JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Self Evaluation Test - Determinats Total Questions - 80

• question_answer1)
  Consider the system of linear equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0,\] \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0,\] \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z+{{d}_{3}}=0,\] Let us denote by \[\Delta (a,b,c)\] the determinant \[\left| \begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\], if \[\Delta \,(a,b,c)\#0,\] then the value of x in the unique solution of the above equations is
  A)
  \[\frac{\Delta (b,c,d)}{\Delta (a,b,c)}\] done clear
  B)
  \[\frac{-\Delta (b,c,d)}{\Delta (a,b,c)}\] done clear
  C)
  \[\frac{\Delta (a,c,d)}{\Delta (a,b,c)}\] done clear
  D)
  \[-\frac{\Delta (a,b,d)}{\Delta (a,b,c)}\] done clear
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• question_answer2)
  For what value of p, is the system of equations : \[{{p}^{3}}x+{{(p+1)}^{3}}y={{(p+2)}^{3}}\] \[px+(p+1)y=p+2\] \[x+y=1\] consistent?
  A)
  \[p=0\] done clear
  B)
  \[p=1\] done clear
  C)
  \[p=-1\] done clear
  D)
  For all \[p>1\] done clear
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• question_answer3)
  If \[C=2cos\theta ,\] then the value of the determinant\[\Delta =\left[ \begin{matrix}    C & 1 & 0  \\    1 & C & 1  \\    6 & 1 & C  \\ \end{matrix} \right]\] is
  A)
  \[\frac{2{{\sin }^{2}}2\theta }{\sin \theta }\]       done clear
  B)
         \[8{{\cos }^{3}}\theta -4\cos \theta +6\] done clear
  C)
  \[\frac{2\sin 2\theta }{\sin \theta }\] done clear
  D)
  \[8{{\cos }^{3}}\theta +4\cos \theta +6\] done clear
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• question_answer4)
  If \[{{e}^{i\theta }}=\cos \theta +i\sin \theta ,\] then the value of \[\left| \begin{matrix}    1 & {{e}^{i\pi /3}} & {{e}^{i\pi /4}}  \\    {{e}^{-i\pi /3}} & 1 & {{e}^{i2\pi /3}}  \\    {{e}^{-i\pi /4}} & {{e}^{-i2\pi /3}} & 1  \\ \end{matrix} \right|\]is
  A)
  \[-2+\sqrt{2}\] done clear
  B)
  \[2-\sqrt{2}\] done clear
  C)
  \[-2-\sqrt{2}\] done clear
  D)
  1 done clear
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• question_answer5)
  If the value of the determinant \[\left| \begin{matrix}    a & 1 & 1  \\    1 & b & 1  \\    1 & 1 & c  \\ \end{matrix} \right|\] is positive, where \[a\ne b\ne c,\] then the value of abc
  A)
  Cannot be less than 1 done clear
  B)
  Is greater than \[-8\] done clear
  C)
  Is less than \[-8\] done clear
  D)
  Must be greater than 8 done clear
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• question_answer6)
  If \[f(x)=\left| \begin{matrix}    1+{{\sin }^{2}}x & {{\cos }^{2}}x & 4\sin 2x  \\    {{\sin }^{2}}x & 1+{{\cos }^{2}}x & 4\sin 2x  \\    {{\sin }^{2}}x & {{\cos }^{2}}x & 1+4\sin 2x  \\ \end{matrix} \right|\]What is the maximum value of \[f(x)\]?
  A)
  2 done clear
  B)
  4 done clear
  C)
  6 done clear
  D)
  8 done clear
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• question_answer7)
  Let \[\Delta =\left| \begin{matrix}    1+{{x}_{1}}{{y}_{1}} & 1+{{x}_{1}}{{y}_{2}} & 1+{{x}_{1}}{{y}_{3}}  \\    1+{{x}_{2}}{{y}_{1}} & 1+{{x}_{2}}{{y}_{2}} & \,1+{{x}_{2}}{{y}_{3}}  \\    1+{{x}_{3}}{{y}_{1}} & 1+{{x}_{3}}{{y}_{2}} & 1+{{x}_{3}}{{y}_{3}}  \\ \end{matrix} \right|\] then value of \[\Delta \] is
  A)
  \[{{x}_{1}}{{x}_{2}}{{x}_{3}}+{{y}_{1}}{{y}_{2}}{{y}_{3}}\] done clear
  B)
  \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{y}_{1}}{{y}_{2}}{{y}_{3}}\] done clear
  C)
  \[{{x}_{2}}{{x}_{3}}{{y}_{2}}{{y}_{3}}+{{x}_{3}}{{x}_{1}}{{y}_{3}}{{y}_{1}}+{{x}_{1}}{{x}_{2}}{{y}_{1}}{{y}_{2}}\] done clear
  D)
  0 done clear
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• question_answer8)
  The rank of the matrix \[\left[ \begin{matrix}    -1 & 2 & 5  \\    2 & -4 & a-4  \\    1 & -2 & a+1  \\ \end{matrix} \right]\] is
  A)
  1 if \[a=6\] done clear
  B)
  2 if \[a=1\] done clear
  C)
  3 if \[a=2\] done clear
  D)
  1 if\[a=4\] done clear
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• question_answer9)
  If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ne -2\] and\[f(x)=\left| \begin{matrix}    (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x  \\    (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x  \\    (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x  \\ \end{matrix} \right|\]then \[f(x)\] is a polynomial of degree
  A)
  \[1\] done clear
  B)
  0 done clear
  C)
  \[3\] done clear
  D)
  \[2\] done clear
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• question_answer10)
  If \[\left| \begin{matrix}    {{x}^{2}}+x & 3x-1 & -x+3  \\    2x+1 & 2+{{x}^{2}} & {{x}^{3}}-3  \\    x-3 & {{x}^{2}}+4 & 3x  \\ \end{matrix} \right|\]\[={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{7}}{{x}^{7}},\] then the value of \[{{a}_{0}}\] is
  A)
  \[25\] done clear
  B)
  \[24\] done clear
  C)
  \[23\] done clear
  D)
  \[21\] done clear
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• question_answer11)
