All Free Quantitative Methods MCQs with Answers
Every Quantitative Methods question in the bank, across all chapters, each with the correct answer and a written explanation. Free and unlimited, with no account needed.
360 questions · page 18 of 18
- A. 1
- B. 2
- C. 3
- D. 4
Explanation: Subtracting twice the second equation from the first gives y = 3. Substituting y = 3 into x + y = 5 gives x = 2.
Correct answer: 2- A. [[2, 2], [3, 20]]
- B. [[3, 1], [3, 9]]
- C. [[3, -1], [3, 9]]
- D. [[1, 1], [3, 5]]
Explanation: Matrix addition is performed by adding entries in corresponding positions. Thus the entries are 1 + 2, 2 + (-1), 3 + 0, and 4 + 5.
Correct answer: [[3, 1], [3, 9]]- A. 15
- B. 17
- C. 19
- D. 21
Explanation: Expanding along the first row gives 1(3) minus 2(0 minus 8). Therefore, the determinant is 3 + 16 = 19.
Correct answer: 19- A. [[5, 9], [2, 4]]
- B. [[3, 9], [2, 5]]
- C. [[7, 6], [3, 4]]
- D. [[7, 9], [2, 4]]
Explanation: Each entry of AB is obtained by multiplying a row of A by a column of B. The first row becomes (1)(3) + (2)(2) = 7 and (1)(1) + (2)(4) = 9.
Correct answer: [[7, 9], [2, 4]]- A. The coefficient determinant is zero
- B. The coefficient determinant is positive
- C. The constant term is positive
- D. The coefficient determinant is one
Explanation: A homogeneous system always has the zero solution. A nonzero solution exists when the coefficient matrix is singular, so its determinant…
Correct answer: The coefficient determinant is zero346. Which elementary row operation always preserves the solution set of a system of linear equations?
- A. Multiplying a row by zero
- B. Interchanging two equations
- C. Deleting a nonzero row
- D. Changing one coefficient only
Explanation: Interchanging two rows only changes the order of the equations, not their mathematical content.
Correct answer: Interchanging two equations- A. A has a unique inverse
- B. A has only positive entries
- C. A must be an identity matrix
- D. A is a singular matrix
Explanation: A square matrix with determinant zero is singular and therefore has no inverse.
Correct answer: A is a singular matrix- A. 13/7
- B. 9/7
- C. 5/7
- D. 15/7
Explanation: The coefficient determinant is 2(-1) minus 3(4), which equals -14. Replacing the x-column gives a determinant of 8(-1) minus 3(6) = -26…
Correct answer: 13/7- A. 3 × 2
- B. 2 × 3
- C. 3 × 3
- D. 2 × 2
Explanation: Transposition changes rows into columns and columns into rows. Therefore, a 3 × 2 matrix becomes a 2 × 3 matrix.
Correct answer: 2 × 3- A. [[2, -3], [4, 1]]
- B. [[3, -6], [9, 0]]
- C. [[6, -3], [4, 1]]
- D. [[2, -9], [4, 3]]
Explanation: To find A, divide every entry of 3A by 3. This gives [[2, -3], [4, 1]], so option A is correct.
Correct answer: [[2, -3], [4, 1]]- A. 1
- B. 2
- C. 3
- D. 4
Explanation: From the second equation, y = 3x - 3. Substituting this into the first equation gives x = 2 and therefore y = 3.
Correct answer: 3- A. 6
- B. 8
- C. 10
- D. 12
Explanation: The required entry is obtained by multiplying the second row of A by the first column of B: (3 × 0) + (4 × 2) = 8.
Correct answer: 8- A. [[0, 0], [0, 0]]
- B. [[1, 1], [1, 1]]
- C. [[1, 0], [0, 1]]
- D. [[0, 1], [1, 0]]
Explanation: The 2 × 2 identity matrix has ones on the main diagonal and zeros elsewhere.
Correct answer: [[1, 0], [0, 1]]- A. -3
- B. -1
- C. 3
- D. 5
Explanation: Expanding along the first row gives 2(3 × 1 - 2 × 2) - 1(1 × 1 - 2 × 0) = -3. Therefore, the determinant is -3.
Correct answer: -3- A. rank(A) = 1 and rank([A|B]) = 1
- B. rank(A) = 2 and rank([A|B]) = 3
- C. rank(A) = 3 and rank([A|B]) = 2
- D. rank(A) = 2 and rank([A|B]) = 2
Explanation: A unique solution requires the rank of the coefficient matrix to equal the rank of the augmented matrix and to equal the number of…
Correct answer: rank(A) = 2 and rank([A|B]) = 2- A. [[2, 1], [2, 1]]
- B. [[3, 1], [2, 2]]
- C. [[2, 2], [1, 1]]
- D. [[4, 2], [4, 2]]
Explanation: Subtract the identity matrix from the right side to obtain [[4, 2], [4, 2]], then divide every entry by 2. This gives X = [[2, 1], [2, 1]].
Correct answer: [[2, 1], [2, 1]]- A. 1/(ad-bc) [[a, -b], [-c, d]]
- B. 1/(ad-bc) [[d, -b], [-c, a]]
- C. 1/(ad+bc) [[d, b], [c, a]]
- D. 1/(ad-bc) [[-d, b], [c, -a]]
Explanation: The inverse is found by interchanging a and d, changing the signs of b and c, and dividing by the determinant ad - bc.
Correct answer: 1/(ad-bc) [[d, -b], [-c, a]]- A. Replace the second column with the constants
- B. Add the two coefficient columns
- C. Replace the first column with the constants
- D. Multiply both coefficient columns by the constants
Explanation: To calculate x with Cramer's rule, replace the coefficient column belonging to x with the constants column.
Correct answer: Replace the first column with the constants- A. It has only the zero solution
- B. It has no solution
- C. It always has exactly two solutions
- D. It has infinitely many solutions
Explanation: A homogeneous system is always consistent because the zero vector is a solution.
Correct answer: It has infinitely many solutions- A. To make every entry equal to one
- B. To reveal pivots and simplify back-substitution
- C. To change a rectangular matrix into a square matrix
- D. To calculate the transpose without multiplication
Explanation: Row-echelon form places the matrix in a triangular pattern that clearly shows pivot positions and free variables.
Correct answer: To reveal pivots and simplify back-substitution