KC Sinha Mathematics Solution Class 12 Chapter 6 सारणिक (Determinants) Exercise 6.2 (Q4-Q6)
k का मान निकालें यदि त्रिभुज का क्षेत्रफल 4 वर्ग इकाई तथा शीर्ष निम्नलिखित है:
[Find the value of k if area of triangle is 4 sq units and vertices are :](i) (-2,0)(0,4)(0,k)
Sol :
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Diagram
4 वर्ग इकाई
$\frac{1}{2}\left|\begin{array}{cccc}-2 & 0 & 1 \\ 0 & 4 & 1 \\ 0 & k & 1\end{array}\right|=\pm 4$
$-2\left|\begin{array}{cc}4 & 1 \\ k & 1\end{array}\right|=\pm 8$
-2(4-k)=±8
-8+2k=±8
$\begin{array}{r|l}-8+2 k=8 &-8+2k=-8\\2 k=16 &k=\frac{0}{2} \\k=\frac{16}{2}&k=0\\k=8\end{array}$
(ii) (k,0)(0,2)(4,0)
Sol :
Question 5
जाँच करें कि निम्नलिखित बिन्दुएँ संरेख है या नही ं
[Examine whether following points are collinear or not :](i) (2,5),(-5,-2),(-1,2)
Sol :
Let A(2,5), B(-5,-2) and C(-1,2) are three points
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$=\frac{1}{2}\left|\begin{array}{ccc}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|$
$=\frac{1}{2}\left|\begin{array}{ccc}2 & 5 & 1 \\ -5 & -2 & 1 \\ -1 & 2 & 1\end{array}\right|$
R1→R1-R2 , R2→R2-R3
$=\frac{1}{2}\left|\begin{array}{rrr}7 & 7 & 0 \\ -4 & -4 & 0 \\ -1 & 2 & 1\end{array}\right|$
C3 के सापेक्ष विस्तार करने पर ,
$=\frac{1}{2}(1)\left|\begin{array}{cc}7 & 7 \\ -4 & -4\end{array}\right|$
$=\dfrac{1}{2}(-28+28)$
=0
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(ii) (-3,2),(-5,-4),(7,-6)
Sol :
(iii) (1,5),(2,4),(3,3)
Sol :
Sol :
Question 6
साबित करें कि बिन्दुएँ (a,b+c),(b,c+a) तथा (c,a+b) संरेख है।
[Prove that the points (a,b+c),(b,c+a) and (c,a+b) are collinear]Sol :
Let the points are (a,b+c),(b,c+a) and (c,a+b)
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$=\frac{1}{2}\left|\begin{array}{ccc}a & b+c & 1 \\ b & c+a & 1 \\ c & a+b & 1\end{array}\right|$
$C_{2} \rightarrow C_{2}+C_{1}$
$=\frac{1}{2}\left|\begin{array}{ccc}a & a+b+c & 1 \\ b & a+b+ c & 1 \\ c & a+b+c & 1\end{array}\right|$
$C_{2} \rightarrow C_{2}+C_{1}$
$=\frac{1}{2}\left|\begin{array}{ccc}a & a+b+c & 1 \\ b & a+b+c & 1 \\ c & a+b+c & 1\end{array}\right|$
$=\frac{1}{2} \times 0=0$
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