parkinson k gre math for winners algebra sets geometry coord

CONTENTS GEOMETRY AND MENSURATION GEOMETRY: LINES, ANGLES & TRIANGLES Lines Angles Triangles Congruence of Triangles Similarity of Triangles Properties of two Similar Triangles Some Important Theorems Basic Pythagorean Triplets GEOMETRY: POLYGONS & QUADRILATERALS PROPERTIES OF POLYGONS SOME IMPORTANT POLYGONS Regular Hexagon Quadrilaterals TYPES OF QUADRILATERALS: Parallelogram Rectangles Rhombus Square Trapezium GEOMETRY: CIRCLES: Properties of Circles Tangents ELLIPSE MENSURATION CUBOID CUBE PRISM CYLINDER PYRAMID CONE SPHERE PRACTICE PROBLEMS SETS & VENN DIAGRAMS BACKGROUND DIFFERENT WAYS OF REPRESENTING A SET TYPES OF SETS Some important results on subsets: OPERATIONS ON SETS Union of Sets Intersection of Sets Difference of Sets Distributive Property of Union and Intersections De-Morgan's Laws Complement of a Set SOME IMPORTANT RESULTS PRACTICE PROBLEMS INEQUALITIES BACKGROUND Solution of an Inequation PROPERTIES OF INEQUALITIES Properties of absolute function SOLVING LINEAR INEQUALITIES IN ONE UNKNOWN SOLVING QUADRATIC INEQUALITIES SYSTEM OF INEQUALITIES IN ONE UNKNOWN PRACTICE PROBLEMS COORDINATE GEOMETRY CARTESIAN COORDINATE SYSTEM Distance Formula: Area of a Triangle Centre of Gravity or Centroid of a Triangle In-centre of a Triangle Circumcentre of a Triangle Orthocentre of a Triangle SLOPE OF A LINE Slope of a line when coordinates of any two points on the line are given: Conditions for parallelism and perpendicularity of lines in terms of their slopes: Angle between two lines: Colinearity of three points: DIFFERENT FORMS OF THE EQUATION OF A LINE: Horizontal and vertical lines: Point-slope form: Two-point form: Slope-intercept form: Slope of a Line Distance of a Point from a Line EQUATIONS SOLUTION OF AN EQUATION Properties of equality used to find the solution of an equation LINEAR EQUATION WITH ONE VARIABLE SIMULTANEOUS EQUATIONS Applications of Linear Equations QUADRATIC EQUATIONS Roots of a quadratic equation Discriminant Relation between roots and coefficients Equations reducible to Quadric Equations GRE Math for Winners Algebra, Sets, Geometry & Coordinate Geometry K Parkinson The author is a PhD in Engineering and has been training students to achieve high GRE scores for the past 10 years GEOMETRY AND MENSURATION The chapters of geometry and mensuration have had their fair share of questions in GRE For doing well in questions based on this chapter, you should familiarize yourself with the basic formulae and visualizations of the various shapes of solids and two dimensional figures based on this chapter GEOMETRY: LINES, ANGLES & TRIANGLES Background Geometry and Mensuration are important areas in the GRE examination Over the past few years the trend has been that there are around -6 questions based on these chapters LINES AB is a line segment A B A line segment extended indefinitely at one end is called a ray Here PQ is a ray P Q A line segment extended indefinitely at both the ends is called a line Here LM is a line L M The points which lie on the same line are called collinear points The points which not lie on the same line are called non-collinear points Plane: A plane is a flat surface It has length and width but no thickness Parallel lines: Two lines which lie in the same plane are said to be parallel lines when they not intersect each other at any point even if they are extended to infinity When two lines are parallel to the third line, then they are also parallel to each other Here line x is | | line y X Y Perpendicular lines: Two lines in a plane which intersect each other at right angles are called perpendicular lines Perpendicular is denoted by the symbol ⊥ If two lines are perpendicular to the same line, they are parallel to each other Some Important Points: