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CAIE P1 2005 June Q6
6 marks Standard +0.3
6 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5. An arithmetic progression has 21 terms and common difference 1.5. Given that the sum of all the terms in the geometric progression is equal to the sum of all the terms in the arithmetic progression, find the first term and the last term of the arithmetic progression.
CAIE P1 2005 June Q7
7 marks Standard +0.3
7 A function f is defined by f : \(x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 - 2 \sin x\), for \(0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }\), where \(A\) is a constant.
  3. State the largest value of \(A\) for which g has an inverse.
  4. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2005 June Q8
8 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-3_438_805_849_669} In the diagram, \(A B C\) is a semicircle, centre \(O\) and radius 9 cm . The line \(B D\) is perpendicular to the diameter \(A C\) and angle \(A O B = 2.4\) radians.
  1. Show that \(B D = 6.08 \mathrm {~cm}\), correct to 3 significant figures.
  2. Find the perimeter of the shaded region.
  3. Find the area of the shaded region.
CAIE P1 2005 June Q9
10 marks Moderate -0.3
9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).
  1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
  2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
CAIE P1 2005 June Q10
10 marks Standard +0.3
10 The equation of a curve is \(y = x ^ { 2 } - 3 x + 4\).
  1. Show that the whole of the curve lies above the \(x\)-axis.
  2. Find the set of values of \(x\) for which \(x ^ { 2 } - 3 x + 4\) is a decreasing function of \(x\). The equation of a line is \(y + 2 x = k\), where \(k\) is a constant.
  3. In the case where \(k = 6\), find the coordinates of the points of intersection of the line and the curve.
  4. Find the value of \(k\) for which the line is a tangent to the curve.
CAIE P1 2005 June Q11
11 marks Standard +0.3
11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$
  1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
  2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
  3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\).
CAIE P1 2006 June Q1
3 marks Easy -1.8
1 A curve has equation \(y = \frac { k } { x }\). Given that the gradient of the curve is - 3 when \(x = 2\), find the value of the constant \(k\).
CAIE P1 2006 June Q2
4 marks Moderate -0.8
2 Solve the equation $$\sin 2 x + 3 \cos 2 x = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2006 June Q3
5 marks Easy -1.2
3 Each year a company gives a grant to a charity. The amount given each year increases by \(5 \%\) of its value in the preceding year. The grant in 2001 was \(\\) 5000$. Find
  1. the grant given in 2011,
  2. the total amount of money given to the charity during the years 2001 to 2011 inclusive.
CAIE P1 2006 June Q4
5 marks Moderate -0.8
4 The first three terms in the expansion of \(( 2 + a x ) ^ { n }\), in ascending powers of \(x\), are \(32 - 40 x + b x ^ { 2 }\). Find the values of the constants \(n , a\) and \(b\).
CAIE P1 2006 June Q5
6 marks Standard +0.3
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 marks Standard +0.8
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
8 marks Standard +0.3
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 marks Standard +0.3
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 marks Standard +0.3
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 marks Standard +0.3
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 marks Standard +0.3
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
4 marks Standard +0.3
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
4 marks Moderate -0.5
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
4 marks Moderate -0.3
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 marks Standard +0.3
4 Find the real roots of the equation \(\frac { 18 } { x ^ { 4 } } + \frac { 1 } { x ^ { 2 } } = 4\).
CAIE P1 2007 June Q5
5 marks Standard +0.3
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
7 marks Moderate -0.5
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 marks Moderate -0.3
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 marks Moderate -0.8
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 \(y = \mathrm { f } ( x )\).