Questions — OCR MEI (4456 questions)

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OCR MEI C2 Q11
3 marks Easy -1.8
Find \(\int 30x^2 dx\). [3]
OCR MEI C2 Q12
4 marks Easy -1.2
Find \(\int (x^5 + 10x^3) dx\). [4]
OCR MEI C2 Q1
4 marks Easy -1.2
Find \(\int (3x^5 + 2x^{-\frac{1}{2}}) dx\). [4]
OCR MEI C2 Q2
11 marks Moderate -0.3
Fig. 11 shows the curve \(y = x^3 - 3x^2 - x + 3\). \includegraphics{figure_11}
  1. Use calculus to find \(\int_{-1}^{3} (x^3 - 3x^2 - x + 3) dx\) and state what this represents. [6]
  2. Find the \(x\)-coordinates of the turning points of the curve \(y = x^3 - 3x^2 - x + 3\), giving your answers in surd form. Hence state the set of values of \(x\) for which \(y = x^3 - 3x^2 - x + 3\) is a decreasing function. [5]
OCR MEI C2 Q3
3 marks Easy -1.2
Find \(\int \left(x - \frac{3}{x^2}\right) dx\). [3]
OCR MEI C2 Q4
4 marks Easy -1.2
Find \(\int (20x^4 + 6x^{-\frac{2}{3}}) dx\). [4]
OCR MEI C2 Q5
10 marks Easy -1.2
Find \(\int (12x^5 + \sqrt[5]{x} + 7) dx\). [5]
OCR MEI C2 Q6
5 marks Moderate -0.8
Find \(\int \left(x^{\frac{1}{2}} + \frac{6}{x^3}\right) dx\). [5]
OCR MEI C2 Q7
4 marks Easy -1.2
Find \(\int \left(x^4 + \frac{1}{x^3}\right) dx\). [4]
OCR MEI C2 Q8
5 marks Easy -1.3
  1. Differentiate \(12\sqrt{x}\). [2]
  2. Integrate \(\frac{6}{x^5}\). [3]
OCR MEI C2 Q1
5 marks Moderate -0.8
An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression. [5]
OCR MEI C2 Q2
12 marks Moderate -0.3
Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants. Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.
  1. How many of Jill's descendants would there be in generation 8? [2]
  2. How many of Jill's descendants would there be altogether in the first 15 generations? [3]
  3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac{\log_{10}2000003}{\log_{10}3} - 1.$$ Hence find the least possible value of \(n\). [4]
  4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? [3]
OCR MEI C2 Q3
5 marks Moderate -0.8
  1. Find \(\sum_{r=1}^{5} \frac{21}{r+2}\). [2]
  2. A sequence is defined by $$u_1 = a, \text{ where } a \text{ is an unknown constant,}$$ $$u_{n+1} = u_n + 5.$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence. [3]
OCR MEI C2 Q4
5 marks Moderate -0.3
The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]
OCR MEI C2 Q5
3 marks Moderate -0.8
\(S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  1. Another geometric progression has first term \(2a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\). [1]
  2. A third geometric progression has first term \(a\) and common ratio \(r^2\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\). [2]
OCR MEI C2 Q6
4 marks Easy -1.8
Find the second and third terms in the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 3.$$ Find also the sum of the first 50 terms of this sequence. [4]
OCR MEI C2 Q7
10 marks Standard +0.3
A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25.
  1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\). [7]
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2^{n-2} : 3^{n-2}\). [3]
OCR MEI C2 Q1
12 marks Standard +0.3
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250.
    1. Find the values of \(A\) and \(D\). [4]
    2. Find the sum of the 21st to 50th terms inclusive of this sequence. [3]
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250. Use the formula for the sum of a geometric progression to show that \(\frac{r^4 - 1}{r^2 - 1} = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\). [5]
OCR MEI C2 Q2
5 marks Moderate -0.8
A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5. Find the value of \(b\) and find also the sum of the first 15 terms of the progression. [5]
OCR MEI C2 Q3
5 marks Moderate -0.3
In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030. Find the first term and the common difference. [5]
OCR MEI C2 Q4
5 marks Moderate -0.3
The second term of a geometric sequence is 6 and the fifth term is \(-48\). Find the tenth term of the sequence. Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence. [5]
OCR MEI C2 Q5
5 marks Moderate -0.3
The third term of an arithmetic progression is 24. The tenth term is 3. Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive. [5] Simplify
OCR MEI C2 Q6
5 marks Easy -1.2
  1. Find the 51st term of the sequence given by $$u_1 = 5,$$ $$u_{n+1} = u_n + 4.$$ [3]
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots$$ [2]
OCR MEI C2 Q7
5 marks Moderate -0.8
An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression. [2]
  2. Find the sum of the 21st to the 50th terms inclusive of this progression. [3]
OCR MEI C2 Q8
5 marks Moderate -0.3
The second term of a geometric progression is 18 and the fourth term is 2. The common ratio is positive. Find the sum to infinity of this progression. [5]