{"id":18419,"date":"2020-01-07T15:15:33","date_gmt":"2020-01-07T09:45:33","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=18419"},"modified":"2021-07-02T15:30:43","modified_gmt":"2021-07-02T10:00:43","slug":"2nd-puc-maths-question-bank-chapter-3-miscellaneous-ex","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-question-bank-chapter-3-miscellaneous-ex\/","title":{"rendered":"2nd PUC Maths Question Bank Chapter 3 Matrices Miscellaneous Exercise"},"content":{"rendered":"

Students can Download Maths Chapter 3 Matrices Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers<\/a>\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.<\/p>\n

Karnataka 2nd PUC Maths Question Bank Chapter 3 Matrices Miscellaneous Exercise<\/h2>\n

Question 1.
\nLet \\(A=\\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix}\\), Show that (al +bA)n<\/sup> + nan – 1 <\/sup>ba where 1 is the identity matrix of order 2 and n \u2208 N.
\nAnswer:
\n\"2nd<\/p>\n

Question 2.
\nIf\\(A=\\left[\\begin{array}{lll}{1} & {1} & {1} \\\\{1} & {1} & {1} \\\\{1} & {1} & {1}\\end{array}\\right]\\)
\n,prove that
\nAn <\/sup>= \\(\\left[\\begin{array}{lll}{3^{n-1}} & {3^{n-1}} & {3^{n-1}} \\\\{3^{n-1}} & {3^{n-1}} & {3^{-1}} \\\\{3^{n-1}} & {3^{n-1}} & {3^{n-1}}\\end{array}\\right], n \\in N\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 3.
\n\\(\\mathbf{A}=\\left[\\begin{array}{ll}{\\mathbf{3}} & {-4} \\\\{\\mathbf{1}} & {\\mathbf{1}}\\end{array}\\right]\\)
\nthen prove that
\nAn <\/sup>= \\(\\left[\\begin{array}{cc}{1+2 n} & {-4 n} \\\\{n} & {1-2 n}\\end{array}\\right]\\) where n is any positive integer.
\nAnswer:
\n\"2nd
\n\u2234 P (k +1) is true
\nP(n) is true for all the values of +ve n.<\/p>\n

Question 4.
\nIf A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
\nAnswer:
\nSince A &B are symmetric matrices A = a’ & B = B’
\nTo prove (AB – BA)’ = – (AB – BA)
\nL.H.S = (AB – BA)’
\n(AB)’ – (BA)’
\n= B’A’-A’B’
\n= BA – AB
\n= – (AB – BA) = R.H.S
\nThus proved.<\/p>\n

\"KSEEB<\/p>\n

Question 5.
\nShow that the matrix g’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
\nAnswer:
\n\"2nd<\/p>\n

Question 6.
\nfind the values of x, y, z if the matrix
\nA = \\(\\left[\\begin{array}{rrr}{0} & {2 y} & {x} \\\\{x} & {y} & {-z} \\\\{x} & {-y} & {z}\\end{array}\\right]\\) satisfy the equation A’ A = 1
\nAnswer:
\n\"2nd<\/p>\n

\"2nd<\/p>\n

Question 7.
\nFor what values of x:
\n\\(\\left[\\begin{array}{lll}{1} & {2} & {1}\\end{array}\\right]\\left[\\begin{array}{rrr}{1} & {2} & {0} \\\\{2} & {0} & {1} \\\\{1} & {0} & {2}\\end{array}\\right]\\left[\\begin{array}{l}{0} \\\\{2} \\\\{x}\\end{array}\\right]=0 ?\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\n\\(\\mathbf{A}=\\left[\\begin{array}{rr}{3} & {1} \\\\{-1} & {2}\\end{array}\\right]\\) show that A2<\/sup> – 5A + 7I = 0
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 9.
\nFind x, if
\n\\(\\left[\\begin{array}{ccc}{\\mathbf{x}} & {-5} & {-1}\\end{array}\\right]\\left[\\begin{array}{ccc}{\\mathbf{1}} & {\\mathbf{0}} & {\\mathbf{2}} \\\\{\\mathbf{0}} & {\\mathbf{2}} &{\\mathbf{1}} \\\\{\\mathbf{2}} & {\\mathbf{0}} &{\\mathbf{3}}\\end{array}\\right]\\left[\\begin{array}{l}{\\mathbf{x}} \\\\{\\mathbf{4}} \\\\ {\\mathbf{1}}\\end{array}\\right]=\\mathbf{0}\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 10.
\nA manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below:
\n\"2nd
\n(a) If unit sale prices of, y and z are \u20b92. 50, \u20b91.50 and \u20b91.00, respectively, find the total revenue in each market with the help of matrix algebra.
\n(b) If the costs of the above three commodities are \u20b9 2.00, \u20b9 1.00 and 50 paise respectively. Find the gross profit.
\nAnswer:
\nMatrix representing the sales is
\n\"2nd
\nProfit in market I is
\n\u20b9 46,000 – \u20b9 31,000
\n= \u20b9 15,000<\/p>\n

