Questions asked by eight mushrooms

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• The parametric equations of a curve are x=cosθ and y=1+sinθ. Show that the curve is a circle with centre(0,1) and radius 1 unit. Find the equation of the chord at the points on the curve where θ=#π/6# and θ=#π/3# respectively. ?
  
• Points P(2#p^2#,#1/p#) and Q(2#q^2#,#1/q#)lies on a curve.(pls see below) ?
  
• Find the equation of straight line having a gradient of -#1/t# and passing through the point(#t^2#,2t).(pls see details below). ?
  
• A curve is given by the parametric equations x=t,y=#1/t#. (pls see details below). ?
  
• The parametric equations of a curve are x=#t^2# and y=#3/t#. Points P and Q with parameters p and q respectively lie on the curve. (pls see below). ?
  
• A polynomial #p(x)# of degree #n>=2# has a remainder of #9# when it is divided by #(x+2)# and a remainder of #-1# when it is divided by #(x-3)#. Find the remainder of #p(x)# when it is divided by #(x^2-x-6)#. ?
  
• Show that x=#1/4# is one of the roots of equation 4#x^3#-#x^2#-4x+1=0 Factorise 4#x^3#-#x^2#-4x+1 completely. Hence,solve (pls see below).?
  
• Find the set of values of k for which the equation #x^2#+2(k-1)x+k-1=0 has no real roots. ?
  
• Show that (I)2#x^2#-8x+11>0 ∀ x (ii)6x<9#x^2#+5 ∀ x. ?
  
• Express 4#x^2#-x+3/#x^3#-1 in partial fractions. ?
  
• In a geometric progression in which all the terms are positive, the third term exceeds the first term by 32 while the fifth term exceeds the first term by 320. Find the common ratio and the first term. ?
  
• Given that the first term of a geometric progression is 8 and the sum of the first three terms is 104, find the possible values of the common ratio. In each case, write down the corresponding first three terms of the series. ?
  
• The 1st term of a geometric progression exceeds the second term by 1 and the sum of the first three terms is #13/5#.If there are positive as well as negative terms in the geometric progression, find the sum of the first five terms. ?
  
• 1)The nth term of a progression is 6(#-4/3##)^2#. Prove that the progression is a geometric progression(G.P.).Obtain the first term and common ratio. (one more question, pls see below). ?
  
• The positive numbers e and f are the 2nd and 14th terms respectively of an arithmetic progression whose first term is 1.It is also given that e,9 and f are three consecutive terms of a geometric progression. Find the values of e and f. ?