79452
The values of a for which the system of equation \(x+y+z=0, x+a y+a z=0, x-a y+z\) \(\mathbf{0}\), possess non-zero solutions, are given by :
1 1,2
2 \(1,-1\)
3 1,0
4 none of these
Explanation:
(B) : According to given summation, The system of equations possess non-zero solutions \(\therefore\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \mathrm{a} & \mathrm{a} \\ 1 & -\mathrm{a} & 1 \end{array}\right|=0\) Operating \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) \(\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \mathrm{a}-1 & \mathrm{a}-1 \\ 0 & -\mathrm{a}-1 & 0 \end{array}\right|=0\) \(\Rightarrow 1\left[0+\left(\mathrm{a}^{2}-1\right)\right]=0\) \(\mathrm{a}^{2}=1\) \(a= \pm 1\)
BCECE-2006
Matrix and Determinant
79453
If \(n\) is an integer and if \(\left|\begin{array}{lll} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right),\) then \(n\) equals
79456
If \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\), then the number of distinct real roots of this equation in the interval \(-\pi / 2
1 2
2 0
3 1
4 3
Explanation:
Exp:(a) \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}\right|=0\) \(\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\) \(\left|\begin{array}{ccc}2 \cos x+\sin x & \cos x & \cos x \\ 2 \cos x+\sin x & \sin x & \cos x \\ 2 \cos x+\sin x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{lll}1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x\end{array}\right|=0\) \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 0 & \cos x-\sin x & 0 \\ 0 & -(\cos x-\sin x) & (\cos x-\sin x) \\ 1 & \cos x & \sin x \end{array}\right|=0\) \((2 \cos x+\sin x)(\cos x-\sin x)^2=0\) \((2 \cos x+\sin x)=0 \text { or } \cos x-\sin x=0\) \(\begin{array}{ll}\text { Case-I } & 2 \cos x+\sin x=0 \\ & \tan x=-2 \\ \text { Case-II } & \cos x-\sin x=0 \\ & \tan x=1\end{array}\)
79452
The values of a for which the system of equation \(x+y+z=0, x+a y+a z=0, x-a y+z\) \(\mathbf{0}\), possess non-zero solutions, are given by :
1 1,2
2 \(1,-1\)
3 1,0
4 none of these
Explanation:
(B) : According to given summation, The system of equations possess non-zero solutions \(\therefore\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \mathrm{a} & \mathrm{a} \\ 1 & -\mathrm{a} & 1 \end{array}\right|=0\) Operating \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) \(\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \mathrm{a}-1 & \mathrm{a}-1 \\ 0 & -\mathrm{a}-1 & 0 \end{array}\right|=0\) \(\Rightarrow 1\left[0+\left(\mathrm{a}^{2}-1\right)\right]=0\) \(\mathrm{a}^{2}=1\) \(a= \pm 1\)
BCECE-2006
Matrix and Determinant
79453
If \(n\) is an integer and if \(\left|\begin{array}{lll} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right),\) then \(n\) equals
79456
If \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\), then the number of distinct real roots of this equation in the interval \(-\pi / 2
1 2
2 0
3 1
4 3
Explanation:
Exp:(a) \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}\right|=0\) \(\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\) \(\left|\begin{array}{ccc}2 \cos x+\sin x & \cos x & \cos x \\ 2 \cos x+\sin x & \sin x & \cos x \\ 2 \cos x+\sin x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{lll}1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x\end{array}\right|=0\) \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 0 & \cos x-\sin x & 0 \\ 0 & -(\cos x-\sin x) & (\cos x-\sin x) \\ 1 & \cos x & \sin x \end{array}\right|=0\) \((2 \cos x+\sin x)(\cos x-\sin x)^2=0\) \((2 \cos x+\sin x)=0 \text { or } \cos x-\sin x=0\) \(\begin{array}{ll}\text { Case-I } & 2 \cos x+\sin x=0 \\ & \tan x=-2 \\ \text { Case-II } & \cos x-\sin x=0 \\ & \tan x=1\end{array}\)
79452
The values of a for which the system of equation \(x+y+z=0, x+a y+a z=0, x-a y+z\) \(\mathbf{0}\), possess non-zero solutions, are given by :
1 1,2
2 \(1,-1\)
3 1,0
4 none of these
Explanation:
(B) : According to given summation, The system of equations possess non-zero solutions \(\therefore\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \mathrm{a} & \mathrm{a} \\ 1 & -\mathrm{a} & 1 \end{array}\right|=0\) Operating \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) \(\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \mathrm{a}-1 & \mathrm{a}-1 \\ 0 & -\mathrm{a}-1 & 0 \end{array}\right|=0\) \(\Rightarrow 1\left[0+\left(\mathrm{a}^{2}-1\right)\right]=0\) \(\mathrm{a}^{2}=1\) \(a= \pm 1\)
BCECE-2006
Matrix and Determinant
79453
If \(n\) is an integer and if \(\left|\begin{array}{lll} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right),\) then \(n\) equals
79456
If \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\), then the number of distinct real roots of this equation in the interval \(-\pi / 2
1 2
2 0
3 1
4 3
Explanation:
