OCR MEI C1 (Core Mathematics 1) 2011 June

Solve the inequality \(6(x + 3) > 2x + 5\). [3]

A line has gradient 3 and passes through the point \((1, -5)\). The point \((5, k)\) is on this line. Find the value of \(k\). [2]

1. Evaluate \(\left(\frac{9}{16}\right)^{-\frac{1}{2}}\). [2]
2. Simplify \(\frac{(2ac^2)^3 \times 9a^2c}{36a^4c^{12}}\). [3]

The point P \((5, 4)\) is on the curve \(y = f(x)\). State the coordinates of the image of P when the graph of \(y = f(x)\) is transformed to the graph of

1. \(y = f(x - 5)\), [2]
2. \(y = f(x) + 7\). [2]

Find the coefficient of \(x^4\) in the binomial expansion of \((5 + 2x)^6\). [4]

Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]

A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

1. Find algebraically the coordinates of the points of intersection of the curve \(y = 4x^2 + 24x + 31\) and the line \(x + y = 10\). [5]
2. Express \(4x^2 + 24x + 31\) in the form \(a(x + b)^2 + c\). [4]
3. For the curve \(y = 4x^2 + 24x + 31\),
   1. write down the equation of the line of symmetry, [1]
   2. write down the minimum \(y\)-value on the curve. [1]

\includegraphics{figure_12} Fig. 12 shows the graph of \(y = \frac{4}{x^2}\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2x + 5\) and hence find graphically the three roots of the equation \(\frac{4}{x^2} = 2x + 5\). [3]
2. Show that the equation you have solved in part (i) may be written as \(2x^3 + 5x^2 - 4 = 0\). Verify that \(x = -2\) is a root of this equation and hence find, in exact form, the other two roots. [6]
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x^3 + 2x^2 - 4 = 0\). [3]

\includegraphics{figure_13} Fig. 13 shows the circle with equation \((x - 4)^2 + (y - 2)^2 = 16\).

1. Write down the radius of the circle and the coordinates of its centre. [2]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. Give your answers in surd form. [4]
3. Show that the point A \((4 + 2\sqrt{2}, 2 + 2\sqrt{2})\) lies on the circle and mark point A on the copy of Fig. 13. Sketch the tangent to the circle at A and the other tangent that is parallel to it. Find the equations of both these tangents. [7]