Questions P1 (1374 questions)

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CAIE P1 2006 June Q5
5 The curve \(y ^ { 2 } = 12 x\) intersects the line \(3 y = 4 x + 6\) at two points. Find the distance between the two points.
CAIE P1 2006 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-2_389_995_1432_575} In the diagram, \(A B C\) is a triangle in which \(A B = 4 \mathrm {~cm} , B C = 6 \mathrm {~cm}\) and angle \(A B C = 150 ^ { \circ }\). The line \(C X\) is perpendicular to the line \(A B X\).
  1. Find the exact length of \(B X\) and show that angle \(C A B = \tan ^ { - 1 } \left( \frac { 3 } { 4 + 3 \sqrt { } 3 } \right)\).
  2. Show that the exact length of \(A C\) is \(\sqrt { } ( 52 + 24 \sqrt { } 3 ) \mathrm { cm }\).
CAIE P1 2006 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_545_759_269_694} The diagram shows a circle with centre \(O\) and radius 8 cm . Points \(A\) and \(B\) lie on the circle. The tangents at \(A\) and \(B\) meet at the point \(T\), and \(A T = B T = 15 \mathrm {~cm}\).
  1. Show that angle \(A O B\) is 2.16 radians, correct to 3 significant figures.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2006 June Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_517_1117_1362_514} The diagram shows the roof of a house. The base of the roof, \(O A B C\), is rectangular and horizontal with \(O A = C B = 14 \mathrm {~m}\) and \(O C = A B = 8 \mathrm {~m}\). The top of the roof \(D E\) is 5 m above the base and \(D E = 6 \mathrm {~m}\). The sloping edges \(O D , C D , A E\) and \(B E\) are all equal in length. 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.
  1. Express the vector \(\overrightarrow { O D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\), and find its magnitude.
  2. Use a scalar product to find angle \(D O B\).
CAIE P1 2006 June Q9
9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { } ( 6 - 2 x ) }\), and \(P ( 1,8 )\) is a point on the curve.
  1. The normal to the curve at the point \(P\) meets the coordinate axes at \(Q\) and at \(R\). Find the coordinates of the mid-point of \(Q R\).
  2. Find the equation of the curve.
CAIE P1 2006 June Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-4_515_885_662_630} The diagram shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant. The curve has a minimum point on the \(x\)-axis.
  1. Find the value of \(k\).
  2. Find the coordinates of the maximum point of the curve.
  3. State the set of values of \(x\) for which \(x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\) is a decreasing function of \(x\).
  4. Find the area of the shaded region.
CAIE P1 2006 June Q11
11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto k - x & \text { for } x \in \mathbb { R } , \text { where } k \text { is a constant, }
\mathrm { g } : x \mapsto \frac { 9 } { x + 2 } & \text { for } x \in \mathbb { R } , x \neq - 2 . \end{array}$$
  1. Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has two equal roots and solve the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) in these cases.
  2. Solve the equation \(\operatorname { fg } ( x ) = 5\) when \(k = 6\).
  3. Express \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
CAIE P1 2007 June Q1
1 Find the value of the constant \(c\) for which the line \(y = 2 x + c\) is a tangent to the curve \(y ^ { 2 } = 4 x\).
CAIE P1 2007 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_633_787_402_680} The diagram shows the curve \(y = 3 x ^ { \frac { 1 } { 4 } }\). The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\). Find the volume of the solid obtained when this shaded region is rotated completely about the \(x\)-axis, giving your answer in terms of \(\pi\).
CAIE P1 2007 June Q3
3 Prove the identity \(\frac { 1 - \tan ^ { 2 } x } { 1 + \tan ^ { 2 } x } \equiv 1 - 2 \sin ^ { 2 } x\).
CAIE P1 2007 June Q4
4 Find the real roots of the equation \(\frac { 18 } { x ^ { 4 } } + \frac { 1 } { x ^ { 2 } } = 4\).
CAIE P1 2007 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-2_586_682_1726_733} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius 12 cm . The lines \(A X\) and \(B X\) are tangents to the circle at \(A\) and \(B\) respectively. Angle \(A O B = \frac { 1 } { 3 } \pi\) radians.