  If A is an orthogonal matrix of order 3 and \[B=\left[ \begin{matrix}    1 & 2 & 3  \\    -3 & 0 & 2  \\    2 & 5 & 0  \\ \end{matrix} \right],\] then which of the following is/are correct? 1. \[|AB|=\pm 47\] 2. \[AB=BA\] Select the correct answer using the code given below:
  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2      done clear
  D)
  Neither 1 nor 2 done clear
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• question_answer12)
  Let \[\lambda \] and \[\alpha \] be real. The set of all values of x for which the system of linear equations \[\lambda x+(\sin \alpha )y+(cos\alpha )z=0\] \[x+(cos\alpha )y+(sin\alpha )z=0\] \[-x+(\sin \alpha )-(\cos \alpha )z=0\] has a non-trivial solution, is
  A)
  \[\left[ 0,\,\sqrt{2} \right]\] done clear
  B)
  \[\left[ -\sqrt{2},0 \right]\] done clear
  C)
  \[\left[ -\sqrt{2},\sqrt{2} \right]\] done clear
  D)
  None of these done clear
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• question_answer13)
  For what values of k, does the system of linear equation \[x+y+z=2,\]  \[2x+y-z=3,\] \[3x+2y+kz=4\] have a unique solution?
  A)
  \[k=0\] done clear
  B)
  \[-1<k<1\] done clear
  C)
  \[-2<k<2\] done clear
  D)
  \[k\ne 0\] done clear
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• question_answer14)
  Let \[x<1,\] then value of \[\left| \begin{matrix}    {{x}^{2}}+2 & 2x+1 & 1  \\    2x+1 & x+2 & 1  \\    3 & 3 & 1  \\ \end{matrix} \right|\] is
  A)
  Non-negative done clear
  B)
  Non-positive done clear
  C)
  Negative done clear
  D)
  Positive done clear
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• question_answer15)
  If \[[a]\] denotes the integral part of a and \[x={{a}_{3}}y+{{a}_{2}}z,\] \[y={{a}_{1}}z+{{a}_{3}}z\] and \[z={{a}_{2}}x+{{a}_{1}}y,\]where x, y, z are not all zero. If \[{{a}_{1}}=m-[m],\] m being a non-integral constant, then \[{{a}_{1}}{{a}_{2}}{{a}_{3}}\] is
  A)
  \[>1\] done clear
  B)
  \[>-1\] done clear
  C)
  \[<1\] done clear
  D)
  \[<-1\] done clear
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• question_answer16)
  If [ ] denotes the greatest integer less than or equal to the real number under consideration and \[-1\le x<0;\] \[0\le y<1;\] \[1\le z<2,\] then the value of the determinant\[\left| \begin{matrix}    [x]+1 & [y] & [z]  \\    [x] & [y]+1 & [z]  \\    [x] & [y] & [z]+1  \\ \end{matrix} \right|\]is
  A)
  \[[z]\] done clear
  B)
  \[[y]\] done clear
  C)
  [x] done clear
  D)
  None of these done clear
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• question_answer17)
  Let \[f(x)=\left| \begin{matrix}    n & n+1 & n+2  \\    ^{n}{{P}_{n}} & ^{n+1}{{P}_{n+1}} & ^{n+2}{{P}_{n+2}}  \\    ^{n}{{C}_{n}} & ^{n+1}{{C}_{n+1}} & ^{n+2}{{C}_{n+2}}  \\ \end{matrix} \right|,\] where the symbols have their usual meanings. The \[f(x)\] is divisible by
  A)
  \[{{n}^{2}}+n+1\] done clear
  B)
  \[(n+1)!\] done clear
  C)
  \[(2n+1)!\] done clear
  D)
  None of the above done clear
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• question_answer18)
  The determinant \[\left| \begin{matrix}    x & \sin \theta  & \cos \theta   \\    -\sin \theta  & -x & 1  \\    \cos \theta  & 1 & x  \\ \end{matrix} \right|\] is independent of
  A)
  x only done clear
  B)
  \[\theta \] only done clear
  C)
  x and \[\theta \] both done clear
  D)
  None of these done clear
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• question_answer19)
  If \[{{A}^{-1}}=\left[ \begin{matrix}    1 & -2  \\    -2 & 2  \\ \end{matrix} \right],\] what is det(A)?
  A)
  \[2\] done clear
  B)
  \[-2\] done clear
  C)
  \[\frac{1}{2}\] done clear
  D)
  \[-\frac{1}{2}\] done clear
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• question_answer20)
  If \[A=\left[ \begin{matrix}    1 & 2 & -1  \\    -1 & 1 & 2  \\    2 & -1 & 1  \\ \end{matrix} \right],\] then det(adj (adj A)) is
  A)
  \[{{(14)}^{4}}\] done clear
  B)
  \[{{(14)}^{3}}\] done clear
  C)
  \[{{(14)}^{2}}\] done clear
  D)
  \[{{(14)}^{1}}\] done clear
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• question_answer21)
  If \[f(x)=\left| \begin{matrix}    \cos x & x & 1  \\    2\sin x & {{x}^{2}} & 2x  \\    \tan x & x & 1  \\ \end{matrix} \right|,\] then \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{f'(x)}{x} \right]\] is
  A)
  \[2\] done clear
  B)
  \[-2\] done clear
  C)
  \[1\] done clear
  D)
  \[-1\] done clear
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• question_answer22)
  The number of values of k for which the system of equations \[(k+1)x+8y=4k;\] \[kx+(k+3)y=3k-1\] has infinitely many solutions is
  A)
  0 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  infinite             done clear
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• question_answer23)
  If the system of linear equations \[x+2ay+az=0;\] \[x+3by+bz=0;\] \[x+4cy+cz=0\] has a non - zero solution, then a, b, c.
  A)
  Satisfy  \[a+2b+3c=0\] done clear
  B)
  Are in A.P done clear
  C)
  Are in G.P done clear
  D)
  Are in H.P. done clear
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• question_answer24)
