
| reasoning that dy/dx < 0 between the two roots | M1 |
| -1 < x < 3 | A1 |
| Question 13[5 marks] | |
|---|---|
| Answer or working | Marks |
| using S1 = a to find the first term a = -1 | M1 |
| finding S2 and using T2 = S2 - S1 | M1 |
| common difference d = 4 | A1 |
| using T15 = a + 14d | M1 |
| T15 = 55 | A1 |
| Final answer: first term = -1; common difference = 4; 15th term = 55 | |
| Question 14[6 marks] | |
|---|---|
| Answer or working | Marks |
| taking out the factor -2 from the x terms: -2(x^2 - 6x) - 7 | M1 |
| completing the square inside the bracket: (x - 3)^2 - 9 | M1 |
| substituting back to get -2[(x - 3)^2 - 9] - 7 | M1 |
| 11 - 2(x - 3)^2 (a = 11, b = 2, c = 3) | A1 |
| identifying the maximum value as 11 | B1 |
| stating this occurs at x = 3 | B1 |
| Final answer: 11 - 2(x - 3)^2; maximum value 11 at x = 3 | |