i) A line which is perpendicular to a line segment i.e., intersects it at the midpoint of the segment is called perpendicular bisector of the segment ii) Every point on the perpendicular bisector of a segment is equidistant from the two endpoints of the segment Conversely, if any point is equidistant from the two endpoints of the segment, then it must lie on the perpendicular bisector of the segment If PO is the perpendicular bisector of segment AB, then, AP = PB Also, if AP = PB, then P lies on the perpendicular bisector of segment AB iii) The ratio of intercepts made by three parallel lines on a transversal is equal to the ratio of the corresponding intercepts made on any other transversal by the same parallel lines If line a | | line b | | line c and line l and line m are two transversals, then, PR/RT = QS/SU ANGLES An angle is the union of two non-collinear rays with common origin The common origin is called the vertex and the two rays are sides of the angle The angle is generally denoted by the symbol '∠' Thus, in the figure, angle ABC is denoted as ∠ABC Angle is measured in degrees The shaded region is known as the interior of the angle Acute angle: An acute angle is less than 90° ∠ABC is acute Obtuse angle: An obtuse angle is more than 90o but less than 180o ∠ABC is obtuse Right Angle: A 90° angle is called a right angle ∠ABC in the adjoining figure is a right angle x + × 2/3 = x + = ∴x = ii) We can solve simultaneous equations by eliminating one of the two variables and obtaining an equation which has only one variable For this, the coefficients of the variable to be eliminated are first made equal in both the equations and then the two equations are either added or subtracted as required For example i) Solve x + y = and x - y = x + y = … (1) x – y = … (2) Adding equations (1) and (2), 2x = 10 ∴x = Putting x = in (1) + y = ∴y = ∴(5, 3) is the solution set ii) Solve 2x + 3y = and 2x – 4y = 2x + 3y = … (1) 2x – 4y = … (2) Subtracting (2) from (1), 7y = ∴y = 6/7 From (1), 2x = – (3/7) = - 18/7 = 45/7 ∴x = 45/14 Thus, if the coefficients of some variable in both the equations are the same, the suitable variable can be eliminated by simple addition or subtraction iii) When the coefficients of either of the variables is not the same in both equations, we either multiply or divide these equations with real numbers other than to obtain new equations equivalent to the original ones, so that one of the variables can be easily eliminated For Example: Solve 4x + 3y = 12 and 3x + 4y = 18 4x + 3y = 12 … (1) 3x + 4y = 18 … (2) Multiplying (1) by and (2) by 4, (4x + 3y) × 3= 12 × (3x + 4y) × 4= 18 ×4 12x + 9y = 36 12x + 16y = 72 7y = -36 ∴ y = 36/7 Substituting value of y in (1) 4x + 3(36/7) = 12 4x = 12 - 108/7 = 84 -108/7 = -24/7 ∴ x = -6/7 Note: i) To solve a system of simultaneous equations, the number of different equations (not equivalent) should be at least equal to the number of variables i.e., if there are three variables then we require at least three non equivalent equations to solve them ii) Before solving a system of simultaneous equations, they should be reduced to their simplest forms APPLICATIONS OF LINEAR EQUATIONS Problems on real life situations can be translated in the form of linear equations by using variables Translate the statements of the problem into mathematical statements For example, i) Consider two sisters whose ages differ by years Then, if we assume age of one sister to be x, then the age of the other sister could be x + or x - ii) If the digits of a two digit number are in the ratio : 1, then two digits can be assumed as x and 2x or as x and y, where x = 2y or 2x = y iii) If x, y, z are the digits of a three digit number, starting from left, then the number formed can be written as l00x + l0y + z The number formed after