Questions — CAIE P1 (1228 questions)

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CAIE P1 2018 June Q1
3 marks Easy -1.2
1 Express \(3 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
CAIE P1 2018 June Q2
3 marks Moderate -0.8
2 Find the coefficient of \(\frac { 1 } { x }\) in the expansion of \(\left( x - \frac { 2 } { x } \right) ^ { 5 }\).
CAIE P1 2018 June Q3
5 marks Standard +0.3
3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.
CAIE P1 2018 June Q4
6 marks Moderate -0.3
4 A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(A ( 3,1 )\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\). Find the \(y\)-coordinate of \(B\).
CAIE P1 2018 June Q5
5 marks Moderate -0.5
5 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.
CAIE P1 2018 June Q6
7 marks Moderate -0.3
6 The coordinates of points \(A\) and \(B\) are \(( - 3 k - 1 , k + 3 )\) and \(( k + 3,3 k + 5 )\) respectively, where \(k\) is a constant ( \(k \neq - 1\) ).
  1. Find and simplify the gradient of \(A B\), showing that it is independent of \(k\).
  2. Find and simplify the equation of the perpendicular bisector of \(A B\).
CAIE P1 2018 June Q7
9 marks Moderate -0.3
7
    1. Express \(\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 }\) in the form \(a \sin ^ { 2 } \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac { \tan ^ { 2 } \theta - 1 } { \tan ^ { 2 } \theta + 1 } = \frac { 1 } { 4 }$$ for \(- 90 ^ { \circ } \leqslant \theta \leqslant 0 ^ { \circ }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-11_549_796_267_717} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\).
    2. Find the \(y\)-coordinate of \(B\).
CAIE P1 2018 June Q8
8 marks Moderate -0.3
8
  1. The tangent to the curve \(y = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3 x\). Find the equation of the tangent at \(A\).
  2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function.
CAIE P1 2018 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.
  1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
  2. Use a scalar product to find angle \(B D O\).
CAIE P1 2018 June Q10
9 marks Moderate -0.8
10 The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } + 2\) for \(x \geqslant c\), where \(c\) is a constant.
  1. State the smallest possible value of \(c\).
    In parts (ii) and (iii) the value of \(c\) is 4 .
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Solve the equation \(\mathrm { ff } ( x ) = 51\), giving your answer in the form \(a + \sqrt { } b\).
CAIE P1 2018 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-18_645_723_258_573} The diagram shows part of the curve \(y = ( x + 1 ) ^ { 2 } + ( x + 1 ) ^ { - 1 }\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.
  1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2 ( x + 1 ) ^ { 3 } = 1\) and find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2019 June Q1
5 marks Moderate -0.8
1 The term independent of \(x\) in the expansion of \(\left( 2 x + \frac { k } { x } \right) ^ { 6 }\), where \(k\) is a constant, is 540.
  1. Find the value of \(k\).
  2. For this value of \(k\), find the coefficient of \(x ^ { 2 }\) in the expansion.
CAIE P1 2019 June Q2
5 marks Moderate -0.3
2 The line \(4 y = x + c\), where \(c\) is a constant, is a tangent to the curve \(y ^ { 2 } = x + 3\) at the point \(P\) on the curve.
  1. Find the value of \(c\).
  2. Find the coordinates of \(P\).
CAIE P1 2019 June Q3
4 marks Moderate -0.8
3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).
CAIE P1 2019 June Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium \(A B C D\) in which the coordinates of \(A , B\) and \(C\) are (4, 0), (0, 2) and \(( h , 3 h )\) respectively. The lines \(B C\) and \(A D\) are parallel, angle \(A B C = 90 ^ { \circ }\) and \(C D\) is parallel to the \(x\)-axis.
  1. Find, by calculation, the value of \(h\).
  2. Hence find the coordinates of \(D\).
CAIE P1 2019 June Q5
6 marks Moderate -0.8
5 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } + 12 x - 3\) for \(x \in \mathbb { R }\).
  1. Express \(- 2 x ^ { 2 } + 12 x - 3\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. State the greatest value of \(\mathrm { f } ( x )\).
    The function g is defined by \(\mathrm { g } ( x ) = 2 x + 5\) for \(x \in \mathbb { R }\).
  3. Find the values of \(x\) for which \(\operatorname { gf } ( x ) + 1 = 0\).
CAIE P1 2019 June Q6
7 marks Standard +0.3
6
  1. Prove the identity \(\left( \frac { 1 } { \cos x } - \tan x \right) ^ { 2 } \equiv \frac { 1 - \sin x } { 1 + \sin x }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \cos 2 x } - \tan 2 x \right) ^ { 2 } = \frac { 1 } { 3 }\) for \(0 \leqslant x \leqslant \pi\).
CAIE P1 2019 June Q7
7 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-12_775_823_260_662} The diagram shows a three-dimensional shape in which the base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal squares. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. The point \(M\) is the mid-point of \(A F\). Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i }\) and \(\overrightarrow { O D } = 3 \mathbf { i } + 10 \mathbf { k }\).
  1. Express each of the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { G M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
  2. Use a scalar product to find angle \(G M A\) correct to the nearest degree.
CAIE P1 2019 June Q8
8 marks Standard +0.3
8
  1. The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
  2. Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme \(A\) is to increase the amount of waste recycled each month by 0.16 tonnes.
    Scheme \(B\) is to increase the amount of waste recycled each month by \(6 \%\) of the amount recycled in the previous month.
    The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme \(A\) Scheme \(B\) \(\_\_\_\_\)
CAIE P1 2019 June Q9
7 marks Moderate -0.3
9 The function f is defined by \(\mathrm { f } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant 2 \pi\).
  1. State the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 2 - 3 \cos x\) for \(0 \leqslant x \leqslant p\), where \(p\) is a constant.
  3. State the largest value of \(p\) for which g has an inverse.
  4. For this value of \(p\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2019 June Q10
9 marks Moderate -0.8
10 A curve for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5\) has a stationary point at \(( 3,6 )\).
  1. Find the equation of the curve.
  2. Find the \(x\)-coordinate of the other stationary point on the curve.
  3. Determine the nature of each of the stationary points. \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve \(y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }\) and a point \(P ( 2,1 )\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).
  4. Show that the \(x\)-coordinate of \(Q\) is \(\frac { 16 } { 9 }\).
  5. Find, showing all necessary working, the area of the shaded region.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P1 2019 June Q1
3 marks Moderate -0.5
1 Find the coefficient of \(x\) in the expansion of \(\left( \frac { 2 } { x } - 3 x \right) ^ { 5 }\).
CAIE P1 2019 June Q2
5 marks Moderate -0.8
2 Two points \(A\) and \(B\) have coordinates \(( 1,3 )\) and \(( 9 , - 1 )\) respectively. The perpendicular bisector of \(A B\) intersects the \(y\)-axis at the point \(C\). Find the coordinates of \(C\).
CAIE P1 2019 June Q3
5 marks Moderate -0.8
3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }\). The point \(P ( 2,9 )\) lies on the curve.
  1. A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\). [2]
  2. Find the equation of the curve.
CAIE P1 2019 June Q4
5 marks Moderate -0.3
4 Angle \(x\) is such that \(\sin x = a + b\) and \(\cos x = a - b\), where \(a\) and \(b\) are constants.
  1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(x\).
  2. In the case where \(\tan x = 2\), express \(a\) in terms of \(b\).