OCR C1 (Core Mathematics 1)

1. Find the value of \(y\) such that

$$4 ^ { y + 1 } = 8 ^ { 2 y - 1 } .$$

1. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
2. A circle has the equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of the circle. Given that the \(x\)-axis is a tangent to the circle,
2. find the value of \(k\).

4. $$f ( x ) = 4 x - 3 x ^ { 2 } - x ^ { 3 }$$

1. Fully factorise \(4 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.

5. (i) Find in exact form the coordinates of the points where the curve \(y = x ^ { 2 } - 4 x + 2\) crosses the \(x\)-axis.
(ii) Find the value of the constant \(k\) for which the straight line \(y = 2 x + k\) is a tangent to the curve \(y = x ^ { 2 } - 4 x + 2\).

6. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 }$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),

1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
3. find the rate at which the area of the stain is increasing when \(t = 15\).

7. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).

1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(( x + p ) ^ { 2 } + q\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
2. Sketch \(l\) and \(C\) on the same set of axes.
3. Find the coordinates of the points where \(I\) and \(C\) intersect.

8. $$f ( x ) \equiv \frac { ( x - 4 ) ^ { 2 } } { 2 x ^ { \frac { 1 } { 2 } } } , x > 0$$

1. Find the values of the constants \(A , B\) and \(C\) such that $$f ( x ) = A x ^ { \frac { 3 } { 2 } } + B x ^ { \frac { 1 } { 2 } } + C x ^ { - \frac { 1 } { 2 } }$$
2. Show that $$f ^ { \prime } ( x ) = \frac { 3 x ^ { 2 } - 8 x - 16 } { 4 x ^ { \frac { 3 } { 2 } } }$$
3. Find the coordinates of the stationary point of the curve \(y = \mathrm { f } ( x )\).

9.
\includegraphics[max width=\textwidth, alt={}, center]{6dc260d0-efec-42b8-995f-5a9e1b964255-3_659_1109_251_404} The diagram shows the parallelogram \(A B C D\).
The points \(A\) and \(B\) have coordinates \(( - 1,3 )\) and \(( 3,4 )\) respectively and lie on the straight line \(l _ { 1 }\).

1. Find an equation for \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
3. Find the area of parallelogram \(A B C D\).