Pre-Pharmacy and Pre-Medicine 1 Mathematics PDF

DR IBRAHIM DUMBUYA HEAD OF INDUSTRIAL TECHNOLOGY NJALA UNIVERSITY Mr.HASSAN E J TURAY LECTURER,COMAHS.USL PRE-PHARMACY AND PRE- MEDICINE 1 MATHEMATICS  PRE – CALCULUS 1. NUMBERS AND EQUATIONS Numbers Classification of numbers in mathematics is a way of grouping numbers...

DR IBRAHIM DUMBUYA HEAD OF INDUSTRIAL TECHNOLOGY NJALA UNIVERSITY Mr.HASSAN E J TURAY LECTURER,COMAHS.USL PRE-PHARMACY AND PRE- MEDICINE 1 MATHEMATICS  PRE – CALCULUS 1. NUMBERS AND EQUATIONS Numbers Classification of numbers in mathematics is a way of grouping numbers into different categories based in their properties and how they are represented. There are many types of numbers but some of the most common ones are: Natural Numbers: These are the positive integers that we use for counting, such as 1,2,3,4, and so on. They are also called counting numbers. Natural numbers are labelled N. Whole Numbers: They are the natural numbers with zero added, such as 0,1,2,3,4 and so on. They are labelled W. Integers: These are the whole numbers with the negative natural numbers added, such as -3, -2, -1, 0, 1, 2, 3 and so on. They are also called singed numbers. They are labelled Z.   Complex Numbers: These are numbers that can be written as a+b¡, where a and b are real numbers and i is an indeterminate satisfying ¡2 = -1. Prime Numbers: These are Numbers that have only two factors, 1 and themselves. Composite Numbers: These are  Simultaneous Linear Equations in Two Variables The solution of a system of linear equations in two variables can be found by graphing the equation and finding the point of intersection of the lines, if any. Alternatively, the solution can be found by algebraic methods, such as substitution or elimination. SOLVING LINEAR EQUATIONS BY SUBSTITUTION METHOD The substitution method is a way of solving a system of linear equations in two variables by expressing one variable in terms of the other from one equation, and then substituting into the other equation.  Examples  Examples  Examples  Examples  Examples  Examples  Examples  SOLVING LINEAR EQUATIONS BY ELIMINATION METHOD The elimination method is a way of solving a system of linear equations in two variables by adding or subtracting the equations to eliminate one variable and obtain an equation in one variable.  Examples  Examples  Examples  Examples  Examples  Examples  Examples Then our new equations are: 2p – 3q = 4 -------(1) 4p + q = 1 -------(2) Multiply (2) by 3  Examples  Examples  Examples  Examples  Consistent and inconsistent Systems  Exercise  Exercise  GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS The graphical method of solving linear equations involves plotting the equations on a graph and finding their point of intersection. Here are the steps: 1. Draw a graph for each of the given linear equations. Each linear equation will be represented as a straight line on  GRAPHICAL METHOD OF SOLVING LINEAR EQUATIONS 2. Find the coordinates of the point of intersection of the two lines drawn. The coordinates of this point will be the common solution of the given equations. 3. Write the values of x and y. These are the solutions to your pair of linear equations.  4. Check the solution by substituting the values of x and y obtained in both the given equations. Let us take two linear equations and solve them using the graphical method. Table of values of 𝑦= 8− 𝑥 𝑓𝑜𝑟 0 ≤ 𝑥≤ 4 𝑥 0 1 2 3 4 8 8 8 8 8 8 −𝑥 0 −1 −2 −3 −4 𝑦 8 7 6 5 4 Next consider y = x + 2. 