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2019-20 2nd Semester MTH004 Final Exam Sample Paper

I. Choose the correct answer.

1. Let u = 2i + 3j, v = i 一 j + k, and w = 3i + j 一 2k, then (u × v)．w = [ ] (A) 17. (B) 一17. (C) 13 (D) 一13

2 Let u, v, w be vectors in 3-space. Which one of the following does NOT make sense? [ ]

(A) ())u))v) × w (B) ))u))v．w (C) (u．v)．w (D) (u × v) × w

x 一 1 z 一 3

that P is the intersection of l and Π . Which one of the following are the equations of the line through P

and Q(1, 1, 1)? [ ]

z 一 1

x 一 1

of the following is the equation of the plane passing through l and perpendicular to Π? [ ]

(A) x + y 一 3z + 6 = 0. (B) x 一 y 一 3z + 10 = 0.

5 Which of the following are the parametric equations of the tangent line to the curve r(t) = e2ti + sin 3tj + ln (t2 + 1)k at (1, 0, 0)? [ ]

(A) x = 2 + t, y = 3, z = 0. (B) x = 2 + t, y = 3t, z = 0.

x x y y

7 Let f (x, y) = arcsin(xy). In which direction, the function value increases most rapidly at (1, )? [ ]

(A)〈^3, 2〉. (B) 一〈^3, 2〉. (C)〈2, 一^3〉 (D)〈一2, ^3〉

8 Let z = z(x, y) be an implicit function determined by z3 一 3xyz = 27, then = [ ]

(A) z2 一 xy . (B) z2 一 xy . (C) 一 z2 一 xy (D) 一 z2 一 xy

9 Let z = eu sin v, u = xy, v = . Then = [ ]

(A) ye塞g sin + e塞g cos . (B) xe塞g sin 一 e塞g cos .

10 Which one of the following conclusion is true for f (x, y) = x3 一 y3 + 3x2 + 3y2 一 9x? [ ]

(A) f (1, 0) is a local maximum (B) f (1, 0) is a local minimum

11 Let f (x, y) be a diﬀerentiable function satisfying f塞 (1, 一2) = 一1, and fg (1, 一2) = 4. Use the total diﬀerential to approximate ∆z = f (1.1, 一2.1) 一 f (1, 一2). Then ∆z s [ ]

(A) 一0.5. (B) 0.5. (C) 一0.3 (D) 0.3

12 Let S be the triangle with vertices (0, 0), (1, 1), (0, 1). Then x2 ydA = [ ]

(A) . (B) . (C) (D)

13 Suppose that S = {(x, y) : x2 + y2 ≤ 1, y ≥ 0}. Which integral equals zero? [ ]

(A) xye塞2 ←g2 dA. (B) e塞2 ←g2 dA. (C) (x + y)e塞2 ←g2 dA (D) (x 一 y)e塞2 ←g2 dA

d d d d

14 Let function f (x, y) be a continuous function and D = {(x, y) : 0 ≤ x ≤ 2, 0 ≤ y ≤ 1}. Assume that 1 1 1 2 1 2

f (x, y)dxdy = 4, f (x, y)dydx = 3 and f (x, y)dxdy = 一2. Which of the following

… … … 1 … 1

equals f (x, y)dA? [ ]

(A) 7. (B) 5. (C) 2. (D) 1.

1 1 xy

… 塞2 ^1 + y3

^2 一 1 ^2 + 1 ^2 + 1 一^2 + 1

(A) 3 . (B) 3 . (C) 一 3 (D) 3

16 Which one of the following positive series converges? [ ]

≈ ≈ ≈ ≈

17 Which one of the following positive series diverges? [ ]

≈ n) ≈ n! ≈ n2 ≈ n)

(A) n! . (B) n) . (C) n! . (D) 4) n! .

18 Which one of the following series is conditionally convergent? [ ]

(A) )去1(≈)(一1)) . (B) )去1(≈)(一1)) . (C) )去1(≈) ~~2 +~~ 一)1)) . (D) )去1(≈)(一1)) .

≈

19 If the power series a) (x + 1)) is conditionally convergent at x = 3, which statement about its )去1

radius of the convergence is true? [ ]

(A) The radius of convergence is 3. (B) The radius of convergence is 4.

≈ ≈

20 If a) and b) both diverge, then [ ] )去1 )去1

≈ ≈

(A) (a) + b) ) diverges (B) a) b) diverges

)去1 )去1

≈ ≈

II. Calculations and comprehensive problems.

21. Given the ellipsoid

x2 y2 z2

a2 b2 c2

(1) Let P (x… , y… , z… ) be a point on this surface in the ﬁrst octant, ﬁnd the tangent plane Π to this ellipsoid at P.

(2) Let V be the volume of the solid bounded by the three coordinate planes and the tangent plane Π, ﬁnd the coordinates of P such that the volume V attains its minimum value.

22. Assume that the solid E is bounded by the circular cone z = ^x2 + y2 and the circular paraboloid z = 2 一 x2 一 y2 .

(1). Sketch the projection D of E onto the xy-plane (z = 0).

(2). Evaluate the volume V of the solid E .

(3). Evaluate the area of the whole boundary surface of the solid E .

≈ n2 x)