Questions — CAIE P1 (1202 questions)

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CAIE P1 2010 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607} The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.
  1. Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
  2. Find the area of the shaded region between the two curves.
  3. The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).
CAIE P1 2011 November Q1
1 Find the term independent of \(x\) in the expansion of \(\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 6 }\).
CAIE P1 2011 November Q2
2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.
CAIE P1 2011 November Q3
3
  1. Sketch, on a single diagram, the graphs of \(y = \cos 2 \theta\) and \(y = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  2. Write down the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(0 \leqslant \theta \leqslant 2 \pi\).
  3. Deduce the number of roots of the equation \(2 \cos 2 \theta - 1 = 0\) in the interval \(10 \pi \leqslant \theta \leqslant 20 \pi\).
CAIE P1 2011 November Q4
4 A function f is defined for \(x \in \mathbb { R }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 6\). The range of the function is given by \(\mathrm { f } ( x ) \geqslant - 4\).
  1. State the value of \(x\) for which \(\mathrm { f } ( x )\) has a stationary value.
  2. Find an expression for \(\mathrm { f } ( x )\) in terms of \(x\).
CAIE P1 2011 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-2_512_903_1302_621} The diagram represents a metal plate \(O A B C\), consisting of a sector \(O A B\) of a circle with centre \(O\) and radius \(r\), together with a triangle \(O C B\) which is right-angled at \(C\). Angle \(A O B = \theta\) radians and \(O C\) is perpendicular to \(O A\).
  1. Find an expression in terms of \(r\) and \(\theta\) for the perimeter of the plate.
  2. For the case where \(r = 10\) and \(\theta = \frac { 1 } { 5 } \pi\), find the area of the plate.
CAIE P1 2011 November Q6
6
  1. The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200 . Find the seventh term.
  2. A geometric progression has first term 1 and common ratio \(r\). A second geometric progression has first term 4 and common ratio \(\frac { 1 } { 4 } r\). The two progressions have the same sum to infinity, \(S\). Find the values of \(r\) and \(S\).
CAIE P1 2011 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .
  1. Find an expression for \(y\) in terms of \(x\).
  2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
  3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.
CAIE P1 2011 November Q8
8 Relative to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 7 \mathbf { j } - p \mathbf { k }\) and the point \(B\) has position vector \(8 \mathbf { i } - \mathbf { j } - p \mathbf { k }\), where \(p\) is a constant.
  1. Find \(\overrightarrow { O A } \cdot \overrightarrow { O B }\).
  2. Hence show that there are no real values of \(p\) for which \(O A\) and \(O B\) are perpendicular to each other.
  3. Find the values of \(p\) for which angle \(A O B = 60 ^ { \circ }\).
CAIE P1 2011 November Q9
9 A line has equation \(y = k x + 6\) and a curve has equation \(y = x ^ { 2 } + 3 x + 2 k\), where \(k\) is a constant.
  1. For the case where \(k = 2\), the line and the curve intersect at points \(A\) and \(B\). Find the distance \(A B\) and the coordinates of the mid-point of \(A B\).
  2. Find the two values of \(k\) for which the line is a tangent to the curve.
CAIE P1 2011 November Q10
10
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-4_799_1390_255_376} The diagram shows the curve \(y = \sqrt { } ( 1 + 2 x )\) meeting the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The \(y\)-coordinate of the point \(C\) on the curve is 3 .
  1. Find the coordinates of \(B\) and \(C\).
  2. Find the equation of the normal to the curve at \(C\).
  3. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
CAIE P1 2011 November Q11
11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 10 & \text { for } 0 \leqslant x \leqslant 2
\mathrm {~g} : x \mapsto x & \text { for } 0 \leqslant x \leqslant 10 \end{array}$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
  2. State the range of f .
  3. State the domain of \(\mathrm { f } ^ { - 1 }\).
  4. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x ) , y = \mathrm { g } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
  5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2011 November Q1
1
  1. Find the first 3 terms in the expansion of \(( 2 - y ) ^ { 5 }\) in ascending powers of \(y\).
  2. Use the result in part (i) to find the coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 2 - \left( 2 x - x ^ { 2 } \right) \right) ^ { 5 }\).
