Questions — CAIE (7646 questions)

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CAIE P1 2012 June Q10
9 marks Moderate -0.3
10 It is given that a curve has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + x\).
  1. Find the set of values of \(x\) for which the gradient of the curve is less than 5 .
  2. Find the values of \(\mathrm { f } ( x )\) at the two stationary points on the curve and determine the nature of each stationary point.
CAIE P1 2012 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-4_636_951_255_596} The diagram shows the line \(y = 1\) and part of the curve \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\).
  1. Show that the equation \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\) can be written in the form \(x = \frac { 4 } { y ^ { 2 } } - 1\).
  2. Find \(\int \left( \frac { 4 } { y ^ { 2 } } - 1 \right) \mathrm { d } y\). Hence find the area of the shaded region.
  3. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the exact value of the volume of revolution obtained.
CAIE P1 2013 June Q1
3 marks Moderate -0.8
1 It is given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ^ { 3 } + x\), for \(x \in \mathbb { R }\). Show that f is an increasing function.
CAIE P1 2013 June Q2
5 marks Moderate -0.3
2
  1. In the expression \(( 1 - p x ) ^ { 6 } , p\) is a non-zero constant. Find the first three terms when \(( 1 - p x ) ^ { 6 }\) is expanded in ascending powers of \(x\).
  2. It is given that the coefficient of \(x ^ { 2 }\) in the expansion of \(( 1 - x ) ( 1 - p x ) ^ { 6 }\) is zero. Find the value of \(p\).
CAIE P1 2013 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{d0074ac8-42d2-49f4-a417-4a348537bccc-2_492_682_708_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 8 cm . Angle \(B O A\) is \(\alpha\) radians. \(O A C\) is a semicircle with diameter \(O A\). The area of the semicircle \(O A C\) is twice the area of the sector \(O A B\).
  1. Find \(\alpha\) in terms of \(\pi\).
  2. Find the perimeter of the complete figure in terms of \(\pi\).
CAIE P1 2013 June Q4
6 marks Moderate -0.5
4 The third term of a geometric progression is - 108 and the sixth term is 32 . Find
  1. the common ratio,
  2. the first term,
  3. the sum to infinity.
CAIE P1 2013 June Q5
7 marks Standard +0.3
5
  1. Show that \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } \equiv \frac { 1 } { \sin ^ { 2 } \theta - \cos ^ { 2 } \theta }\).
  2. Hence solve the equation \(\frac { \sin \theta } { \sin \theta + \cos \theta } + \frac { \cos \theta } { \sin \theta - \cos \theta } = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2013 June Q6
7 marks Standard +0.3
6 Relative to an origin \(O\), the position vectors of three points, \(A , B\) and \(C\), are given by $$\overrightarrow { O A } = \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k } , \quad \overrightarrow { O B } = q \mathbf { j } - 2 p \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = - \left( 4 p ^ { 2 } + q ^ { 2 } \right) \mathbf { i } + 2 p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.
  1. Show that \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O C }\) for all non-zero values of \(p\) and \(q\).
  2. Find the magnitude of \(\overrightarrow { C A }\) in terms of \(p\) and \(q\).
  3. For the case where \(p = 3\) and \(q = 2\), find the unit vector parallel to \(\overrightarrow { B A }\).
CAIE P1 2013 June Q7
9 marks Moderate -0.3
7 A curve has equation \(y = x ^ { 2 } - 4 x + 4\) and a line has equation \(y = m x\), where \(m\) is a constant.
  1. For the case where \(m = 1\), the curve and the line intersect at the points \(A\) and \(B\). Find the coordinates of the mid-point of \(A B\).
  2. Find the non-zero value of \(m\) for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.
CAIE P1 2013 June Q8
10 marks Moderate -0.3
8
  1. Express \(2 x ^ { 2 } - 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  2. The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant k\), where \(k\) is a constant. It is given that f is a one-one function. State the smallest possible value of \(k\). The value of \(k\) is now given to be 7 .
  3. Find the range of f .
  4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2013 June Q9
11 marks Moderate -0.3
9 A curve has equation \(y = \mathrm { f } ( x )\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } + 3 x ^ { - \frac { 1 } { 2 } } - 10\).
  1. By using the substitution \(u = x ^ { \frac { 1 } { 2 } }\), or otherwise, find the values of \(x\) for which the curve \(y = \mathrm { f } ( x )\) has stationary points.
  2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence, or otherwise, determine the nature of each stationary point.
  3. It is given that the curve \(y = \mathrm { f } ( x )\) passes through the point \(( 4 , - 7 )\). Find \(\mathrm { f } ( x )\).
