Vectors. Paper 1 Section A. Each correct answer in this section is worth two marks. 4. The point B has coordinates

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1 PSf Vectors Paper Section A Each correct answer in this section is worth two marks.. A vector v is given b 2. 6 What is the length, in units, of v? A. 7 B D The point B has coordinates (, 0, 5) and AB = 9. 5 What are the coordinates of point A? A. (0,, 0) B. (0,, 0). ( 6,, 0) D. (6,, 0) 2. The point A has coordinates (9, 7, 2) and B(5, 5, ). What is the value of AB? A. B.. 29 D. 2. What is the distance between the points (,, ) and (2, 7, 4)? A. 86 B D. 62 hsn.uk.net Page 5. Vectors p and q are defined b p = 2i k and q = i + j + k. Find 2 p q in component form. A. 5 B D. Questions marked c SQA

2 PSf Higher Mathematics PSf k 8. Parallelogram AB is shown 6. The vector u is given b 2k below. 2k where k > 0 is a constant. A D Given that u is a unit vector, what is the value of k? a B A. 9 B. 5. D. 5 PSf 7. ABDE is a square-based pramid, and X is the centre of the base. E c The point D divides AB in the ratio :. Find D in terms of a and c. A. a 4 c B. a c. 4 c a D. c a A Given that A 4 = 4 and 0 2 E = 2, find XE. 5 4 A B D. 4 5 hsn.uk.net Page 2 B X D 9. The point P has coordinates (4,, 7) and Q(7, 9, 4). The point R divides PQ in the ratio : 2. Find the components of PR. A. B.. D Questions marked c SQA

3 0. The vectors and 7 perpendicular. What is the value of k? A. B.. 0 D. 8 k 2 are PSf a 0. The vectors and 2a are b b perpendicular. Find an epression for a in terms of b. A. a = b 2 B. a = 2 b2. a = 2 b2 2 D. a = b 2 2. For two vectors u and v, u = 4, v = 7 and u.v =. What is the value of u.(u + v)? A. 7 B D. 44 [END F PAPER SETIN A] Paper Section B. Two vectors a and b are given b a = 2 4 A third vector c is defined b c = 2a b. and b = Find the components of c and c.. hsn.uk.net Page Questions marked c SQA

4 PSf 4. The cubic with equation = and the line = + 5 are shown PSfrag in the diagram. The line and curve intersect at the points P, Q and R. P Q (a) Given that the -coordinate of Q is 2, find R the coordinates of P, Q and R. 7 (b) Determine the ratio in which Q divides the line PR The circles centred at A and B have equations = 0 and = 0 respectivel. (a) Write down the coordinates of A and B. 2 (b) Show that the circles touch eternall. 4 (c) The circles touch at point. PSf (i) Find the ratio in which divides AB. B (ii) Hence find the coordinates of. 4 A 6. The point Q divides the line joining P(,, 0) to R(5, 2, ) in the ratio 2 :. Find the coordinates of Q. 7. R is the point (,, 7) and T(8, 4, ). The point S divides RT in the ratio : 2. Find the coordinates of S. PSf 8. A triangle has vertices at the origin, A(8, 6, 2) A(8, 6, 2) and B(0,, ) as shown in the diagram. M A is a median of triangle AB, and the point M divides A B(0,, ) in the ratio 2 :. Find the coordinates of M. 4 hsn.uk.net Page 4 Questions marked c SQA

5 Higher Mathematics 9. VABD is a pramid with a rectangular base ABD. Relative to some appropriate aes, VA represents 7i j k AB represents 6i + 6j 6k AD represents 8i 4j + 4k. PSf PSf K divides B in the ratio :. A B Find VK in component form. D V K k 20. Vectors u and v are given b u = and v = k where k is a constant. k (a) Given that u.v = 2, show that k k 2 = 0. 2 (b) Show that (k + ) is a factor of k k 2 and hence full factorise k k 2. 5 (c) Given that u =, find the value of k. 2 (d) Given that the angle between u and v is θ, find the eact value of cos θ. 2 hsn.uk.net Page 5 Questions marked c SQA

6 PSf 2. (a) Roadmakers look along the tops of a set of T-rods to ensure that straight sections of road are being created. Relative to suitable aes the top left corners of the B T-rods are the points A( 8, PSfrag 0, 2), B( 2,, ) and (6,, 5). A Determine whether or not the section of road AB has been built in a straight line. (b) A further T-rod is placed such that D has coordinates (, 4, 4). Show that DB is perpendicular PSfrag to AB. B A D Find the value of c for which the vectors u = 4 c 2 and v = 5 are perpendicular. 2. Two vectors u and v are such that u = 7, v = 4 and u.v = 4. The vector w is defined b w = 2u + 2 v. Evaluate w.w and hence state w [END F PAPER SETIN B] hsn.uk.net Page 6 Questions marked c SQA

