Question i. Sketch, on the same diagram, the graphs of and for . ii. Verify that is a root of the equation , and state the other root of this equation for which . iii. Hence state the set of values of , for , for which Solution i. We are required to sketch and for . First we sketch for . We […]
Question The diagram shows a quadrilateral ABCD in which the point A is , the point B is and the point C is . The diagonals AC and BD intersect at M. Angle and . Calculate i. the coordinates of M and D, ii. the ratio . Solution i. First we find the coordinates […]
Question The equation of a curve is . Find i. an expression for and the coordinates of the stationary point on the curve, ii. the volume obtained when the region bounded by the curve and the x-axis is rotated through about the x-axis. Solution i. Gradient (slope) of the curve is the derivative of equation of the curve. Hence […]
Question A curve is such that . The line is the normal to the curve at the point on the curve. Given that the x-coordinate of is positive, find i. the coordinates of P, ii. the equation of the curve. Solution i. We are given that equation of the line is; We can rearrange the equation of the line as; […]
Question The diagram shows a circle touching a circle at a point . Circle has centre A and radius 6cm, and circle has centre B and radius 10 cm. Points D and E lie on and respectively and DE is parallel to AB. Angle radians and angle radians. i. By considering the perpendicular distances of D and E […]
Question The equation of a curve is and the equation of a line is , where is a constant. i. In the case where , find the coordinates of the points of intersection of the line and the curve. ii. Find the value of for which the line is a tangent to the curve. Solution i. For the case […]
Question Relative to an origin , the position vectors of points A and B are given by and where is a constant. i. Find the value of for which angle AOB =. ii. In the case where , find the vector which has magnitude 28 and is in the same direction as . Solution i. We recognize […]
Question The functions and are defined for by where and are constants. Given that and , find i. the values of and , ii. an expression for . Solution i. We are given that; We need to find and . First we find ; We have; For ; Now we evaluate ; We are given that; Therefore; Now we find the […]
Question i. Find the first 3 terms in the expansion of , in ascending powers of . ii. Use the result in part (i) to find the coefficient of in the expansion of . Solution i. First rewrite the given expression in standard form. Expression for the Binomial expansion of is: In the given case: Hence; ii. We can […]
Question a) An arithmetic progression contains 25 terms and the first term is −15. The sum of all the terms in the progression is 525. Calculate i. the common difference of the progression ii. the last term in the progression iii. the sum of all the positive terms in the progression. b) A […]