  For all values of A, B, C and P, Q, R the value of the determinant\[{{(x+a)}^{3}}\left| \begin{matrix}    \cos (A-P) & \cos (A-Q) & \cos (A-R)  \\    \cos (B-P) & \cos (B-Q) & \cos (B-R)  \\    \cos (C-P) & \cos (C-Q) & \cos (C-R)  \\ \end{matrix} \right|\] is
  A)
  \[1\] done clear
  B)
  \[0\] done clear
  C)
  \[2\] done clear
  D)
  None of these done clear
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• question_answer25)
  If \[f(x)=a+bx+c{{x}^{2}}\]and \[\alpha ,\beta ,\lambda \] are roots of the equation \[{{x}^{3}}=1,\]then \[\left| \begin{matrix}    a & b & c  \\    b & c & a  \\    c & a & b  \\ \end{matrix} \right|\] is equal to
  A)
  \[f(\alpha )+f(\beta )+f(\lambda )\] done clear
  B)
  \[f(\alpha )f(\beta )+f(\beta )f(\lambda )+f(\gamma )+f(\alpha )\] done clear
  C)
  \[f(\alpha )f(\beta )f(\gamma )\] done clear
  D)
  \[-f(\alpha )f(\beta )f(\gamma )\] done clear
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• question_answer26)
  In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has the value
  A)
  \[1\] done clear
  B)
  \[9\] done clear
  C)
  \[16\] done clear
  D)
  \[24\] done clear
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• question_answer27)
  If A is a square matrix of order n, then adj (adj A) is equal to
  A)
  \[|A{{|}^{n-1}}A\] done clear
  B)
  \[|A{{|}^{n}}A\] done clear
  C)
  \[|A{{|}^{n-2}}A\] done clear
  D)
  None of these done clear
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• question_answer28)
  Suppose \[\Delta =\left| \begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\] and\[\Delta '=\left| \begin{matrix}    {{a}_{1}}+p{{b}_{1}} & {{b}_{1}}+q{{c}_{1}} & {{c}_{1}}+r{{a}_{1}}  \\    {{a}_{2}}+p{{b}_{2}} & {{b}_{2}}+q{{c}_{2}} & {{c}_{2}}+r{{a}_{2}}  \\    {{a}_{3}}+p{{b}_{3}} & {{b}_{3}}+q{{c}_{3}} & {{c}_{3}}+r{{a}_{3}}  \\ \end{matrix} \right|\]. Then
  A)
  \[\Delta '=\Delta \] done clear
  B)
  \[\Delta '=\Delta \,\,(1-pqr)\] done clear
  C)
  \[\Delta '=\Delta \,\,(1+p+q+r)\] done clear
  D)
  \[\Delta '=\Delta \,\,(1+pqr)\] done clear
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• question_answer29)
  Matrix \[{{M}_{r}}\] is defined as \[{{M}_{r}}=\left( \begin{matrix}    r & r-1  \\    r-1 & r  \\ \end{matrix} \right),\]\[r\in N\].  The value of  \[det({{M}_{1}})+\det \,({{M}_{2}})+\det \,({{M}_{3}})+....+\det \,({{M}_{2014}})\] is
  A)
  \[2013\] done clear
  B)
  \[2014\] done clear
  C)
  \[{{(2013)}^{2}}\] done clear
  D)
  \[{{(2014)}^{2}}\] done clear
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• question_answer30)
  If \[a\ne b\ne c\] are all positive, then the value of the determinant \[\left| \begin{matrix}    a & b & c  \\    b & c & a  \\    c & a & b  \\ \end{matrix} \right|\] is
  A)
  Non-negative done clear
  B)
  Non-positive done clear
  C)
  Negative done clear
  D)
  Positive done clear
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• question_answer31)
  If D is determinant of order 3 and D' is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct?
  A)
  \[D'={{D}^{2}}\] done clear
  B)
  \[D'={{D}^{3}}\] done clear
  C)
  \[D'=2{{D}^{2}}\] done clear
  D)
  \[D'=3{{D}^{3}}\] done clear
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• question_answer32)
  If \[A=\left[ \begin{matrix}    1 & 2  \\    0 & 3  \\ \end{matrix} \right]\] is a \[2\times 2\] matrix and \[f(x)={{x}^{2}}-x+2\] is a polynomial, then what is \[f(A)\]?
  A)
  \[\left[ \begin{matrix}    1 & 7  \\    1 & 7  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 8  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 6  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    2 & 6  \\    0 & 7  \\ \end{matrix} \right]\] done clear
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• question_answer33)
  If \[\omega \] is the cube root of unity, then what is one root of the equation \[\left| \begin{matrix}    {{x}^{2}} & -2x & -2{{\omega }^{2}}  \\    2 & \omega  & -\omega   \\    0 & \omega  & 1  \\ \end{matrix} \right|=0?\]
  A)
  \[1\] done clear
  B)
  \[-2\] done clear
  C)
  \[2\] done clear
  D)
  \[\omega \] done clear
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• question_answer34)
  The value of the determinant \[\left| \begin{matrix}    {{\cos }^{2}}54{}^\circ  & {{\cos }^{2}}36{}^\circ  & \cot 135{}^\circ   \\    {{\sin }^{2}}53{}^\circ  & \cot 135{}^\circ  & {{\sin }^{2}}37{}^\circ   \\    \cot 135{}^\circ  & co{{s}^{2}}25{}^\circ  & {{\cos }^{2}}65{}^\circ   \\ \end{matrix} \right|\] is equal to
  A)
  \[-2\] done clear
  B)
  \[-1\]    done clear
  C)
  \[0\] done clear
  D)
  \[1\] done clear
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• question_answer35)
  If \[\left| \begin{matrix}    {{x}^{n}} & {{x}^{n+2}} & {{x}^{2n}}  \\    1 & {{x}^{a}} & a  \\    {{x}^{n+5}} & {{x}^{a+6}} & {{x}^{2n+5}}  \\ \end{matrix} \right|=0\,\forall \,x\,\in R,\] where \[n\in N\] then value of 'a' is
  A)
  \[n\] done clear
  B)
  \[n-1\] done clear
  C)
  \[n+1\] done clear
  D)
  None of these done clear
