AS MATHEMATICS HOMEWORK C1

Transcription

1 Student Teacher AS MATHEMATICS HOMEWORK C September 05 City and Islington Sixth Form College Mathematics Department

2 HOMEWORK INTRODUCTION You should attempt all the questions. If you find the work difficult then get help [lunchtime workshops room 6, online, friends, teacher etc]. You should expect to spend 4 hours on maths homework per week. Complete homework task set by the teacher. Review notes, read your textbook, consult websites. Revise for exams and progress tests. Complete pre learning and watch relevant videos. Use A4 lined or squared paper. Write your name, title of homework, date, teacher. Work in pen or pencil. Show appropriate working then mark your answers with a tick, cross or? If you have any questions or feedback write this on the homework. This is your chance to make a good first impression thoughtful work clearly presented please. Pre- learning is included in each homework this is the work you will need to complete to be ready for the next lesson! Make sure you do it! Topic HW Differentiation Date completed Comment HW HW HW4 HW5 HW6 HW7 HW8 HW9 Graph sketching Quadratic Equations Differentiation tangents and normals Sequences Practice Test Coordinate Geometry Integration Transformations HW0

3 HW DIFFERENTIATION Your first Maths homework! It is important to get into good habits right away. Read the notes you made in class and try to remember what was said about the background to the topic. Exercise A Key words: Function, derivative, gradient, calculus, differentiate. Do some research - who invented (or discovered) calculus? When did they do it? Why did they do it?. Differentiate the following functions to find a) = + b) = +5 HW c) 4 4 y = x d) y = 8x 6 e) y = x + 7x+ f) y = 5x + x. Expand and simplify then find ( ) 4 a) ( )=( 4) b) f ( x) = ( x + )( x + 5)( x 6) 6 7x + 5x c) f ( x) = d) f ( x) = x(x + 5x) x 4. =( +)( ) a) Sketch the curve. [use a graph drawing program is you are unsure] b) Expand the function. c) Find the derivative of the function. d) Complete the table (right). e) What do you notice about the graph when =0? Do your answers for the gradient make sense in terms of your sketch. Are they likely to be correct? Exercise B more challenging!. Differentiate with respect to x (a) 4 + 6, dy (b) (c) ( x + 4) x 6 x + 7x x

4 Exercise C Exam questions. [C Jan 007 Q7] Given that y = 4x + x, x > 0, find. [C Jan 0 Q] d y. (4) The curve C has equation y = x 8 x + 5, x 0. (a) Find d y, giving each term in its simplest form. (). Go to Click C Edexcel -> 6. Differentiation Try: Matching O-test and have a look at some of the other resources. Go to Pre-learning under C tab on and complete week pre-learning Answers EXA. a) dy dy = 9x 8 +6x b) = e) dy dy = x f) = 0x x c) dy 4 = d) dy 5 = 4x x. a) f '( x) = x 8 b) f '( x) = x + 4 x c) f '( x) = 8x + 5 d) 5 f ' ( x) = 5x + x 4. a) b) y = x x + 4 dy c) = x 6x d) x y dy e) dy = 0 at the turning points of the graph. dy EXB a) = 4x + x b) dy = 6 x c) dy 5 + x = x 4

5 EXC..

6 HW GRAPH SKETCHING HW Key words: Function, axis, gradient, intersection, parabola, cubic, reciprocal. Exercise A. Find the equations of the lines from two given points. (a) A(, 4) B(6, 6) (b) A(-,) B(, -) (c) A(, -) B(4, 8) (d) A(-, 0) B(9, ) Exercise B exam questions. [C May 0 Q9] The line L has equation 4y + = x. The point A (p, 4) lies on L. (a) Find the value of the constant p. () The line L passes through the point C (, 4) and is perpendicular to L. (b) Find an equation for L giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5). [C Jan 0 Q5] The curve C has equation y = x(5 x) and the line L has equation y = 5x + 4. (a) Sketch C and L on the same diagram, showing the coordinates of the points at which C and L meet the axes. (4) *(b) Use algebra to show that C and L do not intersect. (4). [Jan 0 Q0] 4x + 8x + a(x + b) + c. (a) Find the values of the constants a, b and c. () (b) Sketch the curve with equation y = 4x + 8x +, showing clearly the coordinates of any points where the curve crosses the coordinate axes. (4) Go to Pre-learning under C tab on and complete week pre-learning Answers EXA.a) y = x + b) y = 4 x + c) 5 x y = 0 d) x 6 y + = 0 EXB. (a) p = 9.5 (b) x + y 8 = 0 6

7 . (a) (b).

