Questions — CAIE P1 (1228 questions)

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CAIE P1 2013 June Q3
5 marks Moderate -0.3
3
  1. Express the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) as a quadratic equation in \(\cos ^ { 2 } \theta\).
  2. Solve the equation \(2 \cos ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 \leqslant \theta \leqslant \pi\), giving solutions in terms of \(\pi\).
CAIE P1 2013 June Q4
6 marks Moderate -0.8
4
  1. Find the first three terms in the expansion of \(( 2 + a x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 + 2 x ) ( 2 + a x ) ^ { 5 }\) is 240 , find the possible values of \(a\).
CAIE P1 2013 June Q5
6 marks Moderate -0.3
5
  1. Sketch, on the same diagram, the curves \(y = \sin 2 x\) and \(y = \cos x - 1\) for \(0 \leqslant x \leqslant 2 \pi\).
  2. Hence state the number of solutions, in the interval \(0 \leqslant x \leqslant 2 \pi\), of the equations
    1. \(2 \sin 2 x + 1 = 0\),
    2. \(\sin 2 x - \cos x + 1 = 0\).
CAIE P1 2013 June Q6
7 marks Standard +0.3
6 The non-zero variables \(x , y\) and \(u\) are such that \(u = x ^ { 2 } y\). Given that \(y + 3 x = 9\), find the stationary value of \(u\) and determine whether this is a maximum or a minimum value.
CAIE P1 2013 June Q7
8 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,
  1. the coordinates of \(X\),
  2. the ratio \(A X : X B\).
CAIE P1 2013 June Q8
8 marks Moderate -0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_716_437_1137_854} The diagram shows a parallelogram \(O A B C\) in which $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)$$
  1. Use a scalar product to find angle \(B O C\).
  2. Find a vector which has magnitude 35 and is parallel to the vector \(\overrightarrow { O C }\).
CAIE P1 2013 June Q9
8 marks Standard +0.8
9
  1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
  2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).
CAIE P1 2013 June Q10
9 marks Standard +0.2
10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x + k , x \in \mathbb { R }\), where \(k\) is a constant.
  1. In the case where \(k = 3\), solve the equation \(\mathrm { ff } ( x ) = 25\). The function g is defined by \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x + 8 , x \in \mathbb { R }\).
  2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has no real solutions. The function \(h\) is defined by \(h : x \mapsto x ^ { 2 } - 6 x + 8 , x > 3\).
  3. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2013 June Q11
9 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-4_643_570_849_790} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } x } - x\) and points \(A ( 1,7 )\) and \(B ( 4,0 )\) which lie on the curve. The tangent to the curve at \(B\) intersects the line \(x = 1\) at the point \(C\).
  1. Find the coordinates of \(C\).
  2. Find the area of the shaded region.
CAIE P1 2014 June Q1
2 marks Easy -1.2
1 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-2_750_1287_258_427} The diagram shows part of the graph of \(y = a + b \sin x\). State the values of the constants \(a\) and \(b\). [2
CAIE P1 2014 June Q2
4 marks Moderate -0.8
2
  1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
  2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).
CAIE P1 2014 June Q3
4 marks Moderate -0.8
3 Find the term independent of \(x\) in the expansion of \(\left( 4 x ^ { 3 } + \frac { 1 } { 2 x } \right) ^ { 8 }\).
CAIE P1 2014 June Q4
5 marks Moderate -0.3
4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.
CAIE P1 2014 June Q5
5 marks Moderate -0.3
5 An arithmetic progression has first term \(a\) and common difference \(d\). It is given that the sum of the first 200 terms is 4 times the sum of the first 100 terms.
  1. Find \(d\) in terms of \(a\).
  2. Find the 100th term in terms of \(a\).
CAIE P1 2014 June Q6
6 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-3_625_897_260_623} The diagram shows triangle \(A B C\) in which \(A B\) is perpendicular to \(B C\). The length of \(A B\) is 4 cm and angle \(C A B\) is \(\alpha\) radians. The arc \(D E\) with centre \(A\) and radius 2 cm meets \(A C\) at \(D\) and \(A B\) at \(E\). Find, in terms of \(\alpha\),
  1. the area of the shaded region,
  2. the perimeter of the shaded region.
CAIE P1 2014 June Q7
6 marks Moderate -0.3
7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).
CAIE P1 2014 June Q8
6 marks Moderate -0.3
8 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 p \\ 4 \\ p ^ { 2 } \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } - p \\ - 1 \\ p ^ { 2 } \end{array} \right)$$
  1. Find the values of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
  2. For the case where \(p = 3\), find the unit vector in the direction of \(\overrightarrow { B A }\).
CAIE P1 2014 June Q9
7 marks Standard +0.3
9
  1. Prove the identity \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } \equiv \frac { 1 } { \tan \theta }\).
  2. Hence solve the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 4 \tan \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2014 June Q10
9 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-4_819_812_255_662} The diagram shows the function f defined for \(- 1 \leqslant x \leqslant 4\), where $$f ( x ) = \begin{cases} 3 x - 2 & \text { for } - 1 \leqslant x \leqslant 1 \\ \frac { 4 } { 5 - x } & \text { for } 1 < x \leqslant 4 \end{cases}$$
  1. State the range of f .
  2. Copy the diagram and on your copy sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\).
  3. Obtain expressions to define the function \(\mathrm { f } ^ { - 1 }\), giving also the set of values for which each expression is valid.
CAIE P1 2014 June Q11
10 marks Standard +0.3
11 A line has equation \(y = 2 x + c\) and a curve has equation \(y = 8 - 2 x - x ^ { 2 }\).
  1. For the case where the line is a tangent to the curve, find the value of the constant \(c\).
  2. For the case where \(c = 11\), find the \(x\)-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.
CAIE P1 2014 June Q12
11 marks Moderate -0.8
12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).
  1. Find the equation of the curve.
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  3. Find the coordinates of the stationary point and determine its nature.
CAIE P1 2014 June Q1
5 marks Moderate -0.5
1 Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to \(( 10,3 )\) meets the \(x\)-axis.
CAIE P1 2014 June Q2
5 marks Standard +0.3
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) \left( \frac { x } { 2 } - \frac { 4 } { x } \right) ^ { 6 }\).
CAIE P1 2014 June Q3
5 marks Moderate -0.8
3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).
  1. Find an expression, in terms of \(k\), for
    1. \(\sin \theta\),
    2. \(\tan \theta\).
    3. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).
CAIE P1 2014 June Q4
5 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-2_358_618_1082_762} The diagram shows a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(O\). The chord \(A B\) divides the sector into a triangle \(A O B\) and a segment \(A X B\). Angle \(A O B\) is \(\theta\) radians.
  1. In the case where the areas of the triangle \(A O B\) and the segment \(A X B\) are equal, find the value of the constant \(p\) for which \(\theta = p \sin \theta\).
  2. In the case where \(r = 8\) and \(\theta = 2.4\), find the perimeter of the segment \(A X B\).