88288
The equation of the plane meets the axes in \(\mathrm{A}\), and \(C\) such that centroid of the \(\triangle \mathrm{ABC}\) is \(\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3},\right)\) is given by
1 \(x+y+z=1\)
2 \(x+y+z=2\)
3 \(\frac{x}{3}+\frac{y}{3}+\frac{z}{3}=3\)
4 \(x+y+z=\frac{1}{3}\)
Explanation:
(A): Given, The centroid of triangle \(\triangle \mathrm{ABC}\) is \(=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\) the planes meets the axes in \(\mathrm{ABC}\) Now, The centroid of the triangle \(\triangle \mathrm{ABC}\) is \(\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{b}}{3}, \frac{\mathrm{c}}{3}\right)\) \(\Rightarrow \quad \mathrm{a}=1, \mathrm{~b}=1 \& \mathrm{c}=1\) \(\therefore\) The equation of plane meet the co-ordinate axes is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) \(\frac{x}{1}+\frac{y}{1}+\frac{z}{1}=1\) \(x+y+z=1\)
BCECE-2014
Co-Ordinate system
88289
If \(A(-1,3,2), B(2,3,5)\) and \(C(3,5,-2)\) are vertices of a \(\triangle \mathrm{ABC}\), then angles of are
88290
If \(G\) and \(G^{\prime}\) are respectively centroid of \(\triangle A B C\) and \(\Delta A^{\prime} B^{\prime} C^{\prime}\), then \(A A^{\prime}+B B^{\prime}+C^{\prime}\) is equal to
1 \(2 \mathrm{GG}^{\prime}\)
2 \(3 \mathrm{GG}^{\prime}\)
3 \(\frac{2}{3} \mathrm{GG}^{\prime}\)
4 \(\frac{1}{3} \mathrm{GG}^{\prime}\)
Explanation:
(B) : Let a.b.c be the position vectors of A,B and C respectively. Then, the position vector of \(G\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) Let the position vectors of \(\mathrm{A}^{\prime} \mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) be \(\mathrm{a}^{\prime} \mathrm{b}^{\prime}\) Then, the position vector of \(G^{\prime}\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) \(\therefore A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}-a\right)+\left(b^{\prime}-b\right)+\left(c^{\prime}-c\right)\) \(\Rightarrow A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}+b^{\prime}+c^{\prime}\right)-(a+b+c)\) \(=\left(\frac{a^{\prime}+b^{\prime}+c^{\prime}}{3}-\frac{a+b+c}{3}\right)=3 G G^{\prime}\)
CG PET-2016
Co-Ordinate system
88291
The orthocenter of the triangle formed by the line \(x=2, y=3\) and \(3 x+2 y=6\) at the point
1 \((2,0)\)
2 \((2,3)\)
3 \((0,3)\)
4 None of these
Explanation:
(B) : Given equation of lines are \(x=2, y=3\) and \(3 x+2 y=6\) Since, \(\triangle \mathrm{ABC}\) is right angled at \(\mathrm{C}\). Now, orthocenter of \(\triangle \mathrm{ABC}\) will the vertex at which right angle is forming. \(\therefore\) Orthocenter of \(\triangle \mathrm{ABC}\) is \((2,3)\).