8 HW QUADRATIC EQUATIONS Key words solve, factors, factorise, equation, quadratic, inequality, discriminant, complete the square, Solve the following quadratic equations by method of factorising. HW For example solve Exercise A x (x )( x 5) = 0 x = 0 x = x + 5 = 0 or or x = 5 x 5 = 0. x + 7 x + = 0. x + 8 x + = 0. x x + 4 = 0 4. x x + 4 = 0 5. x + x 4 = 0 6. x 0 x 9 = 0 7. x 64 = 0 8. x 9 x = 0 9. x + 7 x + 0 = 0 0. x + x = ( x 5). 9 x + 5 =. ( x 5 ) = 8 9 x x Exercise B By sketching a graph solve the following inequalities: >0. 5 <0. 6> <0 Exercise C exam questions. [C May 0 Q4] Solve the simultaneous equations x + y = 4y x = (7). [C May 00 Q4] (a) Show that x + 6x + can be written as (x + p) + q, where p and q are integers to be found. () 8

9 (b) Sketch the curve with equation y = x + 6x +, showing clearly any intersections with the coordinate axes. () (c) Find the value of the discriminant of x + 6x +. (). [C Jan 008 Q8] The equation x + kx + 8 = k has no real solutions for x. (a) Show that k satisfies k + 4k < 0. () (b) Hence find the set of possible values of k. (4) 4. [C Jan 009 Q7] The equation kx + 4x + (5 k) = 0, where k is a constant, has different real solutions. (a) Show that k satisfies k 5k + 4 > 0. () (b) Hence find the set of possible values of k. (4) 5. [C May 0 Q8] where p and q are integers. 4x 5 x = q (x + p), (a) Find the value of p and the value of q. () (b) Calculate the discriminant of 4x 5 x. () (c) Sketch the curve with equation y = 4x 5 x, showing clearly the coordinates of any points where the curve crosses the coordinate axes. () Go to Pre- learning under C tab on and complete week pre learning Answers EXA. -, -4. -, -6., 8 4., 5. 4, , 7. -8, , 9 9., , , 4., 4 EXB. <, >. 0< <5. < 6, >6 4. < < EXC. =, = ; =5, =. p=,q= b. check desmos c. = 8. (b) 8 < k < (b) k < or k > (a) p =, q = - (b) 4 (c) (0, 5)

10 HW4 DIFFERENTIATION tangents and normals HW4 Key words: Derivative, gradient, tangent, normal, differentiate Remember to check your work as you go along. Mark your own work using the following symbols: correct, wrong (but don t know why), no idea?? Exercise A. ( )= 8 +0 (a) Find f '( x) at the point =4 [This is the same as f '(4) ] (b) Write f ( x ) in the form ( )=( + ) + (c) What do your answers to parts a and b tell you about the graph?. =( +)( +)( ) (a) (b) (c) (d) (e) (f) (g) Sketch the graph of, giving the coordinates of where the curve crosses the axes [Your sketch should be half A4 page and carefully drawn] Find the y coordinate where = and show it on your graph. Expand the expression for. Differentiate y in terms of. Find the gradient at =. Work out the equation of the tangent at = and write it in the form = +. Draw the tangent on the graph using the equation you have just found. 4. (a) Find the equation of the tangent to the graph y = + 5x at the point x =. x (b) Find the equation of the normal to the graph = at the point =. Exercise B exam questions. [C May 005] The curve C has equation y = x 4x + 8x +. The point P has coordinates (, 0). (a) Show that P lies on C. () (b) Find the equation of the tangent to C at P, giving your answer in the form = +, where and are constants. (5) Another point Q also lies on C. The tangent to C at Q is parallel to the tangent to C at P. (c) Find the coordinates of Q. (5) 0

