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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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