Questions P1 (1401 questions)

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CAIE P1 2012 November Q5
5 marks Moderate -0.3
5 A curve has equation \(y = 2 x + \frac { 1 } { ( x - 1 ) ^ { 2 } }\). Verify that the curve has a stationary point at \(x = 2\) and determine its nature.
CAIE P1 2012 November Q6
7 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-2_526_659_1336_742} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) on \(O A\) is such that \(B C\) is perpendicular to \(O A\). The point \(D\) is on \(B C\) and the circular arc \(A D\) has centre \(C\).
  1. Find \(A C\) in terms of \(r\) and \(\theta\).
  2. Find the perimeter of the shaded region \(A B D\) when \(\theta = \frac { 1 } { 3 } \pi\) and \(r = 4\), giving your answer as an exact value.
CAIE P1 2012 November Q7
7 marks Standard +0.3
7
  1. Solve the equation \(2 \cos ^ { 2 } \theta = 3 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  2. The smallest positive solution of the equation \(2 \cos ^ { 2 } ( n \theta ) = 3 \sin ( n \theta )\), where \(n\) is a positive integer, is \(10 ^ { \circ }\). State the value of \(n\) and hence find the largest solution of this equation in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2012 November Q8
8 marks Standard +0.8
8 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-3_629_853_251_644} The diagram shows the curve \(y ^ { 2 } = 2 x - 1\) and the straight line \(3 y = 2 x - 1\). The curve and straight line intersect at \(x = \frac { 1 } { 2 }\) and \(x = a\), where \(a\) is a constant.
  1. Show that \(a = 5\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2012 November Q9
9 marks Moderate -0.5
9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. The position vectors of points \(C\) and \(D\) relative to \(O\) are \(3 \mathbf { a }\) and \(2 \mathbf { b }\) respectively. It is given that $$\mathbf { a } = \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { b } = \left( \begin{array} { l } 4 \\ 0 \\ 6 \end{array} \right) .$$
  1. Find the unit vector in the direction of \(\overrightarrow { C D }\).
  2. The point \(E\) is the mid-point of \(C D\). Find angle \(E O D\).
CAIE P1 2012 November Q10
10 marks Moderate -0.8
10 The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \in \mathbb { R }\).
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } + c\) and hence state the coordinates of the vertex of the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \leqslant 1\).
  2. State the range of g .
  3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).
CAIE P1 2012 November Q11
12 marks Moderate -0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{e69332d0-2e45-4a86-a1f9-5d83bca1ad9b-4_885_967_255_589} The diagram shows the curve \(y = ( 6 x + 2 ) ^ { \frac { 1 } { 3 } }\) and the point \(A ( 1,2 )\) which lies on the curve. The tangent to the curve at \(A\) cuts the \(y\)-axis at \(B\) and the normal to the curve at \(A\) cuts the \(x\)-axis at \(C\).
  1. Find the equation of the tangent \(A B\) and the equation of the normal \(A C\).
  2. Find the distance \(B C\).
  3. Find the coordinates of the point of intersection, \(E\), of \(O A\) and \(B C\), and determine whether \(E\) is the mid-point of \(O A\).
CAIE P1 2012 November Q1
3 marks Moderate -0.8
1 In the expansion of \(\left( x ^ { 2 } - \frac { a } { x } \right) ^ { 7 }\), the coefficient of \(x ^ { 5 }\) is - 280 . Find the value of the constant \(a\).
CAIE P1 2012 November Q2
4 marks Moderate -0.8
2 A function f is such that \(\mathrm { f } ( x ) = \sqrt { } \left( \frac { x + 3 } { 2 } \right) + 1\), for \(x \geqslant - 3\). Find
  1. \(\mathrm { f } ^ { - 1 } ( x )\) in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants,
  2. the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2012 November Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-2_485_755_751_696} The diagram shows a plan for a rectangular park \(A B C D\), in which \(A B = 40 \mathrm {~m}\) and \(A D = 60 \mathrm {~m}\). Points \(X\) and \(Y\) lie on \(B C\) and \(C D\) respectively and \(A X , X Y\) and \(Y A\) are paths that surround a triangular playground. The length of \(D Y\) is \(x \mathrm {~m}\) and the length of \(X C\) is \(2 x \mathrm {~m}\).
  1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the playground is given by $$A = x ^ { 2 } - 30 x + 1200$$
  2. Given that \(x\) can vary, find the minimum area of the playground.
CAIE P1 2012 November Q4
6 marks Moderate -0.3
4 The line \(y = \frac { x } { k } + k\), where \(k\) is a constant, is a tangent to the curve \(4 y = x ^ { 2 }\) at the point \(P\). Find
  1. the value of \(k\),
  2. the coordinates of \(P\).
