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MA225B Mock Test 1

TRUE/FALSE. Write 'T' if the statement is true and 'F' if the statement is false. (1 mark each)

1) The vector 1/5, 2/5, 2/5 is a unit vector.

2) Two vectors v and w are parallel if v · w = 0 .

3) If two planes ax + by + cz = d and Ax + By + Cz = D are parallel, then a = A,b = B, and c = C.

4) Every point on the parametric curve r(t) = (t,t 2,-t) lies on the surface xz + y = 0.

5) Proj u v = Proj v u for all vectors u and v.

6) The shape of the curves r(t) = (t,t 2,t 3) and R(t) = (t 2,t 4,t 6) are the same around t=1.

7) The vector -5,4,1 is parallel to the plane -5x + 4y + z = 2.

8) There are vectors u and v such that u · v = u × v .

MULTIPLE CHOICE. (2 marks each)

The position vector of a particle is r(t). Find the requested vector.

9) The velocity at t = 1 for r(t) = (2 - 4t 2)i + (6t + 5)j - e -6tk

A) v(1) = 8i + 6j + 6e -6k

B) v(1) = -8i + 6j + 6e -6k

C) v(1) = -4i +6j + 6e -6k

D) v(1) = -8i + 6j - 6e -6k

Identify the type of surface represented by the given equation.

10) y2 + z2 = 6

A) Parabolic cylinder

B) Ellipsoid

C) Cylinder

D) Paraboloid

11) The following equations each describe the motion of a particle. For which path is the particle's speed constant?

(1) r(t) = t 2i + t 5j

(2) r(t) = cos (8t)i + sin (5t)j

(3) r(t) = ti + tj

(4) r(t) = cos (9t 2)i + sin (9t 2)j

A) Path (3)

B) Path (4) and Path (2)

C) Path (1)

D) Path (2) and Path (3)

Find projv u.

12) v = 3i - j + 3k, u = 11i + 2j + 10k

A) 195 19 i - 65 19 j + 195 19 k

B) 183 19 i - 61 19 j + 183 19 k

C) 671 15 i + 122 15 j + 122 3 k

D) 671 225 i + 122 225 j + 122 45 k

The position vector of a particle is r(t). Find the requested vector.

13) The acceleration at t = 1 for r(t) = (3t - 2t 4)i + (2 - t)j + (6t 2 - 7t)k

A) a(1) = 24i + 12k

B) a(1) = -6i + 12k

C) a(1) = -24i - j + 12k

D) a(1) = -24i + 12k

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors at time t = 0.

14) r(t) = (4t 2 + 7)i + (3t 3 - 2t)k

A) Δ

B) Δ 2

C) Δ 4

D) 0

Find parametric equations for the line described below.

15) The line through the points P(-1, -1, 7) and Q(-7, 4, 3)

A) x = t + 6, y = t - 5, z = 7t + 4

B) x = -6t - 1, y = 5t - 1, z = -4t + 7

C) x = -6t + 1, y = 5t + 1, z = -4t - 7

D) x = t - 6, y = t + 5, z = 7t - 4

Write down the equation for the plane.

16) The plane through the point P(4, -3, 2) and normal to n = 2, 7, 6 .

A) 4x - 3y + 2z = -1

B) -4x + 3y - 2z = -1

C) -2x - 7y - 6z = -1

D) 2x + 7y + 6z = -1

SHORT QUESTIONS.

17) For the smooth curve

r(t) = (6 sin t)i - (9 cos 3t)j +e -10tk,

find the parametric equation for the line that is tangent to r at t = 0.

(5 marks)

18) Compute the first derivatives for the following. (5 marks)

a) ·(t) = (cos(e 2t), sin(e 2t), e2t).

b) (t,2t,3t 2)·(cos(t),sin(t),log(t)) at t=Δ . (log(t)=ln(t) throughout this course.)