{"id":19017,"date":"2020-01-09T11:20:51","date_gmt":"2020-01-09T05:50:51","guid":{"rendered":"https:\/\/kseebsolutions.guru\/?p=19017"},"modified":"2021-07-02T15:30:33","modified_gmt":"2021-07-02T10:00:33","slug":"2nd-puc-maths-question-bank-chapter-4-miscellaneous-ex","status":"publish","type":"post","link":"https:\/\/kseebsolutions.guru\/2nd-puc-maths-question-bank-chapter-4-miscellaneous-ex\/","title":{"rendered":"2nd PUC Maths Question Bank Chapter 4 Determinants Miscellaneous Exercise"},"content":{"rendered":"

Students can Download Maths Chapter 4 Determinants 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 4 Determinants Miscellaneous Exercise<\/h2>\n

Question 1.
\nProve that the determinant
\n\\(\\left| \\begin{array}{rrr} { x } & { \\sin \\theta } & { \\cos \\theta } \\\\ { -\\sin \\theta } & { -x } & { 1 } \\\\ { 1 } & { 1 } & x \\end{array} \\right| is\\quad independent\\quad of\\quad \\theta \\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 2.
\nWith out expanding, prove that
\n\\(\\left|\\begin{array}{lll}{\\mathbf{a}} & {\\mathbf{a}^{2}} & {\\mathbf{b} \\mathbf{c}} \\\\{\\mathbf{b}} & {\\mathbf{b}^{2}} & {\\mathbf{c} \\mathbf{a}} \\\\{\\mathbf{c}} & {\\mathbf{c}^{2}} & {\\mathbf{a} \\mathbf{b}}\\end{array}\\right|=\\left|\\begin{array}{ccc}{\\mathbf{1}} & {\\mathbf{a}^{2}} & {\\mathbf{a}^{3}} \\\\{\\mathbf{1}} & {\\mathbf{b}^{2}} & {\\mathbf{b}^{3}} \\\\{\\mathbf{1}} & {\\mathbf{c}^{2}} & {\\mathbf{c}^{3}}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 3.
\nEvaluate
\n\\(\\left|\\begin{array}{ccc}{\\cos \\alpha \\cos \\beta} & {\\cos \\alpha \\sin \\beta} & {-\\sin \\alpha} \\\\{\\sin \\alpha \\cos \\beta} & {\\sin \\alpha \\sin \\beta} & {\\cos \\alpha}\\end{array}\\right|\\)
\nAnswer:
\nexpand along c3
\n<\/sub>-sin \u03b1 (-sin \u03b1 sin2<\/sup>\u03b2 – sin \u03b1 cos2<\/sup>\u03b2) + cosa (cos\u03b1 cos2<\/sup> \u03b2+cos\u03b1 sin2<\/sup>\u03b2)
\n= sin2<\/sup> \u03b1 (sin2<\/sup> \u03b2 + cos2<\/sup> \u03b2 )+cos2<\/sup> \u03b1 (cos2<\/sup> \u03b2 + sin2<\/sup> \u03b2)
\n= sin2<\/sup>\u03b1 + cos2<\/sup>\u03b1= 1<\/p>\n