CAIE P1 2011 November Q2
2 The functions f and g are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + a
& \mathrm {~g} : x \mapsto b - 2 x \end{aligned}$$ where \(a\) and \(b\) are constants. Given that \(\mathrm { ff } ( 2 ) = 10\) and \(\mathrm { g } ^ { - 1 } ( 2 ) = 3\), find
  1. the values of \(a\) and \(b\),
  2. an expression for \(\mathrm { fg } ( x )\).
CAIE P1 2011 November Q3
3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 5 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + 7 \mathbf { j } + p \mathbf { k }$$ where \(p\) is a constant.
  1. Find the value of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
  2. In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow { A B }\).
CAIE P1 2011 November Q4
4 The equation of a curve is \(y ^ { 2 } + 2 x = 13\) and the equation of a line is \(2 y + x = k\), where \(k\) is a constant.
  1. In the case where \(k = 8\), find the coordinates of the points of intersection of the line and the curve.
  2. Find the value of \(k\) for which the line is a tangent to the curve.
CAIE P1 2011 November Q5
5
  1. Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  2. Verify that \(x = 30 ^ { \circ }\) is a root of the equation \(\sin x = \cos 2 x\), and state the other root of this equation for which \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
  3. Hence state the set of values of \(x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), for which \(\sin x < \cos 2 x\).
CAIE P1 2011 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-3_801_1273_255_434} The diagram shows a circle \(C _ { 1 }\) touching a circle \(C _ { 2 }\) at a point \(X\). Circle \(C _ { 1 }\) has centre \(A\) and radius 6 cm , and circle \(C _ { 2 }\) has centre \(B\) and radius 10 cm . Points \(D\) and \(E\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and \(D E\) is parallel to \(A B\). Angle \(D A X = \frac { 1 } { 3 } \pi\) radians and angle \(E B X = \theta\) radians.
  1. By considering the perpendicular distances of \(D\) and \(E\) from \(A B\), show that the exact value of \(\theta\) is \(\sin ^ { - 1 } \left( \frac { 3 \sqrt { } 3 } { 10 } \right)\).
  2. Find the perimeter of the shaded region, correct to 4 significant figures.
CAIE P1 2011 November Q7
7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }\). The line \(3 y + x = 17\) is the normal to the curve at the point \(P\) on the curve. Given that the \(x\)-coordinate of \(P\) is positive, find
  1. the coordinates of \(P\),
  2. the equation of the curve.
CAIE P1 2011 November Q8
8 The equation of a curve is \(y = \sqrt { } \left( 8 x - x ^ { 2 } \right)\). Find
  1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and the coordinates of the stationary point on the curve,
  2. the volume obtained when the region bounded by the curve and the \(x\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
CAIE P1 2011 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{3fd0b68f-41b1-4eee-8018-bcaf3cf22950-4_767_993_255_575} The diagram shows a quadrilateral \(A B C D\) in which the point \(A\) is ( \(- 1 , - 1\) ), the point \(B\) is ( 3,6 ) and the point \(C\) is (9,4). The diagonals \(A C\) and \(B D\) intersect at \(M\). Angle \(B M A = 90 ^ { \circ }\) and \(B M = M D\). Calculate
  1. the coordinates of \(M\) and \(D\),
  2. the ratio \(A M : M C\).
CAIE P1 2011 November Q10
10
  1. An arithmetic progression contains 25 terms and the first term is - 15 . The sum of all the terms in the progression is 525. Calculate
    1. the common difference of the progression,
    2. the last term in the progression,
    3. the sum of all the positive terms in the progression.
  2. A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be \(
    ) 4000\( in 2012 and will increase by \)5 \%$ each year. Calculate
    1. the value of the grant in 2022,
    2. the total amount the college will receive in the years 2012 to 2022 inclusive.
CAIE P1 2011 November Q1
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( k + \frac { 1 } { 3 } x \right) ^ { 5 }\) is 30 . Find the value of the constant \(k\).
CAIE P1 2011 November Q2
2 The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is
  1. an arithmetic progression,
  2. a geometric progression.
CAIE P1 2011 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{96cc217a-ffb3-4764-946e-e32271784ad7-2_680_977_689_584} The diagram shows the curve \(y = 2 x ^ { 5 } + 3 x ^ { 3 }\) and the line \(y = 2 x\) intersecting at points \(A , O\) and \(B\).
  1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2 x ^ { 4 } + 3 x ^ { 2 } - 2 = 0\).
  2. Solve the equation \(2 x ^ { 4 } + 3 x ^ { 2 } - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form.