Questions — CAIE P1 (1202 questions)

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CAIE P1 2020 June Q1
1 Find the set of values of \(m\) for which the line with equation \(y = m x + 1\) and the curve with equation \(y = 3 x ^ { 2 } + 2 x + 4\) intersect at two distinct points.
CAIE P1 2020 June Q2
2 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 3 x ^ { - \frac { 1 } { 2 } }\). It is given that the point (4,7) lies on the curve. Find the equation of the curve.
CAIE P1 2020 June Q3
3 In each of parts (a), (b) and (c), the graph shown with solid lines has equation \(y = \mathrm { f } ( x )\). The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

  1. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_412_645_367_788} State, in terms of f , the equation of the graph shown with broken lines.

  2. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_650_423_1046_900} State, in terms of f , the equation of the graph shown with broken lines.

  3. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_550_631_1975_804} State, in terms of f , the equation of the graph shown with broken lines.
CAIE P1 2020 June Q4
4
  1. Expand \(( 1 + a ) ^ { 5 }\) in ascending powers of \(a\) up to and including the term in \(a ^ { 3 }\).
  2. Hence expand \(\left[ 1 + \left( x + x ^ { 2 } \right) \right] ^ { 5 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying your answer.
CAIE P1 2020 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre \(O\) and radius 5 cm . The thickness of the cord and the size of the pin \(P\) can be neglected. The pin is situated 13 cm vertically below \(O\). Points \(A\) and \(B\) are on the circumference of the circle such that \(A P\) and \(B P\) are tangents to the circle. The cord passes over the major arc \(A B\) of the circle and under the pin such that the cord is taut. Calculate the length of the cord.
CAIE P1 2020 June Q6
6 A point \(P\) is moving along a curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per minute. The equation of the curve is \(y = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } }\).
  1. Find the rate at which the \(y\)-coordinate is increasing when \(x = 1\).
  2. Find the value of \(x\) when the \(y\)-coordinate is increasing at \(\frac { 5 } { 8 }\) units per minute.
CAIE P1 2020 June Q7
7
  1. Show that \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } \equiv \frac { 2 } { \sin \theta \cos \theta }\).
  2. Hence solve the equation \(\frac { \tan \theta } { 1 + \cos \theta } + \frac { \tan \theta } { 1 - \cos \theta } = \frac { 6 } { \tan \theta }\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P1 2020 June Q8
8 The first term of a progression is \(\sin ^ { 2 } \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\). The second term of the progression is \(\sin ^ { 2 } \theta \cos ^ { 2 } \theta\).
  1. Given that the progression is geometric, find the sum to infinity.
    It is now given instead that the progression is arithmetic.
    1. Find the common difference of the progression in terms of \(\sin \theta\).
    2. Find the sum of the first 16 terms when \(\theta = \frac { 1 } { 3 } \pi\).
CAIE P1 2020 June Q9
9 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 3 \text { for } x > c , \text { where } c \text { is a constant, }
& \mathrm { g } ( x ) = \frac { 1 } { x + 1 } \quad \text { for } x > - 1 \end{aligned}$$
  1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
    It is given that f is a one-one function.
  2. State the smallest possible value of \(c\).
    It is now given that \(c = 5\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  4. Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf .
CAIE P1 2020 June Q10
10
  1. The coordinates of two points \(A\) and \(B\) are \(( - 7,3 )\) and \(( 5,11 )\) respectively.
    Show that the equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = 11\).
  2. A circle passes through \(A\) and \(B\) and its centre lies on the line \(12 x - 5 y = 70\). Find an equation of the circle.
CAIE P1 2020 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-18_387_920_260_609} The diagram shows part of the curve with equation \(y = x ^ { 3 } - 2 b x ^ { 2 } + b ^ { 2 } x\) and the line \(O A\), where \(A\) is the maximum point on the curve. The \(x\)-coordinate of \(A\) is \(a\) and the curve has a minimum point at ( \(b , 0\) ), where \(a\) and \(b\) are positive constants.
  1. Show that \(b = 3 a\).