reversing the digits will be l00z + 10y + x Look for the quantities that are equal and then equate them Solve the equations so formed to find the value of the unknown quantities For example, i) The age of two sisters differ by years Sum of the ages is Find their ages Let the age of one sister be x and that of the other be x + ∴ x + x + = 24 ∴ 2x = 22 ∴x = 11 Hence, the ages of the sisters are 11 and 13 ii) Two numbers are in the ratio 2: The difference of the two numbers is 15 Find two numbers Let the smaller number be x and the larger number be y Now, y = 2x …… (1) y – x = 15 … … (2) Substituting the value of y from (1) and (2) 2x – x = 15 ∴x = 15 y = 2x = × 15 = 30 iii) The sum of a three digit number and its reverse is 444 The unit’s digit is greater than the hundred’s digit by Also, the ten’s digit is twice the hundred’s digit Find original number Let the digit in the hundred’s, ten’s and unit’s place be x, y and z The number formed can be written as l00x + l0y + z The number formed after reversing the digits will be l00z + 10y + x Also, z – x = …… (1) ∴z=x+2 y = 2x …… (2) (100x + 10y + z) + (100z + 10y + x) = 444 …… (3) Substituting the value of z and y from (1) and (2) in (3) 100x + 10 × 2x + x + + 100(x + 2) + 10 × 2x + x = 444 100 x + 20x + x + + 100x + 200 + 20x + x = 444 242x + 202 = 444 ∴242x = 242 ∴ x = 1, y = × = and z = + = Hence, the number is 123 Example: I purchased some apples and oranges for$3.80 Each apple costs 20cents and each orange costs 25cents What is the minimum number of apples that I purchased? i) ii) iii) iv) 12 v) 19 Ans: C Solution: Let the number of apples be a and number of oranges be o 0.20a + 0.25o = $3.80 Now, 3.80 is not divisible by 0.25 so a cannot be zero (Moreover the problem indicates that a is a non-zero quantity) Assuming a to be we get: 0.25o = 3.60 Again, 3.60 is not divisible by 0.25 Proceeding in the same way, the minimum value of a for which the equation is satisfied is a = 4 × 0.20 + 12 × 0.25 = $3.80 QUADRATIC EQUATIONS A quadratic polynomial equated to zero is called a quadratic equation The general form of a quadratic equation is ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ If α and β are the two zeros of the quadratic polynomial p(x), then α and β are the roots of the corresponding quadratic equation p(x) = A quadratic equation cannot have more than two distinct roots Example: x2 + 4x – = 0; x – 1/x = ROOTS OF A QUADRATIC EQUATION i) The roots α and β of a quadratic equation can be found by factoring the corresponding polynomial If ax2 + bx + c = and the polynomial ax2 + bx + c has the factorization (lx + m) and (px + q) then (lx + m)(px + q) = If product of (lx + m) and (px + q) is 0, then either (lx + m) = or (px + q) = Example: i) x2 + 8x + 16 = x2 + 4x + 4x + 16 = x(x + 4) + 4(x + 4) = (x + 4)(x + 4) = ∴x=-4 α = β = - ii) 4x2 – 2x – = ∴ 4x2 + 4x – 6x – = 4x(x+1) – 6(x + 1) = (4x – 6)(x + 1) = 4x – = or x + = x = 6/4 = 3/2 or x = - ∴x = -1, 3/2 α =3/2 and β = - iii) The roots α and β of the quadratic equation are given by the formulae Example: 3x2 + 7x + = a = 3, b = 7, c = ∴ The two roots of the equation 3x2 + 7x + 1= are Note: The factorization method can be used only if the last term is easily factorable, whereas the formula method can be used to solve any quadratic equation DISCRIMINANT In a quadratic equation ax2 + bx + c = 0, the term 'b2 4ac' is called the discriminant It is also denoted by ∆ or D The nature of the roots depends on the discriminant of an equation Case (i): If D is positive; the roots of the equation are real and unequal In particular, if D is positive and a perfect square, the roots are rational and unequal In particular, if D is positive and not a perfect square, the roots are irrational and unequal Case (ii): If D is equal to zero; the roots are real and