CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics
THEME: Algebra
Group
1st Bach
Worksheet
1
1.- Given a = 3, b = ¼, c = 5, d = 2/7 and e = 2, find the value of the following expressions, giving the answer as simply as possible: a)
a+c e
b)
a + 3b 2d
2.- Find the coefficient of x3 in the expansion of (x2 + 3x + 4)( 3x2 - 2x -7) 3.- Simplify by collecting likely terms 30x2 y4 – 12xy3 +18xy2 4.- Cancell to lowest terms
12a 2b 3c 2 8a 3 b 2 c
5.- Factorise where possible a) 3x2 -12
b) 25x2 -9y2
c) y2+ 16
d) 4x2+ 4x
e) y2- 6y + 9
x2 − 2x 6.- Simplify the following algebraic fraction: x2 − 4 3(2 x + 4 ) 2(2 − x ) −x= 7.- Solve the following equation 5 3 8.- Solve the following quadratic equations a) x2 + x -12 = 0
b) z2 - 9 = 0
c) y2+ 2y + 2 = 0
9.- Write down two numbers which sum 100, one of them being 90 units bigger than the other one. 10.- Solve only thinking (without algebraic equations) Write down a three-figure number so that the addition of them is equal to their multiplication.
Worksheet 1
Rodolfo Ucha Piñeiro
CIFP
Ferrol
SUBJECT: Mathematics
THEME: Algebra
Group
1st Bach
Worksheet
2
1.- Given that a = 3, b = ¼, c = 5, d = 2/7 and e = 2, find the value of these, giving the answer as simply as possible: a)
(2b )2
b)
3d
e−c c−e
c)
a − 3b
2.- Find the coefficient of x3 in the expansion of (x3 + 3x2 + 2x - 4)( -3x2 - 2x -7)
3.- Simplify
ab 2 2cd × 4c 2 d ab
4.- Factorise where possible f) 8x3 -27y3
g) z4 -16
h) x2 - 5
5.- Simplify the following algebraic fraction:
6.- Factorise the polynomial
i) 4x2 + 1
j) y - 10y + 25 2
x2 + x x2 + 2x + 1
P( x) = 4 x6 + 8 x5 − 3x 4 − 7 x3 − 2 x 2
7.- Solve the following equation
4(x − 3) 5(x + 8) − = 6( x + 3) − 2 2 6
8.- Solve the following quadratic equations d) 5x2 + 100x = 0
e) z2 –z +8 = 0
f) y4+ 2y2 - 3 = 0
9.- In the last elections there were 5,219 votes and 4 candidates. The winner overcome to the adversaries by 22, 30 y 73 votes, although none of them knew how to calculate the exact number of votes that each one had got. How can we obtain this information? 10.- An excellent bottle of wine costs 11€, but the manufacturer will give back the value of the cristal bottle if you return it to him. Knowing that the wine costs 10€ more than the cristal bottle, How much does the cristal bottle cost?
Worksheet 2
CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics 1.- Expand
THEME: Algebra
Group
1st Bach
Worksheet
3
(2 x − 1) 4
2.- Factorise the polynomial
P ( x) = x 5 + 4 x 4 + x 3 − 6 x 2
3.- Simplify the following algebraic fraction:
a2 x+ x a4 x− 3 x 4.- Solve the following simultaneous equations:
2x − 3y = 1 − x + 2 y = 0 Remember: there are three algebraic methods (substitution, equating, elimination) and the graphic method 5.- Solve the following equation
3 3 = x 2 + 4 x − 1 2 x 2 − 3x + 5
6.- Solve the following equation
x +1 1 −1 = x −1 x
7.- Solve the following equation
x + 3 x + 1 = 17
8.- Peter and John have 1000 € in total, but Peter has triple money than John. How much money has every one got? Pedro y José tienen juntos 1000 euros, pero Pedro tiene el triple de dinero que José. ¿Cuánto tiene cada uno?
9.- Divide the number 10 into two parts. Multiply one part by the other part and, then, multiply one of the parts by itself. This square is four times the previous product. Divide 10 en dos partes. Multiplica una parte por la otra y, entonces, multiplica una parte por si misma. Este cuadrado es cuatro veces aquel producto. (Siglo IX)
10.- A barrel full of wine weighs 35 kg., when it is half full, it weighs 19 kg. How many Kg does the barrel weigh when it is empty? Un tonel, lleno de vino tiene un peso de 35 Kg. Cuando esta lleno hasta la mitad, pesa 19 Kg. ¿Cuánto pesa el tonel vacío?