  2. Show that the area of the shaded region between the line and the curve is \(k a ^ { 4 }\), where \(k\) is a fraction to be found.
    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 2021 June Q1
1 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { x ^ { 4 } } + 32 x ^ { 3 }\). It is given that the curve passes through the point \(\left( \frac { 1 } { 2 } , 4 \right)\). Find the equation of the curve.
CAIE P1 2021 June Q2
2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms is 1410 . Find the 60th term of the progression.
CAIE P1 2021 June Q3
3
  1. Find the first three terms in the expansion of \(( 3 - 2 x ) ^ { 5 }\) in ascending powers of \(x\).
  2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + x ) ^ { 2 } ( 3 - 2 x ) ^ { 5 }\).
CAIE P1 2021 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-05_677_1591_260_278} The diagram shows part of the graph of \(y = a \tan ( x - b ) + c\).
Given that \(0 < b < \pi\), state the values of the constants \(a , b\) and \(c\).
CAIE P1 2021 June Q5
5 The fifth, sixth and seventh terms of a geometric progression are \(8 k , - 12\) and \(2 k\) respectively. Given that \(k\) is negative, find the sum to infinity of the progression.
CAIE P1 2021 June Q6
6 The equation of a curve is \(y = ( 2 k - 3 ) x ^ { 2 } - k x - ( k - 2 )\), where \(k\) is a constant. The line \(y = 3 x - 4\) is a tangent to the curve. Find the value of \(k\).
CAIE P1 2021 June Q7
7
  1. Prove the identity \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \equiv 1 - \tan ^ { 2 } \theta\).
  2. Hence solve the equation \(\frac { 1 - 2 \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } = 2 \tan ^ { 4 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2021 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.
  1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
  2. Find the exact perimeter of the plate.
  3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).
CAIE P1 2021 June Q9
9 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } ( x ) = ( x - 2 ) ^ { 2 } - 4 \text { for } x \geqslant 2 ,
& \mathrm {~g} ( x ) = a x + 2 \text { for } x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a constant.
  1. State the range of f.
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  3. Given that \(a = - \frac { 5 } { 3 }\), solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\).
  4. Given instead that \(\operatorname { ggf } ^ { - 1 } ( 12 ) = 62\), find the possible values of \(a\).
CAIE P1 2021 June Q10
10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).
  1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
  2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).
CAIE P1 2021 June Q11
11 The equation of a curve is \(y = 2 \sqrt { 3 x + 4 } - x\).
  1. Find the equation of the normal to the curve at the point (4,4), giving your answer in the form \(y = m x + c\).
  2. Find the coordinates of the stationary point.
  3. Determine the nature of the stationary point.
  4. Find the exact area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 4\).
    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 2021 June Q1
1
  1. Express \(16 x ^ { 2 } - 24 x + 10\) in the form \(( 4 x + a ) ^ { 2 } + b\).
  2. It is given that the equation \(16 x ^ { 2 } - 24 x + 10 = k\), where \(k\) is a constant, has exactly one root. Find the value of this root.
CAIE P1 2021 June Q2
2
  1. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 2 \mathrm { f } ( x - 1 )\).
    Describe fully the two single transformations which have been combined to give the resulting transformation.
  2. The curve \(y = \sin 2 x - 5 x\) is reflected in the \(y\)-axis and then stretched by scale factor \(\frac { 1 } { 3 }\) in the \(x\)-direction. Write down the equation of the transformed curve.
CAIE P1 2021 June Q3
3 The equation of a curve is \(y = ( x - 3 ) \sqrt { x + 1 } + 3\). The following points lie on the curve. Non-exact values are rounded to 4 decimal places. $$A ( 2 , k ) \quad B ( 2.9,2.8025 ) \quad C ( 2.99,2.9800 ) \quad D ( 2.999,2.9980 ) \quad E ( 3,3 )$$
  1. Find \(k\), giving your answer correct to 4 decimal places.
  2. Find the gradient of \(A E\), giving your answer correct to 4 decimal places.
    The gradients of \(B E , C E\) and \(D E\), rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.
  3. State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point \(E\).