Ratios of Distances between Centroid, Circumcenter, Incenter and Orthocenter of Triangle

Q. A Point $O$ is the centre of a circle circumscribed about a triangle $ABC$. Then $\overrightarrow{OA}\sin 2A+\overrightarrow{OB}\sin 2B+\overrightarrow{OC}\sin 2C$ is equal to

1. $(\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC})\sin 2A$
2. $3\overrightarrow{OG}$, where $G$ is the centroid of triangle $ABC$
3. $\overrightarrow{0}$
4. None of these

Q.

If $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ determine the vertices of a triangle, show that $\frac{1}{2}[\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ are collinear. Also, find the vector normal to the plane of the triangle.

Q. Let $P$ and $Q$ be two points in $xy-$plane on the curve $y=x^7-2x^5+5x^3+8x+5$ such that $\overrightarrow{OP}\cdot \hat{i}=2$ and $\overrightarrow{OQ}\cdot \hat{i}=-2$ and the magnitude of $\overrightarrow{OP}+\overrightarrow{OQ}=2M$ $($where $O$ is origin$).$ Then the value of $M$ is

Q. Let $ABC$ be a triangle whose centroid is $G$, orthocentre is $H$ and circumcentre is the origin $O$. If $D$ is any point in the plane of the triangle such that no three of $O,A,C$ and $D$ are collinear satisfying the relation $\overrightarrow{AD}+\overrightarrow{BD}+\overrightarrow{CH}+3\overrightarrow{HG}=\lambda \overrightarrow{HD}$, then the value of the scalar $\lambda$ is

Q. If a, b, c are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then

1. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$
   
2. $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$
3. ${\overrightarrow{a}}^2={\overrightarrow{b}}^2+{\overrightarrow{c}}^2$
   
4. $Noneofthese$

Q. A curve is represented by $y=\sin x.$ If $x$ is changed from $\frac{\pi }{3}$ to $\frac{\pi }{3}+\frac{\pi }{100},$ find approximately the change in $y.$

Q. Given the vertices of triangle by position vectors $\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{k}and\hat{j}+\hat{k}$ the centroid and Incentre of the triangle will be given by

1. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+1)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$
2. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+2)\hat{i}+(\sqrt{2}+1)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$
3. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+2)\hat{j}+(\sqrt{2}+2)\hat{k}}{2+\sqrt{2}}$
4. $\frac{2\hat{i}+2\hat{j}+3\hat{k}}{3},\frac{(\sqrt{2}+1)\hat{i}+(\sqrt{2}+2)\hat{j}+(\sqrt{2}+1)\hat{k}}{2+\sqrt{2}}$

Q. Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A,B,C$ and $P$ are $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{4}$ respectively, then the position vector of the orthocentre of this trianlge, is:

1. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$
2. $\overrightarrow{0}$
3. $-\left(\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}\right)$
4. $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}$

Q. Given orthocentre $\overline{H}$ and circumcentre $\overline{C}$ for a triangle as $2\hat{i}+3\hat{j}and4\hat{i}+5\hat{k}$.Then the centroid of triangle can be given by,

1. 
2. 
3. 
4. 

Q. In $\Delta ABC,\overrightarrow{AB}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{AC}=3\hat{i}-\hat{j}-2\hat{k}.$
If the bisector of $\angle BAC$ meets $BC$ at $D$ and $G$ is the centroid of $\Delta ABC$, then $|\overrightarrow{GD}|=$

1. $1$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $2$

Q. Given position vectors $\overline{a},\overline{b},\overline{c}$ of points A, B, C for a triangle. The centroid can be given by

1. $\frac{\overline{a}+\overline{b}+\overline{c}}{3}$
2. $\frac{2}{3}(\overline{a}+\overline{b}+\overline{c})$
3. $\overline{a}+\overline{b}+\overline{c}$
4. $\frac{(\overline{a}.\overline{b})\times \overline{c}}{3}$

Q. Given position vectors $\overline{a},\overline{b},\overline{c}$ of points A, B, C for a triangle. The centroid can be given by

1. 
2. 
3. 
4. 

