1. Given ) inl. 61T 3^nr 13. f(x) : (2-3x)-r, lr'. i. find the binomial expansion of(x), in ascending powers ofx, up to and including the term
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1 61T 3^nr Given ) f(x) : (2-3x)-r, lr'. i find the binomial expansion of(x), in ascending powers ofx, up to and including the term inl. Give each coeflicient as a simplified fraction.
2 2. (a) Use integration to find Jl "o' (s) (b) Hence calculate I,*^.0, rtl- rr_ l L \l=-y-2?_.l
3 9x2 + 20x 3. -"- Express -"'"-10 ^" ' in. partial fractions. (x + 2)(3x - l) (4) qx}+lo:c -to 3xl-rSr-2- -
4 4. Figure I Figure 1 shows a sketch of part of the curve with equation, = r;k The frnite region R, shown shaded in Flgure 1, is bounded by the curve, the x-axis, the line with equation x = 1 and the line with equation x = 4. (a) Complete the table with the value ofy conesponding to l: 3, giving your answer to 4 decimal places. (t) x 2, 4 v (b) Use the trapezium rule, with all the values ofy in the completed table, to obtain an estimate of the area of the region R, giving your answer to 3 decimal places. (3) (c) Use the substitution u = 1 + 4x, to find, by integrating, the exact area ofr. d3 ) l.o98r (8) (v ts (3d;)
5 { a_ -* e,- U= l+rl-xl+.jr 4 = L ') dr; 2{i do.. /rcc zrlx-!zlt g1=3 lgil \l=l,,1-:c = u=t z) ac = (.r.=rf zl't@1*3*t ** -u- \r, - r,r 7 JL :Lf( r,-1;a) -[E-t^z)J ='i(tr r^r-rn+ = {+zrni t ar tt -]'\-'l 3+ - Zu'+3* - t^*]',
6 Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations,=t-1r. 2' v:2r-l The curve crosses the Jr-rxis at tle point I and crosses the x-axis at tle point B. (a) Show that I has coordinates (0, 3). (b) Find the x coordinate of the point B. (c) Find an eqriation of the normal to C at the point A. The region R, as showr shaded in Figure 2, is bounded by the curve C the line x = -1 'the r-axis. (d) Use integration to find the exact axea ofr. a)!t-=oo l=ll=>t=2 -'. S=22-t--3 + a) a) (s) and (o =$ z) dtr l
7 Nate S--at => l'.5, l^2t =) l^s. txlnz o frr^5- *e(a*r^d.1 5, i'rr- ) # -5r^L +A = 2r tnz- Gfi[f.l) ( I\^ o. ^ - :;"-:t at I=OrL=?- -.) Me = 3ln2 --; M*= io, ga; Ifi(t-o) => $= *,- + J r).1=l at=o X==t => -l:-,l:9r,, t.= 4 L..l J $'' x.-l t=ọ^ o\.r 4rar ; l? It '-.jl*-+;+[ffi-+*4-
8 Figure 3 2cos x, where x is miasured in radians. The curve crosses the;r-axis at the point I and at the point B. Figure 3 shows a sketch of part of the curve with equation y = I - (a) Find, in terms ofz, the x coordinate of the point A and the x coordinate ofthe point B' (3) The finite region,s enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S is rotated through 2x radians about the x-axis. O) Find, by integration, the exact value of the volume of the solid generated. (6) b=o?) 2 CoS 2L =\ =) GS:c-=!- -) z-= Cos-'( l) ' g,t t({,o Voturqg- = r-lios*)'a>t-
9 = ir f i,^2--4s'^z-+3x- L-E \ r{i6i!o)-, - - L*fa+ r-:f-
10 7. With respect to a fixed origin O, the lines lrandlrarc given by the equations where,1 and p are scalar parameters. /, : r: (9i + 13j - 3k) +,i(i + 4j - 2k) l, : r : (2i-i + k) +t(2i + i + k) (a) Given that /, and /, meet, find the position vector of their point of intersection. (s) (b) Find the acute angle between I, and l, place. giving your answer in degrees to 1 decimal (3) Given that the point,4 has position vector 4i + 16j - 3k and that the point P lies on /, such that lp is perpendicular to /,, (c) find the exact coordinates ofp. Ii q- -3 'ti (6) jl!3r,'+}r:.:l +/w C,A. =:tb
11 J 5+X_\2+l(\+4\=O? 2!A--+ -'- A:,5
12 t (3 8. A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at 3 oc and, minutes after the bottle is placed in the refrigerator the tempefature of the water in the bottle is d oc' The rate of change of the temperature of the water in the bottle is modelled by the differential equation, d0 -(3-0) dt 125 (a) By solving the differential equation, show that, where I is a constant. 0:Aea.008t+3 (4) Given that the temperature of the water in the bottle when it was put in the refrigerator was 16 oc, (b) find the time taken for the temperature of the water in the bottle to fall to 10 "C, giving your answer to the nearcst minute. (s) 0'oo<L +c-
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