Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Riemannian connection on a surface wikipedia , lookup
Line (geometry) wikipedia , lookup
Euler angles wikipedia , lookup
Multilateration wikipedia , lookup
Rule of marteloio wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Integer triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Euclidean geometry wikipedia , lookup
MURANGβA UNIVERSITY COLLEGE (A Constituent College of Jomo Kenyatta University of Agriculture and Technology) DEPARTMENT OF APPLIED SCIENCES END OF SEMESTER EXAMS UNIT CODE: SMB0102 DATE: 17TH August 2015 SUBJECT: GEOMETRY CLASS: BRIDGING MATHEMATICS TIME: 2 HOURS INSTRUCTIONS ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER TWO QUESTIONS QUESTION ONE (30 MARKS) (a) Without using calculators or tables (i) obtain tan 240° leaving your answer in surd form. (ii) solve for π, 0Λ < π < 90° if πΆππ (3π + 20Λ) = πππ(4π). (3 Marks) (2 Marks) (b) Calculate the number of sides of a regular polygon whose (i) interior angle is 135Λ. (ii) exterior angle is 72Λ. (3 Marks) (1 Mark) (c) Draw a triangle ABC without using a protractor so that AB = 5cm, angle ABC = 45Λ and angle BAC = 60Λ. (i) Measure AC and BC. (ii) Drop a perpendicular from A to BC. (4 Marks) (d) Draw a line AB = 6 cm. Without using a protractor, construct the locus of all points P above the line segment AB such that AB = 6 cm and angle APB = 45°. (4 Marks) (e) Joel whose height is 2.1 m observes his shadow to be 5 m on the horizontal ground. Calculate the angle of elevation to the sun at that time. (3 Marks) 1 (f) The interior angles of a hexagon are 2π₯, 2 π₯, π₯ + 40°, 110°, 130° πππ 160°. Calculate the size of the smallest angle. (g) Construct a regular hexagon of sides 5cm by inscribing it in a circle. (4 Marks) (3 Marks) (h) Alice walks from a point A on a bearing of 30° for 5 km and then walks due south to a point 8 km from A. Calculate: i. Aliceβs new bearing from A. ii. Aliceβs total distance covered. (3 Marks) 1 QUESTION TWO (20 MARKS): OPTIONAL (a) (i) Without using a protractor or a setsquare, draw a pentagon ABCDE with AB = 8 cm, BC = 6 cm, CD = 5.2 cm, angles < πΈπ΄π΅ = 150°, < π΄π΅πΆ = 120°, < π΅πΆπ· = 135° πππ < πΆπ·πΈ = 60°. (8 Marks) (ii) Measure DE and angle < π΄πΈπ·. (2 Marks) (b) (i) Without using a protractor, construct triangle ABC in which BC = 6 cm, AB = 8 cm and <ABC = 135°. Measure <BAC, < π΅πΆπ΄ and line AC. (5 Marks) (ii) Draw a line AB = 10 cm. Draw a circle of radius 4.5 cm centered at B. Draw a tangent from A to point P on the circle you have drawn. (3 Marks) Measure the length of the tangent and the angle ABP. (2 Marks) QUESTION THREE (20 MARKS): OPTIONAL (a) A pentagon has the following interior angles: π₯Λ, (2π₯ β 50)Λ, (π₯ + 40)Λ, 2π₯Λπππ(2π₯ β 10)Λ. Calculate the sizes of all the exterior angles. (5 Marks) (b) Solve the following equations: (i) Sin(2A +10)° = Cos(3A), 0 β€ π΄ β€ 90° (ii) 2Sin2π + 1 = 0, 0 β€ π β€ 360° (5 Marks) (c) Solve the following trigonometric equations: i. 2sin2π₯ + 1 = 3sinπ₯. 0° β€ π₯ β€ 360° 3 cos2π β 4 cosπ β 4 = 0 ii. β180° β€ π β€ 180° (8 Marks) (d) Without using tables or calculators, find the value of cos(β210°) and leave your answer in surd form. (2 Marks) QUESTION FOUR (20 MARKS): OPTIONAL (a) Given that AB =(23) and BC = (β2 ), work out: 4 (i) AB + BC 1 2 (ii) π΅πΆ (iii) (iv) β 3AB AB β 2CB (5 Marks) 6 (b) Find the values of x and y if (1βπ₯ ). ) = (2π¦+1 3 2 (2 Marks) (c) Given the points P(-3, -5) and Q(2,9): (i) find vector PQ. (ii) find the length of PQ leaving your answer in surd form. (2 Marks) (d) If β4 (28)πππ(π₯+3 ) are parallel, find the value of x. (4 Marks) (e) (i) What is the image of triangle ABC with A(-3,5), B(2,1), C(-5,0) after a translation vector (43)?. (ii) A translation transforms P(3,9) to Pβ(-2,5). Obtain the translation vector and the image of M(-6,4) after the same translation. (7 Marks) QUESTION FIVE (20 MARKS): OPTIONAL (a) i The three angles of a triangle are 2x, 4x +30° and 10x β 10°. Calculate the size of each angle. What type of triangle is this? (3 Marks) ii The lengths of two sides of a triangle are 10 cm and 16 cm. If the area of the triangle is 80 cm2, find the size of the angle between the two sides . (2 Marks) (b) A, B and C are three points on the surface of the earth. (i) Calculate the distance in nautical miles between π΄(40°π, 20°π)πππ π΅(40°π, 100°π) measured along the circle of latitude. (2 Marks) (ii) Calculate the distance in both kilometers and nautical miles along the circle of latitude between π΄(40°π, 20°π)πππ πΆ(40°π, 30°πΈ). Take the radius of the earth as 6370 km. (3Marks) (c) An aircraft leaves Nairobi (1°15β²π, 36°49β²πΈ)at 0900 hours and flies due west. At 1900 hours Nairobi time, the plane is above the town π΄(1°15β²π, 0°49β²πΈ). Find the speed of the aircraft in: (i) Km/h (4 Marks) (ii) Knots. (Take R = 6370 km) (3 Marks) (1°15β²π, (d) Given that the locations of Nairobi and New York are 36°49β²πΈ)πππ (40°45β²π, 70°0β²π) respectively, find the difference in time between the two cities. (3 Marks) 3