(a) Complete the table of values for \(y = x^3 - 3x - 1\)
\(x\)
−2
−1
0
1
2
3
\(y\)
1
(2)
(b) On the grid, draw the graph of \(y = x^3 - 3x - 1\) for \(-2 \leqslant x \leqslant 3\)(2)
(c) By drawing a suitable straight line on the grid, find an estimate for the solution of the equation \(x^3 - 3x - 6 = 0\) Give your answer correct to 1 decimal place. (2)
(d) For the curve with equation \(y = x^3 - 3x - 1\)
(i) find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\)
(ii) find the gradient of the curve at the point where \(x = 4\)
(4)
Mark scheme (a)
Scheme
Marks
−3, (1), −1, −3, 1, 17
B2
(2)
Notes
B2: for all correct, B1 for 3 or 4 correct
Apart from questions 5, 7, 13c, 16b, 20, 21 and 22 (where the mark scheme states otherwise) the correct answer, unless clearly obtained by an incorrect method, should be taken to imply a correct method.
Mark scheme (b)
Scheme
Marks
All points plotted correctly from their table
B1
Curve
B1
(2)
Notes
B1: B1 ft if at least B1 scored in (a) Plotting tolerance ± ½ sq
B1: B1 ft if B1 scored from plotting points. Must be attempt at a smooth curve & not line segments
Mark scheme (c)
Scheme
Marks
Line segment at \(y = 5\) drawn
M1
2.2 → 2.5 inc
A1
(2)
Notes
M1: M1 for \(x^3 - 3x - 1 = 5\) stated or evidence of reading from \(y = 5\) or \(y\)=5 stated