equal Case (iii): If D is negative; the roots are complex or imaginary The above information can be summarized in the table below RELATION BETWEEN ROOTS AND COEFFICIENTS i) If α and β are the two roots of a quadric equation ax2 + bx + c = then, Sum of the roots = α + β = -b/a = − Coefficient of x/Coefficient of x2 Product of the roots = α × β = c/a = Conatant term/Coefficient of x2 Example: 7p2 – 5p – = a = 7, b = - 5, c = - ∴ Sum of roots = -b/a = 5/7 Product of roots = c/a = -2/7 ii) Forming a quadric equation with the given roots If α, β are two distinct roots of a quadric equation, then the equation can be formed as x2 – (α + β) x + α β = i.e., x2 – (sum of roots) x + product of roots = is the required equation Example: If the roots are and – 3, then Sum of the roots = – = and product of roots = - 12 Quadric equation is x2 – x – 12 = iii) If c = a, the roots are reciprocal Example: 2x2 + x + = a = 2, b = 1, c = ∴ Product of roots = 2/2 = ∴ Roots are reciprocal iv) If b = 0, the roots are equal in magnitude, but opposite in sign Example: x2 – = a = 1, b = 0, c = - ∴ α + β = 0/1 = ∴ Roots are equal in magnitude but opposite in sign v) If one root of a quadric equation with rational coefficients is irrational, the other root must be its irrational conjugate Example: x2 + 4x + = EQUATIONS REDUCIBLE TO QUADRIC EQUATIONS Some equations are not quadric equations but they can be reduced to quadratic equations Few examples of such equations have the form i) ax4 + bx2 + c = ii) a (x2 + 1/x2 ) + b(x+1/x) + c = iii) a (x2 + 1/x2) + b(x - 1/x) + c = Example: consider the equation 2y4 – 5y2 + = The degree of the equation is and the equation will have roots This equation can be reduced to a quadric equation by substituting y2 = x 2x2 – 5x + = 2x2 – 4x – x + = 2x(x-2) – 1(x-2) = (2x – 1)(x – 2) = ∴ x = 2, 1/2 Now y2 = and y2 = 1/2 ∴ y = ±√2 and y = ± 1/√2 y = +√2, −√2,, +1/√2,, −1/√2 Example: (m2 + 1/m2 ) − (m + 1/m) +14 = Let (m + 1/m) be x (m2 + 1/m2 + 2) = x2 ⇒ (m2 + 1/m2 ) = x2 − Substituting in equation, 2(x2 – 2) – 9x + 14 = ⇒ 2x2 – – 9x + 14 = ⇒ 2x2 – 4x – 5x + 10 = ⇒ 2x(x – 2) – 5(x – 2) = ⇒ (2x-5)(x – 2) = ∴ x = 5/2, m + 1/m = ⇒ m2 + = 2m ⇒ m2 – 2m + = (m − 1)2 = ⇒ m = ∴ m = 1/2, 2, Example: If α and β are the roots of equation 2x2 + 2x + = 0, find α/β + β/α Solution : 2x2 + 2x + = Example: Solve √2x + + = √3x + Solution: √2x + + = √3x + Taking squares on both sides, 2x + + 16 + 8√2x+3 = 3x + 2; 8√2x + = x + – – 16 8√2x + = x – 17 Squaring again, 64(2x + 3) = x2 + 289 – 34x; 128x + 192 = x2 + 289 – 34x x2 – 162x + 97 = Example: Find 'a' such that the sum of the roots of the equation ax2 + 4x + 6a = may be equal to their product Solution: ax2 + 4x + 6a = Sum of the roots = -4/a, Product of the roots = 6a/a = -4/a = = -4/6 = -2/3 Example: Find the values of m, for which the equation x2 – 2(5 + 2m) + 3(7 + 100m) = has real and equal roots Solutions : D = (-2)2(5 + 2m)2 – × 3(7 + 10m) = 4(25 + 4m2 + 20m) – 12(7 + 10m) = 100 + 16m2 + 80m – 84 – 120m = 16m2 – 40m + 16 For real and equal roots, D = 16m2 – 40m + 16 = 2m2 – 5m + = ⇒ (2m – 1)(m – 2) = 0; m = 1/2, Example: The difference of a number and twice its positive square root is 675 Find the number Solution: Let the number be x ∴ x - 2√x = 675 ⇒ -2√x = 675 - x Squaring both sides, 4x = x2 + 455625 – 1350x ⇒ x2 – 1354x + 455625 = ⇒ x2 – 729x – 625x + 455625 = ⇒ x(x – 729) – 625(x – 729) = ⇒ (x – 625)(x – 729) = ⇒ ∴ x = 625 or 729 x = 625 does not satisfy the statement i.e., 625 - 2√625 ≠675 ∴ x = 7298 is the number Example: 117 marbles have to be divided among three children A, B and C such that B gets 11 less than what A gets and the product of number of marbles of B and C is equal to thirty times that of A Find the number of marbles with A, B and C Solution: Let b get x marbles Then, A gets = x + 11 