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STPM Maths 109

Apr 05, 2018

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    Numbers and Sets

    1.Determine the value of a if 2 + a i1 + 2i is a real number and ind this real number.[4] 99-T

    2. Solve the equation 2 log3 l o gx = 32. [6] 99-S3. Solve equation 4x1 x = 3.

    [5] 00-S4.Show that a + b 2ab.If x + y + z = c, show that x + y + z 13 c.

    [2], [2] 01-T

    5.Given that z = 1 + 2i and z =34i.Express 1z + 1z 2z in the form a + bi where aand b are real numbers.[3] 01-S

    6.Express 59246 in the form of p2+q3 where p and q are the integers. [7] 027.If(x + i y) = i, ind all the real values of x and y. [6] 038. Using the law of algebra of sets, show that, for any sets A and B,

    (A B) (B A) = (A B) (A B) [6] 049. Using the law of the algebra of sets, show that(A B) (A B) = B

    [4] 0510. The complex number z and z satisfy the equation z = 2 2 3 i(a)Express z and z in the form of a + bi, where a and b are the real numbers.(b)Represent z and z in an Argand diagram.(c)For each z and z, ind the modulus, and its arguments in radians.

    [6], [1], [4]06

    11.If A,B and C are abitrary sets,show that:[(A B) (B C)] (A C) = [4] 06

    12. If log xa =3log2 l og(x 2 a), express x in terms of a.[6] 07

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    13.Simplify(a) (7 3)27 + 3

    (b) 2(1+3i)(13i) where i = 1

    [3], [3] 07

    14.Using deinitions,show that,for any sets A,B and CA (B C) (A B) ( A C )[5] 08

    15. If z is a complex number such that |z| = 1, ind the real part of 11 z.[6] 08

    16.Given that:

    log(3 x 4 a) +log3x = 2loga +log(1 2 a)where 0 < a < 12 , ind x. [7] 09

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    Polynomials

    1. Show that for all real values of x, x + x + 1x + 1 does not lie between 3 and 1.[5] 99-T

    2.Equation ax + bx + c = 0 with a, b and c as non zero constants, has roots and .

    (a)If = , with 0, 1, and equation x + mx + n = 0, with , m and n as a nonzero constants,has roots ( + 1) and ( + 1) show that mn = bac.(b)If > and equations x + px + q = 0, with p and q as non zero constants,has roots + and ,express p and q in terms a,b and c.[7], [8] 99-T3.Given that m and n are constants and the two quadratic equations 3x + mx + 2 = 0 andx + nx + 4 = 0 has one common root. Show that 2m + 3n 7 m n + 5 0 = 0 .

    [5] 99-S

    4. Given that x + 2 is a factor of f(x) = x + (a + 2 b)x + (a 3 b)x + 8, ind a in terms of band ind q(x)so that f(x) = (x + 2)q(x)holds for all values of b.Determine the values of b so that f(x) = 0 has at least two distinct real roots.Sketch on different diagram, the graph of y = f(x)when b = 65 and b = 25.

    [5], [6], [4] 99-S

    5. Function f is deined by f(x) = 1x , with x R and x 0. Determine the set of values of xsuch that f(x) > (x 1).[5] 00-T

    6.Given that x + mx + nx 6 = 0 is divisible by x 3 and x + 2, ind m and n.[5] 00-T

    7. Given that the ax + bx + c = 0 has roots and . Show that: + = ba and = ca(a)If = 2, b = a + c, express a in terms of c(b) Show that cx + (b 3abc)x a = 0 has roots 1 and 1. [4], [5], [6] 00-T8.Determine the values of k so that the quadratic equation x + 2kx + 3k 3 = 0 has twodistinct real roots.

    [4] 00-S

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    9. Show that the real roots of ax + bx + c = 0, a 0 are given by:x = b b 4ac2a Deduce that if m + ni, with m, n R, is a root of this equation, then m ni is another root.

    (a)Show that 2 + i is a root of f(x) = 0 where f(x) = 2x 5x 2x + 15, and ind its other

    roots.(b)Find a polynomial g(x)so that f(x) xg(x) = 15 7x. Express g(x)in the form ofp(x q) + r,with p,q, r R, ind the maximum 1g(x).

    [5], [5], [5] 00-S

    10.Given that and are roots of x + 3x + 1 = 0. Find a quadratic equation with roots + 1

    and + 1

    .

    [5] 01-T11. Given that f(x) = x + px + 7x + q, where p and q are constants. When x = 1,f(x) = 0.When f(x) is divided by x + 1, the remainder is 16. Find the values of p and q.(a)Show that f(x) = 0 only has one real root.Find the set of values of x such that f(x) > 0.(b)Express x + 9f(x) in partial fraction.

