Straight Line. Paper 1 Section A. O xy

PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of a? A. 3 B. 1 3 C. 1 3 D. 3 Ke utcome Grade Facilit Disc. Calculator Content Source C 1.1 C 0.7 0.62 NC G2, G5 HSN 089 PSf hsn.uk.net Page 1 Questions marked c SQA

PSf 2. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0). A. 2 B. 1 2 C. 1 2 D. 2 Ke utcome Grade Facilit Disc. Calculator Content Source C 2.4 C 0.43 0.77 NC G9, G2, G5 HSN 097 PSf Paper 1 Section B [END F PAPER 1 SECTIN A] 3. Three lines have equations 2 + 3 4 = 0, 3 17 = 0 and 3 10 = 0. Determine whether or not these lines are concurrent. 4 hsn.uk.net Page 2 Questions marked c SQA

PSf 4. Find the size of the angle a PSf that the line joining the points A(0, 1) and B(3 3, 2) B(3 3, 2) makes with the positive direction of the -ais. a 3 A(0, 1) Part Marks Level Calc. Content Answer U1 C1 3 C NC G2 30 2000 P1 Q3 1 ss: know how to find gradient or equ. 2 pd: process 3 ic: interpret eact value 1 2 ( 1) 3 3 0 2 tan a = gradient stated or implied b 3 3 a = 30 5. The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form. 2 6. Find the equation of the straight line which is parallel to the line with equation 2 + 3 = 5 and which passes through the point (2, 1). 3 Part Marks Level Calc. Content Answer U1 C1 3 C CN G3, G2 2 + 3 = 1 2001 P1 Q1 1 ss: epress in standard form 2 ic: interpret gradient 3 ic: state equation of straight line 1 = 2 3 + 5 3 stated or implied b 2 2 m line = 2 3 stated or implied b 3 3 ( 1) = 2 3 ( 2) hsn.uk.net Page 3 Questions marked c SQA

PSf 7. 8. Find the coordinates of the point on the curve = 2 2 7 + 10 where the tangent to the curve makes an angle of 45 with the positive direction of the -ais. 4 Part Marks Level Calc. Content Answer U1 C3 4 C NC G2, C4 (2, 4) 2002 P1 Q4 1 sp: know to diff., and differentiate 2 pd: process gradient from angle 3 ss: equate equivalent epressions 4 pd: solve and complete 1 d d = 4 7 2 m tang = tan 45 = 1 3 4 7 = 1 4 (2, 4) hsn.uk.net Page 4 Questions marked c SQA

PSf 9. 10. Calculate, to the nearest degree, the angle between the -ais and the tangent to the curve with equation = 3 4 5 at the point where = 2. 4 hsn.uk.net Page 5 Questions marked c SQA

PSf 11. 12. The point P( 1, 7) lies on the curve with equation = 5 2 + 2. Find the equation of the tangent to the curve at P. 4 hsn.uk.net Page 6 Questions marked c SQA

PSf 13. Find the equation of the tangent to the curve = 4 3 2 at the point where = 1. 4 14. Find the equation of the tangent to the curve = 3 3 + 2 at the point where = 1. 4 15. Find the equation of the tangent to the curve with equation = 5 3 6 2 at the point where = 1. 4 hsn.uk.net Page 7 Questions marked c SQA

PSf 16. hsn.uk.net Page 8 Questions marked c SQA

PSf 22. [END F PAPER 1 SECTIN B] hsn.uk.net Page 14 Questions marked c SQA

PSf Paper 2 1. hsn.uk.net Page 15 Questions marked c SQA

PSf 2. 3. hsn.uk.net Page 16 Questions marked c SQA

PSf 4. 5. Find the equation of the perpendicular bisector of the line joining A(2, 1) and B(8, 3). 4 hsn.uk.net Page 17 Questions marked c SQA

PSf 8. 9. Find the equation of the line through the point (3, 5) which is parallel to the line with equation 3 + 2 5 = 0. 2 hsn.uk.net Page 19 Questions marked c SQA

PSf 10. 11. The vertices of a triangle are P( 1, 1), Q(2, 1) and R( 6, 2). Find the equation of the altitude of triangle PQR, drawn from P. 3 12. Find the equation of the median AD of triangle ABC where the coordinates of A, B and C are ( 2, 3), ( 3, 4) and (5, 2) respectivel. 3 hsn.uk.net Page 20 Questions marked c SQA

PSf 15. Triangle ABC has vertices A( 1, 6), B( 3, 2) and C(5, 2). A( 1, 6) Find (a) the equation of the line PSfrag p, the median from C of triangle ABC. C(5, 2) 3 (b) the equation of the line q, the perpendicular bisector of BC. 4 (c) the coordinates of the point of B( 3, 2) intersection of the lines p and q. 1 Part Marks Level Calc. Content Answer U1 C1 (a) 3 C CN G7 = 2 2002 P2 Q1 (b) 4 C CN G7 = 2 + 2 (c) 1 C CN G8 (0, 2) 1 ss: determine midpoint coordinates 2 pd: determine gradient thro 2 pts 3 ic: state equation of straight line 4 ss: determine midpoint coordinates 5 pd: determine gradient thro 2 pts 6 ss: determine gradient perp. to 5 7 ic: state equation of straight line 1 F = mid AB = ( 2, 2) 2 m FC = 0 stated or implied b 3 3 equ. FC is = 2 4 M = mid BC = (1, 0) 5 m BC = 1 2 6 m = 2 7 0 = 2( 1) 8 pd: process intersection 8 (0, 2) hsn.uk.net Page 23 Questions marked c SQA

