MH1810-Math I 2016-2017 Semester 2
Question 1)
A matrix is only invertible when its determinant is non-zero. Find the determinant of the 3 x 3 matrix with the unknown x inside using (1) Cofactor expansion or (2) 3x3 matrix determinant shortcut whereby ((a11*a22*a33 + a12*a23*a31 + a13*a21*a32) - (a31*a22*a13 + a32*a23*a11 + a33*a21*a12)) (found in Tang Wee Kee Math I notes: "Algebra: Matrices II - Determinants" page 19)
After finding the determinant, we can solve for values of x (x = 5 & x = -5) in which the determinant is zero and exclude those values to find the answer that is required.
Question 2)
Part i of the question is very basic, firstly, "let z be the third roots of -8" and solve accordingly to get 3 different roots (2e^(pi/3)i, 2e^(pi)i, 2e^(-pi/3)i).
For part ii, you can use your GC to help you check your answers. Sketching out the complex number can give you an idea of the argument and modulus of the complex number. For the latter part, convert (1 + i) to its exponential form and simplify accordingly.
(Answer: sqrt(2)*e^(pi/4)i, i)
Question 3)
To find the equation of plane, we can find the normal vector to the plane by cross product of 2 vectors (which can be found between any 2 points (A, B, C)) lying on the plane. By definition, a dot product of 2 vectors that are perpendicular to each other will give zero and the way equation of planes are defined is that we will take the normal vector to dot product with any vector lying on the plane and equate it to zero, i.e. n . AP = 0 (where P is an unknown point and A can be any point on the plane).
(Answer: -x + 7y - 3z = -4)
Note: Finding the normal vector is very important since the later parts will make use of this value.
For this part, we shall find the acute angle between the normal of the plane to line AD before using it to deduct from 90 degrees to find the acute angle between the plane and line AD. To find the angle between 2 vectors, we can use the definition of dot product. And to find the acute angle, we use the absolute value of the dot product on the left hand side, i.e.:
| AD . n | = ||AD|| ||n|| cos (theta)
(Answer: 0.237rad)
To find the equation of a line, we need a point (OD) and the vector parallel to the line (n, normal to the plane).
Question 4)
The left hand limit must be equal to the right hand limit for the limit at x tend to 1 to exist.
(Answer: c = -1)
For f to be continuous at 1, not only the limit at 1 must exist, it must also be equal to the limit at 1, i.e. lim[x -> 1] (f(x)) = f(1)
(Answer: c = -1 & a+5 = b)
To be differentiable at 1, f should be continuous and the graph should not have any kinks, it should be a smooth, continuous curve. Therefore, lim[x-> 1] ( (f(x) - f(1))/(x-1) ) must exist; in other words, the left hand limit must be equal to the right hand limit and we will get a = -4. Checking the conditions for the curve to be continuous at 1, we will have c = -1 and b = 1.
Question 5)
(i) Use substitution u = tan x and the integral becomes very simple.
(Answer: e^(tan x) + C)
(ii) Since it is rotated about the x-axis, it will be easier to calculate the integral using the cross-sectional method whereby you add small volumes of discs with radius y from x = 0 to x = pi.
(Answer: 3/2pi^2 + 4pi)
(iii) Make use of FTC to solve. Swap the variable x to the top, and use substitution to change out ln x to u.
(Answer: -(ln x)^3)
Question 6)
(i) Linearization: L(x) = f(a) + (x-a)f'(a). Approximate sqrt(16.1) at a = 16.
(Answer: 4.0125)
(ii) Note that the width is the diameter of the semi-circle and with that we can write out the area constraint equation: wl - 1/2w^2 + 1/8pi w^2 = 800. Next, we can write out the perimeter equation: P = 2l + 1/2 pi w. Substitute the area constraint equation into the perimeter equation to remove length variable, l. Differentiate the new perimeter equation with respect to width, w. To find the stationary point, let dP/dw = 0, to get w = 29.936. Don't forget to check that at w = 29.936, it is a minimum point. Use the area equation to find the length when w = 29.936.
(Answer: l = 29.936, w = 29.936)
That's all for Mathematics 1 2016-2017 Semester 2 solution. If you have any doubts, opinions or suggestions, feel free to leave them in forum section or email me @ KYX@outlook.sg. Thanks~