    [4], [6], [5] 01-T

    12.Express1 2 x

    x(1 + 2 x) as partial fraction.

    [5] 01-S13. Function f is deined by f(x) = x (p + 1)x + p, where n and p are positive integers.Show that x 1 is a factor of f(x)for all values of p.when p = 4,x 2 is a factor of f(x).Find the value of n and factorise f(x)completely.With the value of n you have obtained,ind the set of values of p such thatf(x) + 2x 2 = 0 has roots which are distinct and real.[3], [5], [7] 01-S

    14. Show that polynomials 2x

    9x

    + 3x + 4 has x 1 as factor.

    Hence,(a)Find all the real roots of 2x 9x + 3x + 4 = 0(b)Determine the set of values of x so that 2x 9x + 3 x + 4 < 1 2 1 2 .[2], [5], [6] 02

    15.Show that1 is the only real root of the equation x + 3x + 5 x + 3 = 0 .[5] 03

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    16.Find the set of values of x such that 2 < x 2x + x 2 < 0 .[7] 03

    17.Sketch,on the same coordinates axes,the graphs of y=2x and y=2+ 1x.Hence, solve the inequality 2 x > 2 + 1x.[4], [4] 04

    18.Find all the solution set of the inequality |x 2| < 1x , where x 0.[7] 0519.The polynomial p(x) = x + ax 7x 4ax + b has a factor of x + 3 and, when divided

    by x 3, the remainder is 60. Find the values of a and b, and factorise p(x)completely.

    Using this substitution y= 1x , solve the equation 12y 8y 7y + 2 y + 1 = 0 .[9], [3] 06

    20.Find the constants A,B,C and D such that:3x + 5x(1 x)(1 + x) = A1 x + B1 + x + C(1 + x) + D(1 + x)[8] 07

    21.Using the substitution y = x + 1x , express f(x) = x 4 x 6 4x + 1x as a polynomialin y.Hence,ind all the real roots of the equation f(x) = 0.[3], [10] 07

    22.The polynomial p(x) = 2x + 4x + 12 x k has factor x + 1.(a) Find the value of k.

    (b) Factorise p(x) completely.

    [2], [4] 0823. Find the solution set of the inequality: 4x 1 > 3 3x[10] 08

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    24.Determine the set of values of x satisfying the inequality:xx + 1 1x + 1[4] 09

    25. Find all the values of x if y = |3 x| and 4y (x 9) =24. [9] 0926.The polynomial p(x) = 6x ax bx + 28x + 12, where a and b are real constants,has factors (x + 2 )and (x 2).(a)Find the values of a and b, and hence, factorise p(x)completely.(b)Give that p(x) = (2 x 3)[q(x) 4 1 + 3 x],ind q(x), and determine its range whenx [2,10].

    [7], [8] 09

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    Sequences and Series

    1.Evaluate 110 .

    Express 0.18 as a rational term in lowest form.

    [2], [2] 99-T2. Find the expansion of (1 + x)(1 x) in ascending powers of x until the terms x.(a)If p = q = 12 ,suggest one value of x that enables 1310 to be estimated using the aboveexpansion.Hence,estimate 1310 and write your answers accurate to 4 decimal places.

    (b)If p = 13 , and q lies in the interval [0,9],ind the largest possible coeficient of x.

    [5], [7], [3] 99-T3.Express 2 1 in the form of a2 + b, where a and b are integers.[3] 99-S4.The sum of the irst 2n terms of a series P is 20n4n .Find in terms of n,the sum of theirst n terms of this series.Show that this series is an arithmetic series. Series Q is an arithmetic series such that the sum of its irst n even terms is more than the sumof its irst n odd terms by 4n.Find the common difference of the series.

    If the irst term of series Q is 1,determine the minimum value of n such that the differencebetweenthe sum of the irst n terms of series P and the sum of the irst n terms of series Q is more than 980.[4], [5], [6] 99-S

    5.Simplify 1 + 23 1 23.[4] 00-T

    6. (a) If S denotes the sum of the irst n terms of a geometric series 31+ 13 andS denotes the sum of ininity of this series.Find the smallest n such that: |S S|

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    7. Given that the sum of the irst n terms of a series is n log pq . Show that:(a)The nth term of the series islogpq(b)This is an arithmetic series.[2], [3] 00-S

    8.Expand (1 + x) in ascending power of x until the terms in x . By taking x = 140 ,indthe approximation for 32.8 correct to four decimal places.If the expansion of 1 + a x1 + b x and (1 + x) are the same until the terms in x , ind the values ofa and b.Hence,show that 32.8 203101.

