User interface language: English | Español

Date May 2018 Marks available 4 Reference code 18M.2.SL.TZ2.T_6
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Sketch Question number T_6 Adapted from N/A

Question

Consider the curve y = 2x3 − 9x2 + 12x + 2, for −1 < x < 3

Sketch the curve for −1 < x < 3 and −2 < y < 12.

[4]
a.

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

[1]
b.

Find dy dx .

[3]
d.

Given that y = 2x3 − 9x2 + 12x + 2 = k has three solutions, find the possible values of k.

[3]
f.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

(A1)(A1)(A1)(A1)

Note: Award (A1) for correct window (condone a window which is slightly off) and axes labels. An indication of window is necessary. −1 to 3 on the x-axis and −2 to 12 on the y-axis and a graph in that window.
(A1) for correct shape (curve having cubic shape and must be smooth).
(A1) for both stationary points in the 1st quadrant with approximate correct position,
(A1) for intercepts (negative x-intercept and positive y intercept) with approximate correct position.

[4 marks]

a.

Rick     (A1)

Note: Award (A0) if extra names stated.

[1 mark]

b.

6x2 − 18x + 12     (A1)(A1)(A1)

Note: Award (A1) for each correct term. Award at most (A1)(A1)(A0) if extra terms seen.

[3 marks]

d.

6 < k < 7     (A1)(A1)(ft)(A1)

Note: Award (A1) for an inequality with 6, award (A1)(ft) for an inequality with 7 from their part (c) provided it is greater than 6, (A1) for their correct strict inequalities. Accept ]6, 7[ or (6, 7).

[3 marks]

f.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
d.
[N/A]
f.

Syllabus sections

Topic 5—Calculus » SL 5.1—Introduction of differential calculus
Show 80 related questions
Topic 5—Calculus » SL 5.2—Increasing and decreasing functions
Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection
Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z
Topic 2—Functions » SL 2.3—Graphing
Topic 5—Calculus » SL 5.4—Tangents and normal
Topic 2—Functions » SL 2.5—Modelling functions
Topic 5—Calculus » SL 5.6—Stationary points, local max and min

View options