Cramer’s rule MCQs With Answer

Cramer’s Rule MCQs With Answer

Cramer’s rule MCQs With Answer provide B.Pharm students a focused way to master linear systems, determinants and their applications in pharmaceutical calculations. These objective questions cover 2×2 and 3×3 systems, determinant properties, conditions for unique or infinite solutions, and real-life uses such as dose distribution, compartment models, and formulation balancing. This Student-friendly set is designed for exam preparation, practice tests, and quick revision, emphasizing conceptual clarity and worked examples. Keywords: Cramer’s rule, MCQs, B.Pharm, determinants, linear equations, pharmaceutical calculations, objective questions, answers, exam preparation. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the essential condition for applying Cramer’s rule to solve a linear system Ax = b?

• Matrix A must be square and det(A) ≠ 0
• Matrix A must be rectangular
• Vector b must be zero
• All coefficients must be positive

Correct Answer: Matrix A must be square and det(A) ≠ 0

Q2. For a 2×2 system Ax = b, the determinant of A = [[a, b], [c, d]] is given by which expression?

• a + d – b – c
• ad – bc
• ac + bd
• ab – cd

Correct Answer: ad – bc

Q3. Using Cramer’s rule for a 3×3 system requires computing how many determinants in total?

• 3 determinants
• 4 determinants
• 6 determinants
• 9 determinants

Correct Answer: 4 determinants

Q4. If det(A) = 0 and at least one numerator determinant (for a variable) ≠ 0, what is the solution status?

• Unique solution
• No solution (inconsistent)
• Infinite solutions
• Solution depends on right-hand side being zero

Correct Answer: No solution (inconsistent)

Q5. If det(A) = 0 and all numerator determinants are also 0, then the system has:

• A unique solution
• No solution
• Infinitely many solutions
• Exactly two solutions

Correct Answer: Infinitely many solutions

Q6. For the system 2x + 3y = 5 and x − y = 1, what is x using Cramer’s rule?

• 5/2
• 8/5
• 3/5
• −8/5

Correct Answer: 8/5

Q7. In Cramer’s rule, the numerator determinant for variable x_i is obtained by:

• Replacing the i-th row of A with b
• Replacing the i-th column of A with b
• Transposing A and multiplying by b
• Adding b to the i-th column of A

Correct Answer: Replacing the i-th column of A with b

Q8. Which of the following properties of determinants is true?

• Swapping two rows leaves determinant unchanged
• Multiplying a row by scalar k multiplies determinant by 1/k
• Adding a multiple of one row to another does not change determinant
• Determinant is always positive

Correct Answer: Adding a multiple of one row to another does not change determinant

Q9. For a homogeneous system Ax = 0, what does det(A) ≠ 0 imply?

• Infinitely many nonzero solutions
• Only the trivial solution x = 0
• No solutions at all
• At least one parameterized solution

Correct Answer: Only the trivial solution x = 0

Q10. Which statement describes computational practicality of Cramer’s rule for large systems?

• Efficient and preferred for very large n
• Computationally expensive and not recommended for large n
• Requires no determinants for n > 3
• Gives approximate solutions only

Correct Answer: Computationally expensive and not recommended for large n

Q11. The determinant of the identity matrix I_n is:

• 0
• n
• 1
• Depends on entries

Correct Answer: 1

Q12. Swapping two columns of matrix A has what effect on det(A)?

• No effect
• Changes sign of det(A)
• Multiplies det(A) by 2
• Sets det(A) to zero

Correct Answer: Changes sign of det(A)

Q13. Which of these connects Cramer’s rule to matrix inverse?

• Cramer’s rule uses eigenvalues of A
• Cramer’s rule computes A^2 to solve Ax = b
• Inverse A^{-1} = (1/det A) adj(A), and x = A^{-1} b relates to Cramer’s formula
• Cramer’s rule requires singular values

Correct Answer: Inverse A^{-1} = (1/det A) adj(A), and x = A^{-1} b relates to Cramer’s formula

Q14. For the 2×2 system with A = [[3, 4], [2, 5]] and b = [7, 3], what is det(A)?

• 7
• 11
• 3
• −7

Correct Answer: 7

Q15. For the same system (A as above and b = [7, 3]), what is x using Cramer’s rule?

• 1
• 2
• 3
• 4

Correct Answer: 1

Q16. In pharmacokinetics, Cramer’s rule can help solve which type of model?