    [7], [8] 00-S

    9.Expand (1 + 8 x) in the ascending power of x until the terms in x . By taking x = 1

    100,

    ind 3 correct to ive decimal places. [4] 01-T10. Given that S = a + a r + a r + + a r, with a 0. Show that S = a(1 r)1 r .Give the condition on r such that lim S exists,and express this limit in terms of a and r.(a)Determine the smallest integer of n such that 1 + 43 + 43

    + + 43 >21.(b)Find the sum of ininity 3(1 x) + 3(1 x) + + 3(1 x) + and determinethe set of values of x such that this sum exists.

    [5], [5], [5] 01-T

    11.The sum and the product of three consecutive terms of an arithmetic progression are3 and 24 repestively.Determine the three possible terms of the arithmetic progression.[5] 01-S

    12.Expand 1 xn where n is a positive integer in ascending power of x until the termsin x. If the coeficient of x is 127 , ind n.With this value of n, obtain the expansion 1 xn (1 x) in ascending powers of x until

    the terms in x

    .

    Hence, by taking x = 110 , ind the approximation of10 accurate to 3 decimal places.[6], [5], [4] 01-S

    13. Determine the set of values of x such that the geometric series 1 + e + e + converges.Find the exact value of x so that the series converges to 2.[6] 02

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    14.Express 14k 1 as partial fraction.Hence, ind a simple expression for S = 14k 1

    and ind lim S.

    [4], [4] 02

    15.Express 1 + x1 + 2 x as a series of ascending powers of x up to terms in x .By taking x = 130 ,ind 62 correct to four decimal places.[6], [3] 03

    16.Express U = 2r + 2r in partial fractions.Using the results obtained,(a)Show that U = 1r + 1r + 1r + 2 + 1(r + 2)(b)Show that U =

    32 1n + 1 1n + 2 ,and determine its values of U

    and U + 13

    .

    [3], [2], [9] 03

    17.Expand (1 x) in ascending powers of x up to the terms in x . Hence, ind the value of7 correct to ive decimal places. [5] 0418. Prove that the sum of the irst n terms of a geometric series a + ar + ar + i sa(1 r)1 r .(a)The sum of the irst ive terms of a geometric series is 33 and the sum of the irst ten terms of a geometric series is 1023. Find the common ratio and the irst term of the geometricseries.(b)The sum of the irst n terms and the sum to ininity of the geometric series

    6 3 +32 are S and S respectively.Determine the smallest value of n such that|S S|

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    [6] 05

    20.Express f(x) = x x 1(x + 2)(x + 3) in partial fractions.

    Hence, obtain an expression of f(x)in ascending order of1x up to term in

    1x.

    Determine the set of values of x for which the expansion is valid. [5], [6], [2] 0521. If x is so small that x and higher powers of x may be neglected,show that:(1 x) 2 + x10 2(27x)

    [4] 06

    22.The nth term of an arithmetic progression is T. Show that U = 52 (2) is the nth term of a geometric progression.If T = 12 (17n14),evaluate U .

    [4], [4] 0623.Express the ininite recurring decimal 0.725 (=0.7252525 )as a fraction in its lowerstterms.[4] 07

    24. In the beginning of this year, Mr. Liu and Miss Dora deposited RM10 000 and RM 2 000

    respectively in a bank. They receive an interest of 4% per annum. Mr. Liu does not make any

    additional deposit nor withdrawal, whereas, Miss Dora continues to deposit RM 2 000 at the

    beginning of each of the subsequent years without any withdrawal.

    (a) Calculate the total saving of Mr. Liu at the end of nth year.

    (b) Calculate the total saving of Miss Dora at the end of nth year.

    (c) Determine in which year the total saving of Miss Dora exceeds the total saving of Mr. Liu.

    [3], [7], [5] 08

    25.For the geometric series 6 + 3 + 32 +,obtain the smallest value of n if the differencebetween the sum of the irst n + 4 terms and the sum of irst n terms is less than 4564.[6] 09

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    Matrices1. If A = 1 42 1 , B = 0 13 2 and C = 3 421 19 , ind matrix X such that AXB = C.[5] 99-S

    2. (a)Given

    M = 2 3 11 0 41 1 1Show that M 3M + 8M 24I = 0. Deduce M.(b)Given matrices:A = 1 2 33 2 13 1 2 and B =

    5 1 71 7 57 5 1 Find AB.Hence solve the simulteneous equation: 5 x + y + 7 z = 8x + 7 y 5 z = 1 6

    7 x 5 y + z = 1 4 [7], [8] 99-S3.Given:P = 1 1 21 2 12 1 1(a)Find R such that R = P 4 P I .(b)Show that PR + 4I = .