𝑥 0 1 2 3 4 +2 +2 +2 +2 +2 +2 𝑦 2 3 4 5 6 Plotting these points on the coordinate plane, we get a graph like this. Coordinates and the straight line  Exercise 1. Find the lengths of the straight lines joining the following pairs of points: (a) A(3, 2) and B(8,14), (b) C( - 1, 3) and D(4, 7), (c) E{p, q) and F(r, ,s). Exercise 2. Find the lengths of the straight lines joining the following pairs of points: (a) A(l, 2) and B(5, 2), (b) C(3,4) and D (7,1), (c) E( - 2, 3) and F(4, 3), (d) G( - 6, 1) and H(6,6), (e) J( —4, — 2) and K(3, - 7), (f) L( - 2, - 4) and M( - 10, - 10) Exercise 3. Three of the following four points lie on a circle whose centre is at the origin. Which are they, and what is the radius of the circle? A( - 1, 7), B(5, - 5), C( - 7, 5), D(7, - 1). 4. A and B are the points (12,0) and (0, — 5) respectively. Find the length of AB, and the length of the median, through the origin O, of the triangle OAB. The mid-point of a straight line   Exercise 1. Find the coordinates of the mid-points of the straight lines joining the following pairs of points: (a) A(4, 2) and B(6,10), (b) C( - 5, 6) and D(3, 2), (c) E( - 6, - 1) and F(3, - 4), (d) G(p, q) and H(r, s). 2.P, Q, R are the points (5, — 3), ( — 6,1), (1, 8) respectively. Show that triangle PQR is isosceles, and find the coordinates of the mid-point of the base. 3. A and B are the points ( —1, —6) and (5, —8) respectively. Which of the following points lie on the perpendicular bisector of AB? (a) P(3, -4 ), (b) Q(4,0), (c) R(5,2), (d) S(6, 5). The gradient of a straight line  1. Find the gradients of the lines joining the following pairs of points: (a) (4, 3) and (8,12), (b) ( - 2, - 3) and (4, 6), (c) (5, 6) and (10,2), (d) ( - 3,4) and (8, - 6), (e) ( - 5, 3) and (2, 3), f (p, q) and (r, s),(g) (0, a) and (a, 0), (h) (0,0) and (a,b). 2. Find the gradients of the straight lines joining the following pairs of points: (a)(4, 6) and (9,15), (b) (5, - 11) and ( - 1, 3), (c) (7,0) and ( - 3, - 2). 3. Show that the three given points are in each case collinear, i.e. they lie on the same straight line: (a) (0,0), (3, 5), (21, 35), (b) ( - 3,1), (1, 2), (9, 4), (c) ( - 3, 4), (1, 2), (7, - 1), (d) (1, 2), (0, - 1), ( - 2, - 7). 4. Show that A( — 3,1), B(l, 2), C(0, —1), D( —4, —2) are the vertices of a parallelogram. 5. Show that P(l, 7), Q(7, 5), R(6, 2), S(0,4) are the vertices of a rectangle. Calculate the lengths of the diagonals, and find their point of intersection. 6. P( — 2, —4), Q( — 5, —2), R(2,1), S are the vertices of a parallelogram. Find the coordinates of M, the point of intersection of the diagonals, and of S. 7. Write down the gradient of the straight line joining (a, b) and (p, q). Write down the two conditions that these points should lie on the line y = 7x — 3. From these deduce the gradient of the line.  GRADIENT OF A LINE JOINING TWO POINTS  EXAMPLES  EXAMPLES SLOPE INTERCEPT FORM OF A LINEAR EQUATION  EXAMPLES  EXAMPLES  EXAMPLES   POINT⎼ SLOPE FORM  EXAMPLES y - 3 = 4(x - 2) Simplify to get the equation in slope- intercept form (y = mx + b): (y = 4x - 5) So, the equation of the line is (y = 4x - 5). Parallel and perpendicular lines Since parallel lines make equal corresponding angles with the x-axis, parallel lines have equal gradients. In general, if two lines are perpendicular, the product of their gradients is — 1. Or in other words, if the gradient of one is m, the gradient of the other is — l/m.  Examples 1. Find if AB is parallel or perpendicular to PQ in the following cases: (a) A(l, 4), B(6, 6), P(2, — 1), Q(12, 3); (b) A( - 1, - 1), B(0,4), P( - 4, 3), Q(6,1); (c) A(0, 3), B(7, 2), P(6, - 1), Q( - 1, - 2).    