CAIE P1 (Pure Mathematics 1) 2018 June

1 Express \(3 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

2 Find the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 5 }\).

3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.

4 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(A ( 3,1 )\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\). Find the \(y\)-coordinate of \(B\).

5
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.

6 The coordinates of points \(A\) and \(B\) are \(( - 3 k - 1 , k + 3 )\) and \(( k + 3,3 k + 5 )\) respectively, where \(k\) is a constant ( \(k \neq - 1\) ).

1. Find and simplify the gradient of \(A B\), showing that it is independent of \(k\).
2. Find and simplify the equation of the perpendicular bisector of \(A B\).

7

1. 1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
   2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
   1. Find the \(x\)-coordinate of \(A\).
   2. Find the \(y\)-coordinate of \(B\).

8

1. The tangent to the curve \(y = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3 x\). Find the equation of the tangent at \(A\).
2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function.

9
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.

1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
2. Use a scalar product to find angle \(B D O\).

10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\).
   In parts (ii) and (iii) the value of \(c\) is 4 .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).

11
\includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.