    [3], [2] 00-S

    4.If M = 4 2 23 4 32 4 2 and N = 2 2 10 2 32 6 5 , ind MN and NM.Hence, ind M.During the school holidays, a supermarket offers three sales packages A,B and C for shirts, long

    pants and shoes with brand name Tampan. The number of each item and the offer price for each

    package are shown in the following table.

    Sales Packages Number of shirts Number of long

    pants (pairs)

    Number of shoes

    (pairs)

    Offer price (RM)

    A 4 2 2 190

    B 3 4 3 295C 2 4 2 250

    By representing the prices of a shirt, a pair of long pants and a pair of shoes as x, y and z

    respectively, obtain a matrix equation representing the information above.

    Solve the matrix equation you have obtained to determine the price of each item.

    [6], [3], [6] 00-S

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    5. If A = 1 23 4 , B = 1 21 3 , ind C so that A = BCB.[3] 01-S6. Matrices A and B are given as:

    A = 3 2 01 1 14 0 1 and B = 1 2 25 3 34 8 1 Find AB and deduce A.In conjunction with the XXI SEA Games which was held in Kuala Lumpur, Syarikat

    Wawasan sold three types of souvenirs, that is key chain, calculator and pen. The company

    ordered its supply of souvenirs in two types of packets. The cost of a packet which consists of

    three key chains and two calculators is RM45, whereas cost of a packet which consists of a key

    chain, a calculator, and a pen is RM40. The cost of a pen is four times the cost of a key chain. f

    the cost of a key chain, a calculator and a pen are RMx, RMy and RMz respectively, obtain a

    matrix equation to represent the above information. Determine the cost of each type of souvenirsupplied to the company.

    The selling price of each packet of souvenirs is fixed at RM80. If the profit from the sale

    of a pen is RM25, find the profit obtained from the sale of a key chain and the sale of a calculator.

    [4], [7], [4] 01-S7. Determine the values of a, b and c so that the matrix:2b 1 a b2a 1 a bc

    b b + c 2c 1

    is a symmetrical matrix. [5] 028. Matrices M and N are given as:M = 10 4 915 4 145 1 6 and N =

    2 3 44 3 11 2 4Find MN and deduce N.Product X, Y and Z are assembled from three components A, B and C according to different

    proportions. Each product of X consists of two components of A, four components of B and one

    component of C; Each product of Y consists of three components of A, three components of Band two components of C; and each product of Z consists of four components of A, one

    component of B and four components of C. A total of 750 components of A, 1000 components

    of B, and 500 components of C are used. With x, y and z representing the number of products of

    X, Y and Z assembled, obtain a matrix equation representing the information given.

    Hence, find the number of products X, Y and Z assembled.

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    [4], [4], [4] 029. Matrix A is given by:A = 1 2 33 1 1

    0 1 2

    (a)Find the matrix B such that B = A 10I, where I is 3 3 matrix.(b)Find (A + I)B,and hence ind (A + I)B.[3], [6] 0310. Matrix A is given by:

    A = 3 3 45 4 11 2 3Find the adjoint of A.Hence,ind A.[6] 04

    11.The matrices P and Q,where PQ=QP,are given by:

    P = 2 2 00 0 2a b c and Q = 1 1 00 0 10 2 2Determine the values of a, b and c.Find the real numbers m and n for which P = mQ + nI, where I is the 3 3 identity matrix.[5], [5] 0412. A, B and C are square matrices such that BA = B and ABC = (AB). Show thatA = B = C.

    If B =

    1 2 0

    0 1 01 0 1 ,ind C and A. [3], [7] 0513.Determine the value of k such that the determinant of the matrix k 1 32k + 1 3 20 k 2 is 0.[4] 06

    14. If P = 5 2 31 4 33 1 2 and Q = a 1 18b 1 1213 1 c and PQ = 2I, where I is 3 3 identitymatrix,determine the values of a,b and c.Hence ind P.

    Two groups of workers have their drinks at a stall. The first group comprising ten

    workers has five cups of tea, two cups of coffee and three glasses of fruit juice at a total cost ofRM 11.80. The second group of six workers has three cups of tea, a cup of coffee and two

    glasses of fruit juice at a total cost of RM7.10. The cost of a cup of tea and three glasses of fruit

    juice is the same as the cost of four cups of coffee. If the costs of a cup of tea, a cup of coffee and

    a glass of fruit juice are RMx, RMy and RMz respectively, obtain a matrix equation to represent

    above information. Hence, determine the cost of each drink.