Points of intersection If the two straight lines x + y — 1 = 0 and 2x — y — 8 = 0 cut at the point P(a, h) then the coordinates of P satisfy the equation of each line, and we may write a+b—1=0 2a — b — 8 = 0 The solution of these equations is a = 3, b = — 2, which tells us that the given lines cut at (3, — 2). In practice we obtain the result by solving the equations simultaneously for x and y. NOTE: y = mx + c is a line of gradient m, passing through (0, c). y = mx is a line of gradient m, passing through the origin. y = c is a line of zero gradient (i.e. parallel to the x-axis). x = k is a line parallel to the y-axis.  Exercise 1.Find the points of intersection of the following pairs of straight lines: (a) 2x — 3y = 6 and 4x + y = 19, (b) y = 3x + 2 and 2x + 3y = 17, (c) y = c and y = mx + c, (d) x = — a and y = mx + c. 2.Can you find the point of intersection of 3x — 2y — 10 = 0  3. The points A, B, C have coordinates (7, 0), (3, — 3), ( — 3, 3) respectively. Find the coordinates of D, E, F, the mid-points of BC, CA, AB respectively. Find the equations of the lines AD, BE, and the coordinates of K, their point of intersection. Prove that C, K, F are in a straight line. 4. Find the equations of the lines passing through the point (4, —2) and respectively (a) parallel, (b) perpendicular to the line 2x — 3y — 4 = 0. Find also the coordinates of the foot of the perpendicular from (4, — 2) to 2x — 3y — 4 = 0.  Exercise 5. Find the equation of the line joining the points (6, 3) and (5, 8). Show also that these two points are equidistant from the point ( — 2,4). 6. What is the equation of the straight line joining the points A(7,0) and B(0, 2)? Obtain the equation of the straight line AC such that the x-axis bisects the 7. The points A, B, C have coordinates (7, 0), (3, — 3), ( — 3, 3) respectively. Find the coordinates of D, E, F, the mid- points of BC, CA, AB respectively. Find the equations of the lines AD, BE, and the coordinates of K, their point of intersection. Prove that C, K, F are in a straight line. 8. Prove that the points (—5, 4), ( — 1, —2), (5, 2) lie at three of the corners of a square. Find the coordinates of the 9. The vertices of a quadrilateral ABCD are A(4,0), B(14,11), C(0,6), D( — 10, — 5). Prove that the diagonals AC, BD bisect each other at right angles, and that the length of BD is four times that of AC. 10. The coordinates of the vertices A, B, C of the triangle ABC are ( — 3, 7), (2,19), (10, 7) respectively. (a) Prove that the triangle is isosceles. (b) Calculate the length of the   14. Find the circumcentre of the triangle with vertices ( — 3,0), (7,0), (9, — 6). Show that the point (1, 2) lies on the circumcircle. 15. ABCD is a rhombus. A is the point (2, — 1), and C is the point (4, 7). Find the equation of the diagonal BD. 16. L(—1, 0), M(3, 7), N(5, —2) are the mid-points of the sides BC, CA, AB respectively of the triangle ABC. Find the equation of AB.  QUADRATIC EQUATIONS  Methods of Solving Quadratic equations Quadratic equations can be solved (finding their roots) by the following methods: i. iv. Factorization ii. Completing the square iii. Formula iv. Graphical  Solution by Factorisation Solution (3𝑥– 2) (5𝑥– 4) = (3𝑥-2)2 ⇒ 3𝑥(5𝑥– 4) -2(5𝑥– 4) = (3𝑥– 2) (3𝑥– 2) 15𝑥 - 12𝑥- 10𝑥+ 8 = 9𝑥 - 6𝑥- 6𝑥+ 4 2 2  (x – 1) (6x – 4) = 0 x – 1 = 0 or 6x – 4 = 0 x = 1 or 6x - 4 = 0 x = 1 or 6x = 4 x = 1 or 2/3.                     RADICAL EQUATIONS (SIMPLE FORM)       SIMULTANEOUS LINEAR AND QUADRATIC EQUATIONS      Roots of quadratic equations   NATURE OF THE ROOTS OF A QUADRATIC EQUATION        Exercise  RELATIONSHIPS BETWEEN THE ROOTS AND COEFFICIENTS OF A QUADRATIC EQUATION      SYMMETRIC FUNCTIONS OF THE ROOTS