Mr. J Gallagher
Algebra - Worksheet Leaving Cert - Higher Level
Add and subtract the like terms
Add and subtract the like terms a) 2x – 3y + 7x – 6y b) 16a – 4c + 10b + 6a – 7b + 3c c) -14x + 11y + 4x – 13y
Evaluating Expressions
Evaluate the following expressions when x = 2, y = -3 a) x2 – 4xy b) 6xy + 3x2y c) x2 – y2
Multiplying Brackets
Simplify the following expressions a) 4(3x - 4) + 3(-2x + 5) b) -3(2x – 6y) – 4(x – y) c) 7(2a + 3b) +6(4a – 2b) - 4(3b – 2a) d) (3x - 4) (-2x + 5) e) (2x – 6y) (x – y) f) (2a + 3b) (3b – 2a)
Simplifying Fractions
Simplify the following expressions (Division)
a) Mr. J Gallagher
b)
c)
d)
e)
f)
Factorising
Common Factors - Factorise the following expressions a) x2 + 2x b) 2y2 + 4y c) 4xy – 12x2 d) 3x2y – 15xy
Difference of two squares - Factorise the following expressions a) x2 – 49 b) y2 – 144 c) 4x2 – 25 d) 36y2 – 16x2 Mr. J Gallagher
Difference / Sum of two Cubes - factorise the following expressions a) y3 – 27 b) y3 + 27 c) 64p3 – 27q3 d) 5x3 – 625y3
Grouping factors - Factorise the following expressions a) ax + ay + bx + by b) 3ax – 2ay + 3bx – 2by c) 4x – 4y + abx – aby
Simplify the following expressions (Division)
a)
b)
Factorising quadratics - Factorise the following expressions a) x2 – x – 20 b) x2 + 3x – 10 c) 4x2 + 4x + 1 d) 14x2 + 3x – 2
Quadratic equations - Solve the following equations a) x2 – x – 20 = 0 b) x2 + 3x – 10 = 0 c) 4x2 + 4x + 1 = 0 d) 14x2 + 3x – 2 = 0 Linear Equations
Solve the following equations a) 4x + 3 = 15 Mr. J Gallagher
b) 3y – 6 = 18 c) 3x – 1 = 2x + 11
Linear equations with brackets - Solve the following equations a) 3(x + 2) = 0 b) -3(4x + 3) = 15 c) 3(2y – 6) = 18 d) 4(x – 2) = 3(2x + 4) e) 3(x – 2) = 7(x + 5) – 13 f) 2(2x + 1) – 3 (x – 1) = 9
Linear equations with fractions - Solve the following equations ()
a)
b)
c) Mr. J Gallagher
d)
Simultaneous Equations
Linear Simultaneous Equations – Solve the following for x & y. a) x + 2y = 8 2x + 3y = 14
b) x – 2y = 9 3x + 7y = 1
c) 3x + 4y = 23 y = 2x + 3y 1 linear and 1 non-linear – Solve the following for x & y a) x2 + y2 = 10 x – y = 4
b) x2 + y2 = 20 x – 2y = 0 c) 3x + 5y = 15 x2 + y2 – 10x – 9 = 0
d) x – 2y = 12 x2 + y2 – 10x – 4y + 4 = 0
e) 2x + y = 3 x2 + xy + y2 = 3
Three variables – Solve the following for x, y, z. a) 3x + 5y – z = -3 2x + y – 3z = -9 x + 3y + 2z = 7 Mr. J Gallagher
b) x + y + z = 1 2x – 3y – 2z = -9 2x – 3z = -16
Inequalities
Linear – Solve the following inequality and graph your solution on a number line a) 2x – 3 5, x R b) 3x – 1 > 8, x N c) 4x + 3 < 3x + 10, x R
Compound – Solve the following inequality and graph your solution on a number line a) -5 < 3x + 1 7, x R b) – 3 2x – 3 < 7, x R
c) – 9 < 4x + 3 15, x R
Quadratic Inequalities a) 2x2 – x – 6 0 b) 2x2 – 11x + 5 > 0 c) (2x – 3)2 > 4
d) < 3
e) > 3/2
f) ½
g) |x+2| 5
h) |2x+3| > 5
i) |x+11| < 5
Graph the functions f(x) = |x+1| and g(x) = 4. i. Find the coordinates when f(x) = g(x) Mr. J Gallagher
ii. Hence solve f(x) g(x) and f(x) > g(x).