Question 4.
\nIt a, b, c are real, find the factors of the determination
\n\\(\\Delta=\\left|\\begin{array}{lll}{\\mathbf{b}+\\mathbf{c}} & {\\mathbf{c}+\\mathbf{a}} & {\\mathbf{a}+\\mathbf{b}} \\\\{\\mathbf{c}+\\mathbf{a}} & {\\mathbf{a}+\\mathbf{b}} & {\\mathbf{b}+\\mathbf{c}} \\\\{\\mathbf{a}+\\mathbf{b}} & {\\mathbf{b}+\\mathbf{c}} & {\\mathbf{c}+\\mathbf{a}}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 5.
\nSolve if a \u2260 0 and
\n\\(\\left|\\begin{array}{ccc}{\\mathbf{x}+\\mathbf{a}} & {\\mathbf{x}} & {\\mathbf{x}} \\\\{\\mathbf{x}} & {\\mathbf{x}+\\mathbf{a}} & {\\mathbf{x}} \\\\{\\mathbf{x}} & {\\mathbf{x}} & {\\mathbf{x}+\\mathbf{a}}\\end{array}\\right|=\\mathbf{0}\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 6.
\nProve that
\n\\(\\left|\\begin{array}{ccc}{\\mathbf{a}^{2}} & {\\mathbf{b} \\mathbf{c}} & {\\mathbf{a c}+\\mathbf{c}^{2}} \\\\{\\mathbf{a}^{2}+\\mathbf{a} \\mathbf{b}} & {\\mathbf{b}^{2}} & {\\mathbf{a c}} \\\\{\\mathbf{a b}} & {\\mathbf{b}^{2}+\\mathbf{b c}} & {\\mathbf{c}^{2}}\\end{array}\\right|=4 \\mathbf{a}^{2} \\mathbf{b}^{2} \\mathbf{c}^{2}\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 7.
\n\\(\\begin{aligned}&\\text { If } \\mathbf{A}^{-1}=\\left[\\begin{array}{rrr}{3} & {-1} & {1} \\\\{-15} & {6} & {-5} \\\\{5} & {-2} & {2}\\end{array}\\right], \\mathbf{B}=\\left[\\begin{array}{rrr}{1} & {2} & {-2} \\\\{-1} & {3} & {0} \\\\{0} & {-2} & {1}\\end{array}\\right] \\text { find }\\\\&(\\mathbf{A} \\mathbf{B})^{-1}\\end{aligned}\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 8.
\n\\(\\text { Let } A=\\left[\\begin{array}{ccc}{1} & {-2} & {1} \\\\{-2} & {3} & {1} \\\\{1} & {1} & {5}\\end{array}\\right], \\text { verify that }\\)
\n(i) (adj A)1<\/sup> = Adj (A1<\/sup>)
\n(ii) (A1<\/sup>)1<\/sup> = A
\nAnswer:
\n\"2nd
\n\"2nd
\n\"2nd
\n\"2nd<\/p>\n

Question 9.
\nEvaluate
\n\\(\\left[\\begin{array}{ccc}{\\mathbf{x}} & {\\mathbf{y}} & {\\mathbf{x}+\\mathbf{y}} \\\\{\\mathbf{y}} & {\\mathbf{x}+\\mathbf{y}} & {\\mathbf{x}} \\\\{\\mathbf{x}+\\mathbf{y}} & {\\mathbf{x}} & {\\mathbf{y}}\\end{array}\\right]\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 10.
\n\\(\\left[\\begin{array}{ccc}{\\mathbf{1}} & {\\mathbf{x}} & {\\mathbf{y}} \\\\{\\mathbf{1}} & {\\mathbf{x}+\\mathbf{y}} & {\\mathbf{y}} \\\\{\\mathbf{1}} & {\\mathbf{x}} & {\\mathbf{x}+\\mathbf{y}}\\end{array}\\right]\\)
\nAnswer:
\n\"2nd<\/p>\n

Prove the following using properties<\/p>\n

Question 11.
\n\\(\\left[\\begin{array}{lll}{\\alpha} & {\\alpha^{2}} & {\\beta+\\alpha} \\\\{\\beta} & {\\beta^{2}} & {\\gamma+\\alpha} \\\\{\\gamma} & {\\gamma^{2}} & {\\alpha+\\beta}\\end{array}\\right]=(\\beta-\\alpha)(\\gamma-\\alpha)(\\alpha-\\beta)\\)
\nAnswer:<\/p>\n