CAIE P1 2012 November Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-3_602_751_255_696} The diagram shows a triangle \(A B C\) in which \(A\) has coordinates ( 1,3 ), \(B\) has coordinates ( 5,11 ) and angle \(A B C\) is \(90 ^ { \circ }\). The point \(X ( 4,4 )\) lies on \(A C\). Find
  1. the equation of \(B C\),
  2. the coordinates of \(C\).
CAIE P1 2012 November Q6
7 marks Moderate -0.3
6
  1. Show that the equation \(2 \cos x = 3 \tan x\) can be written as a quadratic equation in \(\sin x\).
  2. Solve the equation \(2 \cos 2 y = 3 \tan 2 y\), for \(0 ^ { \circ } \leqslant y \leqslant 180 ^ { \circ }\).
CAIE P1 2012 November Q7
8 marks Moderate -0.3
7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k \\ - k \\ 2 k \end{array} \right)$$ where \(k\) is a constant.
  1. In the case where \(k = 2\), calculate angle \(A O B\).
  2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.
CAIE P1 2012 November Q8
9 marks Standard +0.3
8
  1. In a geometric progression, all the terms are positive, the second term is 24 and the fourth term is \(13 \frac { 1 } { 2 }\). Find
    1. the first term,
    2. the sum to infinity of the progression.
  2. A circle is divided into \(n\) sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are \(3 ^ { \circ }\) and \(5 ^ { \circ }\). Find the value of \(n\).
CAIE P1 2012 November Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_502_663_255_740} The diagram shows part of the curve \(y = \frac { 9 } { 2 x + 3 }\), crossing the \(y\)-axis at the point \(B ( 0,3 )\). The point \(A\) on the curve has coordinates \(( 3,1 )\) and the tangent to the curve at \(A\) crosses the \(y\)-axis at \(C\).
  1. Find the equation of the tangent to the curve at \(A\).
  2. Determine, showing all necessary working, whether \(C\) is nearer to \(B\) or to \(O\).
  3. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2012 November Q10
8 marks Moderate -0.8
10 A curve is defined for \(x > 0\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x + \frac { 4 } { x ^ { 2 } }\). The point \(P ( 4,8 )\) lies on the curve.
  1. Find the equation of the curve.
  2. Show that the gradient of the curve has a minimum value when \(x = 2\) and state this minimum value.
CAIE P1 2012 November Q11
10 marks Challenging +1.2
11 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_611_668_1699_737} The diagram shows a sector of a circle with centre \(O\) and radius 20 cm . A circle with centre \(C\) and radius \(x \mathrm {~cm}\) lies within the sector and touches it at \(P , Q\) and \(R\). Angle \(P O R = 1.2\) radians.
  1. Show that \(x = 7.218\), correct to 3 decimal places.
  2. Find the total area of the three parts of the sector lying outside the circle with centre \(C\).
  3. Find the perimeter of the region \(O P S R\) bounded by the \(\operatorname { arc } P S R\) and the lines \(O P\) and \(O R\).
CAIE P1 2012 November Q1
3 marks Moderate -0.8
1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 2 - \frac { 1 } { 2 } x \right) ^ { 7 }\).
CAIE P1 2012 November Q2
3 marks Moderate -0.8
2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.
CAIE P1 2012 November Q3
4 marks Moderate -0.3
3 Solve the equation \(7 \cos x + 5 = 2 \sin ^ { 2 } x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2012 November Q4
4 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-2_478_828_708_660} In the diagram, \(D\) lies on the side \(A B\) of triangle \(A B C\) and \(C D\) is an arc of a circle with centre \(A\) and radius 2 cm . The line \(B C\) is of length \(2 \sqrt { } 3 \mathrm {~cm}\) and is perpendicular to \(A C\). Find the area of the shaded region \(B D C\), giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2012 November Q5
5 marks Moderate -0.8
5 The first term of a geometric progression is \(5 \frac { 1 } { 3 }\) and the fourth term is \(2 \frac { 1 } { 4 }\). Find
  1. the common ratio,
  2. the sum to infinity.
CAIE P1 2012 November Q6
6 marks Moderate -0.3
6 The functions f and g are defined for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\) by $$\begin{aligned} & f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 6 } \pi \\ & g ( x ) = \cos x \end{aligned}$$ Solve the following equations for \(- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  1. \(\operatorname { gf } ( x ) = 1\), giving your answer in terms of \(\pi\).
  2. \(\operatorname { fg } ( x ) = 1\), giving your answers correct to 2 decimal places.
CAIE P1 2012 November Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-3_821_688_255_731}
  1. The diagram shows part of the curve \(y = 11 - x ^ { 2 }\) and part of the straight line \(y = 5 - x\) meeting at the point \(A ( p , q )\), where \(p\) and \(q\) are positive constants. Find the values of \(p\) and \(q\).
  2. The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \begin{cases} 11 - x ^ { 2 } & \text { for } 0 \leqslant x \leqslant p \\ 5 - x & \text { for } x > p \end{cases}$$ Express \(\mathrm { f } ^ { - 1 } ( x )\) in a similar way.