Graph the functions f(x) = |x – 2| and g(x) = 4. i. Find the coordinates when f(x) = g(x) ii. Hence solve f(x) g(x) and f(x) > g(x).
Equations with x as index
Solve for x a) 3x = 27 b) 2x+1 = 16 c) 32x+1 = 243 d) 9x+1 = e) 32x+1 = 3 f) 22x-2 = g) 49x = 72+x Manipulation of formula
Write the following equations in terms of x a) y = 4xb – 3a2 b) a = c) a = d) a =
Long Division
Divide and then factorise the following: a) x3 + 2x2 -7x – 2 divided by x – 2 b) 4x3 – 5x2 - 2x +3 divided by x – 1 c) 36x3 – 18x2 - 10x + 4 divided by 3x – 1
Solve the following cubic equations (factor theorem) d) x3 – 6x2 + 11x – 6 = 0 e) 3x3 – 19x2 + 33x – 9 = 0 f) 2x3 – 9x2 + 2x + 21 = 0 Mr. J Gallagher
Unknown Coefficients a) If C(x – a)2 + b = 3x2 – 6x + 5 for all values of x, find a, b, c.
b) p(x + 1)(x + 2) + q(x + 1) + r = 3x2 + 5x + 7, find values of p, q, r.
c) px2 + 2pqx + pq2 + r = 3(x – 2)2 + 5, find values of p, q, r.
Long Division with letters
a) Let f(x) = 2x3 – ax2 – bx + 42. Given that (x + 3) and (x – 2) are factors of f(x), find the value of a and b.
b) Let f(x) = kx3 – 6x2 + bx – 6. Given that (x + 1) and (x – 3) are factors of f(x), find the value of K and b. Then find all of the roots of f(x).
c) Let f(x) = x3 + kx2 – x – k. Given that (x – 1) is a factor, find the value of k. Hence find the other two factors.
d) If (x + 4) is a factor of f(x) = x3 – kx2 – 22x + 56. Find the value of k. Hence find the roots of f(x).
e) Find the range of value of k for which (2 – 3k)x2 + (4 – k)x + 2 = 0 has no real roots. HINT: use quadratic formula.
f) If 3x2 – 11x + r is a factor of 3x3 – 14x2 +17x – r, find the value of r.
g) x2 - t is a factor of x3 – px2 – qx + r, show that pq = r express p in terms of q
h) If x2 + px + q is a factor of x3 + ax2 + b, prove that b = q(a – p) q = p(p – a) q2 + bp = 0
i) If x2 – px + q is a factor of x3 + 3px2 + 3qx + r, prove that q = -2p2 Mr. J Gallagher
r = - 8p3
Evaluating Functions
If the function f(x) = 5x – 2, find a) f(-1) b) f(3) c) f(x) = 3
If the function f(x) = x2 – 6, find a) f(-1) b) f(3) c) f(x) = 3
Graphing Functions
Graph the following linear functions a) f: x 2x + 1 in the domain, -3 x 3. Find the value when f(x) = 0 Find the value when x = 2.5
b) f: x 3x – 2 in the domain, -2 x 4. Find the value when f(x) = 0 Find the value when x = 2.5
Graph the following quadratic functions a) f: x x2 – 4x – 5 in the domain, -3 x 3. Find the values when f(x) = 0 Find the value when x = 2.5
b) f: x 2x2 + 4 in the domain, -3 x 3. Find the values when f(x) = 0 Find the value when x = 2.5
Graph the following cubic functions Mr. J Gallagher
a) f: x x3 – 3x2 – x + 3 in the domain, -2 x 4. Find the values when f(x) = 0 Find the value when x = 2.5
b) f: x x3 – 5x + 2 in the domain, -3 x 3. Find the values when f(x) = 0 Find the value when x = 2.5
Real Life examples
Equations
1) A teacher is organizing a day trip to a museum. The total cost of entry for the students to the museum is 72euro. Mr. J Gallagher
I. By letting x equal the number of students in the class, write an expression to represent the cost of the trip per student. II. If two students decide not to go to the museum, the total cost of entry would be 70euro. Write an expression for the new cost per student. III. The cost of entry per student in this case would be increased by 1euro. Write an equation to represent the above information and hence solve this equation to find the number of students who were originally meant to go on the trip.
