Higher Mathematics December Revision 1. (a) Given f ( x)= x 3 tan 2x , where 0 < x < π 4 , obtain f ′ ( x) . 3 (b) For y = 1 + x 2 1 + x , where x = −1, determine dy dx in its simplest form. 3 Part Marks Level Calc. Content Answer U1 OC2 (a) 3 C CN D4, D2 2005 Q1 (b) 3 C CN D4 2. [SQA] Differentiate the following functions with respect to x , simplifying your answers where possible. (a) h( x)= sin ( x 2 ) cos(3x) . 3 (b) y = ln( x + 3) x + 3 , x > −3. 3 Part Marks Level Calc. Content Answer U1 OC2 (a) 3 C CN D4, D3, D6, D2 1999 SY1 Q3 (b) 3 C CN D5, D8 hsn .uk.net Page 1 Questions marked ‘[SQA]’ c SQA All others c Higher Still Notes
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DecemberRevision - WordPress.com · Part Marks Level Calc. Content Answer U2 OC4 (a) 2 C CN S1 544 2004 A16 (b) 3 C CN S2 r= 11 (c1) 3 B CN S1, S2 a= 12 11 (c2) 2 A CN n= 7 hsn.uk.net
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Higher Mathematics
December Revision
1. (a) Given f (x) = x3 tan 2x , where 0 < x <π
4 , obtain f′(x) . 3
(b) For y =1+ x2
1+ x, where x 6= −1, determine dy
dxin its simplest form. 3
Part Marks Level Calc. Content Answer U1 OC2
(a) 3 C CN D4, D2 2005 Q1
(b) 3 C CN D4
2.[SQA] Differentiate the following functions with respect to x , simplifying your answerswhere possible.
5. A solid is formed by rotating the curve y = e−2x between x = 0 and x = 1through 360◦ about the x -axis. Calculate the volume of the solid that is formed. 5
8.[SQA] A car manufacturer is planning future production patterns. Based on estimates oftime, cost and labour, he obtains a set of three equations for the numbers x , y , z ofthree new types of car. These equations are
x + 2y + z = 602x + 3y + z = 853x + y + (λ + 2)z = 105,
where the integer λ is a parameter such that 0 < λ < 10.
(a) Use Gaussian elimination to find an expression for z in terms of λ . 5
(b) Given that z must be a positive integer, what are the possible values for z? 2
(c) Find the corresponding values of x and y for each value of z . 2
19.[SQA] The point A represents −5+ 5i on an Argand diagram and ABCD is a square withcentre −2+ 2i . Find the complex numbers represented by the points B, C and D,giving your answers in the form x+ iy . 4
Part Marks Level Calc. Content Answer U2 OC3
4 C CN A15 1996 SY1 Q2
20. (a) Obtain the sum of the series 8+ 11+ 14+ · · · + 56. 2
(b) A geometric sequence of positive terms has first term 2, and the sum of thefirst three terms is 266. Calculate the common ratio. 3
(c) An arithmetic sequence, A , has first term a and a common difference 2, anda geometric sequence, B , has first term a and common ratio 2. The first fourterms of each sequence have the same sum. Obtain the value of a . 3
Obtain the smallest value of n such that the sum to n terms for sequence B ismore than twice the sum to n terms for sequence A . 2
21. The sum, S(n) , of the first n terms of a sequence, u1, u2, u3, . . . is given byS(n) = 8n− n2 , n ≥ 1.Calculate the values of u1, u2, u3 and state what type of sequence it is. 3
Obtain a formula for un in terms of n , simplifying your answer. 2
Part Marks Level Calc. Content Answer U2 OC4
5 C CN S1 2005 Q4
22. Define Sn(x) by
Sn(x) = 1+ 2x+ 3x2+ · · ·+ nxn−1,where n is a positive integer.