OCR C2 (Core Mathematics 2)

A geometric progression has first term 75 and second term \(-15\).

1. Find the common ratio. [2]
2. Find the sum to infinity. [2]

Find the area of the finite region enclosed by the curve \(y = 5x - x^2\) and the \(x\)-axis. [6]

During one day, a biological culture is allowed to grow under controlled conditions. At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times (1.06)^t.$$ Using this model,

1. find the number of bacteria present at 11 a.m., [2]
2. find, to the nearest minute, the time when the initial number of bacteria will have doubled. [4]

\includegraphics{figure_4} The diagram shows the curve with equation \(y = (x - \log_{10} x)^2\), \(x > 0\).

1. Copy and complete the table below for points on the curve, giving the \(y\) values to 2 decimal places.

   \(x\)	2	3	4	5	6
   \(y\)	2.89	6.36

   [2]

The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 6\).

1. Use the trapezium rule with all the values in your table to estimate the area of the shaded region. [3]
2. State, with a reason, whether your answer to part (b) is an under-estimate or an over-estimate of the true area. [2]

1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]

\includegraphics{figure_6} The diagram shows triangle \(ABC\) in which \(AC = 8\) cm and \(\angle BAC = \angle BCA = 30°\).

1. Find the area of triangle \(ABC\) in the form \(k\sqrt{3}\). [4]

The point \(M\) is the mid-point of \(AC\) and the points \(N\) and \(O\) lie on \(AB\) and \(BC\) such that \(MN\) and \(MO\) are arcs of circles with centres \(A\) and \(C\) respectively.

1. Show that the area of the shaded region \(BNMO\) is \(\frac{8}{3}(2\sqrt{3} - \pi)\) cm\(^2\). [4]

1. Expand \((2 + x)^4\) in ascending powers of \(x\), simplifying each coefficient. [4]
2. Find the integers \(A\), \(B\) and \(C\) such that $$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
3. Find the real values of \(x\) for which $$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]

1. The gradient of a curve is given by $$\frac{dy}{dx} = 3 - \frac{2}{x^2}, \quad x \neq 0.$$ Find an equation for the curve given that it passes through the point \((2, 6)\). [6]
2. Show that $$\int_2^3 (6\sqrt{x} - \frac{4}{\sqrt{x}}) \, dx = k\sqrt{3},$$ where \(k\) is an integer to be found. [6]

The polynomial f(x) is given by $$\text{f}(x) = x^3 + kx^2 - 7x - 15,$$ where \(k\) is a constant. When f(x) is divided by \((x + 1)\) the remainder is \(r\). When f(x) is divided by \((x - 3)\) the remainder is \(3r\).

1. Find the value of \(k\). [5]
2. Find the value of \(r\). [1]
3. Show that \((x - 5)\) is a factor of f(x). [2]
4. Show that there is only one real solution to the equation f(x) = 0. [4]