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10 PSf The vectors p, q and r are defined as follows: p = i j + 2k, q = 4i j + k, r = 4i 2j + k. (a) Find 2 p q + r in terms of i, j and k. (b) Find the value of 2 p q + r hsn.uk.net Page 0 Questions marked c SQA

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12 PSf PSf 9. A cuboid measuring cm b 5 cm b 7 cm is placed centrall on top of another cuboid measuring 7 cm b 9 cm b 8 cm. oordinates aes are taken as shown. z 5 7 A B (a) The point A has coordinates (0, 9, 8) and has coordinates (7, 0, 8). Write down the coordinates of B. (b) alculate the size of angle AB The square-based PSf pramid ABDT is shown below. T B A D All of the edges of ABDT have length 4 units. 2. (a) Find the eact value of cos TÂ. 2 (b) Hence find the eact value of AT.( AB + A) hsn.uk.net Page 2 Questions marked c SQA

13 PSf ABD is a quadrilateral with vertices A(4,, ), B(8,, ), (0, 4, 4) and D( 4, 0, 8). (a) Find the coordinates of M, the midpoint of AB. (b) Find the coordinates of the point T, which divides M in the ratio 2 :. (c) Show that B, T and D are collinear and find the ratio in which T divides BD In the diagram, the circle centred at A has equation = 0 and the circle centred at B has equation ( + ) 2 + ( + 5) 2 = 4. (a) Find the coordinates of A and PSfrag B. 2 A (b) Find the shortest distance between the two circles. 4 (c) Points A, B and are collinear, and the circle centred at touches the other two circles B eternall. (i) Given that B = kba, find the value of k. (ii) Hence find the coordinates of. (iii) Write down the equation of the circle centred at. 5 hsn.uk.net Page Questions marked c SQA

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16 PSf Vectors u, v and w are given b u = k, v = 2 and w = 4, k 5 where k is a constant. (a) Given that u + v is perpendicular to w, find the value of k. (b) Hence calculate the smallest angle between u and v. 5 hsn.uk.net Page 6 Questions marked c SQA

17 PSf 5. A bo in the shape of a cuboid is designed with circles of different sizes on each face. The diagram shows three of the circles, where the origin represents one of the corners of the cuboid. The z centres of the circles are A(6, 0, 7), B(0, 5, 6) and (4, 5, 0). B Find the size of angle AB. 7 PSf A PSf 6. The parallelogram PQRS is shown in the diagram below. R v Q S u T P The vectors u and v represent line segments QP and QR respectivel, and are such that u = 5, v = 2 and u.v = 5. The point T divides QP in the ratio 2 :. (a) Epress RP and RT in terms of u and v. (b) Hence evaluate RP. RT. 7. For what value of t are the vectors u = t 2 2 and v = 0 perpendicular? 2 t 8. A(4, 4, 0), B( 2, 4, 2) and ( 8, 0, 0) are the vertices of a right-angled triangle. Determine which angle of the triangle is the right angle. Questions marked c SQA hsn.uk.net Page 7

18 PSf 9. k k 40. The vectors u and v have components 2 and k respectivel. 4 Show that there is no value of k for which u and v are perpendicular The diagram shows a square-based PSf z pramid of height 8 units. Square AB has a side length of 6 units. The coordinates of A and D are (6, 0, 0) and (,, 8). lies on the -ais. (a) Write down the coordinates of B. (b) Determine the components of DA A(6, 0, 0) and DB. 2 (c) alculate the size of angle ADB. 4 D(,, 8) B hsn.uk.net Page 8 Questions marked c SQA

19 PSf alculate the acute angle between the vectors u = and v = hsn.uk.net Page 9 Questions marked c SQA

20 PSf alculate the acute angle between the two vectors p = 2i + 4 j k and q = i j + 2k hsn.uk.net Page 20 Questions marked c SQA

21 PSf PQRS is a parallelogram with vertices P(,, ), Q(4, 2, 2) and R(,, ). Find the coordinates of S. hsn.uk.net Page 2 Questions marked c SQA

22 PSf 52. [END F PAPER 2] hsn.uk.net Page 22 Questions marked c SQA