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Transcript
FP1 Revision Worksheet Number 1 1. Fill in the gaps in the following table: Angle in degrees Angle in radians sine cosine tangent 1 3 1 2 2 1 2. (a) (b) For the equation 7 x3 23x 4 34 x 2 Show there is a root between 0 and 1 Use interval bisection to find the interval of width 0.25 that contains this root 3. Try the following improper integrals. Either find the value, or explain why a value cannot be found. 5 2 0 6 x 2 dx (a) 6 dx 7 1 (b) x 4. Solve the following equation to find z in the form a+bi: zz* 23 15i 5 z 5. Calculate the value of the following: 8 r 2 4r 3 r 1 6. Find the general solution of the following equation: 1 cos 3 x 4 2 7. Write down the matrix that represents a transformation by 60 anticlockwise about the origin, and find the image of the point (2,-1) under this transformation. 8. The equation 2 x 2 3x 1 0 has roots , . Find an equation with integer 1 1 coefficients that has roots 2 and 2 . FP1 Revision Worksheet Number 2 1. Fill in the gaps in the following table: Description of transforma tion 2. 1, 0 0, 1 1, 0 0, 1 Matrix Rotation by 30 clockwise about the origin 4, 0 0, 4 Reflection in the line 1 y x 3 Give the general solution of the following trig equation: 1 sin x 6 2 3. Use Euler’s step-by-step method, with a step length of 0.1 to estimate the value of dy 1 2 y when x=0.3, given that y=1 when x=0 and , giving your answer to dx x 4 3 significant figures. 4. 5. Solve the following quadratic equation: 5x2 8x 5 0 Sketch the following curve: 1 2x y x 1 Hence or otherwise solve the following inequality: 1 2x 2 x 1 6. Find the following sum: 10 3r 3 r 5 7. Sketch the following curve writing down the equations of any asymptotes or points where the curve cuts the co-ordinate axes: 2 y 2 x 2x 3 8. By differentiating FROM FIRST PRINCIPLES find the gradient of the curve y x 2 3x at the point where x 2 FP1 Revision Worksheet Number 3 We want to plot a straight line Y mX c to verify the following laws. Fill in the gaps in the table 1. Law y ax 2 Y X m c b x y ab x y bx a 1 ax bx 2 y 2. Show that the equation x 4 5 x 2 0 has a root between 0 and 1. Use linear interpolation once to get an approximation for this root. 3. Sketch the curve y 4. Find the general solution of the trig equation: tan 5 x 3 6 5. (a) (b) (c) (d) 6. x2 x 1 stating the equations of asymptotes and the x 2 3x 2 coordinates of the points where the curve crosses the axes. 1, 0 0, 4 A and B . Find: 2, 2 3, 1 AB BA 3A-2B 2A+AB Sketch the following curve, giving its name, the equations of any asymptotes and the points at which it crosses the co-ordinate axes: x2 y2 1 4 Now describe the transformations that change the above curve into the curve with equation: ( x 1) 2 2 y 1 1 4 Sketch this new curve. FP1 Revision Worksheet Number 4 1. Fill in the gaps in the following table: 1, 0 0,1 Matrix Description of transforma tion 2, 0 0,1 Rotation by 45 anticlockwi se about the origin 1, 0 0, 4 Reflection in the line yx 2. Show that the equation x 5 2 x has a root between x 1 and x 2 . Starting with an approximation of x 1.5 and using the Newton-Raphson method once, obtain a better approximation to this root. 3. Sketch the curve y 4. Solve the following equation finding z in the form a+bi: x2 stating the equations of any asymptotes and points x2 4 where the curve crosses the axes. 4 z 2 2 z * 12i 5. 