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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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