Standard +0.3 This is a standard C3 wave function question with routine transformations. Part (a) is a textbook R-formula exercise, part (b) requires algebraic manipulation of trig identities (multiplying through by sin x cos x), and part (c) is a direct substitution. While multi-step, each component follows well-practiced techniques with no novel insight required, making it slightly easier than average.
2. Express − as a single fraction in its simplest form.
(y +1)(y + 2) (y +2)(y +3)
(5)
3. The function f is even and has domain
2
(cid:0)
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. For x ≥ 0, f(x) = x2 – 4ax, where a is a positive
constant.
(a) In the space below, sketch the curve with equation y = f(x), showing the coordinates of
all the points at which the curve meets the axes. (3)
(b) Find, in terms of a, the value of f(2a) and the value of f(–2a). (2)
Given that a = 3,
(c) use algebra to find the values of x for which f(x) = 45. (4)
4. f(x) = x3 + x2 − 4x − 1.
The equation f(x) = 0 has only one positive root, α.
(a) Show that f(x) = 0 can be rearranged as
4x+1
x = , x ≠ −1.
x+1
(2)
4x +1
The iterative formula x = n is used to find an approximation to α.
n + 1
x +1
n
(b) Taking x = 1, find, to 2 decimal places, the values of x , x and x
1 2 3 4.
(3)
(c) By choosing values of x in a suitable interval, prove that α = 1.70, correct to 2 decimal
places.
(3)
4x +1
(d) Write down a value of x for which the iteration formula x = n does not
1 n + 1
x +1
n
produce a valid value for x .
2
Justify your answer.
(2)
5. The functions f and g are defined by
Answer
Marks
Guidance
f:xα
x – a
+ a, x ∈
3
(cid:0) ,
g:xα 4x + a, x ∈ (cid:0)
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.
where a is a positive constant.
(a) On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of
any points at which your graphs meet the axes. (5)
(b) Use algebra to find, in terms of a, the coordinates of the point at which the graphs of
f and g intersect. (3)
(c) Find an expression for fg(x). (2)
(d) Solve, for x in terms of a, the equation
fg(x) = 3a. (3)
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6. Figure 1
y
O k x
Figure 1 shows a sketch of the curve with equation y = f(x), where
f(x) = 10 + ln(3x) − 1 ex , 0.1 ≤ x ≤ 3.3.
2
Given that f(k) = 0,
(a) show, by calculation, that 3.1 < k < 3.2. (2)
(b) Find f ′(x). (3)
The tangent to the graph at x = 1 intersects the y-axis at the point P.
(c) (i) Find an equation of this tangent.
(ii) Find the exact y-coordinate of P, giving your answer in the form a + ln b. (5)
4
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7. (a) Express sin x + √3 cos x in the form R sin (x + α), where R > 0 and 0 < α < 90°.
(4)
(b) Show that the equation sec x + √3 cosec x = 4 can be written in the form
sin x + √3 cos x = 2 sin 2x. (3)
(c) Deduce from parts (a) and (b) that sec x + √3 cosec x = 4 can be written in the form
sin 2x – sin (x + 60°) = 0. (1)
END
5
Question 7:
7
Total
1. The function f is given by
x 1
f :x α − , x >1.
x2 −1 x+1
1
(a) Show that f(x) = . (3)
(x−1)(x+1)
(b) Find the range of f. (2)
The function g is given by
2
g : x α , x > 0.
x
(c) Solve gf(x) = 70. (4)
y +3 y +1
2. Express − as a single fraction in its simplest form.
(y +1)(y + 2) (y +2)(y +3)
(5)
3. The function f is even and has domain
2
(cid:0)
PhysicsAndMathsTutor.com
. For x ≥ 0, f(x) = x2 – 4ax, where a is a positive
constant.
(a) In the space below, sketch the curve with equation y = f(x), showing the coordinates of
all the points at which the curve meets the axes. (3)
(b) Find, in terms of a, the value of f(2a) and the value of f(–2a). (2)
Given that a = 3,
(c) use algebra to find the values of x for which f(x) = 45. (4)
4. f(x) = x3 + x2 − 4x − 1.
The equation f(x) = 0 has only one positive root, α.
(a) Show that f(x) = 0 can be rearranged as
4x+1
x = , x ≠ −1.
x+1
(2)
4x +1
The iterative formula x = n is used to find an approximation to α.
n + 1
x +1
n
(b) Taking x = 1, find, to 2 decimal places, the values of x , x and x
1 2 3 4.
(3)
(c) By choosing values of x in a suitable interval, prove that α = 1.70, correct to 2 decimal
places.
(3)
4x +1
(d) Write down a value of x for which the iteration formula x = n does not
1 n + 1
x +1
n
produce a valid value for x .
2
Justify your answer.
(2)
5. The functions f and g are defined by
f:xα | x – a | + a, x ∈
3
(cid:0) ,
g:xα 4x + a, x ∈ (cid:0)
PhysicsAndMathsTutor.com
.
where a is a positive constant.
(a) On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of
any points at which your graphs meet the axes. (5)
(b) Use algebra to find, in terms of a, the coordinates of the point at which the graphs of
f and g intersect. (3)
(c) Find an expression for fg(x). (2)
(d) Solve, for x in terms of a, the equation
fg(x) = 3a. (3)
PhysicsAndMathsTutor.com
6. Figure 1
y
O k x
Figure 1 shows a sketch of the curve with equation y = f(x), where
f(x) = 10 + ln(3x) − 1 ex , 0.1 ≤ x ≤ 3.3.
2
Given that f(k) = 0,
(a) show, by calculation, that 3.1 < k < 3.2. (2)
(b) Find f ′(x). (3)
The tangent to the graph at x = 1 intersects the y-axis at the point P.
(c) (i) Find an equation of this tangent.
(ii) Find the exact y-coordinate of P, giving your answer in the form a + ln b. (5)
4
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7. (a) Express sin x + √3 cos x in the form R sin (x + α), where R > 0 and 0 < α < 90°.
(4)
(b) Show that the equation sec x + √3 cosec x = 4 can be written in the form
sin x + √3 cos x = 2 sin 2x. (3)
(c) Deduce from parts (a) and (b) that sec x + √3 cosec x = 4 can be written in the form
sin 2x – sin (x + 60°) = 0. (1)
END
5
\begin{enumerate}[label=(\alph*)]
\item Express $\sin x + \sqrt{3} \cos x$ in the form $R \sin (x + \alpha)$, where $R > 0$ and $0 < \alpha < 90°$. [4]
\item Show that the equation $\sec x + \sqrt{3} \cosec x = 4$ can be written in the form
$$\sin x + \sqrt{3} \cos x = 2 \sin 2x.$$ [3]
\item Deduce from parts (a) and (b) that $\sec x + \sqrt{3} \cosec x = 4$ can be written in the form
$$\sin 2x - \sin (x + 60°) = 0.$$ [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q7 [8]}}