WJEC Unit 1 (Unit 1) 2024 June

1. Given that \(y = 12 \sqrt { x } - \frac { 27 } { x } + 4\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 9\).
   
2. Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 180 ^ { \circ }\) that satisfy the equation

$$2 \sin 2 \theta = 1$$

3. Find \(\int \left( 5 x ^ { \frac { 1 } { 4 } } + 3 x ^ { - 2 } - 2 \right) \mathrm { d } x\).

4. Given that \(n\) is an integer such that \(1 \leqslant n \leqslant 6\), use proof by exhaustion to show that \(n ^ { 2 } - 2\) is not divisible by 3 .

5. A triangle \(A B C\) has sides \(A B = 6 \mathrm {~cm} , B C = 11 \mathrm {~cm}\) and \(A C = 13 \mathrm {~cm}\). Calculate the area of the triangle.

6. (a) Find the exact value of \(x\) that satisfies the equation $$\frac { 7 x ^ { \frac { 5 } { 4 } } } { x ^ { \frac { 1 } { 2 } } } = \sqrt { 147 }$$ (b) Show that \(\frac { ( 8 x - 18 ) } { ( 2 \sqrt { x } - 3 ) }\), where \(x \neq \frac { 9 } { 4 }\), may be written as \(2 ( 2 \sqrt { x } + 3 )\).

7. (a) The line \(L _ { 1 }\) passes through the points \(A ( - 3,0 )\) and \(B ( 1,4 )\). Determine the equation of \(L _ { 1 }\).
(b) The line \(L _ { 2 }\) has equation \(y = 3 x - 3\).

1. Given that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(C\), find the coordinates of \(C\).
2. The line \(L _ { 2 }\) crosses the \(x\)-axis at the point \(D\). Show that the coordinates of \(D\) are \(( 1,0 )\).
   (c) Calculate the area of triangle \(A C D\).
   (d) Determine the angle \(A C D\).

8. Prove that \(x - 10 < x ^ { 2 } - 5 x\) for all real values of \(x\).

9. (a) Write down the binomial expansion of \(( 2 - x ) ^ { 6 }\) up to and including the term in \(x ^ { 2 }\).
(b) Given that $$( 1 + a x ) ( 2 - x ) ^ { 6 } \equiv 64 + b x + 336 x ^ { 2 } + \ldots$$ find the values of the constants \(a , b\).

10. Water is being emptied out of a sink. The depth of water, \(y \mathrm {~cm}\), at time \(t\) seconds, may be modelled by $$y = t ^ { 2 } - 14 t + 49 \quad 0 \leqslant t \leqslant 7$$

1. Find the value of \(t\) when the depth of water is 25 cm .
2. Find the rate of decrease of the depth of water when \(t = 3\).

11. (a) Sketch the graph of \(y = 3 ^ { x }\). Clearly label the coordinates of the point where the graph crosses the \(y\)-axis.
(b) On the same set of axes, sketch the graph of \(y = 3 ^ { ( x + 1 ) }\), clearly labelling the coordinates of the point where the graph crosses the \(y\)-axis.

12. A curve \(C\) has equation \(y = - x ^ { 3 } + 12 x - 20\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature.
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation.

13. The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf { a } = 4 \mathbf { i } + 7 \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + 3 \mathbf { j }$$ respectively.

1. Find the vector \(\mathbf { A B }\).
2. Determine the distance between the points \(A\) and \(B\).
3. The position vector of the point \(C\) is given by \(\mathbf { c } = - 2 \mathbf { i } + 5 \mathbf { j }\). The point \(D\) is such that the distance between \(C\) and \(D\) is equal to the distance between \(A\) and \(B\), and \(C D\) is parallel to \(A B\). Find the possible position vectors of the point \(D\).

14. The diagram below shows a sketch of the curve \(C\) with equation \(y = 2 - 3 x - 2 x ^ { 2 }\) and the line \(L\) with equation \(y = x + 2\). The curve and the line intersect the coordinate axes at the points \(A\) and \(B\).
\includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-18_775_970_589_543}

1. Write down the coordinates of \(A\) and \(B\).
   

   (b) Calculate the area enclosed by \(C\) and \(L\). [6]

   Examiner only

1. The diagram shows a sketch of part of the curve with equation \(y = 2 \sin x + 3 \cos ^ { 2 } x - 3\). The curve crosses the \(x\)-axis at the points \(O , A , B\) and \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-20_620_1009_516_520}

Find the value of \(x\) at each of the points \(A , B\) and \(C\).

16. (a) Find the range of values of \(k\) for which the quadratic equation \(x ^ { 2 } - k x + 4 = 0\) has no real roots.
(b) Determine the coordinates of the points of intersection of the graphs of \(y = x ^ { 2 } - 3 x + 4\) and \(y = x + 16\).
(c) Using the information obtained in parts (a) and (b), sketch the graphs of \(y = x ^ { 2 } - 3 x + 4\) and \(y = x + 16\) on the same set of axes.

17. A function \(f\) is defined by \(f ( x ) = \log _ { 10 } ( 2 - x )\). Another function \(g\) is defined by \(g ( x ) = \log _ { 10 } ( 5 - x )\). The diagram below shows a sketch of the graphs of \(y = f ( x )\) and \(y = g ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-24_782_1072_559_486}

1. The point \(( c , 1 )\) lies on \(y = f ( x )\). Find the value of \(c\).
2. A point \(P\) lies on \(y = f ( x )\) and has \(x\)-coordinate \(\alpha\). Another point \(Q\) lies on \(y = g ( x )\) and also has \(x\)-coordinate \(\alpha\). The distance between \(P\) and \(Q\) is 1.2 units. Find the value of \(\alpha\), giving your answer correct to three decimal places.

18. (a) A circle \(C\) has centre \(( - 3 , - 1 )\) and radius \(\sqrt { 5 }\). Show that the equation of \(C\) can be written as \(x ^ { 2 } + y ^ { 2 } + 6 x + 2 y + 5 = 0\).
(b) (i) Find the equations of the tangents to \(C\) that pass through the origin \(O\).
(ii) Determine the coordinates of the points where the tangents touch the circle.
Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}