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• question_answer36)
  If the system of equations \[\lambda {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+\lambda {{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+{{x}_{2}}+\lambda {{x}_{3}}=1\] is consistent, then \[\lambda \] can be
  A)
  \[5\] done clear
  B)
  \[-2/3\] done clear
  C)
  \[-3\] done clear
  D)
  None of these done clear
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• question_answer37)
  The maximum and minimum value of \[(3\times 3)\]determinant whose elements belongs to \[\{0,1\}\] is
  A)
  \[1,-1\] done clear
  B)
  \[2,-2\] done clear
  C)
  \[4,-4\] done clear
  D)
  None of these done clear
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• question_answer38)
  If \[f(x),\,\,g(x)\] and \[h(x)\] are three polynomials of degree 2 and \[\Delta (x)=\left| \begin{matrix}    f(x) & g(x) & h(x)  \\    f'(x) & g'(x) & h'(x)  \\    f''(x) & g''(x) & h''(x)  \\ \end{matrix} \right|,\]then \[\Delta (x)\] is a polynomial of degree
  A)
  2 done clear
  B)
  3 done clear
  C)
  At most 2 done clear
  D)
  At most 3 done clear
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• question_answer39)
  If \[\alpha .\beta .\gamma \in R,\]then the determinant\[\Delta =\left| \begin{matrix}    {{({{e}^{i\alpha }}+{{e}^{-i\alpha }})}^{2}} & {{({{e}^{i\alpha }}-{{e}^{-i\alpha }})}^{2}} & 4  \\    {{({{e}^{i\beta }}+{{e}^{-i\beta }})}^{2}} & {{({{e}^{i\beta }}-{{e}^{-i\beta }})}^{2}} & 4  \\    {{({{e}^{i\gamma }}+{{e}^{-i\gamma }})}^{2}} & {{({{e}^{i\gamma }}-{{e}^{-i\gamma }})}^{2}} & 4  \\ \end{matrix} \right|\] is
  A)
  Independent of \[\alpha ,\beta \] and \[\gamma \] done clear
  B)
  Dependent on \[\alpha ,\beta \] and \[\gamma \] done clear
  C)
  Independent of \[\alpha ,\beta \] only done clear
  D)
  Independent of \[\alpha ,\gamma \] only done clear
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• question_answer40)
  If A is a square matrix of order 3 with \[\left| A \right|\ne 0,\] then which one of the following is correct?
  A)
  \[\left| adj\,A \right|=\left| A \right|\] done clear
  B)
  \[\left| adj\,A \right|={{\left| A \right|}^{2}}\] done clear
  C)
  \[\left| adj\,A \right|={{\left| A \right|}^{3}}\] done clear
  D)
  \[{{\left| adj\,A \right|}^{2}}=\left| A \right|\] done clear
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• question_answer41)
  If A is a square matrix such that \[{{A}^{2}}=I\] where I is the identity matrix, then what is \[{{A}^{-1}}\] equal to?
  A)
  \[A+1\] done clear
  B)
  Null matrix done clear
  C)
  A done clear
  D)
  Transpose of A done clear
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• question_answer42)
  Let \[\Delta =\] \[\left| \begin{matrix}    \sin x & \sin (x+h) & \sin (x+2h)  \\    \sin (x+2h) & \sin x & \sin (x+h)  \\    \sin (x+h) & \sin (x+2h) & \sin x  \\ \end{matrix} \right|\]Then, \[\underset{h\to 0}{\mathop{\lim }}\,\,\,\left( \frac{\Delta }{{{h}^{2}}} \right)\] is
  A)
  \[9si{{n}^{2}}x\cos x\] done clear
  B)
  \[3{{\cos }^{2}}x\] done clear
  C)
  \[\sin x{{\cos }^{2}}x\] done clear
  D)
  None of these done clear
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• question_answer43)
  The value of determinant \[\left| \begin{matrix}    {{\sin }^{2}}13{}^\circ  & {{\sin }^{2}}77{}^\circ  & \tan 135{}^\circ   \\    {{\sin }^{2}}77{}^\circ  & \tan 135{}^\circ  & {{\sin }^{2}}13{}^\circ   \\    \tan 135{}^\circ  & {{\sin }^{2}}13{}^\circ  & {{\sin }^{2}}77{}^\circ   \\ \end{matrix} \right|\]is
  A)
  \[-1\] done clear
  B)
  \[0\] done clear
  C)
  \[1\] done clear
  D)
  \[2\] done clear
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• question_answer44)
  If \[y=\left| \begin{matrix}    \sin x & \cos x & \sin x  \\    \cos x & -\operatorname{sinx} & \cos x  \\    x & 1 & 1  \\ \end{matrix} \right|,\]then \[\frac{dy}{dx}\] is
  A)
  \[1\] done clear
  B)
  \[2\] done clear
  C)
  \[3\] done clear
  D)
  0 done clear
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• question_answer45)
  If adj \[B=A,\] \[\left| P \right|=\left| Q \right|=1,\] then adj \[({{Q}^{-1}}B{{P}^{-1}})\]is
  A)
  \[PQ\] done clear
  B)
  \[QAP\] done clear
  C)
  \[PAQ\] done clear
  D)
  \[P{{A}^{-1}}Q\] done clear
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• question_answer46)
  If \[f(x)=\left| \begin{matrix}    {{x}^{n-1}} & \cos x & \frac{1}{x+3}  \\    0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}}  \\    \alpha  & {{\alpha }^{3}} & {{\alpha }^{5}}  \\ \end{matrix} \right|\] then \[\frac{{{d}^{n}}}{d{{x}^{n}}}\,\,{{[f(x)]}_{x=0}}=\]
  A)
  \[1\] done clear
  B)
  \[-1\] done clear
  C)
  \[0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  What is the value of the determinant \[\left| \begin{matrix}    1 & bc & a(b+c)  \\    1 & ca & b(c+a)  \\    1 & ab & c(a+b)  \\ \end{matrix} \right|?\]
  A)
  \[0\] done clear
  B)
  \[abc\] done clear
  C)
  \[ab+bc+ca\] done clear
  D)
  \[abc\,(a+b+c)\] done clear
  View Solution play_arrow

