WJEC Unit 3 (Unit 3) 2024 June

The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$

1. Express \(f(x)\) in terms of partial fractions. [4]
2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]

1. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$3\cot\theta + 4\cosec^2\theta = 5.$$ [5]
2. By writing \(24\cos x - 7\sin x\) in the form \(R\cos(x + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\), solve the equation $$24\cos x - 7\sin x = 16$$ for values of \(x\) between \(0°\) and \(360°\). [6]

The diagram below shows a badge \(ODC\). The shape \(OAB\) is a sector of a circle centre \(O\) and radius \(r\) cm. The shape \(ODC\) is a sector of a circle with the same centre \(O\). The length \(AD\) is \(5\) cm and angle \(AOB\) is \(\frac{\pi}{5}\) radians. The area of the shaded region, \(ABCD\), is \(\frac{13\pi}{2}\) cm\(^2\). \includegraphics{figure_3}

1. Determine the value of \(r\). [4]
2. Calculate the perimeter of the shaded region. [3]

A function \(f\) is given by \(f(x) = |3x + 4|\).

1. Sketch the graph of \(y = f(x)\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis. [3]
2. On a separate set of axes, sketch the graph of \(y = \frac{1}{2}f(x) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A'\) and \(B'\). Clearly label the coordinates of the points \(A'\) and \(B'\). [3]

Prove by contradiction the following proposition: When \(x\) is real and positive, \(x + \frac{81}{x} \geq 18\). [4]

1. Differentiate \(\cos x\) from first principles. [5]
2. Differentiate \(e^{3x}\sin 4x\) with respect to \(x\). [3]
3. Find \(\int x^2\sin 2x dx\). [5]

Showing all your working, evaluate

1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]

The function \(f\) is defined by $$f(x) = x^3 + 4x^2 - 3x - 1.$$

1. Show that the equation \(f(x) = 0\) has a root in the interval \([0, 1]\). [1]
2. Using the Newton-Raphson method with \(x_0 = 0 \cdot 8\),
   1. write down in full the decimal value of \(x_1\) as given in your calculator,
   2. determine the value of this root correct to six decimal places. [4]
3. Explain why the Newton-Raphson method does not work if \(x_0 = \frac{1}{3}\). [2]

The diagram below shows a sketch of the curve \(C_1\) with equation \(y = -x^2 + \pi x + 1\) and a sketch of the curve \(C_2\) with equation \(y = \cos 2x\). The curves intersect at the points where \(x = 0\) and \(x = \pi\). \includegraphics{figure_9} Calculate the area of the shaded region enclosed by \(C_1\), \(C_2\) and the \(x\)-axis. Give your answer in terms of \(\pi\). [9]

The function \(f\) has domain \([4, \infty)\) and is defined by $$f(x) = \frac{2(3x + 1)}{x^2 - 2x - 3} + \frac{x}{x + 1}.$$

1. Show that \(f(x) = \frac{x + 2}{x - 3}\). [4]
2. Determine the range of \(f(x)\). [2]
3. Find an expression for \(f^{-1}(x)\) and write down the domain and range of \(f^{-1}\). [4]
4. Find the value of \(x\) when \(f(x) = f^{-1}(x)\). [4]

A curve is defined parametrically by $$x = 2\theta + \sin 2\theta, \quad y = 1 + \cos 2\theta.$$

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(-\tan\theta\). [6]
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]

1. Given that \(\theta\) is small, show that \(2\cos\theta + \sin\theta - 1 \approx 1 + \theta - \theta^2\). [2]
2. Hence, when \(\theta\) is small, show that $$\frac{1}{2\cos\theta + \sin\theta - 1} \approx 1 + a\theta + b\theta^2,$$ where \(a\), \(b\) are constants to be found. [4]

The diagram below shows a sketch of the graph of \(y = f'(x)\) for the interval \([x_1, x_5]\). \includegraphics{figure_13}

1. Find the interval on which \(f(x)\) is both decreasing and convex. Give reasons for your answer. [2]
2. Write down the \(x\)-coordinate of a point of inflection of the graph of \(y = f(x)\). [1]

1. Given that \(y = \frac{1 + \ln x}{x}\), show that \(\frac{dy}{dx} = \frac{-\ln x}{x^2}\). [2]
2. Hence, solve the differential equation $$\frac{dx}{dt} = \frac{x^2 t}{\ln x},$$ given that \(t = 3\) when \(x = 1\). Give your answer in the form \(t^2 = g(x)\), where \(g\) is a function of \(x\). [5]

Robert wants to deposit \(£P\) into a savings account. He has a choice of two accounts. • Account \(A\) offers an annual compound interest rate of \(1\%\). • Account \(B\) offers an interest rate of \(5\%\) for the first year and an annual compound interest rate of \(0.6\%\) for each subsequent year. After \(n\) years, account \(A\) is more profitable than account \(B\). Find the smallest value of \(n\). [5]