11 . [C Jan 006] y C 4 P O Q x The figure above shows part of the curve C with equation =( )( 4). The curve cuts the x-axis at the points P, (,0) and Q, as shown in the figure above. (a) Write down the x-coordinate of P, and the x-coordinate of Q. () (b) Show that d y = x x 4 () (c) Show that y = x + 7 is an equation of the tangent to C at the point (, 6). () The tangent to C at the point R is parallel to the tangent at the point (, 6). (d) Find the exact coordinates of R. (5). [Jan 0 Q] The curve C has equation y = x 8 x + 5, x 0. (a) Find d y, giving each term in its simplest form. () The point P on C has x-coordinate equal to 4. (b) Find the equation of the tangent to C at the point P, giving your answer in the form y = ax + b, where a and b are constants. (4) The tangent to C at the point Q is parallel to the line with equation x y + 8 = 0. (c) Find the coordinates of Q. (5) Go to Pre- learning under C tab on and complete week 4 pre learning

12 Answers EXA. a) 0 b) f ( x) = ( x 4) 6 c) The gradient at the minimum point is 0 and the minimum point is (4,-6). b) y = c) y = x + 5x 4x 4 dy d) = 9x + 0x 4 dy e) = 5 f) y = 5x. a) (, 9) = 4 x + y b) (,) x 4 y + 6 = 0 EXB. b) y = 7x +. a) P is, Q is. c) 46 5, d) (, 0). (a) = 4 (x > 0) (b) y = 6x + (c) (9, )

13 HW5 SEQUENCES Complete on a separate sheet of paper and show clear working. Mark using the answers below. Key words Sequence, Arithmetic, Geometric, Converge, Diverge, Oscillating, Periodic, Increasing, Decreasing, Recurrence relation Read the chapter on Sequences in your text book p9-04 HW5 Formulae n n = U n = a+( n ) d, S n = [ a + ( n ) d ], S n [ a + l] Exercise A. Write down the first five terms in each sequence a) u n = 5 n+ n =,,, b) u n = n n =,,, n c) u = 6 + ( ) n =,,, d) n u ) n n = ( n =,,,. Write down the first five terms in each sequence a) u n+ = u n + u 4 b) un = u n = + u 5 = n+ n c) u = u u d) = un u n+ = u = 0. Calculate the follow sums written in Sigma notation 5 a) r r= 7 b) r r= r c) 5r d) ( ) r r= r= Exercise B recurrence relations. In the recurrence relations below find u, u, K u5 for different starting values of u. Use your calculator with ANS button! a) = +5 =0 or = un + b) un + = u = or u = 8 0 u n + c) u n u + = u = or u = 8 do you recognise this number? n Exercise C

14 . Write out the proof for the sum of an Arithmetic Sequence (and commit to memory) n S n = [ a + ( n ) d ] see page 07 in text book.. Given the sequence, 7,,. Calculate the 9 th term U9 and the sum of 9 terms S 9. In the arithmetic sequence, 7,, a) Calculate u 0 b) Calculate S 0 4. In the arithmetic sequence, 5, 8,. 50. Calculate the number of terms. 5. The nth term of an arithmetic sequence is given by u n = 5+ n a) Write down the first terms of the sequence. b) Calculate the sum of the first terms. 6. The 5 th term of an arithmetic sequence is and the th term is 80[ Hint - form two simultaneous equations and solve them] a) Calculate the first term a, and the common difference d. b) Calculate the sum of the first terms 7. There are 0 terms in an arithmetic sequence. First term is -7, last term 40 a) Calculate the common difference d. b) Calculate the sum of the first 0 terms S 0 Exercise D exam questions. [C May 008 Q7] Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by km. (a) Show that on the 4th Saturday of training she runs km. () (b) Find an expression, in terms of n, for the length of her training run on the nth Saturday. () (c) Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km. () On the nth Saturday Sue runs 4 km. (d) Find the value of n. () (e) Find the total distance, in km, Sue runs on Saturdays in n weeks of training. () 4

15 . [C Jan 0 Q4] A sequence u, u, u,..., satisfies u n + = u n, n. Given that u = 9, (a) find the value of u and the value of u 4, () (b) evaluate u r. 4 r = () Go to Pre- learning under C tab on and complete week 5 pre learning Answers. a) 7,, 7,, 7 b), /4, /9, /6, / 5 c) 4, 8, 4, 8, 4 d) -,9,-7,8,-4. a)4, 4, 44, 4, 404 b) 5, -7, 7, -, 65 c),, -,, -, d) 0,5,.5,.5, a) 45 b) 0 c) 8 d) /60 Part B 0,5,5,5,75 -,-,,,7,.7,.09,.7,.5 8, 8.7, 9.869,.040, 6.796,.5,.47,.44,.44 8, 4.5,.05,.586,.44,.44 <- it s the square root of Part C. See book. 5, n= ,, a= d= Part D. b) U n = + n d) n = 0 e) 480 km. a) =7, = b) =64

