OCR MEI C1 (Core Mathematics 1)

Make \(r\) the subject of the formula \(A = \pi r^2(x+y)\), where \(r > 0\). [2]

Fig. 8 shows a right-angled triangle with base \(2x + 1\), height \(h\) and hypotenuse \(3x\). \includegraphics{figure_1}

1. Show that \(h^2 = 5x^2 - 4x - 1\). [2]
2. Given that \(h = \sqrt{7}\), find the value of \(x\), giving your answer in surd form. [3]

1. Find the set of values of \(k\) for which the line \(y = 2x + k\) intersects the curve \(y = 3x^2 + 12x + 13\) at two distinct points. [5]
2. Express \(3x^2 + 12x + 13\) in the form \(a(x + b)^2 + c\). Hence show that the curve \(y = 3x^2 + 12x + 13\) lies completely above the \(x\)-axis. [5]
3. Find the value of \(k\) for which the line \(y = 2x + k\) passes through the minimum point of the curve \(y = 3x^2 + 12x + 13\). [2]

Make \(a\) the subject of \(3(a + 4) = ac + 5f\). [4]

Find the coordinates of the point of intersection of the lines \(y = 3x - 2\) and \(x + 3y = 1\). [4]

Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]

Simplify \(\frac{(4x^5 y)^3}{(2xy^2) \times (8x^{10}y^4)}\). [3]

You are given that \(f(x) = x^2 + kx + c\). Given also that \(f(2) = 0\) and \(f(-3) = 35\), find the values of the constants \(k\) and \(c\). [4]

Rearrange the equation \(5c + 9t = a(2c + t)\) to make \(c\) the subject. [4]

Factorise and hence simplify the following expression. $$\frac{x^2 - 9}{x^2 + 5x + 6}$$ [3]

Rearrange the following equation to make \(h\) the subject. $$4h + 5 = 9a - ha^2$$ [3]