Questions — CAIE (7646 questions)

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CAIE P1 2022 June Q10
10 marks Standard +0.8
10 The function f is defined by \(\mathrm { f } ( x ) = ( 4 x + 2 ) ^ { - 2 }\) for \(x > - \frac { 1 } { 2 }\).
  1. Find \(\int _ { 1 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x\).
    A point is moving along the curve \(y = \mathrm { f } ( x )\) in such a way that, as it passes through the point \(A\), its \(y\)-coordinate is decreasing at the rate of \(k\) units per second and its \(x\)-coordinate is increasing at the rate of \(k\) units per second.
  2. Find the coordinates of \(A\).
CAIE P1 2022 June Q11
10 marks Standard +0.8
11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).
  1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
    The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
  2. Find the coordinates of \(Q\) in terms of \(m\).
    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 2023 June Q1
3 marks Moderate -0.3
1 Solve the equation \(4 \sin \theta + \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2023 June Q2
6 marks Moderate -0.8
2
  1. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 2 + 3 x ) ^ { 4 }\).
  2. Find the first three terms in the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 5 }\).
  3. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 2 + 3 x ) ^ { 4 } ( 1 - 2 x ) ^ { 5 }\).
CAIE P1 2023 June Q3
4 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-04_1150_1164_269_484} The diagram shows graphs with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
Describe fully a sequence of two transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to \(y = \mathrm { g } ( x )\).
CAIE P1 2023 June Q4
4 marks Moderate -0.3
4 The diagram shows a sector \(A B C\) of a circle with centre \(A\) and radius 8 cm . The area of the sector is \(\frac { 16 } { 3 } \pi \mathrm {~cm} ^ { 2 }\). The point \(D\) lies on the \(\operatorname { arc } B C\). Find the perimeter of the segment \(B C D\).
CAIE P1 2023 June Q5
5 marks Standard +0.3
5 The line with equation \(y = k x - k\), where \(k\) is a positive constant, is a tangent to the curve with equation \(y = - \frac { 1 } { 2 x }\). Find, in either order, the value of \(k\) and the coordinates of the point where the tangent meets the curve. [5]
CAIE P1 2023 June Q6
5 marks Standard +0.3
6 The first three terms of an arithmetic progression are \(\frac { p ^ { 2 } } { 6 } , 2 p - 6\) and \(p\).
  1. Given that the common difference of the progression is not zero, find the value of \(p\).
  2. Using this value, find the sum to infinity of the geometric progression with first two terms \(\frac { p ^ { 2 } } { 6 }\) and \(2 p - 6\).
CAIE P1 2023 June Q7
5 marks Standard +0.3
7 A curve has equation \(y = 2 + 3 \sin \frac { 1 } { 2 } x\) for \(0 \leqslant x \leqslant 4 \pi\).
  1. State greatest and least values of \(y\).
  2. Sketch the curve. \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-09_1127_1219_904_495}
  3. State the number of solutions of the equation $$2 + 3 \sin \frac { 1 } { 2 } x = 5 - 2 x$$ for \(0 \leqslant x \leqslant 4 \pi\).
CAIE P1 2023 June Q8
8 marks Standard +0.3
8 The functions f and g are defined as follows, where \(a\) and \(b\) are constants. $$\begin{aligned} & \mathrm { f } ( x ) = 1 + \frac { 2 a } { x - a } \text { for } x > a \\ & \mathrm {~g} ( x ) = b x - 2 \text { for } x \in \mathbb { R } \end{aligned}$$
  1. Given that \(\mathrm { f } ( 7 ) = \frac { 5 } { 2 }\) and \(\mathrm { gf } ( 5 ) = 4\), find the values of \(a\) and \(b\).
    For the rest of this question, you should use the value of \(a\) which you found in (a).
  2. Find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2023 June Q9
6 marks Standard +0.3
9 Water is poured into a tank at a constant rate of \(500 \mathrm {~cm} ^ { 3 }\) per second. The depth of water in the tank, \(t\) seconds after filling starts, is \(h \mathrm {~cm}\). When the depth of water in the tank is \(h \mathrm {~cm}\), the volume, \(V \mathrm {~cm} ^ { 3 }\), of water in the tank is given by the formula \(V = \frac { 4 } { 3 } ( 25 + h ) ^ { 3 } - \frac { 62500 } { 3 }\).
  1. Find the rate at which \(h\) is increasing at the instant when \(h = 10 \mathrm {~cm}\).
  2. At another instant, the rate at which \(h\) is increasing is 0.075 cm per second. Find the value of \(V\) at this instant.