16 HW6 PRACTICE FOR WEEK 6 TEST HW6. (a) Expand and simplify (5 + ) (5 ). () (b) Express in the form a + b, where a and b are integers. (). (a) Write down the value of. (b) Simplify. 5 () (). Given that, =4 +5 >0, find (a), If instead, = +, >0 (b) Find, giving each term in its simplest form. () () 4. (a) By eliminating y from the equations = + =5 show that +6 5=0. (b) Hence, or otherwise, solve the simultaneous equations = + =5 giving your answers in the form a ± b 4, where a and b are integers. () (5) 5. (a) Show that x - 4x + 0 can be written as (x + p) + q, where p and q are integers to be found. () (b) Sketch the curve with equation y = x - 4x + 0, showing clearly any intersections with the coordinate axes. () 6. The curve C has equation y = (x - 5) (x + 9). (a) Sketch C, showing the coordinates of the points at which C meets the axes. (4) 6

17 7. Given that the equation kx + kx = 0, where k is a constant, has real roots, (a) show that k + k > 0. (b) Hence find the set of possible values of k. () () 8. Varafoukis invested some Drachmas in the Tsipras bond scheme in 008 (year ) and withdrew his funds in 05 on the collapse of the Drachma (year 8). The payments from the bond scheme formed an arithmetic sequence with first term and common difference. Given that the bond paid out 55 Drachma in 00 and 800 in 05, find (a) (b) (c) the value of. the value of. the total pay outs of all the bonds. 9. A sequence x, x, x,... is defined by () () () x =, x = a n + x n -, n, where a is a constant. (a) Write down an expression for x in terms of a. (b) Show that x = a - a -. Given that x = -4 (c) find the possible values of a. () () () 0. The curve C has equation =0+5, x > 0. The point P on C has x-coordinate equal to. (a) Show that the equation of the tangent to C at the point P is y = 6x + 4. (6) Go to Pre-learning under C tab on and complete week 6 pre-learning

18 Answers a) b) a = 0, b= a) b) 5y a) = + b) 5a) +6 b) check Desmos! 6) check Desmos! 7b) <, >0 8a) d=0 b) a = 49 c) a) = 9c) =, = 4b) a= -, b= 8

19 HW7 COORDINATE GEOMETRY Complete on a separate sheet of paper and show clear working. Mark using the answers below. Key words Differentiation, gradient function, gradient, tangent, normal, gradient, perpendicular, reciprocal, midpoint, distance, gradient, perpendicular, normal This work is covered in your text book pages If you are in difficulties read the book, talk to a friend, visit Maths workshop (lunchtime room 6). Exercise A. Using the points A(,) B(6,4) calculate the following. You may find a sketch helpful. a) Coordinates of the Midpoint b) Distance between points HW7 b) Gradient of the line d) Gradient of perpendicular line. The points A(, ) B(0, 7) C(5, -) are vertices of a triangle. Plot the points on square paper a) Calculate gradient AB and gradient AC. What do you notice? b) Find the length of the shortest side? c) Calculate the area of the triangle as an exact answer. Exercise B exam questions. [Jan 007 Q8] The curve C has equation y = 4x + x x, x > 0. (a) Find an expression for. () (b) Show that the point P(4, 8) lies on C. () (c) Show that an equation of the normal to C at the point P is The normal to C at P cuts the x-axis at the point Q. y = x + 0. (4) (d) Find the length PQ, giving your answer in a simplified surd form. (). [C May 0 Q] The points P and Q have coordinates (, 6) and (9, 0) respectively. The line l is perpendicular to PQ and passes through the mid-point of PQ.

