Questions P1 (1374 questions)

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CAIE P1 2014 November Q6
6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).
  1. Show that angle \(O A B\) is a right angle.
  2. Find the area of triangle \(O A B\).
CAIE P1 2014 November Q7
7
  1. A geometric progression has first term \(a ( a \neq 0 )\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2 r\) and sum to infinity \(3 S\). Find the value of \(r\).
  2. An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 . Find the value of \(n\).
CAIE P1 2014 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-3_408_686_264_731} In the diagram, \(A B\) is an arc of a circle with centre \(O\) and radius 4 cm . Angle \(A O B\) is \(\alpha\) radians. The point \(D\) on \(O B\) is such that \(A D\) is perpendicular to \(O B\). The arc \(D C\), with centre \(O\), meets \(O A\) at \(C\).
  1. Find an expression in terms of \(\alpha\) for the perimeter of the shaded region \(A B D C\).
  2. For the case where \(\alpha = \frac { 1 } { 6 } \pi\), find the area of the shaded region \(A B D C\), giving your answer in the form \(k \pi\), where \(k\) is a constant to be determined.
CAIE P1 2014 November Q9
9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).
  1. Find the equation of the normal to the curve at \(P\).
  2. Find the equation of the curve.
  3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.
CAIE P1 2014 November Q10
10
  1. Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\). The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by $$f : x \mapsto x ^ { 2 } - 2 x - 15$$ The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
  2. State the smallest possible value of \(c\). For the case where \(c = 9\) and \(d = 65\),
  3. find \(p\) and \(q\),
  4. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2014 November Q11
11
\includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-4_995_905_260_621} The diagram shows parts of the curves \(y = ( 4 x + 1 ) ^ { \frac { 1 } { 2 } }\) and \(y = \frac { 1 } { 2 } x ^ { 2 } + 1\) intersecting at points \(P ( 0,1 )\) and \(Q ( 2,3 )\). The angle between the tangents to the two curves at \(Q\) is \(\alpha\).
  1. Find \(\alpha\), giving your answer in degrees correct to 3 significant figures.
  2. Find by integration the area of the shaded region.
CAIE P1 2014 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_668_554_260_797} The diagram shows part of the curve \(y = x ^ { 2 } + 1\). Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
CAIE P1 2014 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-2_313_737_1197_703} The diagram shows a triangle \(A O B\) in which \(O A\) is \(12 \mathrm {~cm} , O B\) is 5 cm and angle \(A O B\) is a right angle. Point \(P\) lies on \(A B\) and \(O P\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(A B\) and \(O Q\) is an arc of a circle with centre \(B\).
  1. Show that angle \(B A O\) is 0.3948 radians, correct to 4 decimal places.
  2. Calculate the area of the shaded region.
CAIE P1 2014 November Q3
3
  1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \(( 1 + x ) ^ { 5 }\). The coefficient of \(x ^ { 2 }\) in the expansion of \(\left( 1 + \left( p x + x ^ { 2 } \right) \right) ^ { 5 }\) is 95 .
  2. Use the answer to part (i) to find the value of the positive constant \(p\).
CAIE P1 2014 November Q4
4 A curve has equation \(y = \frac { 12 } { 3 - 2 x }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
  2. Find the possible \(x\)-coordinates of \(A\).
CAIE P1 2014 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{8952fc09-004a-4fb6-ad80-5312095a7057-4_832_775_258_685} The diagram shows a trapezium \(A B C D\) in which \(A B\) is parallel to \(D C\) and angle \(B A D\) is \(90 ^ { \circ }\). The coordinates of \(A , B\) and \(C\) are \(( 2,6 ) , ( 5 , - 3 )\) and \(( 8,3 )\) respectively.
  1. Find the equation of \(A D\).
  2. Find, by calculation, the coordinates of \(D\). The point \(E\) is such that \(A B C E\) is a parallelogram.
  3. Find the length of \(B E\).
CAIE P1 2014 November Q10
10 A curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 24 } { x ^ { 3 } } - 4\). The curve has a stationary point at \(P\) where \(x = 2\).
  1. State, with a reason, the nature of this stationary point.
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Given that the curve passes through the point \(( 1,13 )\), find the coordinates of the stationary point \(P\).
CAIE P1 2014 November Q11
11 The function f : \(x \mapsto 6 - 4 \cos \left( \frac { 1 } { 2 } x \right)\) is defined for \(0 \leqslant x \leqslant 2 \pi\).
  1. Find the exact value of \(x\) for which \(\mathrm { f } ( x ) = 4\).
  2. State the range of f.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
CAIE P1 2014 November Q1
1 In the expansion of \(( 2 + a x ) ^ { 6 }\), the coefficient of \(x ^ { 2 }\) is equal to the coefficient of \(x ^ { 3 }\). Find the value of the non-zero constant \(a\).
