OCR C1 (Core Mathematics 1) 2016 June

1

1. Simplify \(( 2 x - 3 ) ^ { 2 } - 2 ( 3 - x ) ^ { 2 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 x ^ { 2 } - 3 x + 4 \right) \left( 5 - 2 x - x ^ { 3 } \right)\).

2 Express \(\frac { 3 + \sqrt { 20 } } { 3 + \sqrt { 5 } }\) in the form \(a + b \sqrt { 5 }\).

3 Solve the simultaneous equations $$x ^ { 2 } + y ^ { 2 } = 34 , \quad 3 x - y + 4 = 0$$

4 Solve the equation \(2 y ^ { \frac { 1 } { 2 } } - 7 y ^ { \frac { 1 } { 4 } } + 3 = 0\).

5 Express the following in the form \(2 ^ { p }\).

1. \(\left( 2 ^ { 5 } \div 2 ^ { 7 } \right) ^ { 3 }\)
2. \(5 \times 4 ^ { \frac { 2 } { 3 } } + 3 \times 16 ^ { \frac { 1 } { 3 } }\)

6

1. Express \(4 + 12 x - 2 x ^ { 2 }\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the coordinates of the maximum point of the curve \(y = 4 + 12 x - 2 x ^ { 2 }\).

7

1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).

8 A curve has equation \(y = 2 x ^ { 2 }\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates 5 and \(5 + h\) respectively, where \(h > 0\).

1. Show that the gradient of the line \(A B\) is \(20 + 2 h\).
2. Explain how the answer to part (i) relates to the gradient of the curve at \(A\).
3. The normal to the curve at \(A\) meets the \(y\)-axis at the point \(C\). Find the \(y\)-coordinate of \(C\).

9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.

10
\includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).

1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
2. Find the equation of the tangent to the circle at \(A\).
3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).

11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .