Questions P1 (1374 questions)

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CAIE P1 2015 November Q9
9 A curve passes through the point \(A ( 4,6 )\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1 + 2 x ^ { - \frac { 1 } { 2 } }\). A point \(P\) is moving along the curve in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 3 units per minute.
  1. Find the rate at which the \(y\)-coordinate of \(P\) is increasing when \(P\) is at \(A\).
  2. Find the equation of the curve.
  3. The tangent to the curve at \(A\) crosses the \(x\)-axis at \(B\) and the normal to the curve at \(A\) crosses the \(x\)-axis at \(C\). Find the area of triangle \(A B C\).
CAIE P1 2015 November Q10
10 The function f is defined by \(\mathrm { f } ( x ) = 2 x + ( x + 1 ) ^ { - 2 }\) for \(x > - 1\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence verify that the function f has a minimum value at \(x = 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-4_515_920_959_609} The points \(A \left( - \frac { 1 } { 2 } , 3 \right)\) and \(B \left( 1,2 \frac { 1 } { 4 } \right)\) lie on the curve \(y = 2 x + ( x + 1 ) ^ { - 2 }\), as shown in the diagram.
  2. Find the distance \(A B\).
  3. Find, showing all necessary working, the area of the shaded region. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2016 November Q1
1
  1. Express \(x ^ { 2 } + 6 x + 2\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
  2. Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } + 6 x + 2 > 9\).
CAIE P1 2016 November Q2
2 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { 2 x ^ { 3 } } \right) ^ { 8 }\).
CAIE P1 2016 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-2_515_750_669_699} In the diagram \(O C A\) and \(O D B\) are radii of a circle with centre \(O\) and radius \(2 r \mathrm {~cm}\). Angle \(A O B = \alpha\) radians. \(C D\) and \(A B\) are arcs of circles with centre \(O\) and radii \(r \mathrm {~cm}\) and \(2 r \mathrm {~cm}\) respectively. The perimeter of the shaded region \(A B D C\) is \(4.4 r \mathrm {~cm}\).
  1. Find the value of \(\alpha\).
  2. It is given that the area of the shaded region is \(30 \mathrm {~cm} ^ { 2 }\). Find the value of \(r\).
    \(4 C\) is the mid-point of the line joining \(A ( 14 , - 7 )\) to \(B ( - 6,3 )\). The line through \(C\) perpendicular to \(A B\) crosses the \(y\)-axis at \(D\).
CAIE P1 2016 November Q5
5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.
CAIE P1 2016 November Q6
6
  1. Show that \(\cos ^ { 4 } x \equiv 1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x\).
  2. Hence, or otherwise, solve the equation \(8 \sin ^ { 4 } x + \cos ^ { 4 } x = 2 \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2016 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-3_704_558_258_790} The diagram shows parts of the curves \(y = ( 2 x - 1 ) ^ { 2 }\) and \(y ^ { 2 } = 1 - 2 x\), intersecting at points \(A\) and \(B\).
  1. State the coordinates of \(A\).
  2. Find, showing all necessary working, the area of the shaded region.
CAIE P1 2016 November Q8
8 The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = \frac { 4 } { x } - 2 \quad \text { for } x > 0
& \mathrm {~g} ( x ) = \frac { 4 } { 5 x + 2 } \quad \text { for } x \geqslant 0 \end{aligned}$$
  1. Find and simplify an expression for \(\mathrm { fg } ( x )\) and state the range of fg.
  2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { g } ^ { - 1 }\).
CAIE P1 2016 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-4_724_1488_257_330} The diagram shows a cuboid \(O A B C D E F G\) with a horizontal base \(O A B C\) in which \(O A = 4 \mathrm {~cm}\) and \(A B = 15 \mathrm {~cm}\). The height \(O D\) of the cuboid is 2 cm . The point \(X\) on \(A B\) is such that \(A X = 5 \mathrm {~cm}\) and the point \(P\) on \(D G\) is such that \(D P = p \mathrm {~cm}\), where \(p\) is a constant. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.
  1. Find the possible values of \(p\) such that angle \(O P X = 90 ^ { \circ }\).
  2. For the case where \(p = 9\), find the unit vector in the direction of \(\overrightarrow { X P }\).
  3. A point \(Q\) lies on the face \(C B F G\) and is such that \(X Q\) is parallel to \(A G\). Find \(\overrightarrow { X Q }\).