CAIE P1 2023 June Q10
11 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).
  1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.
CAIE P1 2023 June Q11
9 marks Moderate -0.3
11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).
  1. Find the value of \(a\).
  2. Determine the nature of this stationary point.
  3. Find the equation of the curve.
  4. Find the coordinates of any other stationary points on the curve.
CAIE P1 2023 June Q12
9 marks Standard +0.3
12 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).
  1. Write down the equation of circle \(P\).
  2. Find the equation of the tangent to the circle \(P\) at \(A\).
  3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
  4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
    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 2023 June Q1
4 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-02_778_1061_269_532} The diagram shows the graph of \(y = \mathrm { f } ( x )\), which consists of the two straight lines \(A B\) and \(B C\). The lines \(A ^ { \prime } B ^ { \prime }\) and \(B ^ { \prime } C ^ { \prime }\) form the graph of \(y = \mathrm { g } ( x )\), which is the result of applying a sequence of two transformations, in either order, to \(y = \mathrm { f } ( x )\). State fully the two transformations.
CAIE P1 2023 June Q2
4 marks Standard +0.3
2 The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c\), where \(c\) is a constant. It is given that \(\mathrm { f } ( x ) > 2\) for all values of \(x\). Find the set of possible values of \(c\).
CAIE P1 2023 June Q3
6 marks Standard +0.3
3
  1. Give the complete expansion of \(\left( x + \frac { 2 } { x } \right) ^ { 5 }\).
  2. In the expansion of \(\left( a + b x ^ { 2 } \right) \left( x + \frac { 2 } { x } \right) ^ { 5 }\), the coefficient of \(x\) is zero and the coefficient of \(\frac { 1 } { x }\) is 80 . Find the values of the constants \(a\) and \(b\).
CAIE P1 2023 June Q4
7 marks Standard +0.3
4
  1. Show that the equation $$3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0$$ may be expressed in the form \(a \cos ^ { 4 } x + b \cos ^ { 2 } x + c = 0\), where \(a , b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(3 \tan ^ { 2 } x - 3 \sin ^ { 2 } x - 4 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2023 June Q5
7 marks Standard +0.3
5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).
  1. Find the coordinates of the two points of intersection.
  2. Find an equation of the circle with diameter \(A B\).
CAIE P1 2023 June Q6
7 marks Moderate -0.5
6 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-07_389_552_267_799} The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). Angle \(A O B = \theta\) radians. It is given that the length of the \(\operatorname { arc } A B\) is 9.6 cm and that the area of the sector \(O A B\) is \(76.8 \mathrm {~cm} ^ { 2 }\).
  1. Find the area of the shaded region.
  2. Find the perimeter of the shaded region.
CAIE P1 2023 June Q7
8 marks Moderate -0.3
7 The function f is defined by \(\mathrm { f } ( x ) = 2 - \frac { 5 } { x + 2 }\) for \(x > - 2\).
  1. State the range of f.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
    The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x > 0\).
  3. Obtain an expression for \(\operatorname { fg } ( x )\) giving your answer in the form \(\frac { a x + b } { c x + d }\), where \(a , b , c\) and \(d\) are integers.
CAIE P1 2023 June Q8
10 marks Standard +0.8
8 A progression has first term \(a\) and second term \(\frac { a ^ { 2 } } { a + 2 }\), where \(a\) is a positive constant.
  1. For the case where the progression is geometric and the sum to infinity is 264 , find the value of \(a\).
  2. For the case where the progression is arithmetic and \(a = 6\), determine the least value of \(n\) required for the sum of the first \(n\) terms to be less than - 480 .
CAIE P1 2023 June Q9
10 marks Standard +0.3
9 A curve which passes through \(( 0,3 )\) has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 2 } { ( x - 1 ) ^ { 3 } }\).
  1. Find the equation of the curve.
    The tangent to the curve at \(( 0,3 )\) intersects the curve again at one other point, \(P\).
  2. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(( 2 x + 1 ) ( x - 1 ) ^ { 2 } - 1 = 0\).
  3. Verify that \(x = \frac { 3 } { 2 }\) satisfies this equation and hence find the \(y\)-coordinate of \(P\).
CAIE P1 2023 June Q10
12 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).
  1. Find the coordinates of the maximum point of the curve.
  2. Verify that the line \(A B\) is the normal to the curve at \(A\).
  3. Find 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 2024 June Q1
4 marks Moderate -0.8
1 Find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 2 - 5 x ) ( 1 + 3 x ) ^ { 10 }$$