\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 12.
\n\\(\\begin{aligned}&\\left[\\begin{array}{lll}{\\mathbf{x}} & {\\mathbf{x}^{2}} & {\\mathbf{1}+\\mathbf{p} \\mathbf{x}^{3}} \\\\{\\mathbf{y}} & {\\mathbf{y}^{2}} & {\\mathbf{1}+\\mathbf{p} \\mathbf{y}^{3}} \\\\{\\mathbf{z}} & {\\mathbf{z}^{2}} & {\\mathbf{1}+\\mathbf{p}\\mathbf{z}^{3}}\\end{array}\\right]=(\\mathbf{x}-\\mathbf{y})(\\mathbf{y}-\\mathbf{z})(\\mathbf{z}-\\mathbf{x})\\\\&(1+p x y z)\\end{aligned}\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 13.
\n\\(\\left[\\begin{array}{rrr}{3 a} & {-a+b} & {-a+c} \\\\{-b+a} & {3 b} & {-b+c} \\\\{-c+a} & {-c+b} & {3 c}\\end{array}\\right]=3(a+b+c)(a b+b c+c a)\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 14.
\n\\(\\left|\\begin{array}{ccc}{1} & {1+p} & {1+p+q} \\\\{2} & {3+2 p} & {4+3 p+2 q} \\\\{3} & {6+3 p} & {10+6 p+3 q}\\end{array}\\right|=1\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 15.
\n\\(\\left|\\begin{array}{ccc}{\\sin \\alpha} & {\\cos \\alpha} & {\\cos (\\alpha+\\delta)} \\\\{\\sin \\beta} & {\\cos \\beta} & {\\cos (\\beta+\\delta)} \\\\{\\sin \\gamma} & {\\cos \\gamma} & {\\cos (\\gamma+\\delta)}\\end{array}\\right|=0\\)
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 16.
\nSolve
\n\\(\\frac{2}{x}+\\frac{3}{y}+\\frac{10}{z}=4\\)
\n\\(\\frac{4}{x}-\\frac{6}{y}+\\frac{5}{z}=1\\)
\n\\(\\frac{6}{x}+\\frac{9}{y}-\\frac{20}{z}=2\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

Question 17.
\nIt a, b, c are in AP, then the determinant
\n\\(\\left|\\begin{array}{ccc}{x+2} & {x+3} & {x+2 a} \\\\{x+3} & {x+4} & {x+2 b} \\\\{x+4} & {x+5} & {x+2 c}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 18.
\nIf x, y, z are real, then the inverse of
\n\\(\\mathbf{A}=\\left|\\begin{array}{lll}{\\mathbf{x}} & {\\mathbf{0}} & {\\mathbf{0}} \\\\{\\mathbf{0}} & {\\mathbf{y}} & {\\mathbf{0}} \\\\{\\mathbf{0}} & {\\mathbf{0}} & {\\mathbf{z}}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 19.
\nLet
\n\\({ A }=\\left[ \\begin{array}{ccc} { { 1 } } & { \\sin \\theta } & { { 1 } } \\\\ { -\\sin \\theta } & { { 1 } } & { \\sin \\theta } \\\\ { -1 } & { -\\sin \\theta } & { { 1 } } \\end{array} \\right] { 0 }\\leq \\theta \\leq 2\\pi \\)
\nAnswer:
\n\"2nd<\/p>\n

2nd PUC Maths Chapter 4 Determinants Miscellaneous Exercise Additional Questions and Answers<\/h3>\n

Try These Questions<\/p>\n

Question 1.
\n\\(\\text { If } \\mathbf{A}=\\left[\\begin{array}{lll}{1} & {2} & {2} \\\\{2} & {1} & {2} \\\\{2} & {2} & {1}\\end{array}\\right] \\text { find } A^{-1}\\)
\nand hence prove that A2<\/sup> – 4A – 5 I = 0
\nAnswer:
\n\"2nd<\/p>\n