Worksheet 3
CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics
1.- Expand
2 1 2 − x 2 x
THEME: Algebra
Group
1st Bach
Worksheet
4
3 (Santillana 1º Bach)
2.- Factorise the polynomial
P ( x) = x 4 − 2 x 3 + x 2 − 2 x
3.- Simplify the following algebraic fraction:
(Advanced Maths. As core for Edexcel)
4 x + −3 x 2 x 1 4 − − 6 3 3x 4.- Solve the following simultaneous equations by substitution and graphic methods: Santillana 1º Bach
a)
x + y =1 x + y = 2
5.- Solve the following equation
6.- Solve the following equation
b)
x + y =1 2 x + 2 y = 2
2 x 3 − x 2 − 2 x + 25 = 2x x2 − 1 x2 − 3 = 1
c)
x + y =1 2 x − y = 2
(Santillana 1º Bach)
(Santillana 1º Bach)
7.- Determine the value of k so that the equation 3x2 – 6x + k = 0 doesn’t have any root and represent the set of solutions in the number line (Santillana 1º Bach) 8.- An item costs 100€ without VAT (value-added tax). How much does it cost if we include the VAT? Un artículo cuesta 100 euros sin IVA (del 16%). ¿Cuánto cuesta con el IVA incluido?
9.- An object costs 100€ including VAT. How much does it cost without the VAT? Un artículo cuesta 100 euros IVA incluido (del 16%). ¿Cuánto cuesta sin IVA? (en términos exactos ¿cuál es su base imponible?)
10.- 4 magazines and 2 newspapers cost 10€. On the other hand, 1 magazine and 3 newspapers cost 5€. How much does each magazine and newspaper cost? Cuatro revistas y dos diarios cuestan 10 euros. En cambio, una revista y tres diarios cuestan 5 euros. ¿Cuánto cuesta cada revista y cada diario?
Worksheet 4
CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics
THEME: Algebra
Group
1st Bach
Worksheet
5
10
1.- What term includes x11 in the expansion of
2 2 −x x
?
(Santillana 1º Bach)
2.- Simplify the following algebraic expression:
x − y xy 2
1 − x 1 − y
x y y x
x y − y x − x y y−x − x− y
3.- Solve, by factorising, the following equation 4.- Solve the following simultaneous equations:
x 3 − 3x 2 + 2 x = 0
Santillana 1º Bach
x − 2y + z = 1 2 x − 3 z = −8 x − 5 y + 2 z = 3 5.- Solve the following equation
6.- Solve the following equation
−x x2 − x = 3x + 1 2 x − 1 x 2 + 3x = 2 x
(Santillana 1º Bach)
(Santillana 1º Bach)
7.- Write down a third grade polynomial Q(x), whose roots are 3, -1 and 1 and, besides, Q(2) = -18 (Santillana 1º Bach) 8.- The part of the day left is equal to the third part what has already passed. What time is it? Queda, del día, la tercera parte de lo que ya ha transcurrido. ¿Qué hora es?
9.- What is the height of a small house which is two meters lower than an anthem next to it, and three times taller than the small house? ¿Qué altura tiene una caseta que es dos metros más baja que una antena próxima tres veces más alta que la caseta?
10.- I have covered 7/15 of a path and I am still 1/3 of one kilometer short for reaching the half-way mark. What is the length of the path? Llevo recorridos los 7/15 de un camino y aún me falta 1/3 de kilómetro para llegar a la mitad. ¿Qué longitud tiene el camino?
Worksheet 5
CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics
THEME: Algebra
Group
1.- What is the middle term in the expansion of 2.- Find the roots of the following polynomial
3.- Solve the following equation
4.- Solve the following equation
(3x
2
1st Bach
− 2y
)
12
?
Worksheet
6
(Santillana 1º Bach)
P ( x ) = x 4 − 4 x 3 − 12 x 2 + 32 x + 64
x x −1 3 + = −2 x −4 x+2 x−2 2
2x + 8 − x = 2
(Santillana 1º Bach)
(Santillana 1º Bach)
5.- Write down a forth grade equation whose roots are 1, -3 and -4 (Santillana 1º Bach) 6.- Solve the following simultaneous equations and represent the graphical solution:
Santillana
1º Bach
2 x + y = −1 − x + y = −4 − 4 x − y = −1 7.- Solve the inequation − x + 15 ≤ 3 − 7 x (Santillana 1º Bach) 8.- Find m so that the polynomial x4 –mx2 +36 is divisible by (x-2) 9.- Decompose 38 in two addends so that if you divide one of then by the another one you obtain 3 as the quocient and 6 as the remainder. Descomponer el número 38 en dos sumandos de modo que si divides uno por otro se obtenga 3 de cociente y 6 de resto.