CAIE P1 2014 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{5b558b70-df50-437e-8425-47954f360765-2_618_700_431_721} In the diagram, \(O A D C\) is a sector of a circle with centre \(O\) and radius \(3 \mathrm {~cm} . A B\) and \(C B\) are tangents to the circle and angle \(A B C = \frac { 1 } { 3 } \pi\) radians. Find, giving your answer in terms of \(\sqrt { } 3\) and \(\pi\),
  1. the perimeter of the shaded region,
  2. the area of the shaded region.
CAIE P1 2014 November Q3
3
  1. Express \(9 x ^ { 2 } - 12 x + 5\) in the form \(( a x + b ) ^ { 2 } + c\).
  2. Determine whether \(3 x ^ { 3 } - 6 x ^ { 2 } + 5 x - 12\) is an increasing function, a decreasing function or neither.
CAIE P1 2014 November Q4
4 Three geometric progressions, \(P , Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2,1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \ldots\).
Progression \(Q\) is \(3,1 , \frac { 1 } { 3 } , \frac { 1 } { 9 } , \ldots\).
  1. Find the sum to infinity of progression \(R\).
  2. Given that the first term of \(R\) is 4 , find the sum of the first three terms of \(R\).
CAIE P1 2014 November Q5
5
  1. Show that \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta \equiv 2 \sin ^ { 2 } \theta - 1\).
  2. Hence solve the equation \(\sin ^ { 4 } \theta - \cos ^ { 4 } \theta = \frac { 1 } { 2 }\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
    \(6 \quad A\) is the point \(( a , 2 a - 1 )\) and \(B\) is the point \(( 2 a + 4,3 a + 9 )\), where \(a\) is a constant.
  3. Find, in terms of \(a\), the gradient of a line perpendicular to \(A B\).
  4. Given that the distance \(A B\) is \(\sqrt { } ( 260 )\), find the possible values of \(a\).
CAIE P1 2014 November Q7
7 Three points, \(O , A\) and \(B\), are such that \(\overrightarrow { O A } = \mathbf { i } + 3 \mathbf { j } + p \mathbf { k }\) and \(\overrightarrow { O B } = - 7 \mathbf { i } + ( 1 - p ) \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant.
  1. Find the values of \(p\) for which \(\overrightarrow { O A }\) is perpendicular to \(\overrightarrow { O B }\).
  2. The magnitudes of \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b ^ { 2 } = 2 a ^ { 2 }\).
  3. Find the unit vector in the direction of \(\overrightarrow { A B }\) when \(p = - 8\).
CAIE P1 2014 November Q8
8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 3,7 )\) and is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = 36 x ^ { - 3 }\).
  1. State, with a reason, whether this stationary point is a maximum or a minimum.
  2. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ( x )\).
CAIE P1 2014 November Q9
9
\includegraphics[max width=\textwidth, alt={}, center]{5b558b70-df50-437e-8425-47954f360765-3_842_695_1103_724} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3 \sqrt { } x\) intersecting at points \(A\) and \(B\).
  1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\).
  2. Find by integration the area of the shaded region.
    [0pt] [Question 10 is printed on the next page.]
CAIE P1 2014 November Q10
10
  1. The functions f and g are defined for \(x \geqslant 0\) by $$\begin{aligned} & \mathrm { f } : x \mapsto ( a x + b ) ^ { \frac { 1 } { 3 } } , \text { where } a \text { and } b \text { are positive constants, }
    & \mathrm { g } : x \mapsto x ^ { 2 } . \end{aligned}$$ Given that \(\mathrm { fg } ( 1 ) = 2\) and \(\operatorname { gf } ( 9 ) = 16\),
    1. calculate the values of \(a\) and \(b\),
    2. obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
  2. A point \(P\) travels along the curve \(y = \left( 7 x ^ { 2 } + 1 \right) ^ { \frac { 1 } { 3 } }\) in such a way that the \(x\)-coordinate of \(P\) at time \(t\) minutes is increasing at a constant rate of 8 units per minute. Find the rate of increase of the \(y\)-coordinate of \(P\) at the instant when \(P\) is at the point \(( 3,4 )\).
CAIE P1 2015 November Q1
1 In the expansion of \(\left( 1 - \frac { 2 x } { a } \right) ( a + x ) ^ { 5 }\), where \(a\) is a non-zero constant, show that the coefficient of \(x ^ { 2 }\) is zero.
CAIE P1 2015 November Q2
2 The function f is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } - 7\) and \(\mathrm { f } ( 3 ) = 5\). Find \(\mathrm { f } ( x )\).
CAIE P1 2015 November Q3
3 Solve the equation \(\sin ^ { - 1 } \left( 4 x ^ { 4 } + x ^ { 2 } \right) = \frac { 1 } { 6 } \pi\).