Profit in market 11 is
\n\u20b9 53,000 – \u20b936,000
\n= \u20b9 17,000<\/p>\n

Gross profit
\n= \u20b9 15,000 + \u20b917,000
\n= \u20b9 32,000.<\/p>\n

Question 11.
\nFind the matrix X so that
\n\\(\\mathbf{X}\\left[\\begin{array}{lll}{1} & {2} & {3} \\\\{4} & {5} & {6}\\end{array}\\right]_{2 \\times 3}=\\left[\\begin{array}{rrr}{-7} & {-8} & {-9} \\\\{2} & {4} & {6}\\end{array}\\right]_{2 \\times 3}\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 12.
\nIf A and B are square matrices of the same order such that AB = BA, then prove by induction that ABn<\/sup> = Bn<\/sup>A. Further, prove that (AB)n<\/sup> = An<\/sup> Bn <\/sup>for all n \u2208 N.
\nAnswer:
\n(i) Let P(n) be the statement
\nP(n) – \u21d2 Ab n<\/sup> = Bn<\/sup>A
\nP(1) \u21d2 AB = BA
\n\u2234 P( 1) is true
\nLet P(k) be true
\nP(k) \u21d2 ABk<\/sup> = Bk<\/sup>A
\nP(k +1) \u21d2 ABk+1<\/sup>
\n= A(Bk<\/sup>B) = (ABk<\/sup>)B
\n\u2235 matrix multiplication is associative.
\n\u21d2 (Bk<\/sup>A) B \u21d2 Bk<\/sup> (AB) = Bk<\/sup> (BA)
\n= (Bk<\/sup>B) A = Bk+1<\/sup> A
\nHence P(k + 1) is true.
\n\u2234 P(n) is true for all the values of n \u2208 N<\/p>\n

(ii) Let P(n) be the statement
\nP(n) \u21d2 (AB)n<\/sup> = An<\/sup>Bn<\/sup>
\nP(1) = (AB)1<\/sup> = AB
\n\u2234 P(1) is true Let P(k) be true
\nP(k) \u21d2 (AB)k<\/sup>
\n= Ak<\/sup>Bk <\/sup>P(k+1)
\n\u21d2 (AB)k+1<\/sup> = (AB)k<\/sup>AB = (Ak<\/sup>Bk<\/sup>) AB
\n= Ak<\/sup> (Bk<\/sup>A)B = Ak<\/sup> (ABk<\/sup>) B
\n(\u2235 ABn<\/sup> = Bn<\/sup>A whenever Ab = BA)
\n= A<\/sub>k+1 B<\/sub>k+1
\n\u2234 P (k+1) is true
\nHence P(n) is true for all the natures of n \u2208 N<\/p>\n

Choose correct answer in the following questions:<\/p>\n

Question 13.
\nIf \\(\\mathbf{A}=\\left[\\begin{array}{ll}{\\alpha} & {\\beta} \\\\{\\gamma} & {-\\alpha}\\end{array}\\right]\\)
\nis such that A2<\/sup>= I ,then
\n(A) \\(1+\\alpha^{2}+\\beta \\gamma=0\\)
\n(B) \\(1-\\alpha^{2}+\\beta \\gamma=0\\)
\n(C) \\(1-\\alpha^{2}-\\beta \\gamma=0\\)
\n(D) \\(1+\\alpha^{2}-\\beta \\gamma=0\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 14.
\nIf the matrix A is both symmetric and skew symmetric, then
\n(A) A is a diagonal martix
\n(B) A is a zero matrix
\n(C) A is a square matrix
\n(D) None of these
\nAnswer:
\nLet A be a square matrix such that A is both symmetric & skew-symmetric
\n\u21d2 A’ = A – (i) & A’ = – A -(ii)
\n\u21d2 A + A = 0\u00a0 from (i) & (ii)
\n2A = 0
\n\u21d2 A = o
\n\u2234 (b) is correct<\/p>\n