• question_answer48)
  Let \[A={{[{{a}_{ij}}]}_{m\times m}}\] be a matrix and \[C={{[{{c}_{ij}}]}_{m\times m}}\] be another matrix where \[{{c}_{ij}}\] is the cofactor of \[{{a}_{ij}}\]Then, what is the value of \[|AC|\]?
  A)
  \[|A{{|}^{m-1}}\] done clear
  B)
  \[|A{{|}^{m}}\] done clear
  C)
  \[|A{{|}^{m+1}}\] done clear
  D)
  Zero done clear
  View Solution play_arrow

• question_answer49)
  If \[a\ne p,\] \[b\ne q,\] \[c\ne r\]and \[\left| \begin{matrix}    p & b & c  \\    a & q & c  \\    a & b & r  \\ \end{matrix} \right|=0\] then the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is equal to
  A)
  \[-1\] done clear
  B)
  \[1\]    done clear
  C)
  \[-2\] done clear
  D)
  \[2\] done clear
  View Solution play_arrow

• question_answer50)
  If \[A=\left[ \begin{matrix}    a & 0 & 0  \\    0 & a & 0  \\    0 & 0 & a  \\ \end{matrix} \right],\] then the value of \[|adj\,\,A|\] is
  A)
  \[{{a}^{27}}\] done clear
  B)
  \[{{a}^{9}}\] done clear
  C)
  \[{{a}^{6}}\] done clear
  D)
  \[{{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer51)
  Let \[{{S}_{k}}={{\alpha }^{k}}+{{\beta }^{k}}+{{\gamma }^{k}},\] then \[\Delta =\left| \begin{matrix}    {{S}_{0}} & {{S}_{1}} & {{S}_{2}}  \\    {{S}_{1}} & {{S}_{2}} & {{S}_{3}}  \\    {{S}_{2}} & {{S}_{3}} & {{S}_{4}}  \\ \end{matrix} \right|\] is equal to
  A)
  \[{{S}_{6}}\] done clear
  B)
  \[{{S}_{5}}-{{S}_{3}}\] done clear
  C)
  \[{{S}_{6}}-{{S}_{4}}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer52)
  If \[{{\Delta }_{r}}=\left| \begin{matrix}    r-1 & n & 6  \\    {{(r-1)}^{2}} & 2{{n}^{2}} & 4n-2  \\    {{(r-1)}^{2}} & 3{{n}^{3}} & 3{{n}^{2}}-3n  \\ \end{matrix} \right|,\]then \[\sum\limits_{r=1}^{n}{{{\Delta }_{r}}}\] is.
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[3\]\[-1\] done clear
  D)
  done clear
  View Solution play_arrow