NEET Test Series from KOTA - 10 Papers In MS WORD
WhatsApp Here
Co-Ordinate system
88288
The equation of the plane meets the axes in \(\mathrm{A}\), and \(C\) such that centroid of the \(\triangle \mathrm{ABC}\) is \(\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3},\right)\) is given by
1 \(x+y+z=1\)
2 \(x+y+z=2\)
3 \(\frac{x}{3}+\frac{y}{3}+\frac{z}{3}=3\)
4 \(x+y+z=\frac{1}{3}\)
Explanation:
(A): Given, The centroid of triangle \(\triangle \mathrm{ABC}\) is \(=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\) the planes meets the axes in \(\mathrm{ABC}\) Now, The centroid of the triangle \(\triangle \mathrm{ABC}\) is \(\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{b}}{3}, \frac{\mathrm{c}}{3}\right)\) \(\Rightarrow \quad \mathrm{a}=1, \mathrm{~b}=1 \& \mathrm{c}=1\) \(\therefore\) The equation of plane meet the co-ordinate axes is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) \(\frac{x}{1}+\frac{y}{1}+\frac{z}{1}=1\) \(x+y+z=1\)
BCECE-2014
Co-Ordinate system
88289
If \(A(-1,3,2), B(2,3,5)\) and \(C(3,5,-2)\) are vertices of a \(\triangle \mathrm{ABC}\), then angles of are
88290
If \(G\) and \(G^{\prime}\) are respectively centroid of \(\triangle A B C\) and \(\Delta A^{\prime} B^{\prime} C^{\prime}\), then \(A A^{\prime}+B B^{\prime}+C^{\prime}\) is equal to
1 \(2 \mathrm{GG}^{\prime}\)
2 \(3 \mathrm{GG}^{\prime}\)
3 \(\frac{2}{3} \mathrm{GG}^{\prime}\)
4 \(\frac{1}{3} \mathrm{GG}^{\prime}\)
Explanation:
(B) : Let a.b.c be the position vectors of A,B and C respectively. Then, the position vector of \(G\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) Let the position vectors of \(\mathrm{A}^{\prime} \mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) be \(\mathrm{a}^{\prime} \mathrm{b}^{\prime}\) Then, the position vector of \(G^{\prime}\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) \(\therefore A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}-a\right)+\left(b^{\prime}-b\right)+\left(c^{\prime}-c\right)\) \(\Rightarrow A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}+b^{\prime}+c^{\prime}\right)-(a+b+c)\) \(=\left(\frac{a^{\prime}+b^{\prime}+c^{\prime}}{3}-\frac{a+b+c}{3}\right)=3 G G^{\prime}\)
CG PET-2016
Co-Ordinate system
88291
The orthocenter of the triangle formed by the line \(x=2, y=3\) and \(3 x+2 y=6\) at the point
1 \((2,0)\)
2 \((2,3)\)
3 \((0,3)\)
4 None of these
Explanation:
(B) : Given equation of lines are \(x=2, y=3\) and \(3 x+2 y=6\) Since, \(\triangle \mathrm{ABC}\) is right angled at \(\mathrm{C}\). Now, orthocenter of \(\triangle \mathrm{ABC}\) will the vertex at which right angle is forming. \(\therefore\) Orthocenter of \(\triangle \mathrm{ABC}\) is \((2,3)\).
88288
The equation of the plane meets the axes in \(\mathrm{A}\), and \(C\) such that centroid of the \(\triangle \mathrm{ABC}\) is \(\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3},\right)\) is given by
1 \(x+y+z=1\)
2 \(x+y+z=2\)
3 \(\frac{x}{3}+\frac{y}{3}+\frac{z}{3}=3\)
4 \(x+y+z=\frac{1}{3}\)
Explanation:
(A): Given, The centroid of triangle \(\triangle \mathrm{ABC}\) is \(=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\) the planes meets the axes in \(\mathrm{ABC}\) Now, The centroid of the triangle \(\triangle \mathrm{ABC}\) is \(\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{b}}{3}, \frac{\mathrm{c}}{3}\right)\) \(\Rightarrow \quad \mathrm{a}=1, \mathrm{~b}=1 \& \mathrm{c}=1\) \(\therefore\) The equation of plane meet the co-ordinate axes is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) \(\frac{x}{1}+\frac{y}{1}+\frac{z}{1}=1\) \(x+y+z=1\)
BCECE-2014
Co-Ordinate system
88289
If \(A(-1,3,2), B(2,3,5)\) and \(C(3,5,-2)\) are vertices of a \(\triangle \mathrm{ABC}\), then angles of are
88290
If \(G\) and \(G^{\prime}\) are respectively centroid of \(\triangle A B C\) and \(\Delta A^{\prime} B^{\prime} C^{\prime}\), then \(A A^{\prime}+B B^{\prime}+C^{\prime}\) is equal to
1 \(2 \mathrm{GG}^{\prime}\)
2 \(3 \mathrm{GG}^{\prime}\)
3 \(\frac{2}{3} \mathrm{GG}^{\prime}\)
4 \(\frac{1}{3} \mathrm{GG}^{\prime}\)
Explanation:
(B) : Let a.b.c be the position vectors of A,B and C respectively. Then, the position vector of \(G\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) Let the position vectors of \(\mathrm{A}^{\prime} \mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) be \(\mathrm{a}^{\prime} \mathrm{b}^{\prime}\) Then, the position vector of \(G^{\prime}\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) \(\therefore A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}-a\right)+\left(b^{\prime}-b\right)+\left(c^{\prime}-c\right)\) \(\Rightarrow A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}+b^{\prime}+c^{\prime}\right)-(a+b+c)\) \(=\left(\frac{a^{\prime}+b^{\prime}+c^{\prime}}{3}-\frac{a+b+c}{3}\right)=3 G G^{\prime}\)
CG PET-2016
Co-Ordinate system
88291
The orthocenter of the triangle formed by the line \(x=2, y=3\) and \(3 x+2 y=6\) at the point
1 \((2,0)\)
2 \((2,3)\)
3 \((0,3)\)
4 None of these
Explanation:
(B) : Given equation of lines are \(x=2, y=3\) and \(3 x+2 y=6\) Since, \(\triangle \mathrm{ABC}\) is right angled at \(\mathrm{C}\). Now, orthocenter of \(\triangle \mathrm{ABC}\) will the vertex at which right angle is forming. \(\therefore\) Orthocenter of \(\triangle \mathrm{ABC}\) is \((2,3)\).
