OCR C1 (Core Mathematics 1)

1. \(\quad \mathrm { f } ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }\).

Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.

2. Find in exact form the real solutions of the equation $$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$

3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

4. Express each of the following in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational.

1. \(( 4 - 3 \sqrt { 2 } ) ^ { 2 }\)
2. \(\frac { 1 } { 2 + \sqrt { 2 } }\)

5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,

1. show that $$4 k ^ { 2 } + k < 0 ,$$
2. find the set of possible values of \(k\).

6. The curve with equation \(y = x ^ { 2 } + 2 x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\).
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again.

7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of the circle.
2. Find an equation for the circle.
3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.

8. (i) Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
(ii) Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
(iii) Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.

9. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
3. show that triangle \(A B C\) is isosceles.

10. \includegraphics[max width=\textwidth, alt={}, center]{6ef55dbd-f18d-4264-b80c-d181473ca7b3-3_531_786_246_523} The diagram shows an open-topped cylindrical container made from cardboard. The cylinder is of height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). Given that the area of card used to make the container is \(192 \pi \mathrm {~cm} ^ { 2 }\),

1. show that the capacity of the container, \(\mathrm { V } \mathrm { cm } ^ { 3 }\), is given by $$V = 96 \pi r - \frac { 1 } { 2 } \pi r ^ { 3 } .$$
2. Find the value of \(r\) for which \(V\) is stationary.
3. Find the corresponding value of \(V\) in terms of \(\pi\).
4. Determine whether this is a maximum or a minimum value of \(V\).