• question_answer53)
  Consider the following statements:

  1. If det \[A=0,\]then det \[(adj\,A)=0\]

  2. If A is non- singular, then \[\det \,({{A}^{-1}})={{(\det \,A)}^{-1}}\]

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2      done clear
  D)
  Neither 1 nor 2 done clear
  View Solution play_arrow

• question_answer54)
  If \[|{{A}_{n\times n}}|=3\] and \[|adj\,\,A|=243,\] what is the value of n?
  A)
  \[4\] done clear
  B)
  \[5\] done clear
  C)
  \[6\] done clear
  D)
  \[7\] done clear
  View Solution play_arrow

• question_answer55)
  Let A be an \[n\times n\] matrix. If \[\det \,(\lambda A)={{\lambda }^{s}}\det \,(A),\] what is the value of s?
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[-1\] done clear
  D)
  \[n\] done clear
  View Solution play_arrow

• question_answer56)
  If \[g(x)=\left| \begin{matrix}    {{a}^{-x}} & {{e}^{x{{\log }_{e}}a}} & {{x}^{2}}  \\    {{a}^{-3x}} & {{e}^{3x{{\log }_{e}}a}} & {{x}^{4}}  \\    {{a}^{-5x}} & {{e}^{5x{{\log }_{e}}a}} & 1  \\ \end{matrix} \right|,\] then
  A)
  \[g(x)+g(-x)=0\] done clear
  B)
  \[g(x)-g(-x)=0\] done clear
  C)
  \[g(x)\times g(-x)=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  If \[a,b,c,d>0,x\text{ }\in \text{R}\] and \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{x}^{2}}-2(ab+bc+cd)x+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\le 0.\]Then, \[\left| \begin{matrix}    33 & 14 & \log a  \\    65 & 27 & \log b  \\    97 & 40 & \log c  \\ \end{matrix} \right|\] is equal to
  A)
  \[1\] done clear
  B)
  \[-1\]   done clear
  C)
  \[2\] done clear
  D)
  \[0\] done clear
  View Solution play_arrow

• question_answer58)
  If \[A\left[ \begin{matrix}    1 & 2  \\    3 & 5  \\ \end{matrix} \right],\] then the value of the determinant \[|{{A}^{2009}}-5{{A}^{2008}}|\] is
  A)
  \[-6\] done clear
  B)
  \[-5\] done clear
  C)
  \[-4\] done clear
  D)
  \[4\] done clear
  View Solution play_arrow

• question_answer59)
  The value of \[\left| \begin{matrix}    ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}}  \\    ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}}  \\    ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}}  \\ \end{matrix} \right|=0,\] when m is equal to
  A)
  \[6\] done clear
  B)
  \[5\] done clear
  C)
  \[4\] done clear
  D)
  \[1\] done clear
  View Solution play_arrow

• question_answer60)
  If \[a>0,b>0,c>0\] are respectively the pth, qth,rth terms of GP, then the value of the determinant \[\left| \begin{matrix}    \log a & p & 1  \\    \log b & q & 1  \\    \log c & r & 1  \\ \end{matrix} \right|\] is
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer61)
  If x, y, z are complex numbers, and\[\Delta =\left| \begin{matrix}    0 & -y & -z  \\    {\bar{y}} & 0 & -x  \\    {\bar{z}} & {\bar{x}} & 0  \\ \end{matrix} \right|\] then \[\Delta \] is
  A)
  Purely real done clear
  B)
  Purely imaginary done clear
  C)
  Complex done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer62)
  Let A and B be two matrices of order\[n\times n\]. Let A be non-singular and B be singular. Consider the following:

  1. AB is singular

  2. AB is non-singular

  3. \[{{A}^{-1}}B\] is singular

  4.\[{{A}^{-1}}B\] is non singular

  Which of the above is/ are correct?