88288
The equation of the plane meets the axes in \(\mathrm{A}\), and \(C\) such that centroid of the \(\triangle \mathrm{ABC}\) is \(\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3},\right)\) is given by
1 \(x+y+z=1\)
2 \(x+y+z=2\)
3 \(\frac{x}{3}+\frac{y}{3}+\frac{z}{3}=3\)
4 \(x+y+z=\frac{1}{3}\)
Explanation:
(A): Given, The centroid of triangle \(\triangle \mathrm{ABC}\) is \(=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\) the planes meets the axes in \(\mathrm{ABC}\) Now, The centroid of the triangle \(\triangle \mathrm{ABC}\) is \(\left(\frac{\mathrm{a}}{3}, \frac{\mathrm{b}}{3}, \frac{\mathrm{c}}{3}\right)\) \(\Rightarrow \quad \mathrm{a}=1, \mathrm{~b}=1 \& \mathrm{c}=1\) \(\therefore\) The equation of plane meet the co-ordinate axes is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) \(\frac{x}{1}+\frac{y}{1}+\frac{z}{1}=1\) \(x+y+z=1\)
BCECE-2014
Co-Ordinate system
88289
If \(A(-1,3,2), B(2,3,5)\) and \(C(3,5,-2)\) are vertices of a \(\triangle \mathrm{ABC}\), then angles of are
88290
If \(G\) and \(G^{\prime}\) are respectively centroid of \(\triangle A B C\) and \(\Delta A^{\prime} B^{\prime} C^{\prime}\), then \(A A^{\prime}+B B^{\prime}+C^{\prime}\) is equal to
1 \(2 \mathrm{GG}^{\prime}\)
2 \(3 \mathrm{GG}^{\prime}\)
3 \(\frac{2}{3} \mathrm{GG}^{\prime}\)
4 \(\frac{1}{3} \mathrm{GG}^{\prime}\)
Explanation:
(B) : Let a.b.c be the position vectors of A,B and C respectively. Then, the position vector of \(G\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) Let the position vectors of \(\mathrm{A}^{\prime} \mathrm{B}^{\prime}\) and \(\mathrm{C}^{\prime}\) be \(\mathrm{a}^{\prime} \mathrm{b}^{\prime}\) Then, the position vector of \(G^{\prime}\) is \(\frac{a^{\prime}+b+c^{\prime}}{3}\) \(\therefore A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}-a\right)+\left(b^{\prime}-b\right)+\left(c^{\prime}-c\right)\) \(\Rightarrow A A^{\prime}+B B^{\prime}+C C^{\prime}=\left(a^{\prime}+b^{\prime}+c^{\prime}\right)-(a+b+c)\) \(=\left(\frac{a^{\prime}+b^{\prime}+c^{\prime}}{3}-\frac{a+b+c}{3}\right)=3 G G^{\prime}\)
CG PET-2016
Co-Ordinate system
88291
The orthocenter of the triangle formed by the line \(x=2, y=3\) and \(3 x+2 y=6\) at the point
1 \((2,0)\)
2 \((2,3)\)
3 \((0,3)\)
4 None of these
Explanation:
(B) : Given equation of lines are \(x=2, y=3\) and \(3 x+2 y=6\) Since, \(\triangle \mathrm{ABC}\) is right angled at \(\mathrm{C}\). Now, orthocenter of \(\triangle \mathrm{ABC}\) will the vertex at which right angle is forming. \(\therefore\) Orthocenter of \(\triangle \mathrm{ABC}\) is \((2,3)\).
