WJEC Unit 1 (Unit 1) 2024 June

Given that \(y = 12\sqrt{x} - \frac{27}{x} + 4\), find the value of \(\frac{dy}{dx}\) when \(x = 9\). [4]

Find all values of \(\theta\) in the range \(0° < \theta < 180°\) that satisfy the equation $$2\sin 2\theta = 1.$$ [3]

Find \(\int\left(5x^4 + 3x^{-2} - 2\right)dx\). [3]

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. [3]

A triangle \(ABC\) has sides \(AB = 6\)cm, \(BC = 11\)cm and \(AC = 13\)cm. Calculate the area of the triangle. [4]

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

1. The line \(L_1\) passes through the points \(A(-3, 0)\) and \(B(1, 4)\). Determine the equation of \(L_1\). [3]
2. The line \(L_2\) has equation \(y = 3x - 3\).
   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 \((1, 0)\). [4]
3. Calculate the area of triangle \(ACD\). [2]
4. Determine the angle \(ACD\). [2]

Prove that \(x - 10 < x^2 - 5x\) for all real values of \(x\). [4]

1. Write down the binomial expansion of \((2 - x)^6\) up to and including the term in \(x^2\). [3]
2. Given that $$(1 + ax)(2 - x)^6 = 64 + bx + 336x^2 + \ldots,$$ find the values of the constants \(a\), \(b\). [6]

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

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

1. Sketch the graph of \(y = 3^x\). Clearly label the coordinates of the point where the graph crosses the \(y\)-axis. [2]
2. 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. [2]

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

1. Find the coordinates of the stationary points of C and determine their nature. [7]
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]

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

1. Find the vector \(\overrightarrow{AB}\). [2]
2. Determine the distance between the points A and B. [2]
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 \(\overrightarrow{CD}\) is parallel to \(\overrightarrow{AB}\). Find the possible position vectors of the point D. [4]

The diagram below shows a sketch of the curve C with equation \(y = 2 - 3x - 2x^2\) 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{figure_14}

1. Write down the coordinates of A and B. [2]
2. Calculate the area enclosed by C and L. [6]

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{figure_15} Find the value of \(x\) at each of the points A, B and C. [7]

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

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{figure_17}

1. The point \((c, 1)\) lies on \(y = f(x)\). Find the value of \(c\). [2]
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. [5]

1. A circle C has centre \((-3, -1)\) and radius \(\sqrt{5}\). Show that the equation of C can be written as \(x^2 + y^2 + 6x + 2y + 5 = 0\). [2]
2. 1. Find the equations of the tangents to C that pass through the origin O. [6]
   2. Determine the coordinates of the points where the tangents touch the circle. [4]