H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e
b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also passes through the. Find the values of a, b and c. [4] The sequence of numbers S, S3, S4,... is defined by S n n r for all positive integers of n where n. r ( r )! (i) Write down the values of S, S3 and S 4 in the form, where k is an integer k to be determined. Hence make a conjecture for S in terms of n. [] n (ii) Prove your conjecture by mathematical induction. [4] TheMathsCafe P a g e
H Maths Set D Paper 3 The set of points P in an Argand diagram represents the complex number z that satisfies arg z i, where. (i) Give a geometrical description of the locus of P. [] The set of points Q represents another complex number w that satisfies w 6 i. (ii) Sketch the locus of Q. [] (iii) (iv) Find the range of values of such that the locus of P meets the locus of Q more than once. [] Given that π, find the exact minimum value of z w. [3] 4 3 P a g e
H Maths Set D Paper 4 (i) Express ( r 3) in the form ( r ) Ar B( r ), where A and B are constants to be found. [] (ii) (iii) n r Hence, or otherwise, find ( r 3) in terms of n. [4] r n r Using your answer to (ii), find ( r 5). [3] r TheMathsCafe 4 P a g e
H Maths Set D Paper 5 A circle with radius 0 cm is cut into 0 sectors whose areas follow a geometric progression. The first sector, which is the largest, has an area of a cm. The second sector has an area of ar cm, the third sector has an area of ar cm, and so on. Given also that the total area of the odd-numbered sectors is 0π cm more than that of the remaining 0 sectors, show that r 9r r 9 0. Hence find the area of the smallest sector. [7] Suppose the cutting of the circle is done such that starting with the biggest sector, the subsequent sectors have areas following a geometric progression with common ratio 0.7. Find the maximum area of the biggest sector such that the cutting may be continued infinitely. [] 5 P a g e
H Maths Set D Paper 6 It is given that y tan ln x. Show that x y dy. [] dx By further differentiation of this result, find the Macluarin s series for y up to and including the term in x 3. [4] Deduce the first two non-zero terms in the series expansion of sec ln x. [3] x TheMathsCafe 6 P a g e
H Maths Set D Paper 7 The position vector of the point A is i k, and the equation of the line l is r 7i 5j 5 k (3i 7j 4 k ). Find the position vectors of the points B and C, both lying on l, such that AB AC 0. [3] Given that the point P is the midpoint of BC, show that the equation of the plane, which contains l and is perpendicular to AP is r ( 3i + j + 4k) =, where is an integer to be determined. [] The planes and 3 have equations r (i + j 3k) = 5 and r (i j + k) = respectively and A lies in both and 3. (i) Determine the position vector of D, the point of intersection between, 3. [] (ii) Hence find the volume of the tetrahedron ABCD. [3] and [Volume of tetrahedron = base area height] 3 7 P a g e
H Maths Set D Paper 8 Two planes, and are such that : Find 6 r 5 3 and : 4 5 r 4. 3 (i) the acute angle between the planes and, [] (ii) the values of m and n if the point A with coordinates ( m, n, 6) lies on the planes and, [] (iii) the cartesian equation of the line l, given that and meet in l. [3] (iv) the cartesian equation of the plane 3 which contains point A and is perpendicular to both and. [] 9 Another plane 4 has equation r 40. Comment on the geometrical 5 relationship between the 3 planes, and 4 if the system of equations 6x 5y 4z 3 5x y 3z 4 9x y 5z 40 has an infinite number of solutions. [] TheMathsCafe 8 P a g e
H Maths Set D Paper 9 (a) Using the substitution u a tan x, where a is a constant, show that the exact value a π of the integral du is 0 3. [5] ( a u ) 8a (b) The region R is bounded by the y axis and by the graphs of y x x y. Find 4 (i) the area of R, giving your answer correct 3 decimal places, [] (ii) the volume of revolution formed by rotating R through 4 right angles about the x axis, using a non-calculator method. [4] 9 P a g e
H Maths Set D Paper 0 The curve C has equation ax bx 5 y x c where a, b, c are constants and x c. (i) Given that x is an asymptote of C and C has a turning point on the y-axis, determine the values of b and c. [3] (ii) (iii) (iv) Given also that C has no x-intercept, show that 5 a. [] 4 5 5 Sketch the curve C for a, stating clearly the coordinates of any 4 stationary point, point of intersection with the axes, and the equations of any asymptotes. [3] By adding an additional line on the same diagram, determine in terms of a, the set ax bx 5 5 5 of values of x which satisfies the inequality ax for a. x c 4 [3] ax bx 5 (v) Sketch on a separate diagram, the graph of y f ( x), where f x, x c 5 5 for a. [] 4 TheMathsCafe 0 P a g e
H Maths Set D Paper Functions f and g are defined by (a) Given that a, f : x ln x x 3 for x, x a, g : x 6 for x, x b. x (i) Sketch the graph of f and find the range of f, [] (ii) explain why f does not have an inverse, [] (iii) find the exact least value of b such that fg exists and find the corresponding range of fg. [4] (b) (i) State the least value of a for which the function f exists. Hence find f, stating the domain of f. [4] (ii) Using the value of a obtained in (b)(i), sketch the graphs of f, f and f f on the same diagram. [] P a g e