Exp:(a) \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}\right|=0\) \(\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\) \(\left|\begin{array}{ccc}2 \cos x+\sin x & \cos x & \cos x \\ 2 \cos x+\sin x & \sin x & \cos x \\ 2 \cos x+\sin x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{lll}1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x\end{array}\right|=0\) \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 0 & \cos x-\sin x & 0 \\ 0 & -(\cos x-\sin x) & (\cos x-\sin x) \\ 1 & \cos x & \sin x \end{array}\right|=0\) \((2 \cos x+\sin x)(\cos x-\sin x)^2=0\) \((2 \cos x+\sin x)=0 \text { or } \cos x-\sin x=0\) \(\begin{array}{ll}\text { Case-I } & 2 \cos x+\sin x=0 \\ & \tan x=-2 \\ \text { Case-II } & \cos x-\sin x=0 \\ & \tan x=1\end{array}\)
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Matrix and Determinant
79452
The values of a for which the system of equation \(x+y+z=0, x+a y+a z=0, x-a y+z\) \(\mathbf{0}\), possess non-zero solutions, are given by :
1 1,2
2 \(1,-1\)
3 1,0
4 none of these
Explanation:
(B) : According to given summation, The system of equations possess non-zero solutions \(\therefore\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \mathrm{a} & \mathrm{a} \\ 1 & -\mathrm{a} & 1 \end{array}\right|=0\) Operating \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) \(\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \mathrm{a}-1 & \mathrm{a}-1 \\ 0 & -\mathrm{a}-1 & 0 \end{array}\right|=0\) \(\Rightarrow 1\left[0+\left(\mathrm{a}^{2}-1\right)\right]=0\) \(\mathrm{a}^{2}=1\) \(a= \pm 1\)
BCECE-2006
Matrix and Determinant
79453
If \(n\) is an integer and if \(\left|\begin{array}{lll} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right),\) then \(n\) equals
79456
If \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\), then the number of distinct real roots of this equation in the interval \(-\pi / 2
1 2
2 0
3 1
4 3
Explanation:
Exp:(a) \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}\right|=0\) \(\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\) \(\left|\begin{array}{ccc}2 \cos x+\sin x & \cos x & \cos x \\ 2 \cos x+\sin x & \sin x & \cos x \\ 2 \cos x+\sin x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{lll}1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x\end{array}\right|=0\) \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 0 & \cos x-\sin x & 0 \\ 0 & -(\cos x-\sin x) & (\cos x-\sin x) \\ 1 & \cos x & \sin x \end{array}\right|=0\) \((2 \cos x+\sin x)(\cos x-\sin x)^2=0\) \((2 \cos x+\sin x)=0 \text { or } \cos x-\sin x=0\) \(\begin{array}{ll}\text { Case-I } & 2 \cos x+\sin x=0 \\ & \tan x=-2 \\ \text { Case-II } & \cos x-\sin x=0 \\ & \tan x=1\end{array}\)
79452
The values of a for which the system of equation \(x+y+z=0, x+a y+a z=0, x-a y+z\) \(\mathbf{0}\), possess non-zero solutions, are given by :
1 1,2
2 \(1,-1\)
3 1,0
4 none of these
Explanation:
(B) : According to given summation, The system of equations possess non-zero solutions \(\therefore\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & \mathrm{a} & \mathrm{a} \\ 1 & -\mathrm{a} & 1 \end{array}\right|=0\) Operating \(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}\) and \(\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}\) \(\Rightarrow\left|\begin{array}{ccc} 1 & 1 & 1 \\ 0 & \mathrm{a}-1 & \mathrm{a}-1 \\ 0 & -\mathrm{a}-1 & 0 \end{array}\right|=0\) \(\Rightarrow 1\left[0+\left(\mathrm{a}^{2}-1\right)\right]=0\) \(\mathrm{a}^{2}=1\) \(a= \pm 1\)
BCECE-2006
Matrix and Determinant
79453
If \(n\) is an integer and if \(\left|\begin{array}{lll} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array}\right|=(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right),\) then \(n\) equals
79456
If \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\), then the number of distinct real roots of this equation in the interval \(-\pi / 2
1 2
2 0
3 1
4 3
Explanation:
Exp:(a) \(\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x \end{array}\right|=0\) \(\mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\) \(\left|\begin{array}{ccc}2 \cos x+\sin x & \cos x & \cos x \\ 2 \cos x+\sin x & \sin x & \cos x \\ 2 \cos x+\sin x & \cos x & \sin x\end{array}\right|=0\) \((2 \cos x+\sin x)\left|\begin{array}{lll}1 & \cos x & \cos x \\ 1 & \sin x & \cos x \\ 1 & \cos x & \sin x\end{array}\right|=0\) \(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2} \text { and } \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}\) \((2 \cos x+\sin x)\left|\begin{array}{ccc} 0 & \cos x-\sin x & 0 \\ 0 & -(\cos x-\sin x) & (\cos x-\sin x) \\ 1 & \cos x & \sin x \end{array}\right|=0\) \((2 \cos x+\sin x)(\cos x-\sin x)^2=0\) \((2 \cos x+\sin x)=0 \text { or } \cos x-\sin x=0\) \(\begin{array}{ll}\text { Case-I } & 2 \cos x+\sin x=0 \\ & \tan x=-2 \\ \text { Case-II } & \cos x-\sin x=0 \\ & \tan x=1\end{array}\)