Edexcel C1 (Core Mathematics 1)

1. Solve the inequality $$3x - 8 > x + 13.$$ [2]
2. Solve the inequality $$x^2 - 5x - 14 > 0.$$ [3]

Given that \(2^x = \frac{1}{\sqrt{2}}\) and \(2^y = 4\sqrt{2}\),

1. find the exact value of \(x\) and the exact value of \(y\), [3]
2. calculate the exact value of \(2^{y-x}\). [2]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by \(x\) phones per month for the next 35 months, so that \((280 + x)\) phones will be sold in the second month, \((280 + 2x)\) in the third month, and so on. Using this model with \(x = 5\), calculate

1. 1. the number of phones sold in the 36th month, [2]
   2. the total number of phones sold over the 36 months. [2]

The shop sets a sales target of 17000 phones to be sold over the 36 months. Using the same model,

1. find the least value of \(x\) required to achieve this target. [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\). [1]
2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
3. Show that \(\angle PAQ\) is a right angle. [4]

\includegraphics{figure_2} The points \(A (3, 0)\) and \(B (0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.

1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]

The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.

1. Find the length of \(BC\) in terms of \(k\). [2]

Given that the length of \(BC\) is 10 and using your answer to part (b),

1. find the value of \(k\), [4]
2. find the coordinates of \(D\). [2]

The curve \(C\) has equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 3x^2 - 20x + 29$$ and that \(C\) passes through the point \(P(2, 6)\),

1. find \(y\) in terms of \(x\). [4]
2. Verify that \(C\) passes through the point \((4, 0)\). [2]
3. Find an equation of the tangent to \(C\) at \(P\). [3]

The tangent to \(C\) at the point \(Q\) is parallel to the tangent at \(P\).

1. Calculate the exact \(x\)-coordinate of \(Q\). [5]