• Nonlinear enzyme kinetics only
• Linear compartmental models with simultaneous linear equations
• All differential equations without linearization
• Only empirical curve-fitting models

Correct Answer: Linear compartmental models with simultaneous linear equations

Q17. The cofactor C_ij equals which expression?

• Minor M_ij
• (−1)^{i+j} times the minor M_ij
• The transpose of M_ij
• Sum of row i and column j

Correct Answer: (−1)^{i+j} times the minor M_ij

Q18. Determinant geometrically represents:

• Perimeter of a polygon
• Volume (or area) scaling factor of linear transformation represented by A
• Number of solutions of a system
• Inverse of matrix trace

Correct Answer: Volume (or area) scaling factor of linear transformation represented by A

Q19. For A = [[1,2,3],[0,1,4],[5,6,0]], what is det(A)?

• 0
• 1
• −1
• 10

Correct Answer: 1

Q20. If a row of A is multiplied by scalar k, how does det(A) change?

• Determinant is multiplied by k
• Determinant is divided by k
• Determinant remains same
• Determinant becomes zero

Correct Answer: Determinant is multiplied by k

Q21. For a 2×2 system, if det(A) = 5 and numerator det for x = 10, what is x?

• 5
• 2
• 0.5
• −2

Correct Answer: 2

Q22. Cramer’s rule yields exact rational results when all coefficients and constants are:

• Integers or rationals
• Only irrational
• Only complex numbers
• Only zeros

Correct Answer: Integers or rationals

Q23. Which technique is generally faster than direct Cramer’s rule for solving large linear systems?

• Computing many determinants
• Gaussian elimination (LU decomposition)
• Expanding using cofactors for each variable
• Manual substitution only

Correct Answer: Gaussian elimination (LU decomposition)

Q24. In Cramer’s rule, if A is 3×3 and det(A) = 2, and numerator determinants for x, y, z are 4, −6, and 2 respectively, what is y?

• −3
• −2
• 3
• 2

Correct Answer: −3

Q25. Which of the following is NOT a limitation of using Cramer’s rule in practical pharmacy computations?

• Inefficiency for large n
• Sensitivity to rounding in floating-point arithmetic
• Requires det(A) ≠ 0
• Cannot be used for 2×2 systems

Correct Answer: Cannot be used for 2×2 systems

Q26. If two rows of matrix A are identical, det(A) is:

• Equal to product of diagonal
• Zero
• One
• Negative

Correct Answer: Zero

Q27. For system x + 2y + 3z = 1, 2x + y + z = 2, x − y + 2z = 0, which method can directly produce x, y, z using determinants?

• Cramer’s rule
• Newton’s method
• Interpolation
• Integration by parts

Correct Answer: Cramer’s rule

Q28. The adjugate (adj(A)) matrix is built from which elements?

• Eigenvectors of A
• Transposed cofactors (cofactor matrix transposed)
• Original rows rearranged
• Only diagonal minors

Correct Answer: Transposed cofactors (cofactor matrix transposed)

Q29. For a 2×2 matrix A, if det(A) = d and A’s inverse exists, A^{-1} equals:

• (1/d) times swapped and sign-changed entries
• (1/d) times original matrix
• d times original matrix
• Transpose of A

Correct Answer: (1/d) times swapped and sign-changed entries

Q30. Which determinant expansion method is most practical for hand calculation of a 3×3 matrix?

• Row-reduction only
• Sarrus’ rule or cofactor expansion
• QR decomposition
• Singular value decomposition

Correct Answer: Sarrus’ rule or cofactor expansion

Q31. If A is singular (det(A) = 0), what is true about A^{-1}?

• A^{-1} exists and is unique
• A^{-1} does not exist
• A^{-1} equals adj(A)
• A^{-1} equals zero matrix

Correct Answer: A^{-1} does not exist

Q32. Which of the following changes does NOT alter the determinant of a matrix?

• Adding a multiple of one row to another row
• Multiplying a row by 2
• Swapping two rows
• Replacing a row with a copy of another row

Correct Answer: Adding a multiple of one row to another row

Q33. For system 3x + y = 8 and x − 2y = −3, what is y using Cramer’s rule?

• −1
• 1
• 2
• −2

Correct Answer: 1

Q34. Determinant is multilinear in the rows. This means determinant is linear in each row separately when other rows are held fixed. True or false?