Question 2.
\nuse product
\n\\(\\left[\\begin{array}{ccc}{1} & {-1} & {2} \\\\{0} & {2} & {-3} \\\\{3} & {-2} & {4}\\end{array}\\right]\\left[\\begin{array}{ccc}{-2} & {0} & {1} \\\\{9} & {2} & {-3} \\\\{6} & {1} & {-2}\\end{array}\\right]\\)
\nsolve x – y + 2z = 1
\n2y – 3z = 1
\n3x – 2y + 4z = 2
\nAnswer:
\n\"2nd
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 3.
\nIf the point (x, y), (a, 0), (0, b) are collinear, prove that
\n\\(\\frac{x}{a}+\\frac{y}{b}=1\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 4.
\n\\(\\text { If } \\Delta=\\left|\\begin{array}{ccc}{\\mathbf{1}} & {\\mathbf{1}} & {\\mathbf{1}} \\\\{\\mathbf{1}} & {\\mathbf{1}+\\mathbf{x}} & {\\mathbf{1}} \\\\{\\mathbf{1}} & {\\mathbf{1}} & {\\mathbf{1}+\\mathbf{y}}\\end{array}\\right|, \\text { then } \\Delta \\text { is }\\)
\n(a) Divisible by neither x nor y
\n(b) Divisible by y but not x
\n(c) Divisible by x but not y
\n(d) Divisible by both x and y [AIEEE 2007]
\nAnswer:
\n\"2nd<\/p>\n

Question 5.
\nIf a, b, c are positive and unequal, show that the value of the determinant
\n\\(\\left|\\begin{array}{lll}{\\mathbf{a}} & {\\mathbf{b}} & {\\mathbf{c}} \\\\{\\mathbf{b}} & {\\mathbf{c}} & {\\mathbf{a}} \\\\{\\mathbf{c}} & {\\mathbf{a}} & {\\mathbf{b}}\\end{array}\\right|\\) is always negative [Kerala CET 09]
\nAnswer:
\n\"2nd
\nsince a, b, c are positive a + b + c is positive and (a – b)2<\/sup> is always positive,
\n\u0394 is always negative.<\/p>\n

Question 6.
\nUsing properties of matrices prove that 1+abc is a factor of [CET]
\n\\(\\left|\\begin{array}{lll}{\\mathbf{a}} & {\\mathbf{a}^{2}} &{\\mathbf{1}+\\mathbf{a}^{3}} \\\\{\\mathbf{b}} & {\\mathbf{b}^{2}} & {\\mathbf{1}+\\mathbf{b}^{3}} \\\\{\\mathbf{c}} &{\\mathbf{c}^{2}} &{\\mathbf{1}+\\mathbf{c}^{3}}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 7.
\nFor what values of y is the matrix\u00a0 [CBSE 2011]
\n\\(\\mathbf{A}=\\left|\\begin{array}{rr}{\\mathbf{y}^{2}+6} & {\\mathbf{2} \\mathbf{y}} \\\\{\\mathbf{y}+3} & {\\mathbf{2}} \\end{array}\\right|\\)singular.
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 8.
\nFind y
\n\\(\\left[\\begin{array}{cc}{x} & {x-y} \\\\{2 x+y} & {7}\\end{array}\\right]=\\left[\\begin{array}{ll}{3} & {1} \\\\{8} & {7}\\end{array}\\right]\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 9.
\nIf A be a square matrix of order 3 x 3 then find |KA|
\n|KA| = kn <\/sup>|A| when n is the order [CBSE 2007]
\nAnswer:
\n|KA| = k3 <\/sup>|A|<\/p>\n

Question 10.
\nEvaluate\u00a0 [PUC 2009]
\n\\(\\left|\\begin{array}{cccc}{\\sin 30} & {\\cos } & {30} \\\\{-\\sin 60} & {\\cos } & {60}\\end{array}\\right|\\)
\nAnswer:
\n\"2nd<\/p>\n

