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Subject

Mathematics

Chapter

Ch 1Applications of Matrices and Determinants

Weightage~8marksapprox

1.1IntroductionRead →

1.2Inverse of a Non-Singular Square Matrix15 Q

1.3Elementary Transformations of a Matrix3 Q

1.4Applications of Matrices: Solving System of Linear Equations14 Q

1.5Applications of Matrices: Consistency of System of Linear Equations by Rank Method6 Q

1.6Objective Type Questions25 Q

1.7SummaryRead →

CBSE Sample Papers55 Q

Q1The system of equations ax+y+z=0; x+by+z=0; x+y+cz=0 has a non-trivial solution then dfrac11-a+dfrac11-b+dfrac11-c= (a) 1 (b) 2 (c) -1 (d) 0 Q2In the homogeneous system ρ(A) < the number of unknowns then the system has : (a) only trivial solution (b) trivial solution and infinitely many non-trivial solutions (c) only non-trivial solutions (d) no solution Q3If A is a matrix of order 3, then det(kA) is (a) k³det(A) (b) k²det(A) (c) kdet(A) (d) det(A)Q4If A= 21; 34 , then (adj A)A= (a) frac150; 0frac15 (b) 10; 01 (c) 50; 0-5 (d) 50; 05 Q5Solve the system of equations x+y+2z=4; 2x+2y+4z=8; 3x+3y+6z=10 by using determinant.Q6Show that the adjoint of A= -4-3-3; 101; 443 is A itself.Q7Examine the consistency of the system x-3y-8z=-10; 3x+y-4z=0; 2x+5y+6z-13=0 by using rank method and hence solve the system.Q8In echelon form, which of the following is incorrect ? (a) Every row of A which has all its entries 0 occurs below every row which has a non-zero entry. (b) The first non-zero entry in each non-zero row is 1. (c) The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row. (d) Two rows can have same number of zeros before the first non-zero entry.Q9If A = 0 0; 0 5 , then A¹² is : (a) 0 0; 0 60 (b) 0 0; 0 5¹² (c) 0 0; 0 0 (d) 1 0; 0 1 Q10If A is a matrix of order 3, then det(kA) is : (a) k³ det(A) (b) k² det(A) (c) k det(A) (d) det(A)Q11If aex + bey = c; pex + qey = d and Δ₁ = a b; p q ; Δ₂ = c b; d q ; Δ₃ = a c; p d then the value of (x, y) is : (a) (dfracΔ₂Δ₁, dfracΔ₃Δ₁) (b) (logdfracΔ₂Δ₁, logdfracΔ₃Δ₁) (c) (logdfracΔ₁Δ₃, logdfracΔ₁Δ₂) (d) (logdfracΔ₁Δ₂, logdfracΔ₁Δ₃)Q12Find the rank of the matrix 0 1 2 1; 2 -3 0 -1; 1 1 -1 0 .Q13Find the inverse of the matrix 3 1 -1; 2 -2 0; 1 2 -1 .Q14Solve, x+y+2z=4, 2x+2y+4z=8, 3x+3y+6z=12 by using determinant method.Q15In a system of 3 linear non-homogeneous equations with three unknowns, if Δ = 0; and Δₓ = 0; Δy ≠ 0; Δz = 0 then the system has : (a) infinitely many solutions (b) unique solution (c) no solution (d) two solutions Q16Which of the following statement is correct regarding homogeneous system ? (a) has only non-trivial solutions (b) always inconsistent (c) has only trivial solution only if rank of the coefficient matrix is equal to the number of unknowns (d) has only trivial solution Q17Inverse of 3 1 \5 2 is : (a) 3 -1 \-5 -3 (b) 2 -1 \-5 3 (c) -3 5 \1 -2 (d) -2 5 \1 -3 Q18If A is a scalar matrix with scalar k ≠ 0, of order 3, then A⁻¹ is : (a) dfrac1kI (b) dfrac1k²I (c) kI (d) dfrac1k³I Q19Find the adjoint of the matrix A = 1 2; 3 -5 and verify the result A(adj A) = (adj A)A = |A| · I.Q20Solve the following non-homogeneous equations by using determinant method. 2x + 2y + z = 5 x - y + z = 1 3x + y + 2z = 4 Q21Discuss the solutions of the following system of equations for all values of λ. x + y + z = 2, 2x + y - 2z = 2, λ x + y + 4z = 2 Q22In the homogeneous system ρ(A) is less than the number of unknowns, then the system has : (a) only non-trivial solutions (b) no solution (c) only trivial solution (d) trivial solution and infinitely many non-trivial solutions Q23If A is a scalar matrix with scalar k ≠ 0, of order 3, then A⁻¹ is : (a) dfrac1k I (b) kI (c) dfrac1k² I (d) dfrac1k³ I Q24To find the number of coins, in each category, write the suitable system of equations for the given situation: "A bag contains 3 types of coins namely ₹1, ₹2 and ₹5. There are 30 coins amounting to ₹100 in total."Q25Prove that ρ(A) + ρ(B) ≠ ρ(A+B) by giving the suitable matrices A and B of order 3.Q26(a) For what values of μ the system of homogeneous equations x+y+3z=0; 4x+3y+μ z=0; 2x+y+2z=0 have: (i) only trivial solution (ii) infinitely many solutions **OR** (b) Prove by vector method that sin(A+B) = sin Acos