  A)
  1 and 3 done clear
  B)
  2 and 4 only done clear
  C)
  1 only done clear
  D)
  3 only done clear
  View Solution play_arrow

• question_answer63)
  Suppose the system of equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}}\] \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}\] \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}\] has a unique solution \[({{x}_{0}},{{y}_{0}},{{z}_{0}})\]. If \[{{x}_{0}}=0,\] then which one of the following is correct?
  A)
  \[\left| \begin{matrix}    {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|=0\] done clear
  B)
  \[\left| \begin{matrix}    {{d}_{1}} & {{b}_{1}} & {{c}_{1}}  \\    {{d}_{2}} & {{b}_{2}} & {{c}_{2}}  \\    {{d}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|=0\] done clear
  C)
  \[\left| \begin{matrix}    {{d}_{1}} & {{a}_{1}} & {{c}_{1}}  \\    {{d}_{2}} & {{a}_{2}} & {{c}_{2}}  \\    {{d}_{3}} & {{a}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|=0\] done clear
  D)
  \[\left| \begin{matrix}    {{d}_{1}} & {{a}_{1}} & {{b}_{1}}  \\    {{d}_{2}} & {{a}_{2}} & {{b}_{2}}  \\    {{d}_{3}} & {{a}_{3}} & {{b}_{3}}  \\ \end{matrix} \right|=0\] done clear
  View Solution play_arrow

• question_answer64)
  If \[\left| A \right|=8,\] where.4 is square matrix of order 3, then what is \[\left| adj\,\,A \right|\] equal to?
  A)
  \[16\] done clear
  B)
  \[24\] done clear
  C)
  \[64\] done clear
  D)
  \[512\] done clear
  View Solution play_arrow

• question_answer65)
  If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},............\]are positive numbers in G.P. then the value of \[\left| \begin{matrix}    \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}}  \\    \log {{a}_{n+1}} & \log {{a}_{n+2}} & {{\operatorname{loga}}_{n+3}}  \\    \log {{a}_{n+2}} & \log {{a}_{n+3}} & \log {{a}_{n+4}}  \\ \end{matrix} \right|\]
  A)
  \[1\] done clear
  B)
  \[4\] done clear
  C)
  \[3\] done clear
  D)
  \[0\] done clear
  View Solution play_arrow

• question_answer66)
  If \[\text{l}_{r}^{2}+m_{r}^{2}+n_{r}^{2}=\text{1};\] \[r=1,2,3\] and \[{{\text{l}}_{r}}{{\text{l}}_{s}}+{{m}_{r}}{{m}_{s}}+{{n}_{r}}{{n}_{s}}=0;\]\[r\ne s,\]\[r=1,2,3;\] \[s=1,2,3,\]then the value of \[\left| \begin{matrix}    {{\text{l}}_{1}} & {{m}_{1}} & {{n}_{1}}  \\    {{\text{l}}_{2}} & {{m}_{2}} & {{n}_{2}}  \\    {{\text{l}}_{3}} & {{m}_{3}} & {{n}_{3}}  \\ \end{matrix} \right|\] is
  A)
  \[0\] done clear
  B)
  \[\pm 1\] done clear
  C)
  \[2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer67)
  The equations \[2x+3y+4=0;\] \[3x+4y+6=0\] and \[4x+5y+8=0\]are
  A)
  Consistent with unique solution done clear
  B)
  Inconsistent done clear
  C)
  Consistent with infinitely many solutions done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer68)
  If \[\omega \] is a complex cube root of unity, then value of\[\Delta =\left| \begin{matrix}    {{a}_{1}}+{{b}_{1}}\omega  & {{a}_{1}}{{\omega }^{2}}+{{b}_{1}} & {{c}_{1}}+{{b}_{1}}\bar{\omega }  \\    {{a}_{2}}+{{b}_{2}}\omega  & {{a}_{2}}{{\omega }^{2}}+{{b}_{2}} & {{c}_{2}}+{{b}_{2}}\bar{\omega }  \\    {{a}_{3}}+{{b}_{3}}\omega  & {{a}_{3}}{{\omega }^{2}}+{{b}_{3}} & {{c}_{3}}+{{b}_{3}}\bar{\omega }  \\ \end{matrix} \right|\] is
  A)
  \[0\] done clear
  B)
  \[-1\] done clear
  C)
  \[2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer69)
  If \[{{a}_{r}}={{(\cos 2r\pi +i\sin 2r\pi )}^{\frac{1}{9}}},\]then the value of \[\left| \begin{matrix}    {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\    {{a}_{4}} & {{a}_{5}} & {{a}_{6}}  \\    {{a}_{7}} & {{a}_{8}} & {{a}_{9}}  \\ \end{matrix} \right|\] is
  A)
  \[1\] done clear
  B)
  \[-1\] done clear
  C)
  \[0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer70)
  If \[\left| \begin{matrix}    a & \cot A/2 & \lambda   \\    b & \cot B/2 & \mu   \\    c & \operatorname{cotC}/2 & \gamma   \\ \end{matrix} \right|=0,\] where a, b, c, A, B, and C are elements of a triangle ABC with usual meaning. Then, the value of a \[(\mu -\gamma )+b(\gamma -\lambda )+c(\lambda -\mu )=0\] is
  A)
  \[0\] done clear
  B)
  \[abc\] done clear
  C)
  \[ab+bc+ca\] done clear
  D)
  \[2abc\] done clear
  View Solution play_arrow

