ACS MATHEMATICS GRADE 10 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS

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1 ACS MATHEMATICS GRADE 0 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS DO AS MANY OF THESE AS POSSIBLE BEFORE THE START OF YOUR FIRST YEAR IB HIGHER LEVEL MATH CLASS NEXT SEPTEMBER Write as a single fraction y y Write as a single fraction Write as a single fraction 7 Factorize (a b) (a + b) Factorize ( + + ) + ( + )( + 6) 6 Factorize ( + y + z) ( + y z) 7 Simplify y + y y + y a b a + b Write as one fraction y y + y + y y 9 Write as one fraction y a + a ab a b p + q + 0 Show that Evaluate q p q

2 Review eercises Simplify ( ) Simplify ( 7 ) (ab) Simplify (a b ) a / b / a b Simplify Simplify 7 Rationalize + + ( 7 ) ( ) ( ) Simplify Simplify ( n+ ) n n 0 Find ( + ) Find ( + ) ( ) ( ) n n+ ( ) n Complete the square for + 9 and find its least value Show that < 0 for all values of Find the acute angle between the lines y = + and y = + Find the equation of the line with a slope equal to and through the mid-point of the points ( ) and ( ) 6 Find the equation of the perpendicular bisector of the points ( ) and ( - 6) 7 Find the equation of the straight line with and y ais intercepts and respectively Find the equation of the line through (- ) and parallel to the line y = 9 Find the equation of the line through (- ) and normal to the line y = 0 Solve the simultaneous equations 6y = + 6y = 7 7 6y = 6 + 7y = a + b = a b = a + b = a b = Prove that is irrational How many integers are there between 00 and 000 inclusive? Which is the least positive term in the arithmetic series L? The nth term of a sequence is given by the formulau n = n + with n = 0L Find the sum of the first N terms of this series The sum of the first n terms of a sequence is given by S n = n + Find the nth term of the series and show that the sequence is arithmetic 6 The nth term of a series is given by u n = n + Show that the series is arithmetic and find the common difference 7 The nth term of a series is given by u n = n 7 Show that the series is arithmetic and find the sum of the first 00 terms Find the number of terms in L +

3 Review eercises 9 Find the number of terms in +( 0) + ( ) +L + 0 Find the sum L + Find the sum L 0 Find the sum + + +L+ n n Find the sum + + +L+ 6n n Find the sum n n Find the sum n n 6 Find the sum + + +L + n n Hence find which consecutive odd integers have a sum equal to Find the sum + + +L + n + n The sum to infinity of a geometric sequence is times the first term Find the common ratio 9 The sum to infinity of a geometric series is 6/ The fifth term is /6 Find the sum of the first 0 terms of the series 0 Find the sum to infinity of + +K n Find (a) ( k) and (b) ( k) k= n k= n Find the sum of the multiples of 7 which lie between 000 and 9000 Find the sum of the even numbers between 0 and 90 that have as a factor The quantity is defined to be y ie y = Epress in terms of y the quantities + + The quantity is defined to be y ie y = Epress in terms of y the quantities Find without a calculator (a) (b) (c) (d) 7 6 (e) (f) ln e ln e ln ln e 00

4 Review eercises 7 Epress a single arithm (a) b + b b + b 6 b b + b (b) b 9 b + + Find b b 9 9 Prove that b X = / b X 60 Use arithms to solve for (a) = (b) + = 6 Find the co-ordinates of the points where the curve y = + (a) cuts the - ais and (b) cuts the y-ais (c) Sketch the graph of y = + 6 Find the co-ordinates of the points where the curve y = + (a) cuts the - ais and (b) cuts the y-ais (c) Sketch the graph of y = + 6 Show that the line y = is tangent to the curve y = + Find the coordinates of the point of tangency 6 Show that the line y = + is tangent to the curve y = + Find the coordinates of the point of tangency 6 The line y = is tangent to the curve y = + m + m What are the possible values of m? 66 Find the values of for which (a) (b) > 0 (c) 0 (d) (e) + + > 0 (f) ( )( ) 67 Find the values of for which ( +)( ) > 0 6 Find the values of that satisfy + > 69 Solve the inequality < + 70 (a) Prove that + y y When does the equality hold? (b) If y z 0 use the result of (a) or otherwise to prove that ( + y)(y + z)(z + ) yz 7 (a) Two real numbers and y are such that + y = c where c = constant Prove that the greatest value of the product y is c and that this is obtained when = y (b) If + y = prove that ( + )(+ y ) = + (c) Using the results in (a) and (b) y prove that ( + )(+ y ) 9 7 Epress cos θ + sin θ + in terms of cos θ 7 Epress cos θ sin θ + in terms of sin θ 7 Prove that cos θ sin θ cos θ sin θ 7 Prove that ( secθ + tanθ) + sinθ sinθ 76 Prove that + tan θ + + cot θ 77 Find the sum of the infinite series sinθ + sinθ cosθ +sinθ cos θ +L where 0 < θ < π simplifying your answer as much as possible

5 Review eercises 7 The angle θ is acute and sinθ = Find (a) cosθ and (b) tanθ 79 The angle θ is obtuse and sinθ = Find (a) cosθ and (b) tanθ 0 Solve the equation cosθ = cosθ in the range 0 o θ 60 o Solve the equation sinθ = sinθ in the range 0 o θ 60 o Solve the equation sinθ = cosθ in the range 0 o θ 60 o Find the possible value(s) of k if k is a factor of k The remainder when a + b + + is divided by is + Find a and b A cubic polynomial P() has and as roots P() = and the remainder when it is divided by is Find P() 6 Show that is a factor of + Find the values of for which + 0