1.C1 Jan 2011(a) Find the value of 16β14(b) Simplify π₯οΏ½2π₯β14οΏ½42.Find οΏ½οΏ½12π₯5β 3π₯2+ 4π₯13οΏ½dπ₯giving each term in its simplest form.3.Simplify5β2β3 β3β1giving your answer in the form π+πβ3, where π and π are rational numbers.4.A sequence π1, π2, π3, ...is deοΏ½ined by,π1=2ππ+1=3ππβ cwhere π is a constant(a) Find an expression for π2 in terms ofπ.Given that οΏ½ππ3π=1=0(b) οΏ½ind the value of π(2)(2)(5) (4) (1) (4)www.mymathscloud.com
5.Figure 1 shows a sketch of the curve with equation π¦= f (π₯) where f (π₯) =π₯π₯β2 , π₯ β 2The curve passes through the origin and has two asymptotes, with equation π¦ = 1 and π₯=2, as shown in οΏ½igure 1.(a) In the space below, sketch the curve with equation π¦= f (π₯β1) and state the equations of the asymptotes of this curve.(b) Find the coordinates of the points where the curve with equation π¦= f (π₯β1) crosses the coordinate axes(3) (4) www.mymathscloud.com
6.An arithmetic sequence has οΏ½irst term π and common difference π. The sum of the οΏ½irst 10 termsof the sequence is 162.(a) Show that 10π+45π=162Given also that the sixth term of the sequence is 17,(b) write down a second equation in π and π,(c) οΏ½ind the value of π and the value of π7.The curve with equation π¦=f(π₯) passes through the point (-1, 0).Given thatf β²(π₯) =12π₯2β8π₯+1 οΏ½ind f(π₯).8.The equation π₯2+(πβ3)π₯+(3β2π)=0, where π is a constant, has two distinct realroots. (a) Show that πsatisοΏ½iesπ2+2πβ3 >0(b) Find the set of possible values of π.(2)(1) (4) (5) (3) (4) www.mymathscloud.com
9.The line πΏ1has equation 2π¦β3π₯βπ=0, where π is a constant.Given that the point π΄(1, 4)lies onπΏ1, οΏ½ind(a) the value of π(b) the gradient ofπΏ1The line πΏ2passes through π΄ and is perpendicular toπΏ1.(c) Find an equation of πΏ2giving your answer in the form ππ₯+ππ¦+π=0, where π,π and π are integersThe line πΏ2crosses the π₯-axis at point π΅.(d) οΏ½ind the coordinates of π΅(e) οΏ½ind the exact length of π΄π΅(1) (2) (4) (2) (2) www.mymathscloud.com
10.(a) On the same axes, sketch the graphs of (π)π¦=π₯(π₯+2)(3βπ₯)(ππ)π¦=β2π₯showing clearly the coordinates of all the points where the curve cross the coordinate axes.(b)Using your sketch state, giving a reason, the number of real solutions to the equation π₯(π₯+2)(3βπ₯)+ 2π₯=011.The curve πΆ has equationπ¦=12π₯3β9π₯32+8π₯+30,π₯>0(a) Finddπ¦dπ₯(b)Show that the pointπ(4,β8)lies onπΆ(c)Find an equation of the normal to C at the point P , giving your answer in the formππ₯+ππ¦+π=0, where π, π and π are integers (6) (2) (6) (2) (4) www.mymathscloud.com
