JEE Main & Advanced Mathematics Determinants & Matrices Question Bank

done Mock Test - Determinants Total Questions - 25

• question_answer1)
  If \[p+q+r=0=a+b+c\], then the value of the determinant is
  A)
  0         done clear
  B)
  \[pa+qb+rc\] done clear
  C)
  1                    done clear
  D)
  none of these done clear
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• question_answer2)
  The determinant is equal to
  A)
   \[\left| \begin{matrix}    bx+ay & cx+by  \\    b'x+a'y & c'x+b'y  \\ \end{matrix} \right|\] done clear
  B)
   \[\left| \begin{matrix}    ax+by & bx+cy  \\    a'x+b'y & b'x+c'y  \\ \end{matrix} \right|\] done clear
  C)
   \[\left| \begin{matrix}    bx+cy & ax+by  \\    b'x+c'y & a'x+b'y  \\ \end{matrix} \right|\] done clear
  D)
   \[\left| \begin{matrix}    ax+by & bx+cy  \\    a'x+b'y & b'x+c'y  \\ \end{matrix} \right|\] done clear
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• question_answer3)
  If  \[a=\cos \theta +i\sin \theta ,\,\]\[\,b=\cos 2\theta -i\sin 2\theta ,\,\]\[c=\cos 3\theta +i\sin 3\theta \] and if , then
  A)
  \[\theta =2k\pi ,\,k\in Z\] done clear
  B)
  \[\theta =(2k+1)\pi ,k\in Z\] done clear
  C)
   \[\theta =(4k+1)\pi ,k\in Z\] done clear
  D)
   none of these done clear
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• question_answer4)
  If , then z is
  A)
   purely real done clear
  B)
   purely imaginary done clear
  C)
   \[a+ib,\]where \[a\ne 0,\]\[b\ne 0,\] done clear
  D)
   \[a+ib,\]where b = 4 done clear
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• question_answer5)
  If \[f(x)=a+bx+c{{x}^{2}}\] and \[\alpha ,\beta ,\gamma \]are the roots of the equation\[{{x}^{3}}=1,\]then is equal to
  A)
   \[f(\alpha )+f(\beta )+f(\gamma )\] done clear
  B)
   \[f(\alpha )f(\beta )+f(\beta )\]\[f(\gamma )+f(\gamma )\]\[f(\alpha )\] 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_answer6)
  The value of the determinant\[\left| \begin{matrix}    kb & {{k}^{^{2}}}+{{a}^{2}} & 1  \\    kb & {{k}^{2}}+{{b}^{2}} & 1  \\    kc & {{k}^{2}}+{{c}^{2}} & 1  \\ \end{matrix} \right|\]is
  A)
   \[k(a+b)(b+c)(c+a)\] done clear
  B)
   \[k\,abc({{a}^{2}}+{{b}^{2}}+{{c}^{2}})\] done clear
  C)
   \[k(a-b)(b-c)(c-a)\] done clear
  D)
   \[k(a+b-c)(b+c-a)(c+a-b)\] done clear
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• question_answer7)
  If a, b, and c are nonzero real number then \[\Delta =\left| \begin{matrix}    {{b}^{2}}{{c}^{2}} & bc & b+c  \\    {{c}^{2}}{{a}^{2}} & ca & c+a  \\    {{a}^{2}}{{b}^{2}} & ab & a+b  \\ \end{matrix} \right|\]is equal to
  A)
   abc                  done clear
  B)
   \[{{a}^{2}}{{b}^{2}}{{c}^{2}}\] done clear
  C)
   bc+ca+ab        done clear
  D)
   none of these done clear
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• question_answer8)
  The value of the determinant\[\left| \begin{matrix}    1 & 1 & 1  \\    ^{m}{{C}_{1}} & ^{m+1}{{C}_{1}} & ^{m+2}{{C}_{1}}  \\    ^{m}{{C}_{2}} & ^{m+1}{{C}_{2}} & ^{m+2}{{C}_{2}}  \\ \end{matrix} \right|\]
  A)
   1         done clear
  B)
   -1 done clear
  C)
   0                     done clear
  D)
   none of these done clear
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• question_answer9)
  If \[x\ne 0\], \[y\ne 0\], \[z\ne 0\] and , then \[{{x}^{-1}}+{{y}^{-1}}+{{z}^{-1}}\] is equal to
  A)
   \[-\,1\]              done clear
  B)
   \[-\,2\] done clear
  C)
   \[-\,3\]              done clear
  D)
   none of these done clear
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• question_answer10)
  In triangle ABC, if \[\left| \begin{matrix}    1 & 1 & 1  \\    \cos \frac{A}{2} & \cot \frac{A}{2} & \cot \frac{C}{2}  \\    \tan \frac{B}{2}+\tan \frac{C}{2} & \tan \frac{C}{2}+\tan \frac{A}{2} & \tan \frac{A}{2}+\tan \frac{B}{2}  \\ \end{matrix} \right|=0\] then the triangle must be
  A)
   equilateral done clear
  B)
   isosceles done clear
  C)
   obtuse angled done clear
  D)
   none of these done clear
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• question_answer11)
  If , then y =\[f(x)\]represents
  A)
   a straight line parallel to x-axis done clear
  B)
   a straight line parallel to y-axis done clear
  C)
   parabola done clear
  D)
   a straight line with negative slope done clear
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• question_answer12)
  If , then
  A)
   \[f'(x)=0and\,f''(x)=0\]has one common root done clear
  B)
   \[f(x)=0\,\,and\,f'(x)=0\]has one common root done clear
  C)
   sum of roots of \[f(x)\]=0 is \[-\,3\,a\] done clear
  D)
   none of these done clear
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• question_answer13)