  1. Find the exact length of \(A X\), giving your answer in terms of \(\sqrt { } 3\).
  2. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
CAIE P1 2007 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-3_593_878_269_635} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 2,14 ) , B\) is \(( - 2,8 )\) and \(C\) lies on the \(x\)-axis. Find
  1. the equation of \(B C\),
  2. the coordinates of \(C\) and \(D\).
CAIE P1 2007 June Q7
7 The second term of a geometric progression is 3 and the sum to infinity is 12 .
  1. Find the first term of the progression. An arithmetic progression has the same first and second terms as the geometric progression.
  2. Find the sum of the first 20 terms of the arithmetic progression.
CAIE P1 2007 June Q8
8 The function f is defined by \(\mathrm { f } ( x ) = a + b \cos 2 x\), for \(0 \leqslant x \leqslant \pi\). It is given that \(\mathrm { f } ( 0 ) = - 1\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 7\).
  1. Find the values of \(a\) and \(b\).
  2. Find the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) intersects the \(x\)-axis.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
CAIE P1 2007 June Q9
9 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 4
1
- 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3
2
- 4 \end{array} \right) .$$
  1. Given that \(C\) is the point such that \(\overrightarrow { A C } = 2 \overrightarrow { A B }\), find the unit vector in the direction of \(\overrightarrow { O C }\). The position vector of the point \(D\) is given by \(\overrightarrow { O D } = \left( \begin{array} { l } 1
    4
    k \end{array} \right)\), where \(k\) is a constant, and it is given that \(\overrightarrow { O D } = m \overrightarrow { O A } + n \overrightarrow { O B }\), where \(m\) and \(n\) are constants.
  2. Find the values of \(m , n\) and \(k\).
CAIE P1 2007 June Q10
10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).
  1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
  3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
  4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
CAIE P1 2007 June Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-4_862_892_932_628} The diagram shows the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } : x \mapsto \frac { 6 } { 2 x + 3 }\) for \(x \geqslant 0\).
  1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and explain how your answer shows that f is a decreasing function.
  2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).
  3. Copy the diagram and, on your copy, sketch the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs. The function g is defined by \(\mathrm { g } : x \mapsto \frac { 1 } { 2 } x\) for \(x \geqslant 0\).
  4. Solve the equation \(\operatorname { fg } ( x ) = \frac { 3 } { 2 }\). \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. University of 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 2008 June Q1
1 In the triangle \(A B C , A B = 12 \mathrm {~cm}\), angle \(B A C = 60 ^ { \circ }\) and angle \(A C B = 45 ^ { \circ }\). Find the exact length of \(B C\).
CAIE P1 2008 June Q2
2
  1. Show that the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\) can be written in the form \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\).
  2. Hence solve the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2008 June Q3
3
  1. Find the first 3 terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + x ^ { 2 } \right) ^ { 5 }\).
  2. Hence find the coefficient of \(x ^ { 4 }\) in the expansion of \(\left( 1 + x ^ { 2 } \right) ^ { 2 } \left( 2 + x ^ { 2 } \right) ^ { 5 }\).
CAIE P1 2008 June Q4
4 The equation of a curve \(C\) is \(y = 2 x ^ { 2 } - 8 x + 9\) and the equation of a line \(L\) is \(x + y = 3\).
  1. Find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\).
  2. Show that one of these points is also the stationary point of \(C\).
CAIE P1 2008 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-2_543_883_1274_630} The diagram shows a circle with centre \(O\) and radius 5 cm . The point \(P\) lies on the circle, \(P T\) is a tangent to the circle and \(P T = 12 \mathrm {~cm}\). The line \(O T\) cuts the circle at the point \(Q\).
  1. Find the perimeter of the shaded region.
  2. Find the area of the shaded region.
CAIE P1 2008 June Q6
6 The function f is such that \(\mathrm { f } ( x ) = ( 3 x + 2 ) ^ { 3 } - 5\) for \(x \geqslant 0\).
  1. Obtain an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why f is an increasing function.
  2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
CAIE P1 2008 June Q7
7 The first term of a geometric progression is 81 and the fourth term is 24 . Find
  1. the common ratio of the progression,
  2. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.
  3. Find the sum of the first ten terms of the arithmetic progression.