WJEC Unit 1 (Unit 1) 2023 June

1. Using the binomial theorem, write down and simplify the first three terms in the expansion of \((1 - 3x)^9\) in ascending powers of \(x\). [3]
2. Hence, by writing \(x = 0.001\) in your expansion in part (a), find an approximate value for \((0.997)^9\). Show all your working and give your answer correct to three decimal places. [3]

Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\sin^2 \theta - 5\cos^2 \theta = 2\cos \theta$$ [7]

The point \(A\) has coordinates \((-2, 5)\) and the point \(B\) has coordinates \((3, 8)\). The point \(C\) lies on the \(x\)-axis such that \(AC\) is perpendicular to \(AB\).

1. Find the equation of \(AB\). [3]
2. Show that \(C\) has coordinates \((1, 0)\). [3]
3. Calculate the area of triangle \(ABC\). [4]
4. Find the equation of the circle which passes through the points \(A\), \(B\) and \(C\). [5]

1. Find the remainder when the polynomial \(3x^3 + 2x^2 + x - 1\) is divided by \((x - 3)\). [3]
2. The polynomial \(f(x) = 2x^3 - 3x^2 + ax + 6\) is divisible by \((x + 2)\), where \(a\) is a real constant.
   1. Find the value of \(a\). [3]
   2. Showing all your working, solve the equation \(f(x) = 0\). [4]

Simplify the expression \(\sqrt[3]{512a^7} - \frac{a^{\frac{7}{2}} \times a^{-\frac{1}{3}}}{a^6}\). [4]

The diagram below shows a triangle \(ABC\). \includegraphics{figure_6} Given that \(AB = 3\), \(BC = 2\sqrt{5}\), \(AC = 4 + \sqrt{3}\), find the value of \(\cos ABC\). Show all your working and give your answer in the form \(\frac{(a - b\sqrt{3})}{6\sqrt{5}}\), where \(a\), \(b\) are integers. [7]

The curve \(C\) has equation \(y = 2x^2 + 5x - 12\) and the line \(L\) has equation \(y = mx - 14\), where \(m\) is a real constant.

1. Given that \(L\) is a tangent to \(C\),
   1. show that \(m\) satisfies the equation $$m^2 - 10m + 9 = 0,$$ [5]
   2. find the coordinates of the two possible points of contact of \(C\) and \(L\). [6]
2. Given instead that \(L\) intersects \(C\) at two distinct points, find the range of values of \(m\). [2]

Show, by counter example, that the following statement is false. "For all positive integer values of \(n\), \(n^2 + 1\) is a prime number." [3]

1. Given that \(y = x^2 - 3x\), find \(\frac{dy}{dx}\) from first principles. [5]
2. The function \(f\) is defined by \(f(x) = 4x^{\frac{3}{2}} + \frac{6}{\sqrt{x}}\) for \(x > 0\).
   1. Find \(f'(x)\). [2]
   2. When \(x > k\), \(f(x)\) is an increasing function. Determine the least possible value of \(k\). Give your answer correct to two decimal places. [4]

Solve the following equations for values of \(x\).

1. \(\ln(2x + 5) = 3\) [2]
2. \(5^{2x+1} = 14\) [3]
3. \(3\log_7(2x) - \log_7(8x^2) + \log_7 x = \log_3 81\) [6]

The function \(f\) is defined by \(f(x) = \frac{8}{x^2}\).

1. Sketch the graph of \(y = f(x)\). [2]
2. On a separate set of axes, sketch the graph of \(y = f(x - 2)\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis. [3]
3. Sketch the graphs of \(y = \frac{8}{x}\) and \(y = \frac{8}{(x-2)^2}\) on the same set of axes. Hence state the number of roots of the equation \(\frac{8}{(x-2)^2} = \frac{8}{x}\). [2]

The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf{a} = -3\mathbf{i} + 4\mathbf{j}, \quad \mathbf{b} = 5\mathbf{i} + 8\mathbf{j},$$ respectively.

1. Find the vector \(\overrightarrow{AB}\). [2]
2. 1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
   2. The point \(C\) is such that the vector \(\overrightarrow{OC}\) is in the direction of \(\mathbf{a}\). Given that the length of \(\overrightarrow{OC}\) is 7 units, write down the position vector of \(C\). [1]
3. Calculate the angle \(AOB\). [3]

1. Find \(\int \left(4x^{-\frac{2}{3}} + 5x^3 + 7\right) dx\). [3]
2. The diagram below shows the graph of \(y = x(x + 6)(x - 3)\). \includegraphics{figure_13} Calculate the total area of the regions enclosed by the graph and the \(x\)-axis. [9]

1. Two variables, \(x\) and \(y\), are such that the rate of change of \(y\) with respect to \(x\) is proportional to \(y\). State a model which may be appropriate for \(y\) in terms of \(x\). [1]
2. The concentration, \(Y\) units, of a certain drug in a patient's body decreases exponentially with respect to time. At time \(t\) hours the concentration can be modelled by \(Y = Ae^{-kt}\), where \(A\) and \(k\) are constants. A patient was given a dose of the drug that resulted in an initial concentration of 5 units.
   1. After 4 hours, the concentration had dropped to 1.25 units. Show that \(k = 0.3466\), correct to four decimal places. [2]
   2. The minimum effective concentration of the drug is 0.6 units. How much longer would it take for the drug concentration to drop to the minimum effective level? [3]