    [8], [6] 06

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    15. The matrices A and B are given by:A = 1 2 1 3 1 40 1 2 and B =

    35 19 1827 13 453 12 5

    Find the matrix A

    B and deduce the inverse of A.

    Hence,solve the system of linear equations:x 2 y z = 83 x y 4 z = 1 5y + 2 z = 4[5], [5] 0716. Matrix A is given by:

    A = 1 0 01 1 01 2 1

    (a)Show that A

    =I,where I is 33 identity matrix,and deduce A

    .

    (b)Find matrix B which satisies BA = 1 4 30 2 1 1 0 2.[4], [4] 0817. (a)The matrices P,Q and R are given by:

    P = 1 5 62 2 41 3 2 Q = 13 50 331 6 57 20 15 R =

    4 7 131 5 12 1 11 Find the matrices PQ and PQR,and hence deduce (PQ).(b)Using the result in (a),solve the system of linear equations:6 x + 1 0 y + 8 z = 4 5 0 0

    x 2 y + z = 0x + 2 y + 3 z = 1 0 8 0[5], [5] 09

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    Coordinate Geometry1. Sketch the graph of 100x + 36y = 225. Find the area of quadrilateral formed by joiningthe points of intersection of this curve with the coordinate axis.[4] 99-T

    2. Show that the point Pa(t + 1)2t , b(t 1)2t lies on the curve bx a2y2 = ab.Show that the equation of tangent to curve at P is bx(t + 1) ay(t2 1) = 2abt.This tangent cuts the x axis at A and axis at B. M is the midpoint of OA and O denotesorigin,and H divides BM in the ratio of 2: 1.Find the locus of H where t varies.[3], [4], [8] 99-T3.Find the equations of both straight lines that are inclined at an angle of45 with thestraightline 2x + y 3 = 0 and passing through the point (1,4).

    [5] 00-T

    4.Given two parallel lines, and , passing through (5,0)and(5,0)respectively, and meet theline 4x + 3y = 25 respectively at P and Q. If PQ equals 5 units, ind the possible slopes ofand . [7] 01-T5.Find the equation of the tangent to the curve xy=4 at point P (4,1).Point A is a point on the x axis such that PA is parallel to the y axis. tangent to thecurve xy = 4 at P meets the y axis at point B. The straight line passing through B andparallel to the x axis meets the curve at point Q. Find the coordinates of Q and show thatAQ is a tangent to the curve at Q.

    Find the coordinates of the point of intersection of the tangents to the curve xy = 4 at P and

    Q.[4], [7], [4] 01-T6.Given that PQRS is a parallelogram where P(0,9), Q(2,5), R(7,0)and S(a, b)are points onthe plane. Find a and b.Find the shortest distance from P to QR and the area of the parallelogram PQRS.

    [4], [6] 027. the straight line which passes through the points A(4,0)and B(2,4)intersects they axis at the point P. the straight line is perpendicular to and passes through B. If intersects the xaxis and yaxis at the points Q and R respectively,show thatP R : Q R = 5 3 .

    [8] 038.The sum of the distance of the point P from the point (4,0)and the distance of P from the originis 8 units. Show that the locus of P is ellipse (x 2)16 + y12 =1 and sketch the ellipse.

    [7] 04

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    9. A curve is deined by the parametric equation x = 1 2t, y = 2 + 2t .Find the equationof the normal to the curve at the point A(3,4).The normal to the curve at the point A cuts the curve again at the point B.Find the coordinatesof B. [7], [4] 0410.Find the perpendicular distance from the centre of the circle x + y 8 x + 2 y + 8 = 0to the straight line 3x + 4y = 28. Hence, ind the shortest distance between the circle andthe straight line.

    [7] 0511.Show that x + y 2ax 2by + c = 0 is the equation of the circle with centre (a, b)and radius a + b c.

