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ABU DHABI POLYTECHNIC ACADEMIC SUPPORT DEPARTMENT Challenge Exam Sample version Mathematics II Students answer on the question paper Calculators are allowed No additional materials are required STUDENT NAME STUDENT NUMBER A CRN DEPARTMENT READ THESE INSTRUCTIONS CAREFULLY Write your name, number, CRN and department clearly in the boxes above. Answer all questions. Use the 3 blank pages at the end. Show all your working and use appropriate units. Otherwise, you may lose marks. Total number of correct answers Score You may use a pencil for all your work. Answers that are not clearly readable, if any, will not be marked. All mobile devices are not allowed during examination. There are 40 questions. Each is 2.5 points. Pass if Total Points ≥70, Fail if Total Points < 70 Abu Dhabi Polytechnic considers cheating or attempting to cheat a serious offense that will result in disciplinary action taken against involved individuals. RESULT Pass/Fail Formula Sheet 1. Heros’s formula for the area of a triangle: 1 𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐), where 𝑠 = 2 (𝑎 + 𝑏 + 𝑐) 2. Area of the parallelogram: 𝐴 = 𝑏 × ℎ 3. Trapezoidal Rule: 4. Simpson’s Rule: 5. sin() = sin() , cos() = tan() = tan() , csc() = csc() sec() = , cot() = cot() sec() Trigonometric functions applications: s = r cos() , A = 𝟏 r2 𝟐 , v = r Vectors: Ax = A cos Ay = A sin Law of sines using the following triangle where the sides are given as a,b,c and the angles are given as A, B, C. B 𝒂 𝐬𝐢𝐧 𝑨 = 𝒃 𝐬𝐢𝐧 𝑩 = 𝒄 𝐬𝐢𝐧 𝑪 Law of cosines: a2 = b2 + c2 2bc cos A b2 = a2 + c2 2ac cos B c2 = a2 + b2 2ab cos C . C MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 1) 131.3° A) 491.3°; -228.7° B) 491.3°; -48.7° C) 311.3°, -228.7° D) 311.3°; -48.7° 1) Solve the problem. 2) Find the supplement of 12°. A) 78° B) 348° 2) C) 168° 3) Find the complement of 79°. A) 101° B) 11° C) 191° 4) Two angles of a triangle are 37° and 97°. Find the third angle. A) 226° B) 44° C) 134° D) 258° D) 281° D) 46° Find the area. 5) 3) 4) 5) 34 m 29 m 49.5 m A) 1400 m 2 B) 490 m 2 C) 420 m 2 D) 720 m 2 Find the missing length in the right triangle. 6) The legs of a right triangle are 93.1 cm and 60.4 cm. Find the length of the hypotenuse. A) 70.8 cm B) 111 cm C) 71 cm D) 110 cm 6) Solve the problem. 7) A bike trail is in the shape of a trapezoid. Find the distance around the trail. 15.0 km 7) 3.49 km A) 39.69 km 5.60 km 15.6 km B) 87.36 km C) 19.5 km 1 D) 39.7 km Determine the indicated arc or angle. 8) FindBC. 66° 8) 56° A) 56° B) 114° C) 112° Convert to degree measure. Round to two decimal places, if necessary. 9) 2.002 rad A) 229.12° B) 115.01° C) 114.71° D) 28° D) 229.42° 9) For the given angle in standard position, designate the quadrant in which the terminal side lies. 10) - 4 rad 10) A) IV B) II C) I D) III Find the requested function value of . 3 11) If sin = , find cos . 8 A) 8 55 B) 11) 3 55 C) 12) If sin = 0.2857, find cos . A) 0.04444 B) 3.354 55 8 C) 0.9583 D) 55 3 D) 0.1556 Find the requested part of the triangle. Round your answer to two decimal places. 13) Find the measure of the angle A in degrees. 12) 13) 6.2 A) 57.27° 9.18 B) 55.57° C) 34.03° D) 55.97° Solve the problem. Round results to an appropriate number of significant digits. 14) From a boat on the lake, the angle of elevation to the top of a cliff is 17°48'. If the base of the cliff is 2266 ft from the boat, how high is the cliff (to the nearest foot)? A) 738 ft B) 731 ft C) 741 ft D) 728 ft 2 14) Sketch the graph of the line with the given equation. 15) y = 4x - 4 15) A) B) C) D) Solve by the method of elimination by addition or subtraction. 16) x - 3y = 26 4x - 4y = 48 A) x = -5, y = -6 B) x = 5, y = -7 C) x = 4, y = -6 17) 9x + 62 = 7y -6x - 2y = 8 A) x = -4, y = 6 16) D) Inconsistent 17) B) x = -3, y = 5 C) x = -3, y = 6 3 D) Inconsistent Evaluate the determinant. 18) 1 2 2 1 A) 5 18) B) 0 C) -3 Solve the system of equations by determinants. 