Questions — OCR C2 (296 questions)

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OCR C2 Q8
  1. (a) An arithmetic series has a common difference of 7 .
Given that the sum of the first 20 terms of the series is 530 , find
  1. the first term of the series,
  2. the smallest positive term of the series.
    (b) The terms of a sequence are given by $$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1$$ where \(k\) is a positive constant.
    Given that \(u _ { 2 } = 2 u _ { 1 }\),
  3. find the value of \(k\),
  4. show that \(u _ { 3 } = 11 + 6 \sqrt { 2 }\).
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{61af807c-1f2c-417a-85cf-86f2cf566cb9-3_670_1022_1263_374} The diagram shows the curve \(y = 2 x ^ { 2 } + 6 x + 7\) and the straight line \(y = 2 x + 13\).
  1. Find the coordinates of the points where the curve and line intersect.
  2. Show that the area of the shaded region bounded by the curve and line is given by $$\int _ { - 3 } ^ { 1 } \left( 6 - 4 x - 2 x ^ { 2 } \right) d x$$
  3. Hence find the area of the shaded region.
OCR C2 Q1
  1. A sequence is defined by
$$u _ { n + 1 } = \frac { u _ { n } + 1 } { 3 } , \quad n = 1,2,3 , \ldots$$ Given that \(u _ { 3 } = 5\),
  1. find the value of \(u _ { 4 }\),
  2. find the value of \(u _ { 1 }\).
OCR C2 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{5025c118-e763-424b-b2c1-5452953a43a9-1_550_901_817_468} The diagram shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
OCR C2 Q3
3. (i) Show that the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$ can be written as a quadratic equation in \(\sin \chi ^ { \circ }\).
(ii) Hence solve, for \(0 \leq x < 360\), the equation $$3 \cos ^ { 2 } x ^ { \circ } + \sin ^ { 2 } x ^ { \circ } + 5 \sin x ^ { \circ } = 0$$
OCR C2 Q4
  1. (a) Sketch the curve \(y = 5 ^ { x - 1 }\), showing the coordinates of any points of intersection with the coordinate axes.
    (b) Find, to 3 significant figures, the \(x\)-coordinates of the points where the curve \(y = 5 ^ { x - 1 }\) intersects
    1. the straight line \(y = 10\),
    2. the curve \(y = 2 ^ { x }\).
    3. As part of a new training programme, Habib decides to do sit-ups every day.
    He plans to do 20 per day in the first week, 22 per day in the second week, 24 per day in the third week and so on, increasing the daily number of sit-ups by two at the start of each week.
OCR C2 Q6
6. (i) Write down the exact value of \(\cos \frac { \pi } { 6 }\). The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(ii) Use the trapezium rule with two intervals of equal width to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(iii) Using your answer to part (b), find an estimate for the area of \(S\).
OCR C2 Q7
7. The diagram shows the quadrilateral \(A B C D\) in which \(A B = 6 \mathrm {~cm} , B C = 3 \mathrm {~cm}\), \(C D = 8 \mathrm {~cm} , A D = 9 \mathrm {~cm}\) and \(\angle B A D = 60 ^ { \circ }\).
  1. Show that \(B D = 3 \sqrt { 7 } \mathrm {~cm}\).
  2. Find the size of \(\angle B C D\) in degrees to 1 decimal place.
  3. Find the area of quadrilateral \(A B C D\).
OCR C2 Q8
8. \(\mathrm { p } ( x ) = x ^ { 4 } - ( x - 2 ) ^ { 4 }\).
  1. Show that ( \(x - 1\) ) is a factor of \(\mathrm { p } ( x )\).
  2. Show that $$\mathrm { p } ( x ) = 8 x ^ { 3 } - 24 x ^ { 2 } + 32 x - 16$$
  3. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ).
OCR C2 Q9
9. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$f ^ { \prime } ( x ) = 1 + \frac { 2 } { \sqrt { x } } , \quad x > 0$$ The straight line \(l\) has the equation \(y = 2 x - 1\) and is a tangent to \(C\) at the point \(P\).
  1. State the gradient of \(C\) at \(P\).
  2. Find the \(x\)-coordinate of \(P\).
  3. Find an equation for \(C\).
  4. Show that \(C\) crosses the \(x\)-axis at the point \(( 1,0 )\) and at no other point.
OCR C2 Q1
  1. Find
$$\int \left( 3 x ^ { 2 } + \frac { 1 } { 2 x ^ { 2 } } \right) \mathrm { d } x$$
OCR C2 Q2
2. The diagram shows triangle \(P Q R\) in which \(P Q = x , P R = 7 - x , Q R = x + 1\) and \(\angle P Q R = 60 ^ { \circ }\). Using the cosine rule, find the value of \(x\).