    The above diagram shows three circles C, C and C touching one another,where theircentres lie on a straight line. If C and C have equations x + y 10x 4y + 28 = 0 andx + y 16x + 4y + 52 = 0 respectively, ind the equation of C.[7] 0612.The coordinates of the points P and Q are(x, y)and xx + y , yx + y respectively,wherex 0, y 0. If Q moves on a circle with centre (1,1)and radius 3,show that the locus of P isalso a circle.Find the coordinates of the centre and radius of the circle. [6] 07

    13.The lines y = 2x and y = x intersect the curve y

    + 7xy = 18 at points A and Brespectively,where A and B lie in the irst quadrant.(a)Find the coordinates of A and B.(b)Calculate the perpendicular distance of A to OB,where O is the origin. (c)Find the area of OAB.[4], [2], [3] 08

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    14.The parametric equations of a straight line are given by x = 4t 2 and y = 3 3t.(a)Show that the point A 1, 34lies on line .(b)Find the cartesian equation of line .(c)Given line cuts the x and y axes at P and Q respectively, ind the ratio PA: AQ.[2], [2], [4] 09

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    Functions1. Sketch the graph of y = |1 2 x|, x R and the graph of y = x, x 0 on the samecoordinate system.

    Solve the inequality|1 2 x

    |< x.

    [3], [4] 99-T

    2.The function f is deined as follows:f : x 4 + (x 1), x R(a)Sketch the graph of f.(b)State the range of f.(c)Determine if f exists.[2], [1], [2] 99-S3.Given that f(x) =log(1 5 2 x x), ind the range of x so that f(x)is deined.Find the maximum of 15 2x x and hence deduce the maximum value of f(x).

    [3], [4] 00-T

    4.The function f is deined as follows:f : x 5 x + 2x 5 , x 5(a)Find f and hence deduce f.(b)Find f(2).[3], [3] 00-S5. Function f is deined by:

    f(x) = (x 1), x 1

    1 ax , x > 1

    If f is continuous at x = 1, determine the value of a and sketch the graph of f. [5] 01-T6.Express the function f:x12 x 1 + 12 x + 1 , x R in the form that does not involvethe modulus sign.Sketch the graph of f and determine its range. [7] 01-S7. The function f is deined by:

    f : x 3 x + 1 , x R , x 13Find f and state its domain and range.

    [4] 028. The function f is deined by:f(x) = 1 + e, x < 13 , x = 12 + e x , x > 1(a)Find lim f(x) and lim f(x) . Hence determine whether f is continuous at x = 1.(b)Sketch the graph of f.

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    [4], [3] 039. The function f is deined by:f(x) = x 1x + 2 , 0 x < 2

    ax 1 , x 2

    where a R. Find the value of a if lim f(x) exists. With this value of a, determine whether fis continuous at x = 2.[6] 0410. The function f and g are given by:

    f : x e ee + e and g: x 2e + e(a)State the domain of f and g.(b)Without using differentiation,ind the range of f.(c)Show that [f(x)]

    + [g(x)]

    =1.Hence,ind the range of g.

    [1], [4], [6] 0511.Function f,g and h are deined by:f : x xx + 1 , g : x x + 22 , h : x 3 + 2x(a)State the domain of f and g.(b)Find the composite function g f and state its domain and range.(c)State the domain and range of h.(d)State whether h= gf .Give a reason for your answer.[2], [5], [2], [2] 06

    12. The function f is deined by:

    f(x) = x 1 , 1 x < 1|x| 1,otherwise (a)Find lim f(x) , l i m f(x) , l i m f(x) and lim f(x)(b)Determine whether f is continuous at x = 1 and x = 1.[4], [4] 0713. The function f and g are deined by:

    f : x 1x , x R \{0}

    g : x 2 x 1 , x R

    Find fg and its domain. [4] 0814. Given x > 0, (x) = x,ind lim f(x) f(x + h)h .[4] 09

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    Differentiation

    1.If x = sin2 and y = cos2,ind dydx in terms of .[6] 99-T

    2.If y = (2e 6 x + 5), show that:y dydx + dydx = e[4] 99-S3. A composite solid consists of a cuboid and a semicylindrical top with a common face ABCD.

    The breadth and length of the cuboid is x cm and 2x cm respectively and its height is y cm.Given that the total surface area of this solid is 2400cm. Show that:

    y = 124x [9600 (8 + 5 )x]

    If the volume of this solid is V cm

    , express V in terms of x. Hence, show that V attains itsmaximum when x = 404 + .Find this maximum value.

    [3], [9], [3] 99-S4. Find the gradient of the curve 2x + y + 2xy = 5 at the point (2,1).[3] 00-T5. The equation of a curve is:

    y = x 3 + 2Find the asymptotes and the stationary points of the curve.

    Sketch the curve.Determine the number of real roots of the equation:k(x 1)(x 2) = xwhere k > 0.[8], [4], [3] 00-T

    6.Find dydx in terms of x if x = e abd y = e .[4] 00-S

    7. The equation of a curve is y = e

    1e + 1 where k is a positive constant.