Q. If point $O$ is the centre of a circle circumscribed about a triangle $ABC$. Then $\overline{OA}\sin 2A+\overline{OB}\sin 2B+\overline{OC}\sin 2C=$

1. $(\overline{OA}+\overline{OB}+\overline{OC})\cos 2A$
2. $(\overline{OA}+\overline{OB}+\overline{OC})\sin 2A$
3. $\overline{0}$
4. $(\overline{OA}+\overline{OB}+\overline{OC})\tan 2A$

Q. Given 3 points with position vectors $\overline{p_1},\overline{p_2}and\overline{p_3}$ which form the vertices of a triangle with side lengths $a=|\overline{p_2}-\overline{p_1}|,b=|\overline{p_3}-\overline{p_2}|,c=|\overline{p_1}-\overline{p_3}|$. Then the in-centre is given by

1. $I=\frac{a.\overline{p_1}+a.\overline{p_2}+c.\overline{p_3}}{a+b+c}$
2. $I=\frac{b.\overline{p_1}+c.\overline{p_2}+a.\overline{p_3}}{a+b+c}$
3. $I=\frac{c.\overline{p_1}+a.\overline{p_2}+b.\overline{p_3}}{a+b+c}$
4. $I=\frac{\overline{p_1}+\overline{p_2}+\overline{p_3}}{a+b+c}$

Q. Given 3 points with position vectors $\overline{p_1},\overline{p_2}and\overline{p_3}$ which form the vertices of a triangle with side lengths $a=|\overline{p_2}-\overline{p_1}|,b=|\overline{p_3}-\overline{p_2}|,c=|\overline{p_1}-\overline{p_3}|$. Then the in-centre is given by

1. 
2. 
3. 
4. 

Q. Given orthocentre $\overline{H}$ and circumcentre $\overline{C}$ for a triangle as $2\hat{i}+3\hat{j}and4\hat{i}+5\hat{k}$.Then the centroid of triangle can be given by,

1. $\frac{10\hat{i}-13\hat{j}}{3}$
2. $\frac{-10\hat{i}+13\hat{j}}{3}$
3. $\frac{10\hat{i}+13\hat{j}}{3}$
4. $\frac{-10\hat{i}-13\hat{j}}{3}$

Q. Given the vertices of triangle by position vectors $\hat{i}+\hat{j}+\hat{k},\hat{i}+\hat{k}and\hat{j}+\hat{k}$ the centroid and Incentre of the triangle will be given by

1. 
2. 
3. 
4. 

Q. In $\Delta ABC,\overrightarrow{AB}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{AC}=3\hat{i}-\hat{j}-2\hat{k}.$
If the bisector of $\angle BAC$ meets $BC$ at $D$ and $G$ is the centroid of $\Delta ABC$, then $|\overrightarrow{GD}|=$

1. $1$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $2$

Q. If a, b, c are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, then

1. $\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$
2. $Noneofthese$
3. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$
   
4. ${\overrightarrow{a}}^2={\overrightarrow{b}}^2+{\overrightarrow{c}}^2$
   

Q. Given 3 points with position vectors $\overline{p_1},\overline{p_2}and\overline{p_3}$ which form the vertices of a triangle with side lengths $a=|\overline{p_2}-\overline{p_1}|,b=|\overline{p_3}-\overline{p_2}|,c=|\overline{p_1}-\overline{p_3}|$. Then the in-centre is given by

1. $I=\frac{a.\overline{p_1}+a.\overline{p_2}+c.\overline{p_3}}{a+b+c}$
2. $I=\frac{b.\overline{p_1}+c.\overline{p_2}+a.\overline{p_3}}{a+b+c}$
3. $I=\frac{c.\overline{p_1}+a.\overline{p_2}+b.\overline{p_3}}{a+b+c}$
4. $I=\frac{\overline{p_1}+\overline{p_2}+\overline{p_3}}{a+b+c}$

Q. The directions cosines of the joins of the following pairs of vectors :
(i) $(6,3,2),(5,1,4)$ be $\frac{k}{3},\frac{m}{3},-\frac{m}{3}$

Q. In $\Delta ABC,\overrightarrow{AB}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{AC}=3\hat{i}-\hat{j}-2\hat{k}.$
If the bisector of $\angle BAC$ meets $BC$ at $D$ and $G$ is the centroid of $\Delta ABC$, then $|\overrightarrow{GD}|=$

1. $1$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $2$

Q. If O is the circumcenter of $\Delta ABC$ and $R_1,R_2,$ and $R_3$ are the radii of the circumcircles of triangles OBC, OCA, and OAB, respectively, then $\frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3}$ has the value equal to

1. $\frac{abc}{R^3}$
2. $\frac{R^3}{abc}$
3. $\frac{4\Delta }{R^2}$
4. $\frac{\Delta }{4R^2}$

Q. Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A,B,C$ and $P$ are $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{4}$ respectively, then the position vector of the orthocentre of this trianlge, is:

1. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$
2. $\overrightarrow{0}$
3. $-\left(\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}\right)$
4. $\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}}{2}$

Q. State whether the following statements are true or false:
$a\cos A+b\cos B+c\cos C=4R\sin A\sin B\sin C=(a+b+c)\frac{r}{R}$

1. True
2. False

Q. Consider the following statements :
1. There exists no triangle ABC for which sin A + sin B = sin C.
2. If the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio $1:\sqrt{3}:2$.
Which of the above statements is/are correct ?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither I nor 2