Numerical Methods and Calculus

Question 11

The following system of equations 

[caption width="800"] [/caption]

has a unique solution. The only possible value(s) for α is/are

• either 0 or 1
  

• any real number 
  

• 0
  

• any real number other than 5
  

Question 12

The minimum number of equal length subintervals needed to approximate 

[Tex]\int_{1}^{2}xe^xdx[/Tex]

 to an accuracy of at least 

[Tex] \frac{1}{3} * 10^{-6}[/Tex]

 using the trapezoidal rule is

• 100 l
  

• 100
  

• 1000
  

• 1000 l
  

Question 13

The Newton-Raphson iteration 

 can be used to compute the.

• square root of R 
  

• logarithm of R
  

• reciprocal of R
  

• square of R 
  

Question 14

• P = Q - k
• P = Q + k
• P = Q
• P = Q +2 k

Question 15

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x⁴ - 16x³ + 24x² + 37

• 0
• 1
• 2
• 3

Question 16

Consider the following two statements about the function f(x)=|x|

P. f(x) is continuous for all real values of x
Q. f(x) is differentiable for all real values of x 

Which of the following is TRUE?

• P is true and Q is false.

• P is false and Qis true.

• Both P and Q are true

• Both P and Q are false.

Question 17

Consider the series X_n+1 = X_n/2 + 9/(8 X_n), X₀ = 0.5 obtained from the Newton-Raphson method. The series converges to

• 1.5
• sqrt(2)
• 1.6
• 1.4

Question 18

Consider the polynomial p(x) = a0 + a1x + a2x² + a3x³ , where ai ≠ 0 ∀i. The minimum number of multiplications needed to evaluate p on an input x is:

• 3
• 4
• 6
• 9

Question 19

Consider the following system of equations:

3x + 2y = 1
4x + 7z = 1
x + y + z = 3
x – 2y + 7z = 0 

The number of solutions for this system is __________________

• 1
• 0
• 2
• 3

Question 20

The value of the dot product of the eigenvectors corresponding to any pair of different eigenvalues of a 4-by-4 symmetric positive definite matrix is _____________________.

• 0
• 1
• -1
• 2