WJEC Unit 1 (Unit 1) 2019 June

Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\tan\theta + 2\cos\theta = 0$$ [6]

Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]

Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]

The line \(L_1\) passes through the points \(A(-1, 3)\) and \(B(2, 9)\). The line \(L_2\) has equation \(2y + x = 25\) and intersects \(L_1\) at the point \(C\). \(L_2\) also intersects the \(x\)-axis at the point \(D\).

1. Show that the equation of the line \(L_1\) is \(y = 2x + 5\). [3]
2. 1. Find the coordinates of the point \(D\).
   2. Show that \(L_1\) and \(L_2\) are perpendicular.
   3. Determine the coordinates of \(C\). [5]
3. Find the length of \(CD\). [2]
4. Calculate the angle \(ADB\). Give your answer in degrees, correct to one decimal place. [5]

Given that \(n\) is an integer such that \(1 \leq n \leq 4\), prove that \(2n^2 + 5\) is a prime number. [3]

\(OABC\) is a parallelogram with \(O\) as origin. \includegraphics{figure_6} The position vector of \(A\) is \(\mathbf{a}\) and the position vector of \(C\) is \(\mathbf{c}\). The midpoint of \(AB\) is \(D\). The point \(E\) divides the line \(CB\) such that \(CE : EB = 2 : 1\).

1. Find, in terms of \(\mathbf{a}\) and \(\mathbf{c}\),
   1. the vector \(\overrightarrow{AC}\),
   2. the position vector of \(D\),
   3. the position vector of \(E\). [3]
2. Determine whether or not \(\overrightarrow{DE}\) is parallel to \(\overrightarrow{AC}\), clearly stating your reason. [2]

Given that \(a\), \(b\) are integers, simplify the following. Show all your working.

1. \(\frac{2\sqrt{3} + a}{\sqrt{3} - 1}\) [3]
2. \(\frac{2\sqrt{6b^2} - \sqrt{27} + \sqrt{192}}{\sqrt{2}}\) [3]

1. Given that \(y = 2x^2 - 5x\), find \(\frac{dy}{dx}\) from first principles. [5]
2. Given that \(y = \frac{16}{5}x^4 + \frac{48}{x}\), find the value of \(\frac{dy}{dx}\) when \(x = 16\). [3]

The points \(A(-2, 4)\) and \(B(6, 10)\) are such that \(AB\) is the diameter of a circle.

1. Show that the centre of the circle has coordinates \((2, 7)\). [1]
2. The equation of the circle is \(x^2 + y^2 + ax + by + c = 0\). Determine the values of \(a\), \(b\), \(c\). [3]

A straight line, with equation \(y = x + 6\), passes through the point \(A\) and cuts the circle again at the point \(C\).

1. Find the coordinates of \(C\). [5]
2. Calculate the exact area of the triangle \(ABC\). [3]

1. Solve the following simultaneous equations. $$3^{3x} \times 9^y = 27$$ $$2^{-3x} \times 8^{-y} = \frac{1}{64}$$ [6]
2. Find the value of \(x\) satisfying the equation $$\log_a 3 + 2\log_a x - \log_a(x - 1) = \log_a(5x + 2).$$ [7]

Two quantities are related by the equation \(Q = 1.25P^3\). Explain why the graph of \(\log_{10} Q\) against \(\log_{10} P\) is a straight line. State the gradient of the straight line and the intercept on the \(\log_{10} Q\) axis of the graph. [4]

In the binomial expansion of \((2 - 5x)^8\), find

1. the number of terms, [1]
2. the \(4^{\text{th}}\) term, when the expansion is in ascending powers of \(x\), [2]
3. the greatest positive coefficient. [3]

A curve \(C\) has equation \(y = \frac{1}{9}x^3 - kx + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient \(-9\). The \(x\)-coordinate of \(Q\) is \(3\).

1. Show that \(k = 12\). [3]
2. Find the coordinates of each of the stationary points of \(C\) and determine their nature. [6]
3. Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. [2]

The diagram below shows a triangle \(ABC\) with \(AC = 5\) cm, \(AB = x\) cm, \(BC = y\) cm and angle \(BAC = 120°\). The area of the triangle \(ABC\) is \(14\) cm\(^2\). \includegraphics{figure_14} Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. [6]

Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]

The diagram below shows a curve with equation \(y = (x + 2)(x - 2)(x + 1)\). \includegraphics{figure_16} Calculate the total area of the two shaded regions. [8]