19) x + 4y = 17 5x + y = 8 5 11 A) x = - , y = 7 3 C) x = - 19) 15 77 ,y= 19 19 Determine the quadrant containing the terminal side of 20) csc > 0 and sec > 0 A) Quadrant III B) Quadrant II 21) cot > 0 and sin < 0 A) Quadrant III D) 3 B) x = 15 77 ,y= 19 19 D) x = 15 77 ,y=19 19 under the given conditions. B) Quadrant IV C) Quadrant IV D) Quadrant I C) Quadrant II D) Quadrant I 20) 21) Determine the indicated component or components of the vector. Round to an appropriate number of significant digits. 22) Magnitude = 188.5, = 195.8° 22) Find the horizontal and vertical components of V. A) 181.4, 51.3 B) -181.4, 51.3 C) -181.4, -51.3 D) 181.4, -51.3 With the given set of components, find R and . 23) Rx = 8.0, Ry = -5.0 23) A) R = 13.0, = 328° C) R = 9.4, = 328° B) R = 9.4, = -32° D) R = 13.0, = -32° Find the missing parts of the triangle. 24) B = 27.8° C = 107.8° b = 13.17 A) A = 42.4°, a = 28.89, c = 21.76 C) A = 44.4°, a = 21.76, c = 28.89 24) B) A = 42.4°, a = 26.89, c = 19.76 D) A = 44.4°, a = 19.76, c = 26.89 25) B = 63°30' a = 12.20 c = 7.80 A) No triangle satisfies the given conditions. C) b = 11.17, A = 77°49', C = 38°41' 25) B) b = 12.17, A = 75°07', C = 41°23' D) b = 13.17, A = 73°07', C = 37°23' Find the missing parts of the triangle. (Find angles to the nearest hundredth of a degree.) 26) a = 7.8 b = 13.9 c = 15.5 A) No triangle satisfies the given conditions. B) A = 30.15°, B = 63.52°, C = 86.33° C) A = 32.15°, B = 61.52°, C = 86.33° D) A = 28.15°, B = 63.52°, C = 88.33° 4 26) Find the amplitude, period or displacement. 1 27) Find the period of y = 5 sin ( x - ). 3 2 A) 5 27) B) 3 C) 2 D) 6 Use the trapezoidal rule to find the area. 28) A pool was measured every 2.25 yd. The distances across the pool (in yards) are given in the diagram. Find the area. 28) 3.94 3.85 3.27 3.71 3.42 3.37 3.97 3.55 0.00 A) 57.0 yd2 B) 85.0 yd2 C) 93.6 yd2 D) 61.0 yd2 Use Simpson's Rule to find the area. 29) A pool was measured every 2.50 yd. The distances across the pool (in yards) are given in the diagram. Find the area. 29) 2.91 2.98 3.59 2.53 2.87 3.41 2.73 3.62 0.00 A) 53.4 yd2 B) 58.0 yd2 C) 59.5 yd2 D) 54.0 yd2 Solve the problem. 30) The perimeter of a rectangular room is 62 ft. The width is 15 ft. Find the length. A) 47 ft B) 15 ft C) 17 ft D) 16 ft 30) For the given angle in standard position, designate the quadrant in which the terminal side lies. 31) 4 rad A) III 31) B) I C) II 5 D) IV Find the requested function value of . 13 , find cot . 32) If csc = 2 A) 2 165 B) 32) 165 2 C) 13 165 D) 165 13 Solve the right triangle. Round results to an appropriate number of significant digits. 33) B = 59.7°, c = 0.697 cm A) A = 30.3°, b = 1.193 cm, a = 0.352 cm B) A = 30.3°, b = 1.193 cm, a = 0.602 cm C) A = 30.3°, b = 0.602 cm, a = 0.352 cm D) A = 30.3°, b = 0.352 cm, a = 0.602 cm Find the requested part of the triangle. Round your answer to two decimal places. 34) Find the measure of the angle A in degrees. 33) 34) 7.27 A) 24.79° 6.6 B) 22.99° C) 65.21° Determine the sign of the trigonometric function. 35) cos(-401°) A) Positive D) 26.89° 35) B) Negative For the given function value determine the quadrants in which the terminal side of the angle can lie. 36) cos = -0.5 A) II, III B) II, IV C) I, IV D) I, III 36) Determine the quadrant containing the terminal side of 37) tan < 0 and sin < 0 A) Quadrant IV B) Quadrant III 37) under the given conditions. C) Quadrant I D) Quadrant II Find the missing parts of the triangle. (Find angles to the nearest hundredth of a degree.) 38) a = 8.9 b = 13.4 c = 15.9 A) A = 32.03°, B = 57.41°, C = 90.56° B) A = 34.03°, B = 57.41°, C = 88.56° C) No triangle satisfies the given conditions. D) A = 36.03°, B = 55.41°, C = 88.56° Find the missing parts of the triangle. 39) A = 48°30' B = 36°20' a = 10.53 A) C = 96°10', b = 8.33, c = 14.00 C) C = 95°10', b = 8.33, c = 14.00 38) 39) B) C = 96°10', b = 14.00, c = 8.33 D) C = 95°10', b = 14.00, c = 8.33 6 Solve the problem. 40) A beam of light is partly reflected, and the remainder of the beam passes straight through the surface. Find the angle (angle O) between the surface and the part that passes through. 42° A) 42° B) 48° C) 87° 7 D) 132° 40)