  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_answer14)
  Suppose and. Then
  A)
  D'=D done clear
  B)
  D'=D(1-pqr) done clear
  C)
  D'=D(1+p+q+r) done clear
  D)
  D'=D(1+pqr) done clear
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• question_answer15)
  If \[{{a}_{1}},{{a}_{2}}...{{a}_{n}}....\]form a G.P. and \[{{a}_{i}}\]>0, for all \[i\ge 1\], then\[\left| \begin{matrix}    \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}}  \\    \log {{a}_{n+3}} & \log {{a}_{n+4}} & \log {{a}_{n+5}}  \\    \log {{a}_{n+6}} & \log {{a}_{n+7}} & \log {{a}_{n+8}}  \\ \end{matrix} \right|\]is equal to
  A)
   0                     done clear
  B)
   1 done clear
  C)
   2         done clear
  D)
   3 done clear
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• question_answer16)
  For the equation ,
  A)
   There are exactly two distinct roots done clear
  B)
   There is one pair of equation real roots done clear
  C)
   There are three pairs of equal roots done clear
  D)
   Modulus of each root is 2 done clear
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• question_answer17)
  The value of the determinant \[\left| \begin{matrix}    ^{n}{{C}_{r-1}} & ^{n}{{C}_{r}} & (r+1) & ^{n+2}{{C}_{r+1}}  \\    ^{n}{{C}_{r}} & ^{n}{{C}_{r+1}} & (r+2) & ^{n+2}{{C}_{r+2}}  \\    ^{n}{{C}_{r+1}} & ^{n}{{C}_{r+2}} & (r+3) & ^{n+2}{{C}_{r+3}}  \\ \end{matrix} \right|\] is
  A)
  \[{{n}^{2}}+n-1\]       done clear
  B)
   0 done clear
  C)
  \[^{n+3}{{C}_{r+3}}\]          done clear
  D)
  \[^{n}{{C}_{r-1}}{{+}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r+1}}\] done clear
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• question_answer18)
  If \[{{l}^{2}}_{1}+{{m}_{1}}^{2}+{{n}_{1}}^{2}=1\], etc. and \[{{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}}=0\], etc. and \[\Delta =\left| \begin{matrix}    {{l}_{1}} & {{m}_{1}} & {{n}_{1}}  \\    {{l}_{2}} & {{m}_{2}} & {{n}_{2}}  \\    {{l}_{3}} & {{m}_{3}} & {{n}_{3}}  \\ \end{matrix} \right|\], then
  A)
   \[\left| \Delta  \right|\]=3            done clear
  B)
   \[\left| \Delta  \right|\]=2 done clear
  C)
   \[\left| \Delta  \right|\]=1            done clear
  D)
   \[\Delta \]=0 done clear
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• question_answer19)
  If the value of the determinant \[\left| \begin{matrix}    a & 1 & 1  \\    1 & b & 1  \\    1 & 1 & c  \\ \end{matrix} \right|\] is positive, then (a, b, c > 0)
  A)
   \[abc>1\]          done clear
  B)
   \[abc>-\,8\] done clear
  C)
   \[abc<-\,8\]       done clear
  D)
   \[abc>-\,2\] done clear
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• question_answer20)
  If \[{{\Delta }_{1}}=\left| \begin{matrix}    x & b & b  \\    a & x & b  \\    a & a & x  \\ \end{matrix} \right|\] and \[{{\Delta }_{2}}=\left| \begin{matrix}    x & b  \\    a & x  \\ \end{matrix} \right|\]are the given determinants, then
  A)
   \[{{\Delta }_{1}}=3{{({{\Delta }_{2}})}^{2}}\] done clear
  B)
   \[\frac{d}{dx}({{\Delta }_{1}})=3({{\Delta }_{2}})\] done clear
  C)
   \[\frac{d}{dx}({{\Delta }_{1}})=3{{({{\Delta }_{2}})}^{2}}\] done clear
  D)
   \[{{\Delta }_{1}}=3{{\Delta }_{2}}^{3/2}\] done clear
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• question_answer21)
  If x, y, z are different from zero and \[\Delta =\left| \begin{matrix}    a & b-y & c-z  \\    a-x & b & c-z  \\    a-x & b-y & c  \\ \end{matrix} \right|=0\] then the value of the expression \[\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\] is ____.
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• question_answer22)
  The number of positive integral solutions of the equation\[\left| \begin{matrix}    {{x}^{3}}+1 & {{x}^{2}}y & {{x}^{2}}z  \\    x{{y}^{2}} & {{y}^{3}}+1 & {{y}^{2}}z  \\    x{{z}^{2}} & y{{z}^{2}} & {{z}^{3}}+1  \\ \end{matrix} \right|=11\]is ______.
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• question_answer23)
  The maximum degree of x by which determinant\[\left| \begin{matrix}    x+y & x & x  \\    5x+4y & 4x & 2x  \\    10x+8y & 8x & 3x  \\ \end{matrix} \right|\]divisible is ______.
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• question_answer24)
  In a third order determinant, each element of 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 _______.  
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• question_answer25)
  If l, m, and n are the\[{{p}^{th}}\],\[{{q}^{th}}\], and \[{{r}^{th}}\]terms of G.P. and are all positive, then \[\left| \begin{matrix}    \log l & p & 1  \\    \log m & q & 1  \\    lonn & r & 1  \\ \end{matrix} \right|\] equals ______.
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