Q1. Answer the following questions:
1) If u = 2i - j + 3k and v = i + 2k, then |u x v| =
2) If θ is the angle between the vectors u = ⟨2,1,2⟩ and v = ⟨2,6,-3⟩, then cos θ =
3) If u = 10i + 5j and v = 3i - 4j, then proj_v u =
4) Which of the following vectors is normal to the plane 2(x - 1) + y - 3(z - 2) = 0?
5) Which of the following vectors is parallel to the line x = -1 + t, y = 2t, z = 3?
6) Which of the following points lies on the line x = -1 + t, y = 2 + 3t, z = 5 - 2t?
7) Which of the following points lies on the intersection of the line x = -1 + t, y = 2 + 3t, z = 5 - 2t and the plane x - 2y + 3z = -1?
8) If the position of a particle at any time t is given by r(t) = 2t²i - 3tj, then the particle's speed at t = 1 equals
9) Let v(t) = 3t²i + 2tj be the velocity of a particle at any time t. If r(0) = i + j, then r(2) =
10) If r(t) = 2 cos(t)i + 2 sin(t)j, then the curvature κ =
11) The value of lim_{(x,y)→(2,2)} (x+y-4)/(√(x+y)-2) equals
12) Along the path y = x, the value of lim_{(x,y)→(0,0)} (xy)/(x²+y²) equals
13) If f(x,y) = x²eʸ + sin(x), then f_{xyx}(0,0) =
14) If z = 2xy², where x = cos(t) and y = 2t + 1, then dz/dt at t = 0 equals
15) If w = u² and u = x + 2y + 3z, then the value of ∂w/∂x + ∂w/∂y + ∂w/∂z at the point (x,y,z) = (1,1,0) equals
16) If f(x,y) = x²eʸ and g(x,y) = x²y + x cos(y), then (∇f) · (∇g) at the point (-2,0) equals
17) The directional derivative of f(x,y,z) = 3xy²z³ at the point (2,-1,1) in the direction of the unit vector u = 1/3i + 2/3j - 2/3k equals
18) Suppose that f(x,y) has continuous second order partial derivatives everywhere and that the point (0,0) is a critical point of f. If f_{xx}(0,0) = 2, f_{yy}(0,0) = 3, and f_{xy}(0,0) = 4, then at (0,0) the function f has a
19) The equation of the tangent plane to the surface x²y - 4z² = -7 at the point (-3,1,-2) is
20) Which of the following are parametric equations of the normal line to the surface z = xe⁻ʸ through the point (1,0,1)?
21) ∫₀² ∫₀¹ (1 - 6xy²)dxdy
22) After reversing the orders of integration, the integral ∫₀¹ ∫₀²ˣ f(x,y)dydx becomes