20 Find an equation for l, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5). [C Jan 0 Q6] Figure The line l has equation x y + = 0. (a) Find the gradient of l. () The line l crosses the x-axis at the point A and the y-axis at the point B, as shown in Figure. The line l is perpendicular to l and passes through B. (b) Find an equation of l. () The line l crosses the x-axis at the point C. (c) Find the area of triangle ABC. (4) Go to Pre- learning under C tab on and complete week 7 pre learning Answers Part A. a) 4, b) 5 c) d).a) /, - (AB and AC are perpendicular lines) b) 5 c) 0 sq units Part B. (a) x 4x (d) x +y -9 =0. (a) (b) y 4 = x (c) 7 0

21 HW 8 INTEGRATION HW8 Key words: Differential equation, integrate, anti-derivative, general solution, particular solution Read pages This topic is called Integration but is also about solving differential equations. This is some of the notation you will see. =, h = + =, h = + + = + Exercise A family of solutions Find an expression for y.. =. =5. =8 4. Solve to find =6 5. = 5 6. ( )=0 Integrate the following 7. ( 5) 8. 7 Exercise B Solve to find the particular solution y passing through the given point.. Exercise C = +5, (,7). = 8, (,8). [C Jan 008 Q9] 8 The curve C has equation y = f(x), x > 0, and f (x) = 4x 6 x + x. Given that the point P(4, ) lies on C,

22 (a) find f(x) and simplify your answer. (6) (b) Find an equation of the normal to C at the point P(4, ). (4). [June 008 Q] The gradient of a curve C is given by ( x + d y = x ), x 0. (a) Show that d y = x x. () The point (, 0) lies on C. (b) Find an equation for the curve C in the form y = f(x). (6) Differential Equations You have just been solving your first differential equations. More complicated versions of this are used for modelling in Engineering, Physics, Chemistry, Economics and Meteorology. Here are some important examples that you can read about. This list is taken from the Wikipedia article on Differential Equations. Physics and engineering Newton's Second Law in dynamics (mechanics) Radioactive decay in nuclear physics Newton's law of cooling in thermodynamics The wave equation Maxwell's equations in electromagnetism The heat equation in thermodynamics Einstein's field equation in general relativity The Schrödinger equation in quantum mechanics The Navier Stokes equations in fluid dynamics The Lorenz equations whose solutions exhibit chaotic flow. Economics The Black Scholes PDE Malthusian growth model The Vidale Wolfe advertising model Use a computer graph package like Autograph to investigate the family of curves generated by the following differential equations. Ask your teacher!. = 4. =. = +

23 Go to Pre- learning under C tab on and complete week 8 pre learning Answers A ) = + ) = + ) = + 4) = + 5) f(x)=6x 5x+c 6) ( )= ) ) 4 + B ) = +5 ) = C )(a) ( )= 4 + (b) x + 9y 7 = 0 )(b) = +6 4 Further questions see Exercise 5D page 50

24 HW 9 TRANSFORMATIONS HW9 Complete homework on a separate sheet of paper. Show appropriate working out and mark answers. Remember to write Your Name, Title, Date. Key words: Transformation, scale factor, translation, reflection. Transformations Read pages Write a short summary to remind you of the key points. Exercise A [typical exam questions]. This diagram shows a sketch of the curve with equation y y = f(x). The curve crosses the x-axis at the points (, 0) and (4, 0). The minimum point on the curve is P(, ). In separate diagrams sketch the curve with equation (a) y = f(x), () O 4 (b) y = f(x). () P(, ) On each diagram, give the coordinates of the points at which the curve crosses the x-axis, and the coordinates of the image of P under the given transformation. x. The figure above shows a sketch of the curve with equation y = f(x). The curve passes through the points (0, ) and (4, 0) and touches the x-axis at the point (, 0). On separate diagrams sketch the curve with equation (a) y = f(x + ) () (b) y = f(x) () (c) y = f x. () On each diagram show clearly the coordinates of all the points where the curve meets the axes.. Given that f(x) = x, x 0, (a) Sketch the graph of y = f ( x) + and state the equations of the asymptotes (b) Find the coordinates of the point where y = f ( x) + crosses a coordinate axis. 4. [C May 0 Q8] y (0, ) O (, 0) (4, 0) x 4

25 Above is a sketch of the curve C with equation y = f(x). The curve C passes through the origin and through (6, 0). The curve C has a minimum at the point (, ). On separate diagrams, sketch the curve with equation (a) y = f(x), (b) y = f(x), () () (c) y = f(x + p), where p is a constant and 0 < p <. (4) On each diagram show the coordinates of any points where the curve intersects the x-axis and of any minimum or maximum points. Go to Pre- learning under C tab on and complete week 9 pre learning Answers Transformations ExA.a) (, ) 4 b) (, ) a) b) c)

26 6 y. a) y=. x=0 4 b) (-/,0) x 4. 6