OCR C1 (Core Mathematics 1) 2005 January

1

1. Express \(11 ^ { - 2 }\) as a fraction.
2. Evaluate \(100 ^ { \frac { 3 } { 2 } }\).
3. Express \(\sqrt { 50 } + \frac { 6 } { \sqrt { 3 } }\) in the form \(a \sqrt { } 2 + b \sqrt { } 3\), where \(a\) and \(b\) are integers.

2 Given that \(2 x ^ { 2 } - 12 x + p = q ( x - r ) ^ { 2 } + 10\) for all values of \(x\), find the constants \(p , q\) and \(r\).

3

1. The curve \(y = 5 \sqrt { } x\) is transformed by a stretch, scale factor \(\frac { 1 } { 2 }\), parallel to the \(x\)-axis. Find the equation of the curve after it has been transformed.
2. Describe the single transformation which transforms the curve \(y = 5 \sqrt { } x\) to the curve \(y = ( 5 \sqrt { } x ) - 3\).

4 Solve the simultaneous equations $$x ^ { 2 } - 3 y + 11 = 0 , \quad 2 x - y + 1 = 0$$

5 On separate diagrams,

1. sketch the curve \(y = \frac { 1 } { x }\),
2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
3. sketch the curve \(y = - \sqrt { } x\).

6

1. Calculate the discriminant of \(- 2 x ^ { 2 } + 7 x + 3\) and hence state the number of real roots of the equation \(- 2 x ^ { 2 } + 7 x + 3 = 0\).
2. The quadratic equation \(2 x ^ { 2 } + ( p + 1 ) x + 8 = 0\) has equal roots. Find the possible values of \(p\).

7 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = \frac { 1 } { 2 } x ^ { 4 } - 3 x\),
2. \(y = \left( 2 x ^ { 2 } + 3 \right) ( x + 1 )\),
3. \(y = \sqrt [ 5 ] { x }\).

8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.

1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).

9

1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
4. Deduce the value of \(k\).

10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.

1. Calculate the gradient of \(D E\).
2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
4. Calculate the length of \(D F\).
5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).