Question 15.
\nIf A is square matrix such that A2<\/sup> = A, then (I + A)3<\/sup> – 7A is equal to
\n(A) A
\n(B) I – A
\n(C) I
\n(D) 3A
\nAnswer:
\nA2<\/sup> = A
\n(I + A)3<\/sup> – 7A = (I + A) (I+A)2<\/sup> – 7A
\n= (I + A) {(I + A) (I+A)} -7A
\n= (I + A) (I2<\/sup> + 2A+A2<\/sup>) – 7A
\n[v IA = AI = A]
\n= (I + A) (I+2A+A) – 7A (A2<\/sup>= A)
\n= (I + A) (I + 3A) – 7A
\n= I2<\/sup> + AI + 3IA + 3A2<\/sup> – 7A
\n= I + A + 3A + 3A – 7A
\n[\u2235 A2<\/sup> = A & IA = AI = A]
\n=1
\n\u2234 (C) is correct.<\/p>\n

2nd PUC Maths Matrices Miscellaneous Exercise Additional Questions and Answers<\/h3>\n

One Mark Questions:<\/p>\n

Question 1.
\nIf \\(\\mathbf{A}=\\left(\\begin{array}{ll}{\\mathbf{0}} & {\\mathbf{1}} \\\\{\\mathbf{1}} & {\\mathbf{0}}\\end{array}\\right), \\mathbf{F} \\text { ind } \\mathbf{A}^{4}\\)
\n(UP CET 2010)
\nAnswer:
\n\"2nd<\/p>\n

Question 2.
\nIf
\n\\(\\mathbf{A}=\\left(\\begin{array}{rr}{0} & {2} \\\\{3} & {-4}\\end{array}\\right), \\mathbf{K A}=\\left(\\begin{array}{ll}{0} & {3 \\mathbf{a}} \\\\{2 \\mathbf{b}} & {24}\\end{array}\\right)\\),find k, a and b.
\nAnswer
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 3.
\nFind X and Y
\n\\(2 \\mathbf{X}-\\mathbf{Y}=\\left[\\begin{array}{rrr}{6} & {-6} & {0} \\\\{-4} & {2} & {1} \\end{array}\\right]\\)
\n\\(\\mathbf{X}+2 \\mathbf{Y}=\\left[\\begin{array}{rrr}{3} & {2} & {5} \\\\{-2} & {1} & {-7}\\end{array}\\right]\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 4.
\n\\(\\left[\\begin{array}{rrr}{1} & {1} & {1} \\\\{1} & {-2} & {-2} \\\\{1} & {3} &{2}\\end{array}\\right]\\left[\\begin{array}{l}{x} \\\\{y} \\\\{z}\\end{array}\\right]=\\left[\\begin{array}{l}{0} \\\\{3} \\\\{4}\\end{array}\\right], \\text { then }\\left[\\begin{array}{l}{x} \\\\{y} \\\\{z}\\end{array}\\right]\\)(I I T 2006)
\nAnswer:<\/p>\n

\"2nd<\/p>\n

Question 5.
\n\\(\\mathbf{A}=\\left[\\begin{array}{lll}{6} & {8} & {7} \\\\{4} & {2} & {3} \\\\{9} & {7} & {1}\\end{array}\\right]\\)
\nis the sum of symmentric and skew symmentric matrix, Find B
\nAnswer:
\n\"2nd<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can Download Maths Chapter 3 Matrices Miscellaneous Exercise Questions and Answers, Notes Pdf, 2nd PUC Maths Question Bank with Answers\u00a0helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations. Karnataka 2nd PUC Maths Question Bank Chapter 3 Matrices Miscellaneous Exercise Question 1. Let , Show that (al …<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[82],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/18419"}],"collection":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/comments?post=18419"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/18419\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=18419"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=18419"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=18419"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}