OCR C1 (Core Mathematics 1)

1. Find the set of values of the constant \(k\) such that the equation

$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.

2. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively. Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

3. (i) Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
(ii) Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.

4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$

1. Given that

$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).

6.
\includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
4. Determine whether each stationary point is a maximum or a minimum point.

8. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$

1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
2. State the maximum value of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
4. Sketch the curve \(y = \mathrm { f } ( x )\).

9. The circle \(C\) has centre \(( - 3,2 )\) and passes through the point \(( 2,1 )\).

1. Find an equation for \(C\).
2. Show that the point with coordinates \(( - 4,7 )\) lies on \(C\).
3. Find an equation for the tangent to \(C\) at the point ( - 4 , 7). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

10. A curve has the equation \(y = ( \sqrt { x } - 3 ) ^ { 2 } , x \geq 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 - \frac { 3 } { \sqrt { x } }\). The point \(P\) on the curve has \(x\)-coordinate 4 .
2. Find an equation for the normal to the curve at \(P\) in the form \(y = m x + c\).
3. Show that the normal to the curve at \(P\) does not intersect the curve again.