WJEC Unit 1 (Unit 1) 2022 June

Write down the inverse function of \(y = e^x\). On the same set of axes, sketch the graphs of \(y = e^x\) and its inverse function, clearly labelling the coordinates of the points where the graphs cross the \(x\) and \(y\) axes. [3]

Showing all your working, simplify the following expression. [6] $$5\sqrt{48} + \frac{2+5\sqrt{3}}{5+3\sqrt{3}} - (2\sqrt{3})^3$$

The line \(L_1\) passes through the points \(A(0, 5)\) and \(B(3, -1)\).

1. Find the equation of the line \(L_1\). [3]

The line \(L_2\) is perpendicular to \(L_1\) and passes through the origin \(O\).

1. Write down the equation of \(L_2\). [1]

The lines \(L_1\) and \(L_2\) intersect at the point \(C\).

1. Calculate the area of triangle \(OAC\). [4]
2. Find the equation of the line \(L_3\) which is parallel to \(L_1\) and passes through the point \(D(4, 2)\). [2]
3. The line \(L_3\) intersects the \(y\)-axis at the point \(E\). Find the area of triangle \(ODE\). [1]

Solve the inequality \(x^2 + 3x - 6 > 4x - 4\). [4]

The curve \(C_1\) has equation \(y = -x^2 + 2x + 3\) and the curve \(C_2\) has equation \(y = x^2 - x - 6\). The two curves intersect at the points \(A\) and \(B\).

1. Determine the coordinates of \(A\) and \(B\). [4]
2. On the same set of axes, sketch the graphs of \(C_1\) and \(C_2\). Clearly label the points where the two curves intersect. [3]
3. In the diagram drawn in part (b), shade the region satisfying the following inequalities: [2] $$x > 0,$$ $$y < -x^2 + 2x + 3,$$ $$y > x^2 - x - 6.$$

In each of the two statements below, \(x\) and \(y\) are real numbers. One of the statements is true while the other is false. A: \(x^2 + y^2 \geqslant 2xy\), for all real values of \(x\) and \(y\). B: \(x + y \geqslant 2\sqrt{xy}\), for all real values of \(x\) and \(y\).

1. Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]

A circle \(C\) has centre \(A\) and equation \(x^2 + y^2 - 4x - 6y = 3\).

1. Find the coordinates of \(A\) and the radius of \(C\). [3]

The line \(L\) with equation \(y = x + 5\) intersects \(C\) at the points \(P\) and \(Q\).

1. Determine the coordinates of \(P\) and \(Q\). [4]
2. The point \(B\) is on \(PQ\) and is such that \(AB\) is perpendicular to \(PQ\). Find the length of \(PB\). [2]
3. Show that the area of the smaller segment enclosed by \(C\) and \(L\) is \(4\pi - 8\). [2]

1. The graph \(G\) shows the relationship between the variables \(y\) and \(x\), where \(y \propto x\). Sketch the graph \(G\). [1]
2. Mary and Jeff work for a company which pays its employees by hourly rates. Mary's hourly rate is twice Jeff's hourly rate. On a certain day, Jeff worked three times as long as Mary and was paid £120. Calculate Mary's earnings on that day. [3]
3. Atmospheric pressure, \(P\) units, decreases as the height, \(H\) metres, above sea level increases. The rate of decrease is 12% for every 1000m. At sea level, the pressure \(P\) is 1013 units. Write down the model for \(P\) in terms of \(H\) and find the pressure at the top of Mount Everest, which is 8848m above sea level. [3]

Find the range of values of \(k\) for which the quadratic equation \(x^2 + 2kx + 8k = 0\) has no real roots. [4]

Showing all your working, solve the equation \(2^x = 53\). Give your answer correct to two decimal places. [3]

The diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}

1. Find the coordinates of \(D\). [5]
2. Find the area of the shaded region. [6]
3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]

1. Solve the equation \(2x^3 - x^2 - 5x - 2 = 0\). [6]
2. Find all values of \(\theta\) in the range \(0° < \theta < 180°\) satisfying $$\cos(2\theta - 51°) = 0.891.$$ [3]

Find the term which is independent of \(x\) in the expansion of \(\frac{(2-3x)^5}{x^3}\). [4]

A curve \(C\) has equation \(f(x) = 3x^3 - 5x^2 + x - 6\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature. [8]
2. Without solving the equations, determine the number of distinct real roots for each of the following:
   1. \(3x^3 - 5x^2 + x + 1 = 0\),
   2. \(6x^3 - 10x^2 + 2x + 1 = 0\). [4]

Solve the simultaneous equations $$3\log_u(x^2y) - \log_u(x^2y^2) + \log_u\left(\frac{9}{x^2y^2}\right) = \log_u 36,$$ $$\log_u y - \log_u(x + 3) = 0.$$ [8]

The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are defined by \(\mathbf{a} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\).

1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
2. Determine the angle \(\mathbf{b}\) makes with the \(x\)-axis. [2]
3. The vector \(\mu\mathbf{a} + \mathbf{b}\) is parallel to \(4\mathbf{i} - 5\mathbf{j}\).
   1. Find the vector \(\mu\mathbf{a} + \mathbf{b}\) in terms of \(\mu\), \(\mathbf{i}\) and \(\mathbf{j}\). [1]
   2. Determine the value of \(\mu\). [4]