PSf 18. 19. P, Q and R have coordinates (1, 2), (6, 3) and (9, 14) respectivel and are three vertices of a kite PQRS. (a) Find the equations of the diagonals of this kite and the coordinates of the point where the intersect. 7 (b) Find the coordinates of the fourth verte S. 2 hsn.uk.net Page 26 Questions marked c SQA

PSf 29. Find the equation of the tangent at the point (3, 4) on the circle 2 + 2 + 2 4 15 = 0. 4 hsn.uk.net Page 36 Questions marked c SQA

PSf 32. Triangle ABC has vertices A(2, 2), B(12, 2) and C(8, 6). PSf C(8, 6) (a) Write down the equation of l 1, the perpendicular bisector of AB. 1 A(2, 2) B(12, 2) (b) Find the equation of l 2, the perpendicular bisector of AC. 4 (c) Find the point of intersection of lines l 1 and l 2. 1 (d) Hence find the equation of the circle passing through A, B and C. 2 Part Marks Level Calc. Content Answer U2 C4 (a) 1 C CN G3, G7 = 7 2001 P2 Q7 (b) 4 C CN G7 3 + 2 = 23 (c) 1 C CN G8 (7, 1) (d) 2 A/B CN G8, G9, G10 ( 7) 2 + ( 1) 2 = 26 1 ic: state equation of a vertical line pd: process coord. of a midpoint 3 ss: find gradient of AC 4 ic: state gradient of perpendicular 5 ic: state equation of straight line 2 6 pd: find pt of intersection 7 ss: use standard form of circle equ. 8 ic: find radius and complete 1 = 7 2 midpoint = (5, 4) 3 m AC = 2 3 4 m = 3 2 5 4 = 3 2 ( 5) 6 = 7, = 1 7 ( 7) 2 + ( 1) 2 8 ( 7) 2 + ( 1) 2 = 26 or 7 2 + 2 14 2 + c = 0 8 c = 24 hsn.uk.net Page 39 Questions marked c SQA

PSf 34. (a) Find the equation of AB, the perpendicular bisector of the line Q(1, 9) joing the points P( 3, 1) and Q(1, 9). PSf 4 (b) C is the centre of a circle passing through P and Q. Given that QC is C parallel to the -ais, determine the equation of the circle. 3 (c) The tangents at P and Q intersect at T. Write down (i) the equation of the tangent at Q (ii) the coordinates of T. 2 A P( 3, 1) Part Marks Level Calc. Content Answer U2 C4 (a) 4 C CN G7 + 2 = 9 2000 P2 Q2 (b) 3 C CN G10 ( 1) 2 + ( 4) 2 = 25 (c) 2 C CN G11, G8 (i) = 9, (ii) T( 9, 9) B 1 ss: know to use midpoint 2 pd: process gradient of PQ 3 ss: know how to find perp. gradient 4 ic: state equ. of line 5 ic: interpret parallel to -ais 6 pd: process radius 7 ic: state equ. of circle 8 ic: interpret diagram 9 ss: know to use equ. of AB 1 midpoint = ( 1, 5) 2 m PQ = 9 1 1 ( 1) 3 m = 1 2 4 5 = 1 2 ( ( 1)) 5 C = 4 stated or implied b 7 6 radius = 5 or equiv. stated or implied b 7 7 ( 1) 2 + ( 4) 2 = 25 8 = 9 9 T= ( 9, 9) hsn.uk.net Page 41 Questions marked c SQA

PSf 42. ABCD is a quadrilateral with vertices A(4, 1, 3), B(8, 3, 1), C(0, 4, 4) and D( 4, 0, 8). (a) Find the coordinates of M, the midpoint of AB. 1 (b) Find the coordinates of the point T, which divides CM in the ratio 2 : 1. 3 (c) Show that B, T and D are collinear and find the ratio in which T divides BD. 4 hsn.uk.net Page 48 Questions marked c SQA

Higher Mathematics 45. PSf The results of an eperiment give rise to the graph shown. PSf (a) Write down the equation of the line in 1 8 terms of P and Q. 2 3 Q P It is given that P = log e p and Q = log e q. (b) Show that p and q satisf a relationship of the form p = aq b, stating the values of a and b. 4 Part Marks Level Calc. Content Answer U3 C3 (a) 2 A/B CR G3 P = 0 6Q + 1 8 2000 P2 Q11 (b) 4 A/B CR A33 a = 6 05, b = 0 6 1 ic: interpret gradient 2 ic: state equ. of line 3 ic: interpret straight line 4 ss: know how to deal with of log 5 ss: know how to epress number as log 6 ic: interpret sum of two logs 1 m = 1 8 3 = 0 6 2 P = 0 6Q + 1 8 Method 1 3 log e p = 0 6 log e q + 1 8 4 log e q 0 6 5 log e 6 05 6 p = 6 05q 0 6 Method 2 ln p = ln aq b ln p = ln a + b ln q 4 ln p = 0 6 ln q + 1 8 stated or implied b 5 or 6 5 ln a = 1 8 6 a = 6 05, b = 0 6 3 hsn.uk.net Page 51 Questions marked c SQA

PSf 47. [END F PAPER 2] hsn.uk.net Page 53 Questions marked c SQA