CAIE Mathematics (9709)

Find the coordinates of the points of intersection of the curve and the line with equations xyy25 24 += and xy24 0++=.

The coefficient of x in the expansion of px +eo is 1280. Find the value of the constant p.

A point P is moving along the curve with equation yaxx12 2=- in such a way that the x-coordinate of P is increasing at a constant rate of 5 units per second.

Find the rate at which the y-coordinate of P is changing when x9=. Give your answer in terms of the constant a.

Given that the curve has a minimum point when x =, find the value of a.

The equation of a curve is cosyx42 3=+ for x02GG r.

State the greatest and least possible values of y.

Sketch the curve. [2] r 2r rr r

Hence determine the number of solutions of the equation cosxx42 32 1+= - for x02GG r. [1] The diagram shows the curve with equationx54 and the line yx63=- . The line and the curve intersect at the point P which has y-coordinate 3. Find the area of the shaded region.

Prove the identitytan tans incos cosc os ii i

Hence solve the equationcos sincos cos ii i tani A B The diagram shows the circle with equation xy xy1483 60 +- ++ = and the line y2=-. The line intersects the circle at the points A and B. The centre of the circle is C.

Find the coordinates of A, B and C .

Find the angle ACB in radians. Give your answer correct to 3 significant figures.

The chord AB divides the circle into two segments. Find the area of the larger segment.

The equation of a curve is such that =-. It is given that the curve has a stationary point at (,)219- .

Find an expression for

Find the x-coordinate of the other stationary point of the curve, and determine the nature of this stationary point.

Find the equation of the curve.

Find the equation of the normal to the curve at the point where =- and x is positive. Express your answer in the form pxqyr0++ =, where p, q and r are integers.

The first, second and third terms of an arithmetic progression are 4k, k and 8k respectively, where k is a non-zero constant.

Find the value of k.

Find the sum of the first 20 terms of the progression.

The fourth and sixth terms of a geometric progression are 36 and 6 respectively. The common ratio of the progression is positive. Find the sum to infinity of the progression. Give your answer in the form , where a, b and c are integers.

Express xx42 ++ in the form ()xa b ++ , where a and b are integers. [2] The functions f and g are defined as follows. ()xx x42f =+ + for x2G- ()xx 4g=-- for x2H-

(i) Find an expression for ()xf

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