Worked examples




OCR AS-level June 2007


Core Maths papers 1-4

OCR GCE Core Maths 1 (4721): questions 5, 7;
OCR GCE Core Maths 2 (4722): question 6;
OCR GCE Core Maths 3 (4723): questions 1, 4, 5, 6, 8;
OCR GCE Core Maths 4 (4724): questions 1, 2, 3, 5, 6, 7, 8;


Questions by kind permission OCR.
All other content © copyright ballooncalculus.org 2008.





See also: Balloon Calculus Home




In Balloon Calculus, we use three shapes:
the three shapes

A continuous straight line differentiates going down, integrates going up. The dashed/dotted straight line (in the chain-rule shape) is similar, except that the differentiation or integration is 'with respect to' the expression in the dashed/dotted balloon. The product-rule shape is different, in that the two lines in the zoomed-out version of this shape don't operate independently, but only so as to emphasise the second assumption...

In any shape, the whole of the lower level is the derivative of the whole of the top (i.e. the top is integral of the bottom).





OCR AS-level Core Maths 1 June 2007


question: OCR GCE Maths 4721/01 q5 (i)
working: OCR GCE Maths 4721/01 q5 (i)
question: OCR GCE Maths 4721/01 q5 (ii)
working: OCR GCE Maths 4721/01 q5 (ii)




question: OCR GCE Maths 4721/01 q7 (a)
working: OCR GCE Maths 4721/01 q7 (a)
question: OCR GCE Maths 4721/01 q7 (b)
working: OCR GCE Maths 4721/01 q7 (b)







OCR AS-level Core Maths 2 June 2007



question: OCR GCE Maths 4722/01 q6(a)(i)
working: OCR GCE Maths 4722/01 q6(a)(i)
question: OCR GCE Maths 4722/01 q6(a)(ii)
working: OCR GCE Maths 4722/01 q6(a)(ii)
question: OCR GCE Maths 4722/01 q6(b)
working: OCR GCE Maths 4722/01 q6(b)







OCR AS-level Core Maths 3 June 2007



question: OCR GCE Maths 4723/01 q1(i)
working: OCR GCE Maths 4723/01 q1(i)
question: OCR GCE Maths 4723/01 q1(ii)
working: OCR GCE Maths 4723/01 q1(ii)




question: OCR GCE Maths 4723/01 q4(i)
Here we use the chain rule shape in reverse
chain-shape picture
working to show I = 30




question: OCR GCE Maths 4723/01 q5(i)
working: OCR GCE Maths 4723/01 q5(i)
question: OCR GCE Maths 4723/01 q5(ii)
working: OCR GCE Maths 4723/01 q5(ii)




question: OCR GCE Maths 4723/01 q6(i)
working: OCR GCE Maths 4723/01 q6(i)
working to show a = 1/2 ln(15 - 1/6a^2)
question: OCR GCE Maths 4723/01 q6(ii)
working: OCR GCE Maths 4723/01 q6(ii)




question: OCR GCE Maths 4723/01 q8(i)
Here we use the chain rule shape in reverse
working: OCR GCE Maths 4723/01 q8(i) zoomed-in
From this the formula. Alternatively...
working: OCR GCE Maths 4723/01 q8(i)
working to get required derivative
question: OCR GCE Maths 4723/01 q8(ii)
working: OCR GCE Maths 4723/01 q8(ii)
question: OCR GCE Maths 4723/01 q8(iii)
working: OCR GCE Maths 4723/01 q8(iii)







OCR AS-level Core Maths 4 June 2007



question: OCR GCE Maths 4724/01 q1(i)
working: OCR GCE Maths 4724/01 q1(i)
question: OCR GCE Maths 4724/01 q1(ii)
working: OCR GCE Maths 4724/01 q1(ii)




question: OCR GCE Maths 4724/01 q2
working: OCR GCE Maths 4724/01 q2
working: OCR GCE Maths 4724/01 q2
working: OCR GCE Maths 4724/01 q2




question: OCR GCE Maths 4724/01 q3
working: OCR GCE Maths 4724/01 q3




question: OCR GCE Maths 4724/01 q5(i)
working: OCR GCE Maths 4724/01 q5(i)
question: OCR GCE Maths 4724/01 q5(ii)
working: OCR GCE Maths 4724/01 q5(ii)
question: OCR GCE Maths 4724/01 q5(iii)
working: OCR GCE Maths 4724/01 q5(iii)




question: OCR GCE Maths 4724/01 q6
working: OCR GCE Maths 4724/01 q6




question: OCR GCE Maths 4724/01 q7(i)
working: OCR GCE Maths 4724/01 q7(i)
working: OCR GCE Maths 4724/01 q7(i)
question: OCR GCE Maths 4724/01 q7(ii)
working: OCR GCE Maths 4724/01 q7(ii)
question: OCR GCE Maths 4724/01 q7(iii)
working: OCR GCE Maths 4724/01 q7(iii)
working: OCR GCE Maths 4724/01 q7(iii)




question: OCR GCE Maths 4724/01 q8(i)
working: OCR GCE Maths 4724/01 q8(i)
diagram: OCR GCE Maths 4724/01 q8(i)
working: OCR GCE Maths 4724/01 q8(i)
diagram: OCR GCE Maths 4724/01 q8(i)
working: OCR GCE Maths 4724/01 q7(iii)
question: OCR GCE Maths 4724/01 q8(ii)
working: OCR GCE Maths 4724/01 q8(ii)
question: OCR GCE Maths 4724/01 q8(iii)
working: OCR GCE Maths 4724/01 q8(iii)
question: OCR GCE Maths 4724/01 q8(iv)
working: OCR GCE Maths 4724/01 q8(iv)






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