OCR C2 Q3
3.
\includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-1_474_863_1283_520} The diagram shows the curve with equation \(y = \frac { 4 x } { ( x + 1 ) ^ { 2 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 1\).
  1. Use the trapezium rule with four intervals, each of width 0.25 , to find an estimate for the area of the shaded region.
  2. State, with a reason, whether your answer to part (a) is an under-estimate or an over-estimate of the true area.
OCR C2 Q4
4. The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + k x ) ^ { 7 }\), where \(k\) is a positive constant, is 525.
  1. Find the value of \(k\). Using this value of \(k\),
  2. show that the coefficient of \(x ^ { 3 }\) in the expansion is 4375 ,
  3. find the first three terms in the expansion in ascending powers of \(x\) of $$( 2 - x ) ( 1 + k x ) ^ { 7 }$$
OCR C2 Q5
  1. (i) Given that
$$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0$$ (ii) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0$$
OCR C2 Q6
6. $$f ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 6 x + 1$$
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 1\) ).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\).
    2. Hence, or otherwise, solve the equation $$2 x ^ { 3 } + 3 x ^ { 2 } - 6 x - 8 = 0$$
OCR C2 Q7
  1. (i) Given that
$$\log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x$$ show that $$y = 2 x + 1$$ (ii) Solve the simultaneous equations $$\begin{aligned} & \log _ { 2 } ( y - 1 ) = 1 + \log _ { 2 } x
& 2 \log _ { 3 } y = 2 + \log _ { 3 } x \end{aligned}$$
OCR C2 Q8
  1. The first two terms of an arithmetic progression are \(( t - 1 )\) and \(\left( t ^ { 2 } - 5 \right)\) respectively, where \(t\) is a positive constant.
    1. Find and simplify expressions in terms of \(t\) for
      1. the common difference,
      2. the third term.
    Given also that the third term is 19 ,
  2. find the value of \(t\),
  3. show that the 10th term is 75,
  4. find the sum of the first 40 terms.
OCR C2 Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447} The diagram shows the curves \(y = 2 x ^ { 2 } - 6 x - 3\) and \(y = 9 + 3 x - x ^ { 2 }\).
  1. Find the coordinates of the points where the two curves intersect.
  2. Find the area of the shaded region bounded by the two curves.
OCR C2 Q1
  1. A sequence of terms is defined by
$$u _ { n } = 3 ^ { n } - 2 , \quad n \geq 1 .$$
  1. Write down the first four terms of the sequence. The same sequence can also be defined by the recurrence relation $$u _ { n + 1 } = a u _ { n } + b , \quad n \geq 1 , \quad u _ { 1 } = 1 ,$$ where \(a\) and \(b\) are constants.
  2. Find the values of \(a\) and \(b\).
OCR C2 Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{27703044-8bb3-4809-9454-ae6774fec060-1_485_808_973_520} The diagram shows the curve with equation \(y = \sqrt { 4 x - 1 }\).
  1. Use the trapezium rule with four intervals of equal width to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
  2. Explain briefly how you could use the trapezium rule to obtain a more accurate estimate of the area of the shaded region.
OCR C2 Q3
3. (i) Expand \(( 2 + y ) ^ { 6 }\) in ascending powers of \(y\) as far as the term in \(y ^ { 3 }\), simplifying each coefficient.
(ii) Hence expand \(\left( 2 + x - x ^ { 2 } \right) ^ { 6 }\) in ascending powers of \(x\) as far as the term in \(x ^ { 3 }\), simplifying each coefficient.
OCR C2 Q4
4. $$\mathrm { f } ( x ) = \frac { 4 } { 2 + \sin x ^ { \circ } }$$
  1. State the maximum value of \(\mathrm { f } ( x )\) and the smallest positive value of \(x\) for which \(\mathrm { f } ( x )\) takes this value.
  2. Solve the equation \(\mathrm { f } ( x ) = 3\) for \(0 \leq x \leq 360\), giving your answers to 1 decimal place.
OCR C2 Q5
5. (a) Given that \(t = \log _ { 3 } x\),
  1. write down an expression in terms of \(t\) for \(\log _ { 3 } x ^ { 2 }\),
  2. show that \(\log _ { 9 } x = \frac { 1 } { 2 } t\).
    (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4$$
OCR C2 Q6
  1. Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 3 } - 4 } { x ^ { 3 } } , \quad x \neq 0$$ and that \(y = 0\) when \(x = - 1\), find the value of \(y\) when \(x = 2\).