6. 2 x 2 2 x 1 For the curve y , show that the curve does not exist for x2 x 2 x 2 . Hence find the coordinates of the turning points of the curve. Prove the following: n 6r 2 2r 1 n(2n 2 4n 3) r 1 7. (a) (b) (c) (d) 8. On the same diagram sketch clearly the curves: y2 x y2 x 4 ( y 4)2 x y2 4x Give the general solution of the following trig equation: cos x 0 4 FP1 Revision Worksheet Number 5 1. By differentiating from first principles, find the gradient function for the curve: y 2 x2 7 2. Find the general solution of the following trig equation: 1 sin 2 x 2 3. 4. (a) 5x 4 Sketch the curve y and hence or otherwise, solve the inequality 4 x 5x 4 0 4 x Either give a value for the following integrals or a reason why the integral does not exist: 5 1 0 x3 dx 2 (b) 4 x dx 5 5. Find a matrix that does the composite transformation of first rotating by 60 clockwise about the origin and then reflecting in the line y tan 22.5 x 6. Evaluate the following sum: 7 r 1 2 r 2 7. Use Euler’s method with a step length of 0.05 to estimate y when x=2.1 given dy 20 that y=10 when x=2 and dx x 8. Show that the equation x3 3x 1 0 has a root between 1 and 2. Use interval bisection to find an interval of width 0.125 which contains the root. 9. (a) (b) The quadratic equation x 2 2 x 3 0 has roots and . Find the following: (c) 2 2 1 1 (d) 10. Sketch the following curve, and write down its name and the co-ordinates of the centre, and any points where it crosses the co-ordinate axes: x 4 25 2 y 2 100 2 1 FP1 Revision Sheet Answers Sheet 1: Q1 30, 6 , 1 2 , 3 2 45, 4 , 1 2 , 3 Q2 (0,0.25) Q3 (a) undefined (b) 1 Q5 84 Q8 Q6 36 90,1,0, 2 n3 , 367 2 n3 180, ,0,0 Q4 2+3i or -7+3i 1, 3 Q7 2 2 3,1 2 2 x2 5x 4 0 23 , 12 12 , 23 Sheet 2: Q1 reflection in x axis, , rotation by 180, , enlargement 1 , 3 3 , 1 2 2 2 2 scale factor 4 centre the origin Q2 512 2n , 1112 2n Q3 1.07 Q4 453i Q5 Asymptotes x=-1, y=-2 Points (0,1) (0.5,0) 1 x 41 Q6 8775 Q7 Asymptotes x=-1, x=3, y=0 Does not cross x-axis; crosses y axis 2 at 0, 3 Q8 1 Sheet 3: Q1 Possible answers are yx, x3 , a, b 1 log y, log x, a, log b Q2 0.5 yx , x, b, a log y, x, log b, log a Q3 Asymptotes are x=1,x=2,y=1 Doesn’t cross x-axis. Crosses y-axis at (0,0.5) Q4 x 30 n5 0, 4 8,8 3, 8 2, 4 Q5 ' , ' 6, 10 5, 2 12,8 2, 6 Q6 cross y-axis but crosses x-axis at (2,0) and(-2,0) Asymptotes are y Hyperbola Does not x 2 Shifts it to the right by 1 and down by 1 12 , 21 0,1 Sheet 4: Q1 reflection in y-axis, 1 1 , stretch by factor 2 in x direction, , , 1, 0 2 2 stretch by factor 4 in y direction Q2 -1.33 Q3 x=2, x=-2, y=0, (-2,0), (0,-0.5) Q4 1-2i Q5 (-0.5,2) is only turning point Q8 34 2n , 4 2n Sheet 5: Q1 4x Q2 Q3 x=4, y=-5 (0,-1) 45 360n ,135 360n , 45 360n , 215 360n 4 5 ,0 x 4 ,x 4 5 Q4 (a) undefined (b) 1 3 , 1 3 2 2 2 2 Q5 Q6 91 Q7 11.0 Q8 (1.5,1.625) 1 3 , 3 1 2 2 2 2 2 Q9 2, 3,10, Q10 ellipse with centre (4,-2) which crosses axes at (0,4), 3 (0,-8), 4 2 6, 0 , 4 2 6, 0 1 16