OCR C1 (Core Mathematics 1) 2010 June

2

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. Sketch the curve \(y = 3 - \frac { 1 } { x ^ { 2 } }\).
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor 2 . State the equation of the transformed curve.

3

1. Express \(\frac { 12 } { 3 + \sqrt { 5 } }\) in the form \(a - b \sqrt { 5 }\), where \(a\) and \(b\) are positive integers.
2. Express \(\sqrt { 18 } - \sqrt { 2 }\) in simplified surd form.

4

1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.

5 Find the real roots of the equation \(4 x ^ { 4 } + 3 x ^ { 2 } - 1 = 0\).

6 Find the gradient of the curve \(y = 2 x + \frac { 6 } { \sqrt { x } }\) at the point where \(x = 4\).

7 Solve the simultaneous equations $$x + 2 y - 6 = 0 , \quad 2 x ^ { 2 } + y ^ { 2 } = 57 .$$

8

1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
3. State the equation of the normal to the curve at its minimum point.
4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).

9

1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\).
   \(A B\) is the diameter of a circle.
2. Find the radius of the circle.
3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).

10

1. Find the coordinates of the stationary points of the curve \(y = 2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\).
2. State the set of values for \(x\) for which \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\) is a decreasing function.
3. Show that the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 }\) is \(10 x - 4 y - 7 = 0\).
4. Hence, with the aid of a sketch, show that the equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x = \frac { 5 } { 2 } x - \frac { 7 } { 4 }\) has two distinct real roots.