Simplify √75, giving your answer in the form ksqrt(3).
(Total for Question 2 is 1 mark)
3
Write down the value of 4-1/2.
(Total for Question 3 is 1 mark)
4
Factorise x2 - 64.
(Total for Question 4 is 1 mark)
5
Solve x2 = 121, giving both values of x.
(Total for Question 5 is 1 mark)
6
Simplify (x3)(x6)/x4.
(Total for Question 6 is 1 mark)
7
f(x) = 4x - 9. Find f(5).
(Total for Question 7 is 1 mark)
8
g(x) = x2 + 3, x is real. State the range of g.
(Total for Question 8 is 1 mark)
9
Simplify (3x + 15)/3.
(Total for Question 9 is 1 mark)
10
Write 272/3 as an integer.
(Total for Question 10 is 1 mark)
11
Solve the equation x2 - 7x + 10 = 0.
(Total for Question 11 is 2 marks)
12
Express 10/√5 in the form ksqrt(5), stating the value of k.
(Total for Question 12 is 2 marks)
13
f(x) = 3x - 8. Find f-1(x).
(Total for Question 13 is 2 marks)
14
Solve the inequality 3x2 - 11x - 4 ≤ 0.
(Total for Question 14 is 3 marks)
15
Simplify fully (x2 - 16)/(x2 - 2x - 8), stating any values of x for which the expression is undefined.
(Total for Question 15 is 3 marks)
16
Solve the equation |3x - 4| = 11.
(Total for Question 16 is 3 marks)
17
Find the set of values of x for which x2 + 3x - 10 > 0.
(Total for Question 17 is 3 marks)
18
The curve C has equation y = x2 - 4x + 1. The line l has equation y = 2x - 7. Find the coordinates of the point(s) where l intersects C.
(Total for Question 18 is 4 marks)
19
The function f is defined by f(x) = (4x + 3)/(x - 4), x is real, x not equal to 4. Show that f is a self-inverse function, and hence write down the value of ff(6).
(Total for Question 19 is 4 marks)
20
Prove that x2 - 10x + 30 > 0 for all real values of x, and state the minimum value of the expression and the value of x at which it occurs.
(Total for Question 20 is 4 marks)
Mark scheme · P2D Pure: Algebra and Functions: Fluency and Exam Drill
Question 1
B1 9x8 cao
Answer: 9x8
Question 2
B1 5sqrt(3) cao
Answer: 5sqrt(3)
Question 3
B1 1/2 cao
Answer: 1/2
Question 4
B1 (x - 8)(x + 8) cao
Answer: (x - 8)(x + 8)
Question 5
B1 x = 11 or x = -11, both values cao
Answer: x = 11 or x = -11
Question 6
B1 x5 cao
Answer: x5
Question 7
B1 11 cao
Answer: 11
Question 8
B1 g(x) ≥ 3 oe
Answer: g(x) ≥ 3
Question 9
B1 x + 5 cao
Answer: x + 5
Question 10
B1 9 cao
Answer: 9
Question 11
M1 factorises (x - 2)(x - 5)
A1 x = 2 or x = 5 cao
Answer: x = 2 or x = 5
Question 12
M1 multiplies numerator and denominator by √5
A1 2sqrt(5), i.e. k = 2 cao
Answer: 10/√5 = 2sqrt(5), k = 2
Question 13
M1 sets y = 3x - 8 and rearranges to make x the subject
A1 f-1(x) = (x + 8)/3 cao
Answer: f-1(x) = (x + 8)/3
Question 14
M1 factorises 3x2 - 11x - 4 as (3x + 1)(x - 4)
A1 critical values x = -1/3 and x = 4
A1 -1/3 ≤ x ≤ 4 cao, correct inequality direction
Answer: -1/3 ≤ x ≤ 4
Question 15
M1 factorises numerator as (x - 4)(x + 4)
M1 factorises denominator as (x - 4)(x + 2)
A1 (x + 4)/(x + 2) cao, x not equal to 4 or -2
Answer: (x + 4)/(x + 2), x not equal to 4 or -2
Question 16
M1 sets up two linear equations 3x - 4 = 11 and 3x - 4 = -11
A1 x = 5
A1 x = -7/3
Answer: x = 5 or x = -7/3
Question 17
M1 factorises x2 + 3x - 10 as (x + 5)(x - 2)
A1 critical values x = -5 and x = 2
A1 x < -5 or x > 2 cao
Answer: x < -5 or x > 2
Question 18
M1 sets x2 - 4x + 1 = 2x - 7
M1 rearranges to x2 - 6x + 8 = 0
A1 x = 2 and x = 4 (factorises or uses the quadratic formula)
A1 coordinates (2, -3) and (4, 1), both correct
Answer: (2, -3) and (4, 1)
Question 19
M1 sets y = (4x+3)/(x-4) and cross-multiplies: y(x-4) = 4x+3
M1 collects x terms on one side: x(y-4) = 4y+3
A1 f-1(x) = (4x+3)/(x-4), identical to f(x), so f is self-inverse, cso
B1ft ff(6) = 6, using the self-inverse property (ft from part shown)
Answer: f is self-inverse (shown); ff(6) = 6
Question 20
M1 completes the square: (x - 5)2 + k
A1 correct completed square (x - 5)2 + 5
A1 valid conclusion: (x-5)2 ≥ 0 for all real x, so the expression ≥ 5 > 0, cso
B1 minimum value 5, occurring at x = 5
Answer: x2 - 10x + 30 = (x-5)2 + 5 ≥ 5 > 0 (proved); minimum value 5 at x = 5