WJEC Unit 3 (Unit 3) 2023 June

The 12th term of an arithmetic series is 41 and the sum of the first 16 terms is 488. Find the first term and the common difference of the series. [5]

1. Differentiate each of the following with respect to \(x\).
   1. \(\left(\sin x + x^2\right)^5\) [2]
   2. \(x^3 \cos x\) [2]
   3. \(\frac{e^{3x}}{\sin 2x}\) [3]
2. Find the equation of the tangent to the curve $$4y^2 - 7xy + x^2 = 12$$ at the point \((2, 4)\). [6]

1. Express \(\frac{9}{(1-x)(1+2x)^2}\) in terms of partial fractions. [4]
2. Using your answer from part (a), find the expansion of \(\frac{9}{(1-x)(1+2x)^2}\) in ascending powers of \(x\) as far as the term in \(x^2\). State the values of \(x\) for which the expansion is valid. [7]

A function \(f\) with domain \((-\infty,\infty)\) is defined by \(f(x) = 6x^3 + 35x^2 - 7x - 6\).

1. Determine the number of roots of the equation \(f(x) = 0\) in the interval \([-1, 1]\). [2]
2. Use the Newton-Raphson method to find a root of the equation \(f(x) = 0\). Starting with \(x_0 = 1\),
   1. write down the value of \(x_1\),
   2. determine the value of the root correct to one decimal place. [4]
3. It is suggested that another iterative sequence $$x_{n+1} = \sqrt{\frac{7x_n + 6 - 6x_n^3}{35}},$$ starting with \(x_0 = -3\), could be used to find a root of the equation \(f(x) = 0\). Explain why this method fails. [2]

A tree is 80 cm in height when it is planted. In the first year, the tree grows in height by 32 cm. In each subsequent year, the tree grows in height by 90% of the growth of the previous year.

1. Find the height of the tree 10 years after it was planted. [4]
2. Determine the maximum height of the tree. [2]

1. Using the trigonometric identity \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), show that the exact value of \(\cos 75°\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\). [3]
2. Solve the equation \(2\cot^2 x + \cosec x = 4\) for values of \(x\) between \(0°\) and \(360°\). [6]
3. 1. Express \(7\cos\theta - 24\sin\theta\) in the form \(R\cos(\theta + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\).
   2. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$7\cos\theta - 24\sin\theta = 5.$$ [6]

1. The graphs of \(y = 5x - 3\) and \(y = 2x + 3\) intersect at the point A. Show that the coordinates of A are \((2, 7)\). [2]
2. On the same set of axes, sketch the graphs of \(y = |5x - 3|\) and \(y = |2x + 3|\), clearly indicating the coordinates of the points of intersection of the two graphs and the points where the graphs touch the \(x\)-axis. [4]
3. Calculate the area of the region satisfying the inequalities $$y \geqslant |5x - 3| \quad \text{and} \quad y \leqslant |2x + 3|.$$ [4]

The function \(f\) is defined by \(f(x) = \frac{4x^2 + 12x + 9}{2x^2 + x - 3}\), where \(x > 1\).

1. Show that \(f(x)\) can be written as \(2 + \frac{5}{x-1}\). [3]
2. Hence find the exact value of \(\int_3^7 f(x)\,dx\). [4]

The aerial view of a patio under construction is shown below. \includegraphics{figure_9} The curved edge of the patio is described by the equation \(9x^2 + 16y^2 = 144\), where \(x\) and \(y\) are measured in metres. To construct the patio, the area enclosed by the curve and the coordinate axes is to be covered with a layer of concrete of depth 0.06 m.

1. Show that the volume of concrete required for the construction of the patio is given by \(0.015 \int_0^4 \sqrt{144 - 9x^2}\,dx\). [3]
2. Use the trapezium rule with six ordinates to estimate the volume of concrete required. [4]
3. State whether your answer in part (b) is an overestimate or an underestimate of the volume required. Give a reason for your answer. [1]

Two real functions are defined as $$f(x) = \frac{8}{x-4} \quad \text{for} \quad (-\infty < x < 4) \cup (4 < x < \infty),$$ $$g(x) = (x-2)^2 \quad \text{for} \quad -\infty < x < \infty.$$

1. 1. Find an expression for \(fg(x)\). [2]
   2. Determine the values of \(x\) for which \(fg(x)\) does not exist. [3]
2. Find an expression for \(f^{-1}(x)\). [3]

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

1. Find the \(x\)-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary. [5]
2. Determine the range of values of \(x\) for which C is concave. [2]

The rate of change of a variable \(y\) with respect to \(x\) is directly proportional to \(y\).

1. Write down a differential equation satisfied by \(y\). [1]
2. When \(x = 1\) and \(y = 0.5\), the rate of change of \(y\) with respect to \(x\) is 2. Find \(y\) when \(x = 3\). [6]

The curve \(C_1\) has parametric equations \(x = 3p + 1\), \(y = 9p^2\). The curve \(C_2\) has parametric equations \(x = 4q\), \(y = 2q\). Find the Cartesian coordinates of the points of intersection of \(C_1\) and \(C_2\). [7]

1. Use integration by parts to evaluate \(\int_0^1 (3x-1)e^{2x}\,dx\). [4]
2. Use the substitution \(u = 1 - 2\cos x\) to find \(\int \frac{\sin x}{1 - 2\cos x}\,dx\). [4]