• question_answer71)
  The determinant \[\left| \begin{matrix}    a+b+c & a+b & a  \\    4a+3b+2c & 3a+2b & 2a  \\    10a+6b+3c & 6a+3b & 3a  \\ \end{matrix} \right|\] is independent of which one of the following?
  A)
  a and b done clear
  B)
  b and c done clear
  C)
  a and c done clear
  D)
  All of these done clear
  View Solution play_arrow

• question_answer72)
  If a, b, c are in GP, then what is the value of \[\left| \begin{matrix}    a & b & a+b  \\    b & c & b+c  \\    a+b & b+c & 0  \\ \end{matrix} \right|?\]
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer73)
  If \[{{A}_{1}}{{B}_{1}}{{C}_{1}},\] \[{{A}_{2}}{{B}_{2}}{{C}_{2}}\] and \[{{A}_{3}}{{B}_{3}}{{C}_{3}}\]are three digit numbers, each of which is divisible by k, then \[\Delta =\left| \begin{matrix}    {{A}_{1}} & {{B}_{1}} & {{C}_{1}}  \\    {{A}_{2}} & {{B}_{2}} & {{C}_{2}}  \\    {{A}_{3}} & {{B}_{3}} & {{C}_{3}}  \\ \end{matrix} \right|\] is
  A)
  Divisible by k done clear
  B)
  Divisible by \[{{k}^{2}}\] done clear
  C)
  Divisible by \[{{k}^{3}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer74)
  If \[f(x)=\left| \begin{matrix}    {{2}^{-x}} & {{e}^{x{{\log }_{e}}2}} & {{x}^{2}}  \\    {{2}^{-3x}} & {{e}^{3x{{\log }_{e}}2}} & {{x}^{4}}  \\    {{2}^{-5x}} & {{e}^{5x{{\log }_{e}}2}} & 1  \\ \end{matrix} \right|,\] then
  A)
  \[f(x)+f(-x)=0\] done clear
  B)
  \[f(x)-f(-x)=0\] done clear
  C)
  \[f(x)+f(-x)=2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer75)
  If in a triangle ABC, \[\left| \begin{matrix}    1 & \sin A & {{\sin }^{2}}A  \\    1 & \sin B & {{\sin }^{2}}B  \\    1 & \sin C & {{\sin }^{2}}C  \\ \end{matrix} \right|=0\] then the triangle is
  A)
  Equilateral or isosceles done clear
  B)
  Equilateral or right-angled done clear
  C)
  Right angled or isosceles done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer76)
  If A, B, and C are the angles of a triangle and \[\left| \begin{matrix}    1 & 1 & 1  \\    1+\sin A & 1+\sin B & 1+\sin C  \\    \sin A+{{\sin }^{2}}A & \sin B+{{\sin }^{2}}B & \sin C+{{\sin }^{2}}C  \\ \end{matrix} \right|=0,\]then the triangle must be
  A)
  Isosceles done clear
  B)
  Equilateral done clear
  C)
  Right-angled done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer77)
  If the matrix B is the adjoint of the square matrix A and \[\alpha \] is the value of the determinant of A, then what is AB equal to?
  A)
  \[\alpha \] done clear
  B)
  \[\left( \frac{1}{\alpha } \right)I\] done clear
  C)
  \[I\] done clear
  D)
  \[\alpha I\] done clear
  View Solution play_arrow

• question_answer78)
  If \[A=\left[ \begin{matrix}    3 & 2  \\    1 & 4  \\ \end{matrix} \right],\] then what is A (adj A) equal to?
  A)
  \[\left[ \begin{matrix}    0 & 10  \\    10 & 0  \\ \end{matrix} \right]\] done clear
  B)
  \[\left[ \begin{matrix}    10 & 0  \\    0 & 10  \\ \end{matrix} \right]\] done clear
  C)
  \[\left[ \begin{matrix}    1 & 10  \\    10 & 1  \\ \end{matrix} \right]\] done clear
  D)
  \[\left[ \begin{matrix}    10 & 1  \\    1 & 10  \\ \end{matrix} \right]\] done clear
  View Solution play_arrow

• question_answer79)
  \[A=\left| \begin{matrix}    2a & 3r & x  \\    4b & 6s & 2y  \\    -2c & -3t & -z  \\ \end{matrix} \right|=\lambda \left| \begin{matrix}    a & r & x  \\    b & s & y  \\    c & t & z  \\ \end{matrix} \right|,\] then what is the value of \[\lambda \]?
  A)
  \[12\] done clear
  B)
  \[-12\] done clear
  C)
  \[7\] done clear
  D)
  \[-7\] done clear
  View Solution play_arrow

• question_answer80)
  Given \[a=x/(y-z),\]\[b=y/(z-x)\]and \[c=z/(x-y),\] where x, y and z are not all zero, Then the value of \[ab+bc+ca\]
  A)
  \[0\] done clear
  B)
  \[1\] done clear
  C)
  \[-1\] done clear
  D)
  None of these done clear
  View Solution play_arrow