CAIE P1 2013 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{d0074ac8-42d2-49f4-a417-4a348537bccc-4_521_809_258_669} The diagram shows part of the curve \(y = ( x - 2 ) ^ { 4 }\) and the point \(A ( 1,1 )\) on the curve. The tangent at \(A\) cuts the \(x\)-axis at \(B\) and the normal at \(A\) cuts the \(y\)-axis at \(C\).
  1. Find the coordinates of \(B\) and \(C\).
  2. Find the distance \(A C\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.
  3. Find the area of the shaded region.
CAIE P1 2013 June Q1
3 marks Easy -1.3
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 2 } }\) and \(( 2,9 )\) is a point on the curve. Find the equation of the curve.
CAIE P1 2013 June Q2
5 marks Moderate -0.3
2 Find the coefficient of \(x ^ { 2 }\) in the expansion of
  1. \(\left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\),
  2. \(\left( 1 + x ^ { 2 } \right) \left( 2 x - \frac { 1 } { 2 x } \right) ^ { 6 }\).
CAIE P1 2013 June Q3
5 marks Moderate -0.3
3 The straight line \(y = m x + 14\) is a tangent to the curve \(y = \frac { 12 } { x } + 2\) at the point \(P\). Find the value of the constant \(m\) and the coordinates of \(P\).
CAIE P1 2013 June Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-2_645_652_1023_744} The diagram shows a square \(A B C D\) of side 10 cm . The mid-point of \(A D\) is \(O\) and \(B X C\) is an arc of a circle with centre \(O\).
  1. Show that angle \(B O C\) is 0.9273 radians, correct to 4 decimal places.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2013 June Q5
7 marks Standard +0.3
5 It is given that \(a = \sin \theta - 3 \cos \theta\) and \(b = 3 \sin \theta + \cos \theta\), where \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(\theta\).
  2. Find the values of \(\theta\) for which \(2 a = b\).
CAIE P1 2013 June Q6
7 marks Moderate -0.3
6 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.
  1. State the values of \(p\) and \(q\) for which \(\overrightarrow { O A }\) is parallel to \(\overrightarrow { O B }\).
  2. In the case where \(q = 2 p\), find the value of \(p\) for which angle \(B O A\) is \(90 ^ { \circ }\).
  3. In the case where \(p = 1\) and \(q = 8\), find the unit vector in the direction of \(\overrightarrow { A B }\).
CAIE P1 2013 June Q7
7 marks Challenging +1.2
7 The point \(R\) is the reflection of the point \(( - 1,3 )\) in the line \(3 y + 2 x = 33\). Find by calculation the coordinates of \(R\).
CAIE P1 2013 June Q8
8 marks Standard +0.3
8 The volume of a solid circular cylinder of radius \(r \mathrm {~cm}\) is \(250 \pi \mathrm {~cm} ^ { 3 }\).
  1. Show that the total surface area, \(S \mathrm {~cm} ^ { 2 }\), of the cylinder is given by $$S = 2 \pi r ^ { 2 } + \frac { 500 \pi } { r }$$
  2. Given that \(r\) can vary, find the stationary value of \(S\).
  3. Determine the nature of this stationary value.
CAIE P1 2013 June Q9
8 marks Moderate -0.3
9 A function f is defined by \(\mathrm { f } ( x ) = \frac { 5 } { 1 - 3 x }\), for \(x \geqslant 1\).
  1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
  2. Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2013 June Q10
8 marks Moderate -0.3
10
  1. The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
  2. The third term of a geometric progression is four times the first term. The sum of the first six terms is \(k\) times the first term. Find the possible values of \(k\).
CAIE P1 2013 June Q11
10 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-4_598_789_255_678} The diagram shows the curve \(y = \sqrt { } ( 1 + 4 x )\), which intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(B\) meets the \(x\)-axis at \(C\). Find
  1. the equation of \(B C\),
  2. the area of the shaded region.
CAIE P1 2013 June Q1
4 marks Moderate -0.8
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.
CAIE P1 2013 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-2_501_641_461_753} The diagram shows a circle \(C\) with centre \(O\) and radius 3 cm . The radii \(O P\) and \(O Q\) are extended to \(S\) and \(R\) respectively so that \(O R S\) is a sector of a circle with centre \(O\). Given that \(P S = 6 \mathrm {~cm}\) and that the area of the shaded region is equal to the area of circle \(C\),
  1. show that angle \(P O Q = \frac { 1 } { 4 } \pi\) radians,
  2. find the perimeter of the shaded region.