R cm T 60 K

Transcription

1 Mathematics(NSC)/Grade 11/ P 19 Eemplar Grade 11 Mathematics: Question Paper MARKS: 150 TIME: 3hours QUESTION A triangle is drawn with vertices A (0;) ; B (4;5) and C (4;-4) Find the length of AB. () 1.1. Find the equation of the line through B and C. (1) Find the equation of the line through A and B. () Find the inclination of the line through A and B. () 1. W hat wil be the gradient of the line perpendicular to 3 7 0? () 1.3 Using a calculator find the values of the folowing if 4 and 178, : sin 3 (1) 1.3. cos 3( ) () 1.4 Simplif: A A sin180 sin Find the solution to 3tan 1on the interval 0[, 70] (4) 1.6 Consider the diagram below: P 7 cm 5 cm R T 60 K Find the length of KT Find the length of PT. 1.7 Draw, on the graph paper on our diagram sheet, a bo and whisker diagram for a set of data with the folowing characteristics: o Median is 17 o Upper quartile is 0 o Lower quartile is 11 o Maimum value is 30 o Range is 0 (5)

2 Mathematics(NSC)/Grade 11/ P 0 Eemplar 1.8 Find the volume of the right cone with slant height 13 mm and with radius of base 5 mm [33] QUESTION Consider parallelogram ABCD with co-ordinates as shown in the diagram below. A (0;0) B (11;5) D (10;0) C (1;5).1 List two properties that are true of a rectangle which are not true of all parallelograms. (). Find the lengths of AC and BD (leave our answers in surd form). (4).3 Find the gradients of AB and AD. (4).4 Is ABCD a rectangle? Give two detailed reasons using our answers from. and.3. () [1]

3 Mathematics(NSC)/Grade 11/ P 1 Eemplar QUESTION 3 A triangle with vertices A (3;5); B (8;4) and C (;) is drawn. A cop of this diagram appears on our diagram sheet A (3;5) B (8;4) C (;) Give the co-ordinates of the vertices of the image AB ' ' C' of ABC when it is rotated 90 degrees anti-clockwise around the origin. (5) 3. Complete the generalisation of this transformation: ( ; ) (... ;...) () 3.3 Find the equation of the perpendicular bisector of BB '. (5) 3.4 Given that the equation of AA ' is = -4, show that AA ' and BB ' intersect at the centre of rotation. 3.5 Give the co-ordinates of the vertices of A' ' B'' C'' if it is the image of ABC when it is rotated through an angle of 180. [18]

4 Mathematics(NSC)/Grade 11/ P Eemplar QUESTION 4 Square PQRS has vertices with co-ordinates as shown in the diagram below. This diagram is reproduced on our diagram sheet. PQRS is to be enlarged b a scale factor of P (1;) Q (4;4) R (6;1) S (3;-1) 4.1 On the diagram sheet draw this enlargement and indicate the vertices and the co-ordinates of these vertices. P' Q' R' S' (7) 4. Calculate the length of a side of both PQRS and P' Q' R' S' and hence determine the relationship between the increased length of the sides and the increased area of the squares. Work in surd form. (6) [13]

5 Mathematics(NSC)/Grade 11/ P 3 Eemplar QUESTION 5 Throughout this question a calculator ma not be used and all working must be clearl shown. 5.1 Simplif the following: - tan.sin sin cos tan 5 tan 180 cos 90 (6) (4) 5. Consider the equation: cos cos Factorise the left hand side of the equation. (1) 5.. Find the general solution to the equation. (5) 5.3 If sin 58 k, then find the following: sin 38 () 5.3. cos 58 [1] QUESTION 6 Four learners are arguing about whose trigonometric epression best describes a particular situation. Sipho Ra Lorraine Vishnu 1 1 tan tan 1 sin 1 sin cos The each substitute 30 into their epression. The all get the same value. What is it? (1) 6. The each substitute 50. What value do the each get? (4) 6.3 Using our knowledge of trigonometric identities, show that three of the learners epressions are eactl the same. (7) [1]

6 Mathematics(NSC)/Grade 11/ P 4 Eemplar QUESTION 7 You go down to the beach for a few das. As an eperiment ou place a metre stick in the sand to measure the height of the water. As the tide comes in, the height rises and then falls as the tide goes out. You record the heights for 48 hours the height fluctuates between 0 cm and 50 cm. This is shown in the graph below: 50 (5;50) (5;50) (45;50) (48;a) The equation for the height of the water is: 5sin18 5 where is the height of the water and is the time in hours from the beginning of the eperiment. 7.1 Calculate the height of the water when = 48. (1) 7. Calculate the times when the height of the water is 10 cm b solving the equation 5sin for the 48 hour interval rounded off to the nearest hour. (6) [7]

7 Mathematics(NSC)/Grade 11/ P 5 Eemplar QUESTION 8 While ou are on the beach ou stand at the base of a life-guards tower, B, and measure the angle of elevation of a lighthouse, PT, to be. From the top of the 5 m high lifeguards tower, A, the angle of elevation of P is. P A 5m B T 8.1 Find the size of AP ˆ B in terms of and. () 8. Show that PB 5cos sin( ) 8.3 Hence show that PT 5cos.sin sin( ) () [7]

8 Mathematics(NSC)/Grade 11/ P 6 Eemplar QUESTION 9 The diagram alongside looks ver similar to that for the Theorem of Pthagoras, ecept that the central triangle is not right-angled. On each side of a triangle (with angles, and z) a square has been drawn. The outer corners of the three squares have been joined as shown to make three more triangles. D E A c B z b F a C G K H Complete the formula: area of ABC bc (1) 9. Show that area DAK area ABC. 9.3 State with reasons what will be the relationship between the areas of DAK and EBF. () [6] QUESTION 10 On the beach ou find 10 shells and measure their lengths. These lengths are given in the table below. Length (cm) 3, 3,6 5,0 4,1 4,3 4,7 3,4 5, 4,6 4,3 i i 10.1 Calculate the mean length of these ten shells. ()

9 Mathematics(NSC)/Grade 11/ P 7 Eemplar 10. Complete the cop of our table on our diagram sheet and use it to calculate the standard deviation of the length of our sample of shells. (6) 10.3 You also measure the widths of the ten shells. The graph of each shell s length plotted against its width is shown below. The graph is reproduced on the diagram sheet. 4 Length to width comparison of 10 shells 3 W idth (mm) Length (mm) Draw an approimate line of best fit for the data and find its equation. [11] QUESTION 11 A thousand pebbles from the beach are collected and their lengths are measured. The length of the smallest pebble is 1 mm and the largest is 95 mm. The lengths of the pebbles are summarised in the table below: Length of pebble (mm) Number of pebbles Cumulative frequenc < Complete the cumulative frequenc column in the cop of the table on our diagram sheet. () 11. Draw the ogive for this set of data using the graph paper on our diagram sheet. (5) 11.3 Find the median, upper quartile and lower quartile of the data using our graph. [10] End of Paper