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Q1. The following system of equations has 2x + 3y = 10 4x + 6y = 20

• Unique Solution
• No Solution
• Infinitely Many Solution
• Trivial Solution

Solution

Q2. Inverse of a matrix A exists, if 

• Matrix A is singular
• Matrix A is non-singular
• A^T = A
•  | A | = 0

Solution

We know that So, inverse of a matrix exist, if, and a matrix is non-singular, if .

Q3. A determinant may be considered to be a function which associates each square matrix with a unique number , either real or complex .Which of the following represents the  correct definition of this function, if M denotes a matrix and c a real or complex number ?

•     _{f(M) = c}
• f( c)  = M
• _f ^-1_{(M) = c}
•     _f^-1(_{C) =M}

Solution

 

Q4. The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if  

• k = 0
• For all non zero k
• –1 < k < 1
• – 2 < k < 2

Solution

The system of linear equations has a unique solution, if

Q5. If the system of equations x + 2y – 3z = 2 (k + 3) z = 3 (2k + 1) y + z = 2 is inconsistent, then k is:

• – 3 and -½
• -½
• 3
• 2

Solution

Q6. The solution of the following system of equation is 2x + 3y = 5 5x - 2y = 3

• x = 1, y = 1
• x = 3, y = 2
• x = -1, y = -1
• x = 2, y = 3

Solution

The system of equations can be written in the form AX = B, where Thus, A is non singular. Therefore, Thus, x = 1, y = 1

Q7. If the area of triangle is 6 square units and vertices of triangle are (0, k), (0, 6) and (2, 0), then the value of k is:

• 1
• - 1
• 2
• 0,12

Solution

Q8. Let A = [k ] be the matrix of order 1, then determinant of A is equal to _________

• _A
• _|A|
• _k
• ₁

Solution

The value of  the determinant of a matrix of order 1 is the element of the matrix.That is, If A = [a ] be the matrix of order 1, then determinant of A is defined to be equal to a

Q9. If A³ = I then A^-1 = ..............., where A is an invertible matrix

• A
• A²
• I
• A^-1

Solution

A³ = I A^-1A³ = IA^-1 = A^-1 A² = A^-1

Q10. To every square matrix A = [aij] of order n , we can associate a number  called ___________ of the square matrix A, where aij = (i, j)th element of A.

•   determinant
•   Real number  
•   Real or Complex number 
•   None of these 

Solution

  The defination of determinant is : To every square matrix A = [aij] of order n, we can associate a number called determinant of the square matrix A, where aij = (i, j)th element of A.      

Q11. If A is a square matrix of order 3 and |A| = 5 then |3A| = ?

• 15
• 45
• 32
• 135

Solution

In general, if A = kB, where A and B are square matrices of order n, then |A| = kⁿ|B|, where n = 1, 2, 3.....

Q12. The system of equations kx + 2y – z = 1, (k – 1)y – 2z = 2 (k + 2)z = 3 has a unique solution, if k is 

• 0
• 1
• – 2
• – 1

Solution

The system of equations kx + 2y – z = 1, (k – 1)y – 2z = 2 and (k + 2)z = 3 has a unique solution, if So, system has a unique solution for k = – 1.

Q13. The value of a determinant is

• Always a real number
•   never an irrational number
•   _{Never a complex number}
•   _{a Real or a complex number.}

Solution

_{The elements of a matrix to which a determinant is associated may be either real or complex. Therefore, the value of a determinant is a Real or a complex number.}      

Q14. The following system of equations has x + 3y + 3z = 2 x + 4y + 3z = 1 x + 3y + 4z = 2

• Unique Solution
• Trivial Solution
• No Solution
• Infinite Solution

Solution

The system of equation can be written in the form AX = B, where |A|= 1(16 - 9) - 3(4 - 3) + 3(3 - 4) = 7 - 3 - 3 = 1  A is non -singular matrix,inverse exists. Thus, System of equations has unique solution given by 

Q15. Let A and B are invertible matrices of the same order, then (AB^-1) =

• A^-1 B
• BA
• BA^-1
• AB

Solution

Since (AB)^-1 = B^-1A^-1 So, (AB^-1) = (B^-1)^-1A^-1 = BA^-1

Q16. If area of triangle is 35 sq. units with vertices (2, -6), (5, 4) and (K, 4). Then K is

• 12
• – 2
• – 12, – 2
• 12, – 2

Solution

The area of the triangle = 35 unit²                   

Q17. Points (a,a,c),(1,0,1) and (c,c,b) are collinear if

• C = ab
• a,b,c in AP
• C² = ab
• a,b,c in GP

Solution

Q18. The area of the quadrilateral PQRS whose vertices are P (-5, 7), Q (-4, -5), R (-1, -6) and S (4, 5) is

• 54 square units
• 109 square units
• 72 square units
• 17.5 square units

Solution

By joining PR we get two triangle PSR. We find the areas of these triangles separately and add then in order to get the area of quadrilateral. Area of PSR                       Area of PQR                        = 17.5 sq units Area of quadrilateral ABCD = Area of PSR + Area of PQR                                         = (54.5 + 17.5) sq uints                                         = 72 sq units

Q19. The minor M_ij of an element a_ij of a determinant is defined as the value of the determinant obtained after deleting the

• i^th row of the determinant
•  j^th row of the determinant
• i^th row and j^th column of the determinant
• i^th column and j^th row of the determinant

Solution

Minor of an element a_ij of a determinant is a matrix obtained by deleting i^th row and j^th column.

Q20. For a square matrix A in a matrix equation AX = B, if │A│≠ 0, then

• There exists a unique solution
• There exists no solution
• There exists infinite number of solutions
• The system may or may not be consistent

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