10.- Find two numbers knowing that if you add 3 units up to the first number you’ll obtain the second number, and if you add 2 units up to the second number you’ll get twice the first number. Hallar dos números sabiendo que si añades 3 unidades al primero se obtiene el segundo, y que si añades 2 unidades al segundo se obtiene el doble del primero.
Worksheet 6
CIFP
Rodolfo Ucha Piñeiro Ferrol
SUBJECT: Mathematics
THEME: Algebra
1.- What is the middle term in the expansion of
Group
(2 x
2
1st Bach
)?
+1
Worksheet
7
9
(Santillana 1º Bach)
2.- What binomial and exponent do you have to expand to obtain 27 x − 54 x + 236 x − 8 ? 3
3.- Solve the following equation
x+4 1 − 2x − 2 =0 x−3 x − x−6
4.- Solve the following equation
3 5− x = 3 1+ x
2
(Santillana 1º Bach)
(Santillana 1º Bach)
5.- Solve the following simultaneous equations and represent the graphical solution: (Santillana 1º Bach)
2x + y + z = 4
− x + 3 y − 2 z = 1 x + 2 y + z = 1 2 6.- Solve the inequation 2 x + 5 x − 3 > 0 (Santillana 1º Bach) 7.- Find m so that the division of x3 +mx2 -3x -m by (x-4) has as remainder 64. (Santillana 1º Bach)
8.- Suppose you have a division of two numbers. If you add 64 up to the dividend and 5 up to the divisor, the quotient and remainder don’t vary. Find the quotient. Tenemos una división. Si ahora sumamos 65 al dividendo y 5 al divisor los cocientes y restos no varían. Calcula el cociente de la división
9.- For an object, we have paid 9€ more than what the object actually cost, How much did it cost? Por un objeto se pagan 9 pesetas más que la mitad de lo que vale. ¿Cuánto vale?
10.- A father says to his son: “For each exercise you solve correctly I will give you 15€ and for each one that you don’t know how to solve, you will give me 10€”. After doing 25 exercises the boy has 25€. How many exercises did he solve? Un padre dice a su hijo: “Por cada ejercicio que resuelvas bien, te daré 15 euros y por cada uno que hagas mal me darás 10 euros”. Después de 25 ejercicios el muchacho tiene 25 euros. ¿Cuántos ejercicios ha resuelto bien?
Worksheet 7
Rodolfo Ucha Piñeiro
CIFP
Ferrol
SUBJECT: Mathematics
THEME: Algebra
Group
1st Bach
Worksheet
8
1.- Carry out, simplifying the answer, the following algebraic expression:
6x + 3 x2 − 1 2 2 + x +1 x −1
(
1
1
2.- Expand the following expression a 2 − b 2
)(
1
1
a2 + b2
)
3.- Solve the following equation: 3 x − 12 x − 48 x + 168 x − 51x + 180 x = 0 4.- Perform the following operation with algebraic fractions 1 x−2 x 8x + 8 − 2 + 2 ÷ 2 x + 1 3x + 6 x + 3 x − 1 x + 1 5.- Solve the following simultaneous linear system of inequations (Santillana 1º Bach) 7
6
5
4
3
2
2x + y y + 6 < 3 5 4− x 2− y + < 2 3 5 6.- Solve the following equation
1 x + x −1 = 2 x −1
7.- Solve the following nonlinear simultaneous equations and represent the graphical solution: (Santillana 1º Bach)
x + y − 2 = 0 y = x2
8.- Determine the values of k for which the equation 3x2 -6x +K has no solution and make a sketch of the solution set on the real line. 9.- Find a three figure number so that, the sum of its figures is 20 and the unit figure is one unit lower than the hundred figure, and if we substract from the initial number the number obtained inverting their figures, it results 99, calculate the initial number. Un número de tres cifras es tal que, la suma de sus cifras es 20; la cifra de las unidades es una unidad inferior a la de las centenas; y si, al número dado, se le resta el obtenido invirtiendo el orden de sus cifras, se obtiene 99, calcular el número inicial.
10.- El coste totald e 3 prodcutos A, B y C es de 135€.Si se descuenta un 4% en el rpecio de A, un 5% en el de B y un 6% en el de C se ahorran 7.10€. Y si se compran 3 productos de tipo A, 5 de tipo B y 1 de tipo C, el importe toral es 385€. Calcular el preciod e cada producto. (Santillana 1º Bach. Guía y Recursos pg 206)
Worksheet 8