    (a)Show that dydx > 0 .(b)Show that dydx + kx = k. Hence, show that dydx 0 for x 0 and dydx 0 for x 0.(c)Sketch the curve.

    [3], [8], [4] 00-S

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    8. Given that y = sinkx1+coskx , where k is a positive integer. Show that:(sinkx) dydx = ky

    [4] 01-T

    9.The graph y = x + ax +bx+c passes through (3,21)and has stationary points whenx = 2 and x = 2. Find a, b and c.Find the coordinates of these stationary points and determine if they are local extremums. Find also the point of inlexion of the curve.Determine the set of values of x so that dydx < 0.

    [5], [7], [3] 01-T10. A curve has parametric equations x = e 2t and y = e + t. Find the gradient of thecurve at the point with t = ln 2.[5] 01-S

    11.A curve with equation y = x + px + qx + r cuts the y axis at y = 34 and hasstationary points at x = 3 and x = 5. Find the values of p, q and r.Show that the curve cuts the xaxis only at x=1,and ind the gradient of the curve at thatpoint.Sketch the curve.[6], [7], [2] 01-S

    12. Given that y = e cosx,ind dydx and dydx when x = 0.[4] 02

    13. Function f is deined by:f(x) = 2x(x + 1)(x2)Show that f(x) < 0 for all values of x in the domain of f.Sketch the graph of y = f(x). Determine if f is a one to one function. Give reason to youranswer.Sketch the graph y= |f(x)|.Explain how the number of roots of the equation

    |f(x)| = k(x 2)depends on k.

    [5], [6], [4] 02

    14.I f y = ln xy , ind the value of dydx when y = 1.[5] 03

    15. Sketch, on the same coordinate axes, the graphs y = e and y = 21 + x . Show that theequation (1 + x)e 2 = 0 has a root in the interval [0,1].

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    Use the Newton Raphson method with the initial estimate x =0.5 to estimate the rootcorrect to three decimal places.[7], [6] 0316. Using the sketch graphs of y = x and x + y = 1, show that the equation x + x 1 = 0

    has only one real root and state the sucessive integers a and b such that the real root lies in

    the interval (a, b).Use the Newton Raphson method to ind the real root correct to three decimal places.[4], [5] 04

    17. If y = cosxx , where x 0, show that x dydx + 2 dydx + x y = 0 .[4] 05

    18.Find the coordinates of the stationary point on the curve y=x + 1x , where x > 0;

    give the xcoordinate and ycoordinate correct to three decimal places.Determinewhether the stationary point is a minimum point or maximum point. The x coordinate of the point of intersection of ther curves y = x + 1x and y = 1x,where x > 0, . 0.5 < < 1.Use the Newton Raphson method to determinethe value of p correct to three decimal placesand hence, ind the point of intersection.[5], [9] 0519. If y = x ln (x + 1),ind an approximation for the increase in y when x increases by x. Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931.

    [6] 06

    20. The function f is deined by f(t) = 4e 14e + 1 ,where k is a positive constant.(a)Find the value of f(0).(b)Show that f(t) > 0.(c) Show that k{1 [f(t)]} = 2f(t)and hence, show that f(t) < 0.(d)Find lim f(t).(e)Sketch the graph of f.[1], [5], [6], [2], [2] 06

    21. If y = x1 + x , show that x dydx = (1 x)y

    [4] 07

    22. Find the coordinates of the stationary points on the curve y = x 1 and determinetheir nature.Sketch the curve.Determine the number of real roots of the equation x = k(x 1), where k R, when k

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    varies.

    [10], [4], [3] 07

    23. If y = sinxcosxsinx+cosx , show that dydx = 2y dydx.[6] 08

    24.Show that the gradient of the curve y= xx 1 is always decreasing.Determine the coordinates of the point of inlexion of the curve ,and state the intervals whichfor curve is concave upward.Sketch the curve.[3], [5], [3] 0815. The line y + x + 3 = 0 is a tangent to the curve y = px + qx, where p 0 at the pointx = 1. Find the values of p and q.

    [6] 09

    16.A curve is deined by the parametric equations:

    x = t 2t and y = 2t + 1t where t 0.(a)Show that dydx = 2 5t + 2 and hence, 12 < dydx < 2.(b)Find the coordinates of points when dydx = 13.