CAIE P1 2016 November Q10
10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } }\). The point \(A\) is the only point on the curve at which the gradient is - 1 .
  1. Find the \(x\)-coordinate of \(A\).
  2. Given that the curve also passes through the point \(( 4,10 )\), find the \(y\)-coordinate of \(A\), giving your answer as a fraction.
CAIE P1 2016 November Q11
11 The point \(P ( 3,5 )\) lies on the curve \(y = \frac { 1 } { x - 1 } - \frac { 9 } { x - 5 }\).
  1. Find the \(x\)-coordinate of the point where the normal to the curve at \(P\) intersects the \(x\)-axis.
  2. Find the \(x\)-coordinate of each of the stationary points on the curve and determine the nature of each stationary point, justifying your answers. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE P1 2016 November Q1
1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { \sqrt { } ( 4 x + 1 ) }\). The point \(( 2,5 )\) lies on the curve. Find the equation of the curve.
CAIE P1 2016 November Q2
2
  1. Express the equation \(\sin 2 x + 3 \cos 2 x = 3 ( \sin 2 x - \cos 2 x )\) in the form \(\tan 2 x = k\), where \(k\) is a constant.
  2. Hence solve the equation for \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P1 2016 November Q3
3 A curve has equation \(y = 2 x ^ { 2 } - 6 x + 5\).
  1. Find the set of values of \(x\) for which \(y > 13\).
  2. Find the value of the constant \(k\) for which the line \(y = 2 x + k\) is a tangent to the curve.
CAIE P1 2016 November Q4
4 In the expansion of \(( 3 - 2 x ) \left( 1 + \frac { x } { 2 } \right) ^ { n }\), the coefficient of \(x\) is 7 . Find the value of the constant \(n\) and hence find the coefficient of \(x ^ { 2 }\).
CAIE P1 2016 November Q5
5 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, intersects the \(x\) - and \(y\)-axes at the points \(A\) and \(B\) respectively. The mid-point of \(A B\) lies on the line \(2 x + y = 10\) and the distance \(A B = 10\). Find the values of \(a\) and \(b\).
CAIE P1 2016 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{3a631b88-5ba5-49e7-a312-dfd8a6d8a24e-2_615_809_1535_667} The diagram shows a metal plate \(A B C D\) made from two parts. The part \(B C D\) is a semicircle. The part \(D A B\) is a segment of a circle with centre \(O\) and radius 10 cm . Angle \(B O D\) is 1.2 radians.
  1. Show that the radius of the semicircle is 5.646 cm , correct to 3 decimal places.
  2. Find the perimeter of the metal plate.
  3. Find the area of the metal plate.
CAIE P1 2016 November Q7
7 The equation of a curve is \(y = 2 + \frac { 3 } { 2 x - 1 }\).
  1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Explain why the curve has no stationary points. At the point \(P\) on the curve, \(x = 2\).
  3. Show that the normal to the curve at \(P\) passes through the origin.
  4. A point moves along the curve in such a way that its \(x\)-coordinate is decreasing at a constant rate of 0.06 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).
CAIE P1 2016 November Q8
8
  1. A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km . He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km .
    1. How far will he travel on May 15th?
    2. On what date will he finish the event?
  2. A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is \(31 \frac { 1 } { 2 }\). Find
    1. the first term of the progression,
    2. the sum to infinity of the progression.
CAIE P1 2016 November Q9
9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2
- 2
- 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 2
3
6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2
6
5 \end{array} \right)$$
  1. Use a scalar product to find angle \(A O B\).
  2. Find the vector which is in the same direction as \(\overrightarrow { A C }\) and of magnitude 15 units.
  3. Find the value of the constant \(p\) for which \(p \overrightarrow { O A } + \overrightarrow { O C }\) is perpendicular to \(\overrightarrow { O B }\).
CAIE P1 2016 November Q10
10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
  4. State the largest value of \(k\) for which g has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2016 November Q1
1 Find the set of values of \(k\) for which the curve \(y = k x ^ { 2 } - 3 x\) and the line \(y = x - k\) do not meet.
CAIE P1 2016 November Q2
2 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 3 x ) ^ { 6 } + ( 1 + a x ) ^ { 5 }\) is 100 . Find the value of the constant \(a\).
CAIE P1 2016 November Q3
3 Showing all necessary working, solve the equation \(6 \sin ^ { 2 } x - 5 \cos ^ { 2 } x = 2 \sin ^ { 2 } x + \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).