WJEC Unit 1 (Unit 1) 2018 June

Showing all your working, simplify a) \(\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }\),
b) \(\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }\).

The points \(A\) and \(B\) have coordinates \(( - 1,10 )\) and \(( 5,1 )\) respectively. The straight line \(L\) has equation \(2 x - 3 y + 6 = 0\). a) The line \(L\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
b) Determine the ratio in which the line \(L\) divides the line \(A B\).
c) The line \(L\) crosses the \(x\)-axis at the point \(D\). Find the coordinates of \(D\).
d) i) Show that \(L\) is perpendicular to \(A B\).
ii) Calculate the area of the triangle \(A C D\).

Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$ a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\). The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
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0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\). Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$

Given that \(( x - 2 )\) and \(( x + 2 )\) are factors of the polynomial \(2 x ^ { 3 } + p x ^ { 2 } + q x - 12\), a) find the values of \(p\) and \(q\),
b) determine the other factor of the polynomial.

The triangle \(A B C\) is such that \(A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}\) and \(A \widehat { B C } = 32 ^ { \circ }\). Find two possible values for the area of the triangle \(A B C\). a) Use the binomial theorem to expand \(( a + \sqrt { b } ) ^ { 4 }\).
b) Hence, deduce an expression in terms of \(a\) and \(b\) for \(( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }\). a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }\) and \(\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }\). Determine the range of values for \(\mu\) such that \(\mu | \mathbf { v } | > | \mathbf { u } |\).
b) The point \(A\) has position vector \(11 \mathbf { i } - 4 \mathbf { j }\) and the point \(B\) has position vector \(21 \mathbf { i } + \mathbf { j }\). Determine the position vector of the point \(C\), which lies between \(A\) and \(B\), such that \(A C : C B\) is \(2 : 3\). Find the values of \(m\) for which the equation \(4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )\) has real roots. [5]

A curve \(C\) has equation \(y = x ^ { 3 } - 3 x ^ { 2 }\). a) Find the stationary points of \(C\) and determine their nature.
b) Draw a sketch of \(C\), clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\) is positive or
negative, giving a reason for your answer. In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d . \\ & \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true. The value of a car, \(\pounds V\), may be modelled as a continuous variable. At time \(t\) years, the value of the car is given by \(V = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. When the car is new, it is worth \(\pounds 30000\). When the car is two years old, it is worth \(\pounds 20000\). Determine the value of the car when it is six years old, giving your answer correct to the nearest \(\pounds 100\). The curve \(C\) has equation \(y = 7 + 13 x - 2 x ^ { 2 }\). The point \(P\) lies on \(C\) and is such that the tangent to \(C\) at \(P\) has equation \(y = x + c\), where \(c\) is a constant. Find the coordinates of \(P\) and the value of \(c\). a) Solve \(2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2\).
b) Solve \(3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }\).
c) Express \(4 ^ { x } - 10 \times 2 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\). Hence solve the equation \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\).

The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.