    [8], [3] 09

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    Integration

    1. Show that: x(x 1) dx = 118 +lnx .[6] 99-T

    2. Function f is deined by:

    f(x) = x(x ), 0 x < 2 sin(x),2x3(a) Sketch the graph of f.(b) Find the range of f.(c)Determine whether f is a one to one function.Give reasons for your answer.(d)Find the area of the region bounded by graph f and the x axis.[4], [3], [2], [6] 99-T

    3. If f(x) = x

    2 5,ind f(x).Hence,evaluate:

    x(x5)2 x 5 dx [5] 99-S4.Sketch the curve y=x(x 3)(x + 2)If A and A respectively denote the area of the regions bounded by the curve and thex axis above and below x axis, ind A: A.[5] 00-T

    5.Sketch the graphs y = 4,y = 8x and y = 1x.Calculate the volume of the solid of revolution when the region bounded by the above graphs

    is rotated through 360 about yaxis. [5] 00-S6. The function f is deined by:f(x) = 2 |x 1|, x < 3x 9 x + 1 8 , x 3(a)Sketch the graph of f.

    (b)Evaluate f(x)dx

    (c)Determine the set of values of x such that f(x) > 1 x6.

    [5], [5], [5] 00-S7. A curve has equation y = x(4 x).Show that for any point (x, y)lying on the curve, then 2 x 2 and 2 y 2.Sketch the curve.Calculate the area of the region bounded by the curve.Calculate the volume of solid of revolution when the region bounded by this curve and

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    y=x in the irst quadrant is rotated through 360 about the xaxis.[3], [3], [4], [5] 01-T

    8. Find the value of x

    (5x

    + 1) dx

    .

    [6] 01-S9. Sketch on the same coordinate system the curve y = 4x , and y = 4(x 1).Find thecoordinates of the point of intersection of the two curves.Show that the area of the region bounded by y = 4x , y = 4(x 1)and y = 4 is203 4ln2.Calculate the volume of the solid of revolution when the region is rotated through 360 about

    yaxis.

    [6], [4], [5] 01-S10. By using suitable substitution, ind 3 x 1x + 1 dx.[5] 0211. Find the point of intersection of the curves y = x + 3x and y = 2x x 5x.Sketchon the same coordinate system these two curves.Calculate the area of the region bounded by the curve y = x + 3x and y = 2x x 5x.

    [5], [6] 02

    12. Using the substitution u = 3 + 2 sin , evaluate cos

    (3 + 2 s i n ) d

    [5] 0313.The curve y= a2 x(b x),where a0,has a turning point at the point (2,1).Determinethe values of a and b.Calculate the area of the region bounded by the x axis and the curve.Calculate the volume of the solid formed by revolving the region about the xaxis. [4], [4], [4] 03

    14. Show that lnxdx

    = 1.[4] 0415. Sketch, on the same coordinate axes, the line y = 12 x and the curve y = x.Find thecoordinates of the point of intersection.

    Find the area of the region bounded by the line y = 12 x and the curve y = x.Find the volume of the solid formed when the region is rotated through 2 radians about the

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    yaxis.[5], [4], [4] 0416. Sketch, on the same coordinate axes, the curves y = e and y = 2 + 3e.Calculate the area of the region bounded by the y axis and the curves.

    [2], [6] 05

    17. Using trapezium rule, with ive ordinates, evaluate 4 xdx. [4] 06

    18.Express 2 x + 1(x + 1)(2 x) in the form A x + Bx + 1 + C2 x ,where A,B and C are constants.Hence,evaluate 2 x + 1(x + 1)(2 x) dx .

    [3], [4] 06

    19.Find (a) x

    + x + 2x + 2 dx (b) xe dx [3], [4] 0720.The gradient of the tanget to a curve at any point(x, y)is given by dydx = 3 x 52x ,where x > 0. (1,4),(a)Find the equation of the curve.(b)sketch the curve.(c)Calculate the area of the region bounded by the curve and x axis.[4], [2], [5] 07

    21. Show that (x 2)x dx = 53 + 4 l n 23.[4] 0822.Sketch, on the same coordinate axes,the curves y = 6 e and y = 5e, and ind thecoordinates of the point of intersection.Calculate the area of the region bounded by the curves.Calculate the volume of solid formed when the region is rotated through 2 radians aboutthe xaxis.

    [7], [4], [5] 08

    23. Using an appropriate substitution, evaluate x(x 1)dx

    . [7] 0924. Given a curve y = x 4 and a straight line y = x 2,(a)Sketch,on the same coordinate axes,the curve and the straight line.(b)Determine the coordinate of their point of intersection.(c)Calculate the area of the region R bounded by the curve and the straight line.

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    (d)Find the volume of the solid formed when R is rotated through 360 about xaxis. [2], [2], [4], [5] 09