Question 11.
\nFrom the matrix equation AB = AC, we can conclude that B = C provided
\n(A) A is singular
\n(B) A is non singular
\n(C) A is symmetric
\n(D) A is skew symmetric [IIT]
\nAnswer:
\nAB = AC
\nA-1<\/sup> (AB) = A-1<\/sup> (AC) \u21d2 (A-1<\/sup> A) B = (A-1<\/sup> A) C \u21d2 IB = IC \u21d2 B = C
\nAB = AC \u21d2 B = C if A1<\/sup> exist
\nie,. A is non singular
\ncorrect Answer is (B)<\/p>\n

Question 12.
\nA2<\/sup> – A + I = 0, Find the inverse if A [IIT]
\nAnswer:
\nA2<\/sup> – A + I = 0
\nPre – multiply by A-1
\n<\/sup>A-1<\/sup> A2<\/sup> – A-1<\/sup> A + A-1 <\/sup>\u00a0I = 0
\nA -1 + A1<\/sup> = 0
\nA-1<\/sup>= I – A<\/p>\n

\"KSEEB<\/p>\n

Question 13.
\nIt is a triangle
\n\\(\\mathbf{A B C},\\left|\\begin{array}{lll}{\\mathbf{1}} & {\\mathbf{a}} & {\\mathbf{b}} \\\\{\\mathbf{1}} & {\\mathbf{c}} & {\\mathbf{a}} \\\\{\\mathbf{1}} & {\\mathbf{b}} & {\\mathbf{c}}\\end{array}\\right|=\\mathbf{0}\\)
\nthen find the value of sin2<\/sup>A+ sin2<\/sup>B+ sin2<\/sup>C [Karnataka CET 2002]
\nAnswer:
\n\"2nd<\/p>\n

Question 14.
\nLet A, B be two square matrices such that A + B = AB, then\u00a0 [Karnataka CET]
\n(A) AB = BA
\n(B) AB = -BA
\n(C) AB + 2BA = 0
\n(D) None of these
\nAns:
\nA + B = AB A + B – AB = 0
\nA + B – AB + I = 0+1
\n(A – I)(B – I) = 1
\n\u21d2 A -1 inverse of B -1
\n(B -1) (A -1) = I
\nBA – B – A + = I
\nBA – (B + A) \u21d2 BA = B + A
\n= A + B\u00a0 \u2234 AB = BA
\nAnswer (A)<\/p>\n

Question 15.
\n\\(\\mathbf{A}(\\alpha)=\\left[\\begin{array}{cc}{\\cos \\alpha} & {\\sin \\alpha} \\\\{-\\sin \\alpha} & {\\cos \\alpha}\\end{array}\\right]\\),
\nthen show that A(\u03b1)A(\u03b2) = A (\u03b1+\u03b2) [Kerala CET]
\nAnswer:
\n\"2nd<\/p>\n

\"KSEEB<\/p>\n

Question 16.
\n\\(\\mathbf{A}=\\left[\\begin{array}{ll}{\\mathbf{1}} & {-\\mathbf{1}} \\\\{\\mathbf{2}} & {-\\mathbf{1}}\\end{array}\\right], \\mathbf{B}=\\left[\\begin{array}{ll}{\\mathbf{a}} & {-\\mathbf{1}} \\\\{\\mathbf{b}} & {-\\mathbf{1}}\\end{array}\\right] \\text { and }(\\mathbf{A}+\\mathbf{B})^{2}\\)
\n= A2<\/sup> + B2<\/sup> Find the values of a, b
\nAnswer:
\n\"2nd<\/p>\n","protected":false},"excerpt":{"rendered":"

Students can Download Maths Chapter 4 Determinants 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 4 Determinants Miscellaneous Exercise Question 1. Prove that the determinant Answer: …<\/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\/19017"}],"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=19017"}],"version-history":[{"count":0,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/posts\/19017\/revisions"}],"wp:attachment":[{"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/media?parent=19017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/categories?post=19017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kseebsolutions.guru\/wp-json\/wp\/v2\/tags?post=19017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}