• True
• False
• Only for square matrices of even order
• Only for triangular matrices

Correct Answer: True

Q35. In Cramer’s rule for a 4×4 system, how many numerator determinants must you compute to find all unknowns?

• 3
• 4
• 5
• 16

Correct Answer: 4

Q36. If det(A) is very small but nonzero, solving via Cramer’s rule numerically can lead to:

• Perfectly stable solutions
• Ill-conditioned, numerically unstable solutions
• No solutions
• Only integer solutions

Correct Answer: Ill-conditioned, numerically unstable solutions

Q37. For A = [[2,0],[0,3]], what is det(A)?

• 5
• 6
• −6
• 0

Correct Answer: 6

Q38. Which of the following best describes the numerator determinant for a particular variable in Cramer’s rule?

• Determinant of A with the corresponding column replaced by b
• Determinant of A with the corresponding row replaced by b
• Product of diagonal elements of A
• Transpose of A times b

Correct Answer: Determinant of A with the corresponding column replaced by b

Q39. Which is a direct application of determinants in pharmaceutical formulation?

• Solving linear balances of excipient quantities in multi-component mixtures
• Measuring pH with a pH meter
• Determining melting point experimentally
• Performing microbiological assays

Correct Answer: Solving linear balances of excipient quantities in multi-component mixtures

Q40. For system x + y = 4 and x − y = 2, Cramer’s rule gives x = ?

• 1
• 2
• 3
• 4

Correct Answer: 3

Q41. Which matrix type guarantees a nonzero determinant?

• Singular matrix
• Invertible (non-singular) matrix
• Matrix with a zero row
• Matrix with duplicate rows

Correct Answer: Invertible (non-singular) matrix

Q42. In the context of solving simultaneous linear equations for mixing three solutions, Cramer’s rule is most useful when:

• Only approximate answers are acceptable
• Explicit exact symbolic answers are needed and system size is small
• The coefficient matrix is singular
• The system is highly nonlinear

Correct Answer: Explicit exact symbolic answers are needed and system size is small

Q43. Which operation on rows multiplies determinant by −1?

• Multiplying a row by 2
• Swapping two rows
• Adding a multiple of one row to another
• Replacing a row by itself

Correct Answer: Swapping two rows

Q44. If the determinant of A equals 4 and numerator determinants for variables are 8 and 12 for a 2-variable system, what are x and y?

• x = 2, y = 3
• x = 1/2, y = 1/3
• x = 32, y = 48
• x = 4, y = 6

Correct Answer: x = 2, y = 3

Q45. Which statement is correct about Cramer’s rule and matrix rank?

• Cramer’s rule works only when rank(A) < n
• Cramer’s rule works when rank(A) = n (full rank)
• Cramer’s rule is independent of matrix rank
• Cramer’s rule requires rank(A) = 0

Correct Answer: Cramer’s rule works when rank(A) = n (full rank)

Q46. The minor M_ij of a matrix is defined as:

• Determinant obtained by deleting row i and column j
• Element a_ij multiplied by i+j
• Sum of elements in row i
• Trace of the submatrix

Correct Answer: Determinant obtained by deleting row i and column j

Q47. For system 2x + y + z = 3, x + 2y + z = 4, x + y + 2z = 5, if det(A) = 4 and numerator for z = 8, what is z?

• 2
• 4
• 0.5
• −2

Correct Answer: 2

Q48. Which of the following is true about determinant of triangular (upper or lower) matrices?

• Determinant equals product of diagonal entries
• Determinant equals sum of diagonal entries
• Determinant is always zero
• Determinant equals number of nonzero entries

Correct Answer: Determinant equals product of diagonal entries

Q49. When would you prefer Cramer’s rule over Gaussian elimination in a B.Pharm exam setting?

• When the system is large (n > 10)
• When a short, exact symbolic solution for small n is requested
• When numerical stability is the biggest concern
• When determinant is zero

Correct Answer: When a short, exact symbolic solution for small n is requested

Q50. Which statement summarizes Cramer’s rule?

• It uses eigenvalues to solve Ax = b
• It expresses each variable as ratio of determinants: x_i = det(A_i)/det(A)
• It only applies to nonlinear systems
• It avoids computing determinants entirely

Correct Answer: It expresses each variable as ratio of determinants: x_i = det(A_i)/det(A)

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