Questions — CAIE (7279 questions)

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CAIE P1 2014 June Q1
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
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
3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).
  1. Find an expression, in terms of \(k\), for
    (a) \(\sin \theta\),
    (b) \(\tan \theta\).
  2. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).
CAIE P1 2014 June Q4
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 \(A X B\).
CAIE P1 2014 June Q5
5
  1. Prove the identity \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } \equiv \tan \theta\).
  2. Solve the equation \(\frac { 1 } { \cos \theta } - \frac { \cos \theta } { 1 + \sin \theta } + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2014 June Q6
6 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is \(r\), where \(r \neq 1\). Find
  1. the value of \(r\),
  2. the 4th term of each progression.
CAIE P1 2014 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3
4
0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1
3
2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4
5
6 \end{array} \right)$$
  1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
  2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).
CAIE P1 2014 June Q8
8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.
CAIE P1 2014 June Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_849_565_1466_790} The diagram shows part of the curve \(y = 8 - \sqrt { } ( 4 - x )\) and the tangent to the curve at \(P ( 3,7 )\).
  1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
  2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\).
  3. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2014 June Q10
10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 3 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \mapsto x ^ { 2 } + 4 x , \quad x \in \mathbb { R } . \end{aligned}$$
  1. Solve the equation \(\mathrm { ff } ( x ) = 11\).
  2. Find the range of g .
  3. Find the set of values of \(x\) for which \(\mathrm { g } ( x ) > 12\).
  4. Find the value of the constant \(p\) for which the equation \(\mathrm { gf } ( x ) = p\) has two equal roots. Function h is defined by \(\mathrm { h } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant k\), and it is given that h has an inverse.
  5. State the smallest possible value of \(k\).
  6. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
CAIE P1 2014 June Q1
1 Find the coefficient of \(x\) in the expansion of \(\left( x ^ { 2 } - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2014 June Q2
2 The first term in a progression is 36 and the second term is 32 .
  1. Given that the progression is geometric, find the sum to infinity.
  2. Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0 .
CAIE P1 2014 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-2_485_623_790_760} The diagram shows part of a circle with centre \(O\) and radius 6 cm . The chord \(A B\) is such that angle \(A O B = 2.2\) radians. Calculate
  1. the perimeter of the shaded region,
  2. the ratio of the area of the shaded region to the area of the triangle \(A O B\), giving your answer in the form \(k : 1\).
CAIE P1 2014 June Q4
4
  1. Prove the identity \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } \equiv \sin x + \cos x\).
  2. Hence solve the equation \(\frac { \tan x + 1 } { \sin x \tan x + \cos x } = 3 \sin x - 2 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).
CAIE P1 2014 June Q5
5 A function f is such that \(\mathrm { f } ( x ) = \frac { 15 } { 2 x + 3 }\) for \(0 \leqslant x \leqslant 6\).
  1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and use your result to explain why f has an inverse.
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2014 June Q6
6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { \sqrt { } ( 4 x + a ) }\), where \(a\) is a constant. The point \(P ( 2,14 )\) lies on the curve and the normal to the curve at \(P\) is \(3 y + x = 5\).
  1. Show that \(a = 8\).
  2. Find the equation of the curve.
CAIE P1 2014 June Q7
7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2
1
3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6
- 1
7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2
4
7 \end{array} \right)$$
  1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
  2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).
CAIE P1 2014 June Q8
8
  1. Express \(2 x ^ { 2 } - 10 x + 8\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and use your answer to state the minimum value of \(2 x ^ { 2 } - 10 x + 8\).
  2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } - 10 x + 8 = k x\) has no real roots.
CAIE P1 2014 June Q9
9 The base of a cuboid has sides of length \(x \mathrm {~cm}\) and \(3 x \mathrm {~cm}\). The volume of the cuboid is \(288 \mathrm {~cm} ^ { 3 }\).
  1. Show that the total surface area of the cuboid, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = 6 x ^ { 2 } + \frac { 768 } { x }$$
  2. Given that \(x\) can vary, find the stationary value of \(A\) and determine its nature.
CAIE P1 2014 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-3_812_720_1484_715} The diagram shows the curve \(y = - x ^ { 2 } + 12 x - 20\) and the line \(y = 2 x + 1\). Find, showing all necessary working, the area of the shaded region.
CAIE P1 2014 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram \(A B C D\), in which the equation of \(A B\) is \(y = 3 x\) and the equation of \(A D\) is \(4 y = x + 11\). The diagonals \(A C\) and \(B D\) meet at the point \(E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)\). Find, by calculation, the coordinates of \(A , B , C\) and \(D\).
CAIE P1 2015 June Q1
1 Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for
  1. \(\cos \theta\),
  2. \(\tan \theta\),
  3. \(\sin ( \theta + \pi )\).
CAIE P1 2015 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{c8925c7a-cb3b-43b8-9d09-8adc800c6887-2_627_828_641_657} The diagram shows the curve \(y = 2 x ^ { 2 }\) and the points \(X ( - 2,0 )\) and \(P ( p , 0 )\). The point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.
  1. Express the area, \(A\), of triangle \(X P Q\) in terms of \(p\). The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(P Q\) remains parallel to the \(y\)-axis.
  2. Find the rate at which \(A\) is increasing when \(p = 2\).
CAIE P1 2015 June Q3
3
  1. Find the first three terms, in ascending powers of \(x\), in the expansion of
    (a) \(\quad ( 1 - x ) ^ { 6 }\),
    (b) \(( 1 + 2 x ) ^ { 6 }\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \([ ( 1 - x ) ( 1 + 2 x ) ] ^ { 6 }\).
CAIE P1 2015 June Q4
4 Relative to the origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3
0
- 4 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r }