1、11. Find the first 3 terms in ascending powers of x of226xgiving each term in its simplest form.2. f(x) ,x8=43+xx1x 02Giving your answers in their simplest form, find(a) f(x)(3)(b) ()fdxx(4)3.f (x) = 10x3+ 27x2 13x 12(a) Find the remainder when f (x) is divided by (i) x 2(ii) x + 3(3)(b) Hence facto
2、rise (f x) completely.(4)(4)4. Answer this question without the use of a calculator and show all your working.(i) Show that422 622(2+3= )(4)(ii) Show that 27 2163=83+7 (3)25. A sequence is defined by1 3uuunn+12=4, n.1Find the exact values of(a) u2, u3and u4(3)(b) u61(1)(c) uii=199(3)6. Given that a
3、and b are positive constants, solve the simultaneous equationsab = 25log4a log4b = 3Show each step of your working, giving exact values for a and b.(6)7. (a) Show that12sin2x cos x 11 = 0may be expressed in the form12cos2x + cos x 1 = 0(1)(b) Hence, using trigonometry, find all the solutions in the
4、interval 0 - x - 360 of 12sin2x cos x 11 = 0Give each solution, in degrees, to 1 decimal place.(4)8. Find the range of values of k for which the quadratic equationkx2+ 8x + 2(k + 7) = 0has no real roots. (7)39. In the first month after opening, a mobile phone shop sold 300 phones. A model for future
5、sales assumes that the number of phones sold will increase by 5% per month, so that300 1.05 will be sold in the second month, 300 1.052in the third month, and so on.Using this model, calculate(a) the number of phones sold in the 24th month,(2)(b) the total number of phones sold over the whole 24 mon
6、ths.(2)This model predicts that, in the Nth month, the number of phones sold in that month exceeds 3000 for the first time.(c) Find the value of N. (3)10. The curve C has equation y =cos3x, 0 - x - 2(a) In the space below, sketch the curve C.(2)(b) Write down the exact coordinates of the points at w
7、hich C meets the coordinate axes.(3)(c) Solve, for x in the interval 0 - x - 2,1cos3 2x=giving your answers in the form k, where k is a rational number.(4)11. The first three terms of an arithmetic series are 60, 4p and 2p 6 respectively.(a) Show that p = 9(2)(b) Find the value of the 20th term of t
8、his series.(3)(c) Prove that the sum of the first n terms of this series is given by the expression12n (6 n) (3)412.OCABD10 m10 m15 m22 mDiagram NOT drawn to scaleFigure 1Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled ABCD. The pond and pl
9、atform together form a circle of radius 22 m with centre O.OA and OD are radii of the circle. Point B lies on OA such that the length of OB is 10 m and point C lies on OD such that the length of OC is 10 m. The length of BC is 15 m.The platform is bounded by the arc AD of the circle, and the straigh
10、t lines AB, BC and CD.Find(a) the size of the angle BOC, giving your answer in radians to 3 decimal places,(3)(b) the perimeter of the platform to 3 significant figures, (4)(c) the area of the platform to 3 significant figures. (4)513. The curve C has equationy =(xxx ),33)(2x50(a) Find ddyxin a full
11、y simplified form.(3)(b) Hence find the coordinates of the turning point on the curve C.(4)(c) Determine whether this turning point is a minimum or maximum, justifying your answer.(2)The point P, with x coordinate 212, lies on the curve C.(d) Find the equation of the normal at P, in the form ax + by
12、 + c = 0, where a, b and c are integers.(5)14.Figure 2Figure 2 shows part of the line l with equation y = 2x 3 and part of the curve C with equation y = x2 2x 15ORABy = x2 2x 15yy = 2x 3xClDiagram NOT drawn to scaleThe line l and the curve C intersect at the points A and B as shown.(a) Use algebra t
13、o find the coordinates of A and the coordinates of B.(5)In Figure 2, the shaded region R is bounded by the line l, the curve C and the positive x-axis.(b) Use integration to calculate an exact value for the area of R.(7)615.OMZCyxXYDiagram NOT drawn to scaleFigure 3The points X and Y have coordinate
14、s (0, 3) and (6, 11) respectively. XY is a chord of a circle C with centre Z, as shown in Figure 3.(a) Find the gradient of XY.(2)The point M is the midpoint of XY.(b) Find an equation for the line which passes through Z and M.(5)Given that the y coordinate of Z is 10,(c) find the x coordinate of Z,(2)(d) find the equation of the circle C, giving your answer in the formx2+ y2+ ax + by + c = 0where a, b and c are constants.(5)
