Pure math 2/3

The curve with equation y = e^-ax tan x, where a is a positive constant, has only one point in the interval 0 < x < 0.5π at which the tangent is parallel to the x-axis. Find the value of a and state the exact value of the x-coordinate of this point.

• a=2 x=0.25π

• a=4 x=0.5π

• a= 2 x=0.75π

• a=4 x=2.12

• 11.1

• 12.1

• 10.1

• 13.1

Express f(x) in partial fractions. Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x².

• x(6-x)(2+x)^-1(4+x²)^-1

• x(6-x)(2+x)^-1(0.4+x²)^-1

• x(6-x)(6+x)^-1(8+x²)^-1

• x(7-x)(9+x)^-1(2+x²)^-1

• 1.29

• 1.35

• 1.65

• 1.72

The diagram shows the curve y = sin x cos² 2x for 0 ≤ x ≤0.25π and its maximum point M. Using the substitution u = cos x, find by integration the area of the shaded region bounded by the curve and the x-axis.

• 0.0895

• 0.0705

• 0.0612

• 0.0986

• x=0.587

• x=0.646

• x=0.457

• x=0.687

• 0.289

• 0.259

• 0.279

• 0.569