IB Questionbank Test - St Leonard's College



ST LEONARD’S COLLEGEIBHL TEST March 2017INSTRUCTIONS TO CANDIDATESDo not open this examination paper until instructed to do so.You are not permitted access to any calculator for this paper. Section A: answer all questions in the spaces provided Section B: answer all questions in the spaces provided. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required for this paper. The maximum mark for this examination paper is [76 marks]. SECTION A:1a. [2 marks]Use the identity cos2θ=2cos2θ-1 to prove that cos12x=1+cosx2, 0≤x≤π .1b. [2 marks] Find a similar expression for sin12x, 0≤x≤π .1c. [4 marks] Hence find the value of 0π21+cosx+1-cosxdx.2. [6 marks] Paint is poured into a tray where it forms a circular pool with a uniform thickness of 0.5 cm. If the paint is poured at a constant rate of 4cm3s-1 , find the rate of increase of the radius of the circle when the radius is 20 cm.SECTION B:INSTRUCTIONS TO CANDIDATESDo not open this examination paper until instructed to do so. A graphic display calculator is required for this paper.Section A: answer all questions in the spaces provided.Section B: answer all questions in the spaces provided. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures.A clean copy of the Mathematics HL and Further Mathematics HL formula booklet is required for this paper.The maximum mark for this examination paper is [76 marks]. 1a. [4 marks] The function f is defined as fx=e3x+1, x∈R.Find f-1(x) . State the domain of f-1(x) .1b. [5 marks] The function g is defined as gx=lnx, x∈R+ .The graph of y=g(x) and the graph of y=f-1(x) intersect at the point P Find the coordinates of P .1c. [3 marks] The graph of y=g(x) intersects the -axis at the point Q Show that the equation of the tangent Q to the graph of y=g(x) at the point Q is y=x-1.1d. [5 marks] A region R is bounded by the graphs of y=g(x) , the tangent T and the line x=e. Find the area of the region R.1e. [6 marks] The region R is bounded by the graphs of y=g(x), the tangent T and the line x=e .Show that gx≤x-1, x∈R+.(ii) By replacing x with 1x in part (e) (i), show that x-1x≤gx, x∈R+..2a. [3 marks] The shaded region S is enclosed between the curve y=x+2cosx, for 0≤x≤2π , and the line y=x , as shown in the diagram below.Find the coordinates of the points where the line meets the curve.2b. [5 marks] The region S is rotated by 2π about the x -axis to generate a solid.Write down an integral that represents the volume V of the solid.(ii) Find the volume V3. [9 marks] A curve has equation arctanx2+arctany2=π4 .Find dydx in terms of x and y.Find the gradient of the curve at the point where x=12 and y<0 .4a. [14 marks] A function f is defined by fx=12ex+e-x, x∈R .Explain why the inverse function f-1 does not exist.Show that the equation of the normal to the curve at the point P where x=ln3 is given by 9x+12y-9ln3-20=0 .Find the x-coordinates of the points Q and R on the curve such that the tangents at Q and R pass through (0, 0).4b. [8 marks] The domain of f is now restricted to x≥0 .Find an expression for f-1(x) .(ii) Find the volume generated when the region bounded by the curve y=f(x) and the lines x=0 and y=5 is rotated through an angle of 2π radians about the y-axis. ................
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