B + cos A sin B.Q27If ρ(A) = ρ([A mid B]), then the system AX = B of linear equations is : (a) inconsistent (b) consistent and has a unique solution (c) consistent (d) consistent and has infinitely many solutions Q28If (AB)⁻¹= 12 -17; -19 27 and A⁻¹= 1 -1; -2 3 , then B⁻¹= (a) 8 -5; -3 2 (b) 2 -5; -3 8 (c) 8 5; 3 2 (d) 3 1; 2 1 Q29If A= 2 9; 1 7 then prove that (AT)⁻¹=(A⁻¹)T.Q30(a) Test the consistency of the following system of linear equations by rank method. x-y+z=-9 2x-y+z=4 3x-y+z=6 4x-y+2z=7 **OR** (b) If 2cosα=x+dfrac1x and 2cosβ=y+dfrac1y, show that : (i) dfracxmyⁿ-dfracyⁿxm=2isin(mα-nβ) (ii) xmyⁿ+dfrac1xmyⁿ=2cos(mα+nβ)Q31If A= 2 3; 5 -2 be such that λ A⁻¹=A, then λ is : (a) 19 (b) 17 (c) 21 (d) 14 Q32Which one of the following is incorrect ? (a) If A is a square matrix of order n, and λ is a scalar, then Adj (λ A)=λⁿ (Adj A). (b) Adjoint of a symmetric matrix is also a symmetric matrix. (c) A(Adj A) = (Adj A)A = |A|I. (d) Adjoint of a diagonal matrix is also a diagonal matrix.Q33Show that the rank of the matrix 1 2 -1; 3 -1 2; 1 -2 3; 1 -1 1 is 3.Q34Solve the following system of linear equations, using matrix inversion method : 5x+2y=3, 3x+2y=5.Q35(a) Cramer's rule is not applicable to solve the system 3x+y+z=2, x-3y+2z=1, 7x-y+4z=5. Why ? **OR** (b) Prove that the local minimum values for the function f(x)=4x⁶-6x⁴ attain at -1 and 1.Q36A square matrix A of order n has inverse if and only if : (a) ρ(A) > n (b) ρ(A) = n (c) ρ(A) ≠ n (d) ρ(A) < n Q37|adj(adjA)|=|A|¹⁶, then the order of the square matrix A is : (a) 2 (b) 3 (c) 5 (d) 4 Q38Let A= 0 1; 1 1 , B= 1 1; 0 1 be any two Boolean matrices of the same type. Find Avee B and Awedge B.Q39Prove that cosθ -sinθ; sinθ cosθ is orthogonal.Q40If F(α)= cosα 0 sinα; 0 1 0; -sinα 0 cosα , show that [F(α)]⁻¹=F(-α)Q41(a) A boy is walking along the path y=ax²+bx+c through the points (-6, 8), (-2, -12) and (3, 8). He wants to meet his friend at P(7, 60). Will he meet his friend ? (Use Gaussian Elimination method) **OR** (b) Prove that the ellipse x²+4y²=8 and the hyperbola x²-2y²=4 intersect orthogonally.Q42If A is a non-singular matrix such that A⁻¹= 5 3; -2 -1 , then (AT)⁻¹= (a) -1 -3; 2 5 (b) -5 3; 2 1 (c) 5 -2; 3 -1 (d) 5 3; -2 -1 Q43If A, B and C are invertible matrices of some order, then which one of the following is not true ? (a) det A⁻¹=(det A)⁻¹ (b) adjA=|A|A⁻¹ (c) (ABC)⁻¹=C⁻¹B⁻¹A⁻¹ (d) adj(AB)=(adjA)(adjB)Q44Find the rank of the matrix -1 3; 4 -7; 3 -4 .Q45If A= 2 -1 3; -5 3 1; -3 2 3 , then find |adj(adjA)|.Q46If A is a non-singular matrix of order 3×3 and |A|=5 then |A⁻¹| is : (a) 5² (b) 5 (c) dfrac15² (d) dfrac15 Q47If A= 2 3; 5 -2 be such that λ A⁻¹=A, then λ is : (a) 19 (b) 17 (c) 21 (d) 14 Q48If adjA= -1 2 2; 1 1 2; 2 2 1 , find A⁻¹.Q49Solve the system of linear equations 2x+5y=-2, x+2y=-3 by matrix inversion method.Q50(a) Solve, by Cramer's rule, the system of equations x₁-x₂=3, 2x₁+3x₂+4x₃=17, x₂+2x₃=7 **OR** (b) Find the equation of tangent and normal to the curve given by x=7cos t and y=2sin t, t∈ R at any point on the curve.Q51If A is a 3×3 non-singular matrix such that AAT=ATA and B=A⁻¹AT, then BBT= (a) I₃ (b) A (c) BT (d) B Q52If ρ(A)=ρ([Amid B]), then the system AX=B of linear equations is : (a) consistent and has infinitely many solutions (b) consistent and has a unique solution (c) inconsistent (d) consistent Q53If A is a non-singular matrix of odd order, prove that |adjA| is positive.Q54Find the rank of the matrix 2 -2 4 3; -3 4 -2 -1; 6 2 -1 7 by reducing it to an echelon form.Q55(a) Solve the following system of equations, using matrix inversion method. 2x₁+3x₂+3x₃=5 x₁-2x₂+x₃=-4 3x₁-x₂-2x₃=3 **OR** (b) Show that the equation z³+2bar z=0 has five solutions.

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Ch 1 — Applications of Matrices and Determinants

Question 65 of 118

\rho(A) <

(a) only trivial solution

(b) trivial solution and infinitely many non-trivial solutions

(c) only non-trivial solutions

(d) no solution

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←Previous questionQ1 — The system of equations ax+y+z=0; x+by+z=0; x+y+cz=0 has a non-trivial…→Next questionQ3 — If A is a matrix of order 3, then det(kA) is (a) k³det(A) (b) k²det(A)…