MAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists

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1 Data provided: Formula sheet MAS53/MAS59 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics (Materials Mathematics For Chemists Spring Semester hours All questions are compulsory. The marks awarded to each question or section of question are shown in italics. Solve the equation 3 tan 2 x 2 sec 2 x = for 0 < x < π. (5 marks 2 Factorise x 2 + 8x + 5. (2 marks 3 Solve 0.x 2 +.x 2.8 = 0 for x. (4 marks 4 (i A line with gradient 2 passes through the point P(5,7. Find the x- 3 coordinate of the point Q (x, 3 on the line. (3 marks (ii (iii Show that the line through the points A(8,0 and B(4,8 is perpendicular to line through the points C (6,0 and D(0,2. (3 marks Show that x 2 + y 2 x y = is an equation of a circle. Find the centre and radius of the circle. (5 marks (iv Find the equations of the tangent and the normal to the circle x 2 + y 2 5x + 2y = 3 at the point (2,-3. (8 marks MAS53/MAS59 Turn Over

2 MAS53/MAS59 5 Solve log 2 (x + + log 2 (x = (3 marks 6 Find the inverse function, f (x, of the one-to-one function f(x = x 3 2 valid for all real x. State the domain and range of f (x. (5 marks 7 A car makes a round trip journey from point A to point B, these points being separated by 40 km. The car drives at 40 km per hour from point A to point B, but returns at 60 km per hour. Find the average speed of the car. Show that the average speed is given by the harmonic mean of the speeds in each direction. (5 marks 8 If y = ( + x 2 + 4x 3 / sin x, then nd dy/dx. (4 marks 9 Evaluate (x + 2 dx (3 marks 0 (i Showing your working clearly, nd the coecient of x 2 in the expansion of ( + x 4. (2 marks (ii (iii Use the rst four non-zero terms in the binomial theorem to nd an approximation to 3.03, keeping eight decimal places in your answer. (3 marks Use the binomial theorem to evaluate [ lim x + 2 ] x 2 + 4x + 8. (3 marks x A line L passes through the points (,, 2 and (3,,, and a line L 2 passes through the points ( 2, 0, and (4, 8, 2. Show that the lines intersect, and nd the coordinates of the point of intersection. (6 marks 2 Let y = sinh x and z = e y. Show that and hence show that z 2 2xz = 0, ( sinh x = ln x + x 2 +. (5 marks MAS53/MAS59 2 Continued

3 MAS53/MAS59 3 (i If y = ln(cosh x, show that (ii If y = dy dx = tanh x. sinh x cosh 2x cosh 2 x + sinh 2 x dy nd dx. (2 marks (2 marks 4 Evaluate 4x 2 3x + 3 dx. (x + (x 2 (7 marks 4x 5 5 Find the Maclaurin series for cosh 2x, as far as the term in x 4. (4 marks 6 Use de Moivre's theorem to show that cos 3θ = 4 cos 3 θ 3 cos θ, and to express sin 3θ in terms of sin θ. (7 marks 7 A set of linear equations can be written as x A y = z 0 6 4, where A = Find the inverse, A, of A, and use it to nd the values of x, y and z which satisfy the equations. (9 marks End of Question Paper MAS53/MAS59 3

4 Formula Sheet for MAS53/MAS57/MAS59 Examination These results may be quoted without proof, unless proofs are asked for in the question. Trigonometry For any angles A and B sin 2 A + cos 2 A = cos(a ± B = cos A cos B sin A sin B sin(a ± B = sin A cos B ± cos A sin B tan(a ± B = sin 2A = 2 sin A cos A tan A ± tan B tan A tan B cos 2A = 2 cos 2 A = 2 sin 2 A Coordinate Geometry The acute angle α between lines with gradients m and m 2 satisfies tan α = m m 2 + m m 2 (m m 2 while the lines are perpendicular if m m 2 =. The equation of a circle centre (x 0, y 0 and radius a is (x x (y y 0 2 = a 2. Hyperbolic Functions cosh 2 x sinh 2 x = sech 2 x + tanh 2 x = cosh 2 x + sinh 2 x = cosh 2x 2 sinh x cosh x = sinh 2x cosh 2 x = ( + cosh 2x/2 sinh 2 x = ( cosh 2x/2 4

5 Differentiation Function (y x n sin ax cos ax tan ax e ax ln(ax ln f(x sinh x cosh x tanh x sin x Derivative (dy/dx nx n a cos ax a sin ax a sec 2 ax ae ax x f (x f(x cosh x sinh x sech 2 x x 2 cos x x 2 tan x sinh x + x 2 x2 + cosh x x2 tanh x x 2 NB. It is assumed that x takes only those values for which the functions are defined. 5

6 For u and v functions of x, and with u = du dx, v = dv dx, d dx (uv = uv + vu, while For y = y(t, t = t(x, d ( u = vu uv. dx v v 2 dy dx = dy dt dt dx. Integration In the following table the constants of integration have been omitted. Function f(x Integral f(x dx x n x n+ n + n ae ax x a sin ax a cos ax e ax ln x cos ax sin ax a tan ax ln sec ax a 2 + x 2 a tan a a 2 x 2 a tanh a sin a2 x 2 a x2 + a 2 x2 a 2 f (x f(x sinh a cosh a ln f(x 6

7 Integration by parts or uv dx = (integral of V u (integral of V du dx dx u dv dx = uv dx v du dx dx. Series Binomial Theorem: ( + x n = + nx + where ( n = r n(n x ! n(n (n 2 (n r + r! ( n x r + r If n is a positive integer, the series terminates and is convergent for all x. If n is not a positive integer, the series is infinite and converges for x <. Taylor expansion of f(x about x = a is f(a + (x af ( (a + Maclaurin expansion of f(x is (x a2 f (2 (a + + 2! f(0 + xf ( (0 + x2 2! f (2 (0 + + xn n! f (n (0 + Alternating Series Test (x an f (n (a + n! The series a a 2 + a 3 a 4 +, where a, a 2, a 3, a 4,... are all positive, converges if a > a 2 > a 3 > and a n 0 as n. Ratio Test If the series a + a 2 + a 3 + a 4 + satisfies lim a n+ n a n = λ, then. if λ >, the series diverges, 2. if λ <, the series converges. 7

8 Powered by TCPDF ( Vectors If vectors a and b are given in Cartesian component form by a = (a, a 2, a 3 and b = (b, b 2, b 3, then the scalar product a b is given by a b = a b + a 2 b 2 + a 3 b 3 and the vector product a b is given by i j k a b = a a 2 a 3 = (a 2 b 3 a 3 b 2, a 3 b a b 3, a b 2 a 2 b. b b 2 b 3 If a plane passes through a point with position vector a, and is normal to the vector n, then the equation of the plane is where r = (x, y, z. r n = a n, 8