Questions P1 (1374 questions)

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CAIE P1 2012 June Q6
6 The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135 .
  1. Find the common difference of the progression. The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.
  2. Find the common ratio of the geometric progression and the value of \(n\).
CAIE P1 2012 June Q7
7 The curve \(y = \frac { 10 } { 2 x + 1 } - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).
  1. Show that the equation of \(A C\) is \(5 y + 4 x = 8\).
  2. Find the distance \(A C\).
CAIE P1 2012 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{1b5d8cb1-fd1b-4fcf-8975-b5d020991c9a-3_554_385_641_879} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(X B\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(O X\).
  1. Show that angle \(A O B = \frac { 1 } { 3 } \pi\) radians. Express each of the following in terms of \(r , \pi\) and \(\sqrt { } 3\) :
  2. the perimeter of the shaded region,
  3. the area of the shaded region.
CAIE P1 2012 June Q9
9 A curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 4 x\). The curve has a maximum point at (2,12).
  1. Find the equation of the curve. A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.
  2. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing.
CAIE P1 2012 June Q10
10 The equation of a line is \(2 y + x = k\), where \(k\) is a constant, and the equation of a curve is \(x y = 6\).
  1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(A B\).
  2. Find the set of values of \(k\) for which the line \(2 y + x = k\) intersects the curve \(x y = 6\) at two distinct points.
CAIE P1 2012 June Q11
11 The function f is such that \(\mathrm { f } ( x ) = 8 - ( x - 2 ) ^ { 2 }\), for \(x \in \mathbb { R }\).
  1. Find the coordinates and the nature of the stationary point on the curve \(y = \mathrm { f } ( x )\). The function g is such that \(\mathrm { g } ( x ) = 8 - ( x - 2 ) ^ { 2 }\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
  2. State the smallest value of \(k\) for which g has an inverse. For this value of \(k\),
  3. find an expression for \(\mathrm { g } ^ { - 1 } ( x )\),
  4. sketch, on the same diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2013 June Q1
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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 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
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 \(\mathrm { f } ^ { - 1 }\).