Straight Line. Paper 1 Section A. O xy
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1 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 = 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 NC G2, G5 HSN 089 PSf hsn.uk.net Page 1 Questions marked c SQA
2 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 NC G9, G2, G5 HSN 097 PSf Paper 1 Section B [END F PAPER 1 SECTIN A] 3. Three lines have equations = 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
3 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 G P1 Q3 1 ss: know how to find gradient or equ. 2 pd: process 3 ic: interpret eact value 1 2 ( 1) tan a = gradient stated or implied b 3 3 a = The points A and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of AB in its simplest form Find the equation of the straight line which is parallel to the line with equation = 5 and which passes through the point (2, 1). 3 Part Marks Level Calc. Content Answer U1 C1 3 C CN G3, G = P1 Q1 1 ss: epress in standard form 2 ic: interpret gradient 3 ic: state equation of straight line 1 = 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
4 PSf Find the coordinates of the point on the curve = 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 = m tang = tan 45 = = 1 4 (2, 4) hsn.uk.net Page 4 Questions marked c SQA
5 PSf Calculate, to the nearest degree, the angle between the -ais and the tangent to the curve with equation = at the point where = 2. 4 hsn.uk.net Page 5 Questions marked c SQA
6 PSf The point P( 1, 7) lies on the curve with equation = Find the equation of the tangent to the curve at P. 4 hsn.uk.net Page 6 Questions marked c SQA
7 PSf 13. Find the equation of the tangent to the curve = at the point where = Find the equation of the tangent to the curve = at the point where = Find the equation of the tangent to the curve with equation = at the point where = 1. 4 hsn.uk.net Page 7 Questions marked c SQA
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14 PSf 22. [END F PAPER 1 SECTIN B] hsn.uk.net Page 14 Questions marked c SQA
15 PSf Paper 2 1. hsn.uk.net Page 15 Questions marked c SQA
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17 PSf 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
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19 PSf Find the equation of the line through the point (3, 5) which is parallel to the line with equation = 0. 2 hsn.uk.net Page 19 Questions marked c SQA
20 PSf 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 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
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23 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 = P2 Q1 (b) 4 C CN G7 = (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 = m = = 2( 1) 8 pd: process intersection 8 (0, 2) hsn.uk.net Page 23 Questions marked c SQA
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26 PSf 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
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36 PSf 29. Find the equation of the tangent at the point (3, 4) on the circle = 0. 4 hsn.uk.net Page 36 Questions marked c SQA
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39 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 = P2 Q7 (b) 4 C CN G = 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 = m = = 3 2 ( 5) 6 = 7, = 1 7 ( 7) 2 + ( 1) 2 8 ( 7) 2 + ( 1) 2 = 26 or c = 0 8 c = 24 hsn.uk.net Page 39 Questions marked c SQA
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41 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 = 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 = ( 1) 3 m = = 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
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48 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
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51 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 P2 Q11 (b) 4 A/B CR A33 a = 6 05, b = 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 = = P = 0 6Q Method 1 3 log e p = 0 6 log e q log e q log e 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 stated or implied b 5 or 6 5 ln a = a = 6 05, b = hsn.uk.net Page 51 Questions marked c SQA
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Polynomials Past Papers Unit 2 Outcome 1
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