marbles Let C get y marbles ∴ y = 117 – (x + x + 11) = 106 – 2x x(106 – 2x) = 30(x + 11); 106x – 2x2 = 30x + 130 ∴ 2x2 – 76x + 330 = ⇒ x2 – 38x + 165 = ⇒ x2 – 5x – 33x + 165 = ⇒ (x – 33)(x – 5) = 0, ∴ x = 33, So, marbles can be divided in ways i.e., A = 16, B = 5, C = 96 or A = 44, B = 33, C = 40 Example: 'A' and 'B' attempt to solve a quadratic equation of the form ax2 + bx + c = 'A' starts with a wrong value of 'c' and gets the roots as and B starts with a wrong value of b and gets roots as - and - Find the correct roots Solution: The equation is of the form ax2 + bx + c = For A, roots are and ∴ Sum of roots = and product of roots = 12 ∴ Equation is x2 – 8x + 12 = 0, but the value of c is wrong 5x2 – 3x – = For B, roots are – and – ∴ Sum of roots = - and product of roots = 15 ∴ Equation is x2 + 8x + 15 = 0, but value of b is wrong ∴ Correct equation is x2 – 8x + 15 = ⇒ x2 – 5x – 3x + 15 = ⇒ x(x – 5) – 3(x – 5) = ⇒ (x – 3)(x – 5) = ∴ x = 3, So the correct roots are and Example: A garment trader charges the same price for each garment If he increases the price of each garment by $1.5, 30 fewer garments can be purchased for $650 What is the price of each garment? Solution : Let the price of each garment be $x and let n be the number of garments that can be purchased for $650 ∴ nx = 650 New price = $(x + 1.5) New number = n – 30 ∴ (x + 1.5)(n – 30) = 650 ∴ nx + 1.5n – 30x – 45 = 650 Substituting nx = 650 and n = 650/x, 650 + 1.5 × 650/x – 30x – 45 = 650 ∴ 975/x – 30x – 45 = ⇒ 65/x – 2x – = ⇒ 65 – 2x2 – 3x = ⇒ 2x2 + 3x – 65 = ⇒ 2x2 – 10x + 13x – 65 = ⇒ 2x(x - 5) + 13(x – 5) = ⇒ (x – 5) (2x + 13) = ∴ x = 5, -13/2 But price cannot be negative ∴ Price of each garment = $5 Example: If one of the roots of a quadric equation with real coefficients is + √3, find the equation Solution: One root is + √3 It is an irrational root So the other root has to be its conjugate ∴ Its other root is – √3 Sum of roots = + √3 + – √3 = 14 Product of roots = (7 + √3) (7 – √3) = 49 – = 46 ∴ Equation if x2 – 14x + 46 = Example: P(x) = x2 - 26x + 120 is a quadratic polynomial and q(x) = 5(x -α)(x - β) is a quadrant equation with roots α and β such that α, β > When p(x) is divided by x - (α + β) the remainder is 72 and when it is divided by x - (α - β) the remainder reduces by 112 Find q(x) if p(x) = and q(x) have no common root (i.e., no root of q(x) is a root of p(x) = 0) Solution: Let α + β = y and α - β = z Then, p(y) = y2 – 26y + 120 = 72 ∴ y2 – 26y + 48 = ⇒ (y – 24)(y – 2) = ⇒ y = 24 or y = p(z) = z2 – 26z + 120 = 72 – 112 ∴ z2 – 26z + 260 = ⇒ (z – 16)(y – 10) = y = 16 or y = 10 Hence, we have combinations i) α + β = 24 α - β = 16 ⇒ α = 20 or β = ii) α + β = 24 α - β = 10 ⇒ α = 17 or = iii) α + β = α - β = 16 ⇒ α = or β = - iv) α + β = α - β = 10 ⇒ α = or β = - case (iii) and (iv) are not possible as it is mentioned that α, β>0 case (i) is also not possible as it has one common root (α = 20) with p(x) = Hence α = 17 or β = ∴ The required quadric equation is 5(x – 17)(x – 7) ∴ q(x) = 5x2 – 120x + 595 We hope you enjoyed studying from this book In case you would like to provide us with feedback or point out any errors/omissions, kindly get in touch with us at: admin@my-gre.com ###### ... Equations GRE Math for Winners Algebra, Sets, Geometry & Coordinate Geometry K Parkinson The author is a PhD in Engineering and has been training students to achieve high GRE scores for the past... PROBLEMS SETS & VENN DIAGRAMS BACKGROUND DIFFERENT WAYS OF REPRESENTING A SET TYPES OF SETS Some important results on subsets: OPERATIONS ON SETS Union of Sets Intersection of Sets Difference of Sets. .. both the axes If the distance from O to C is equal to k, what is the radius of the circle, in terms of k? i) k ii) k/ √2 iii) k/ √3 iv) k/ 2 v) k/ 3 Ans: ii) Solution The horizontal distance from C