CAIE P1 (Pure Mathematics 1) 2018 November

1 Find the coefficient of \(\frac { 1 } { x ^ { 3 } }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 7 }\).

2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.

3
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-04_467_401_260_872} The diagram shows an arc \(B C\) of a circle with centre \(A\) and radius 5 cm . The length of the arc \(B C\) is 4 cm . The point \(D\) is such that the line \(B D\) is perpendicular to \(B A\) and \(D C\) is parallel to \(B A\).

1. Find angle \(B A C\) in radians.
2. Find the area of the shaded region \(B D C\).

4 Two points \(A\) and \(B\) have coordinates \(( - 1,1 )\) and \(( 3,4 )\) respectively. The line \(B C\) is perpendicular to \(A B\) and intersects the \(x\)-axis at \(C\).

1. Find the equation of \(B C\) and the \(x\)-coordinate of \(C\).
2. Find the distance \(A C\), giving your answer correct to 3 decimal places.

5 In an arithmetic progression the first term is \(a\) and the common difference is 3 . The \(n\)th term is 94 and the sum of the first \(n\) terms is 1420 . Find \(n\) and \(a\).

6
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-08_743_897_260_623} The diagram shows a solid figure \(O A B C D E F G\) with a horizontal rectangular base \(O A B C\) in which \(O A = 8\) units and \(A B = 6\) units. The rectangle \(D E F G\) lies in a horizontal plane and is such that \(D\) is 7 units vertically above \(O\) and \(D E\) is parallel to \(O A\). The sides \(D E\) and \(D G\) have lengths 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. Use a scalar product to find angle \(O B F\), giving your answer in the form \(\cos ^ { - 1 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

7

1. Show that \(\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } \equiv \frac { 2 ( \tan \theta - \cos \theta ) } { \sin ^ { 2 } \theta }\).
2. Hence, showing all necessary working, solve the equation $$\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } = 0$$ for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

8 A curve passes through \(( 0,11 )\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants.

1. Find the equation of the curve in terms of \(a\) and \(b\).
2. It is now given that the curve has a stationary point at \(( 2,3 )\). Find the values of \(a\) and \(b\).

9 A curve has equation \(y = 2 x ^ { 2 } - 3 x + 1\) and a line has equation \(y = k x + k ^ { 2 }\), where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve and the line meet.
2. State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve.

10
\includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-16_648_823_262_660} The diagram shows part of the curve \(y = 2 ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\) and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\). The curve and the line \(x = \frac { 2 } { 3 }\) intersect at the point \(A\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the equation of the normal to the curve at \(A\), giving your answer in the form \(y = m x + c\).

11

1. Express \(2 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 11\) for \(x \leqslant k\).
2. State the largest value of the constant \(k\) for which f is a one-one function.
3. For this value of \(k\) find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x \leqslant p\).
4. With \(k\) now taking the value 1 , find the largest value of the constant \(p\) which allows the composite function fg to be formed, and find an expression for \(\mathrm { fg } ( x )\) whenever this composite function exists.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.