TJC 2024 IP3 Integrated and Advanced Math June Hol HW

Temasek Junior College (IP) 2024 IP3 Intermediate & Advanced Mathematics June Holiday Homework Instructions to all students 1. Complete all working on foolscap paper. Present all working clearly. 2. After completing all questions, self-mark using the answers provided on the last page. 3. For all incorrect answers, you are to re-attempt the question s by doing the necessary corrections. 5. If you require clarification for any errors made, p lease take the initiative to consult your Mathematics tutor via email or when you are back in College. 6. Submit your solutions and corrections to your tutor for checking during the first Mathematics lesson in Term 3 2024. Intermediate Mathematics 1. Kesamet Bank offers two types of deposit accounts, the SimSaver account and the ComSavers account. The SimSavers account offers a simple interest at 5% per annum and the ComSavers account offers an interest at 4.8% per annum, compounded half-yearly. Jay opened and deposited $10 000 into a ComSavers account. (i) Find, to the nearest dollar, the amount of money Jay will have at the end of 4 years. Tee opened and deposited $10 000 into a SimSavers account. (ii) Find, to the nearest dollar, the amount of money Tee will have at the end of 4 years. At the end of 4 years, Tee intends to transfer all the money in her SimSaver account to the ComSaver account for another year. (iii) Explain, with supporting calculations, whether it is worthwhile for Tee to do so as compared to leaving her money in her SimSaver account. Show all working clearly. 2. Sketch the following graphs. (a) ( ) 2 3 2 3yx= − − (b) 2 3yx=− − (c) 2 6 13y x x= + + This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

3. (i) A quadratic graph has a minimum point at ( )3, 2− and cuts the y-axis at ( )0,16 . Find the equation of the curve. (ii) Another quadratic graph cuts the x-axis at ( )4, 0− and ( )6, 0 and passes through 213, 2   . Find the equation of the curve. 4. Show that the expression 2 22 5 25xx− + − is always negative. Hence, explain the implications of your results on the number of possible solutions of x for the equation 2 22 05 25xx− + − = . 5. In 2018, the price of water was $x per litre and a factory spent $100 on its water bill in January. (i) Express, in terms of x, the volume of water used in January. In February, the price of water was reduced by $0.14 per litre but the factory continued to spend $100 on its water bill. (ii) Express, in terms of x, the volume of water used in February. (iii) If the difference in the volumes of water used in both months is 6 litres, formulate an equation in x and show that it simplifies to 26 0.84 14 0xx− − = . (iv) Hence, find the price of water per litre in February, showing your working clearly. In 2019, a water conservation tax of a% was introduced such that the price of water is now $1.97 per litre. (v) Find the value of a, giving your answer correct to the nearest whole number. This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

6. Morph made several different objects from modelling clay. He used 500 cm3 of clay for each object. (a) He made a square-based cuboid of height 2 cm. Calculate the exact length of a side of the square base. (b) He made a pyramid with a base area of 150 cm2. Calculate the height of the pyramid. (c) He made a sphere. Calculate the radius of the sphere, correct to 2 decimal places. (d) He made a cone. Refer to the figure on the right. Then he cut through the cone, parallel to its base, to obtain a small cone and a frustum. The height of the small cone was two -fifths of the height of the full cone. Use a property of the volumes of similar objects to calculate the volume of clay in the small cone. 7. In the triangle ABC, angle 90ABC= and BC is produced to D. (a) Write down the value of cos ACD . (b) Calculate the perpendicular distance from B to AC. [Nov 2007/EM/Paper 1/Qn 12] 6 8 10 A B C D This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

8. Three points, A, B and C, lie on a horizontal field. Angle 75BAC= and the bearing of C from A is 217 . 72 mAB= and 60 mAC = . (a) Calculate (i) the bearing of B from A, (ii) BC, (iii) angle ABC, (iv) the bearing of C from B. (b) A girl standing at B is flying a kite. The kite, K, is vertically above A. The string, BK, attached to the kite is at 24 to the horizontal. Calculate the angle of elevation of the kite when viewed from C. [Nov 2005/EM/Paper 2/Qn 8] 9. The cross-section of a tunnel, shaded in the figure below, is a major segment of a circle, centre O and radius 8 m. The total perimeter of the major segment POQR is 44 m. Calculate (i) the magnitude, in radians, of reflex angle POQ, (ii) the area of triangle POQ, (iii) the total area of the cross-section of the tunnel. [Nov 2010/EM/Paper 2/Qn 8(a)] North 72 60 A B C 8 8 P Q R O This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

10. PQRS is a parallelogram. P is ( )4, 0− , Q is ( )1, 0 and R is ( )9, 4 . (a) Find the coordinates of S. (b) Find the coordinates of the midpoint of PR. (c) Find the equation of the line RS. (d) Find the equation of the line QR. (e) Calculate the area of the parallelogram PQRS. [Nov 2005/EM/Paper 1/Qn 21] 11. The diagram below shows a parallelogram ABCD with AD produced to E. F is the point of intersection of CD and BE. Angle 68BAD= and angle 97BFD= . (a) Find angle ABF. (b) Prove that triangles BCF and EDF are similar. (c) State another triangle that is similar to BCF and EDF. (d) The ratio AD : DE = 3 : 2. (i) Find the ratio AB : CF. (ii) Given that the area of triangle 29.72 cmEDF= , find the total area of the shape ABCFE. [Nov 2014/EM/Paper 2/Qn 4] A B C D E F This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

Advanced Mathematics 1. Solve the following equations. (a) 2 3 4 0xx+ − = (b) ( ) 1 2 22 4 4 9 4xx −+ +=   (c) 14 16 66xx− += 2. Solve the following simultaneous equations. ( ) 218 4 2 3 3 81 yx yx −= = 3. Without the use of a calculator, f ind the value of k for which 1 12 8 2 3 216 33226 k − + = . 4. A triangle ABC in which AB AC= has an area of 46 cm2. Given that its base BC is ( )8 3 2 2 cm− , find the exact height and perimeter of the triangle. 5. (a) Find the smallest value of the integer a for which 2 52ax x++ is positive for all values of x. [Nov 2008/AM/Paper 1/Qn 10(a)] (b) Find the set of values of m for which the curve 22y x x=− and the line 1y mx=+ do not intersect. [Nov 2012/AM/Paper 2/Qn 8(i)] 6. Express 32 2 2 5 9 11 26 x x x xx + − − −− in partial fractions. [modified Nov 2009/AM/Paper 2/Qn 2(i)] This study source was downloaded by 100000857398866 from CourseHero.com on 11-15-2024 00:10:30 GMT -06:00 https://www.coursehero.com/file/237245029/2024-IP3-IMAM-June-Hol-HW-v10-1pdf/

7. The function ( ) 32f6 x x x ax b= − + + , where a and b are constants, is exactly divisible by 3x− and leaves a remainder of 55− when divided by 2x+ . (i) Find the value of a and of b. (ii) Solve the equation ( )f0 x = [Nov 2005/AM/Paper