WJEC Unit 3 (Unit 3) 2022 June

Solve the equation $$6 \sec ^ { 2 } x - 8 = \tan x$$ for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

Differentiate the following functions with respect to \(x\). a) \(x ^ { 3 } \ln ( 5 x )\) b) \(( x + \cos 3 x ) ^ { 4 }\)

The diagram below shows a plan of the patio Eric wants to build. The walls \(O A\) and \(O C\) are perpendicular. The straight line \(A B\) is of length 4 m and is perpendicular to \(O A\). The shape \(O B C\) is a sector of a circle with centre \(O\) and radius OC.
The angle \(B O C\) is \(\frac { \pi } { 3 }\) radians. Calculate the area of the patio \(O A B C\). Give your answer correct to 2 decimal places. The sum to infinity of a geometric series with first term \(a\) and common ratio \(r\) is 120 . The sum to infinity of another geometric series with first term \(a\) and common ratio \(4 r ^ { 2 }\) is \(112 \frac { 1 } { 2 }\). Find the possible values of \(r\) and the corresponding values of \(a\). The function \(f ( x )\) is defined by $$f ( x ) = \frac { 6 x + 4 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) }$$ a) Express \(f ( x )\) in terms of partial fractions.
b) Find \(\int \frac { 3 x + 2 } { ( x - 1 ) ( x + 1 ) ( 2 x + 3 ) } \mathrm { d } x\), giving your answer in the form \(a \ln | g ( x ) |\), where \(a\) is a real number and \(g ( x )\) is a function of \(x\). Geraint opens a savings account. He deposits \(\pounds 10\) in the first month. In each subsequent month, the amount he deposits is 20 pence greater than the amount he deposited in the previous month.
a) Find the amount that Geraint deposits into the savings account in the 12th month.
b) Determine the number of months it takes for the total amount in the savings account to reach \(\pounds 954\).
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The diagram below shows a sketch of the curves \(y = x ^ { 2 }\) and \(y = 8 \sqrt { x }\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-3_508_869_2094_623} Find the area of the region bounded by the two curves.

Find the first three terms in the binomial expansion of \(\frac { 2 - x } { \sqrt { 1 + 3 x } }\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac { 1 } { 22 }\) in your expansion, find an approximate value for \(\sqrt { 22 }\) in the form \(\frac { a } { b }\), where \(a , b\) are integers whose values are to be found.

For each of the following sequences, find the first 5 terms, \(u _ { 1 }\) to \(u _ { 5 }\). Describe the behaviour of each sequence. a) \(\quad u _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\) b) \(u _ { 6 } = 33 , u _ { n } = 2 u _ { n - 1 } - 1\)

Solve the equation $$\frac { 6 x ^ { 5 } - 17 x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } } { ( 3 x + 2 ) } = 0$$

a) Express \(9 \cos x + 40 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). b) Find the maximum possible value of \(\frac { 12 } { 9 \cos x + 40 \sin x + 47 }\).

The diagram below shows a sketch of the graph of \(y = f ( x )\), where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.

A function is defined by \(f ( x ) = 2 x ^ { 3 } + 3 x - 5\). a) Prove that the graph of \(f ( x )\) does not have a stationary point.
b) Show that the graph of \(f ( x )\) does have a point of inflection and find the coordinates of the point of inflection.
c) Sketch the graph of \(f ( x )\). Evaluate the integral \(\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x\).

A rectangle is inscribed in a semicircle with centre \(O\) and radius 4. The point \(P ( x , y )\) is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area \(A\) of the rectangle as a function of \(x\).
b) Show that the maximum value of \(A\) occurs when \(y = x\).

The parametric equations of the curve \(C\) are $$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$ a) Find the coordinates of the points where \(C\) meets the \(y\)-axis.
b) Show that the \(x\)-axis is a tangent to the curve \(C\). a) Prove that $$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$ b) i) Hence show that \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }\) can be expressed as \(\cos \theta + \sin \theta\).
ii) Explain why \(\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta\) when \(\theta = \frac { 3 \pi } { 16 }\).

a) Use a suitable substitution to find $$\int \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$$ b) Hence evaluate \(\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x\). END OF PAPER \end{document}