2) To promote a school play, a class undertook to make 100 posters, with each pupil making the same number of posters. I. If there were x pupils in the class, express the number of posters made by each in terms of x. II. On the day they were making the posters, five pupils were absent and each pupil present had to make one extra poster. Express the number of posters now made by each student in terms of x. III. Calculate the number of students normally in the class.
3) Jane rode her bicycle for x km at 10 km/hr and then rode (x+4) km further at 8km/hr. I. Represent, in terms of x, the total time that she rode. II. If she rode for 2hrs 18mins, find the total distance she travelled.
4) A number of people share equally in a prize of 300euro. The following week, 10 people more share equally the prize of 300euro. Each share the second week was 2.50 less than each share the first week. Find how many people shared the prize the first week.
5) 72 people attended the first night of a concert. They were seated in rows, each of which contained x people. The following evening, 80 people attended the concert and they were seated in rows containing (x + 2) people. If on the second night there was one less row needed than on the first night, calculate the number of rows needed on each night.
Simultaneous Equations
1) The combined cost of a television and a DVD player is 1460euro. The television costs 330euro more than the DVD player. I. Use two equations in x and y to represent the situation II. Hence, find the cost of the television and the cost of the DVD player. Mr. J Gallagher
2) Two numbers have a difference of 13. Twice the bigger number added to 19 times the smaller number makes 110. I. If x is the bigger number and y is the smaller number, write down two equations in x and y. II. Hence, solve for the two numbers.
3) Let the cost of a meal for an adult be x euro and the cost of a meal for a child be y euro. The cost of a meal for three adults and two children amounts to 125euro. The cost of a meal for two adults and three children amounts to 115euro. I. Write down two equations in x and y to represent this information. II. Solve these equations to find the cost of an adults meal and the cost of a child’s meal.
4) A builders’ supplier sells two types of copper pipes. One has a narrow diameter and costs x euro per length. The other has a wider diameter and costs y euro per length. Tony buys 14 length of the narrow pipes and 10 lengths of the wider pipes at a cost of 555euro. Gerry buys 12 lengths of the narrow pipes and 5 lengths of the wider pipes at a cost of 390euro. I. Write two equations to represent the above information. II. Solve these equations for find the cost of the narrow and wider pipes.
5) A school has 82 students in classes P, Q and R. There are x students in class P, y students in class Q and x students in class R. one sixth of class P, one third of class Q and half of class R study maths. One tenth of class P, three eights of class Q and one quarter of class R study chemistry. 27students study maths and 19 study chemistry. Find the number of students in each of the three classes.
6) An examinations paper consists of 30 questions. Four marks are given for each correct answer, two marks are deducted for each incorrect answer and one mark is given for no answer. Sean has x answers correct, y answers incorrect and the remaining z questions with no answer. His total score for the examinations was 54 marks. Eileen had (x – 1) answers correct, (y – 3) answers incorrect and the remaining 2z questions with no answer. Calculate Eileen’s score.
Quadratics
1) A ball is thrown vertically upwards. The height, h metres, of the ball t seconds after it is thrown is given by h = 20t – 5t2. Find the two times when the height of the ball is 15m. Mr. J Gallagher
2) A ball is thrown vertically upwards. The height, h metres, of the ball t seconds after it is thrown is given by h = 36t – 6t2. Find the two times when the height of the ball is 18m.
3) Two consecutive positive odd numbers are squared and the results are added to give 290. Write down an equation to represent this information and hence find the two numbers.
4) The product of two positive consecutive odd integers is 1 less than six times their sum. Find the integers.
Inequalities
1) A gardener is given the following instructions about laying a rectangular lawn. The length x metres, must be 4 metres longer than the width. The width must be greater han 3 metres and the area must be less than 60metres2. Find the range of possible values of x.
2) A boy throws a ball from the top of a building. The ball leaves his hands 2 metres above the building. The height h metres, reached by the ball above the ground after t seconds is given by h = 9 + 8t – t2 I. Find the height of the building. II. Find the time taken for the ball to hit the ground for the first time. III. Find the range of alues of t for which the balls is more than 21metres above the ground.
3) A company maufactures envelopes in different sizes. All the envelopes have lengths and widths in the ratio 2:1. The smallest envelope that the company produces has an area f 144cm2 and the largest envelope has an area of 400cm2. I. Find, in surd form, the smallest and largest values of x. II. The production team decides to stay in this range, but wants to produce envelopes of integer length. List all possible lengths which could be manufactured.