Question
Let \(f(x) = 4{\tan ^2}x – 4\sin x\) , \( – \frac{\pi }{3} \le x \le \frac{\pi }{3}\) .
On the grid below, sketch the graph of \(y = f(x)\) .
Solve the equation \(f(x) = 1\) .
Answer/Explanation
Markscheme
A1A1A1 N3
Note: Award A1 for passing through \((0{\text{, }}0)\), A1 for correct shape, A1 for a range of approximately \( – 1\) to 15.
[3 marks]
evidence of attempt to solve \(f(x) = 1\) (M1)
e.g. line on sketch, using \(\tan x = \frac{{\sin x}}{{\cos x}}\)
\(x = – 0.207\) , \(x = 0.772\) A1A1 N3
[3 marks]
Question
Let \(f(x) = x\cos x\) , for \(0 \le x \le 6\) .
Find \(f'(x)\) .
On the grid below, sketch the graph of \(y = f'(x)\) .
Answer/Explanation
Markscheme
evidence of choosing the product rule (M1)
e.g. \(x \times ( – \sin x) + 1 \times \cos x\)
\(f'(x) = \cos x – x\sin x\) A1A1 N3
[3 marks]
A1A1A1A1 N4
Note: Award A1 for correct domain, \(0 \le x \le 6\) with endpoints in circles, A1 for approximately correct shape, A1 for local minimum in circle, A1 for local maximum in circle.
[4 marks]
Question
Consider \(f(x) = 2 – {x^2}\) , for \( – 2 \le x \le 2\) and \(g(x) = \sin {{\rm{e}}^x}\) , for \( – 2 \le x \le 2\) . The graph of f is given below.
On the diagram above, sketch the graph of g.
Solve \(f(x) = g(x)\) .
Write down the set of values of x such that \(f(x) > g(x)\) .
Answer/Explanation
Markscheme
A1A1A1 N3
[3 marks]
\(x = – 1.32\) , \(x = 1.68\) (accept \(x = – 1.41\) , \(x = 1.39\) if working in degrees) A1A1 N2
[2 marks]
\( – 1.32 < x < 1.68\) (accept \( – 1.41 < x < 1.39\) if working in degrees) A2 N2
[2 marks]
Question
The velocity v ms−1 of an object after t seconds is given by \(v(t) = 15\sqrt t – 3t\) , for \(0 \le t \le 25\) .
On the grid below, sketch the graph of v , clearly indicating the maximum point.
(i) Write down an expression for d .
(ii) Hence, write down the value of d .
Answer/Explanation
Markscheme
A1A1A1 N3
Note: Award A1 for approximately correct shape, A1 for right endpoint at \((25{\text{, }}0)\) and A1 for maximum point in circle.
[3 marks]
(i) recognizing that d is the area under the curve (M1)
e.g. \(\int {v(t)} \)
correct expression in terms of t, with correct limits A2 N3
e.g. \(d = \int_0^9 {(15\sqrt t } – 3t){\rm{d}}t\) , \(d = \int_0^9 v {\rm{d}}t\)
(ii) \(d = 148.5\) (m) (accept 149 to 3 sf) A1 N1
[4 marks]
Question
Let \(g(x) = \frac{1}{2}x\sin x\) , for \(0 \le x \le 4\) .
Sketch the graph of g on the following set of axes.
Hence find the value of x for which \(g(x) = – 1\) .
Answer/Explanation
Markscheme
A1A1A1A1 N4
Note: Award A1 for approximately correct shape, A1 for left end point in circle, A1 for local maximum in circle, A1 for right end point in circle.
[4 marks]
attempting to solve \(g(x) = – 1\) (M1)
e.g. marking coordinate on graph, \(\frac{1}{2}x\sin x + 1 = 0\)
\(x = 3.71\) A1 N2
[2 marks]
Question
Consider the expansion of \({(x + 2)^{11}}\) .
Write down the number of terms in this expansion.
Find the term containing \({x^2}\) .
Answer/Explanation
Markscheme
12 terms A1 N1
[1 mark]
evidence of binomial expansion (M1)
e.g. \(\left( \begin{array}{l}
n\\
r
\end{array} \right){a^{n – r}}{b^r}\) , an attempt to expand, Pascal’s triangle
evidence of choosing correct term (A1)
e.g. 10th term , \(r = 9\) , \(\left( {\begin{array}{*{20}{c}}
{11}\\
9
\end{array}} \right)\) , \({(x)^2}{(2)^9}\)
correct working A1
e.g. \(\left( {\begin{array}{*{20}{c}}
{11}\\
9
\end{array}} \right){(x)^2}{(2)^9}\) , \(55 \times {2^9}\)
\(28160{x^2}\) A1 N2
[4 marks]