Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of $$\Delta$$ABC, then (R + r) is equal to :

The point P (a, b) undergoes the following three transformations successively : (a) reflection about the line y = x. (b) translation through 2 units along the positive direction of x-axis. (c) rotation through angle {\pi \over 4} about the origin in the anti-clockwise direction. If the co-ordinates of the final position of the point P are \left( { - {1 \over {\sqrt 2 }},{7 \over {\sqrt 2 }}} \right), then the value of 2a + b is equal to :

Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :

Let the equation of the pair of lines, y = px and y = qx, can be written as (y $$-$$ px) (y $$-$$ qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 $$-$$ 4xy $$-$$ 5y2 = 0 is :

Let ABC be a triangle with A($$-$$3, 1) and $$\angle$$ACB = $$\theta$$, 0 < $$\theta$$ < {\pi \over 2}. If the equation of the median through B is 2x + y $$-$$ 3 = 0 and the equation of angle bisector of C is 7x $$-$$ 4y $$-$$ 1 = 0, then tan$$\theta$$ is equal to :

Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :

If p and q are the lengths of the perpendiculars from the origin on the lines, x cosec $$\alpha$$ $$-$$ y sec $$\alpha$$ = k cot 2$$\alpha$$ and x sin$$\alpha$$ + y cos$$\alpha$$ = k sin2$$\alpha$$ respectively, then k2 is equal to :

Let A be the set of all points ($$\alpha$$, $$\beta$$) such that the area of triangle formed by the points (5, 6), (3, 2) and ($$\alpha$$, $$\beta$$) is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is :

In an octagon ABCDEFGH of equal side, what is the sum of $$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}+\overrightarrow{AE}+\overrightarrow{AF}+\overrightarrow{AG}+\overrightarrow{AH}$$, if, $$\overrightarrow{AO}=2\hat{i}+3\hat{j}-4\hat{k}$$

If $$\overrightarrow{A}$$ and $$\overrightarrow{B}$$ are two vectors satisfying the relation $$\overrightarrow{A}$$ . $$\overrightarrow{B}$$ = $$\left|\overrightarrow{A}\times \overrightarrow{B}\right|$$. Then the value of $$\left|\overrightarrow{A}-\overrightarrow{B}\right|$$ will be :

Two vectors $$\overrightarrow{P}$$ and $$\overrightarrow{Q}$$ have equal magnitudes. If the magnitude of $$\overrightarrow{P}+\overrightarrow{Q}$$ is n times the magnitude of $$\overrightarrow{P}-\overrightarrow{Q}$$, then angle between $$\overrightarrow{P}$$ and $$\overrightarrow{Q}$$ is :

What will be the projection of vector $$\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}$$ on vector $$\overrightarrow{B}=\hat{i}+\hat{j}$$ ?

Match List - I with List - II Choose the correct answer from the options given below :

Two vectors $$\overrightarrow{X}$$ and $$\overrightarrow{Y}$$ have equal magnitude. The magnitude of ($$\overrightarrow{X}$$ $$-$$ $$\overrightarrow{Y}$$) is n times the magnitude of ($$\overrightarrow{X}$$ + $$\overrightarrow{Y}$$). The angle between $$\overrightarrow{X}$$ and $$\overrightarrow{Y}$$ is :

Assertion A : If A, B, C, D are four points on a semi-circular are with centre at 'O' such that $$\left|\overrightarrow{AB}\right|=\left|\overrightarrow{BC}\right|=\left|\overrightarrow{CD}\right|$$, then $$\overrightarrow{AB}+\overrightarrow{AC}+\overrightarrow{AD}=4\overrightarrow{AO}+\overrightarrow{OB}+\overrightarrow{OC}$$ Reason R : Polygon law of vector addition yields $$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{AD}=2\overrightarrow{AO}$$ In the light of the above statements, choose the most appropriate answer from the options given below :

The magnitude of vectors $$\overrightarrow{OA}$$, $$\overrightarrow{OB}$$ and $$\overrightarrow{OC}$$ in the given figure are equal. The direction of $$\overrightarrow{OA}$$ + $$\overrightarrow{OB}$$ $$-$$ $$\overrightarrow{OC}$$ with x-axis will be :

The angle between vector $$\left(\overrightarrow{A}\right)$$ and $$\left(\overrightarrow{A}-\overrightarrow{B}\right)$$ is :

The resultant of these forces $$\overrightarrow{OP},\overrightarrow{OQ},\overrightarrow{OR},\overrightarrow{OS}$$ and $$\overrightarrow{OT}$$ is approximately .......... N. [Take $$\sqrt{3}=1.7$$, $$\sqrt{2}=1.4$$ Given $$\hat{i}$$ and $$\hat{j}$$ unit vectors along x, y axis]

Statement I : Two forces $$\left(\overrightarrow{P}+\overrightarrow{Q}\right)$$ and $$\left(\overrightarrow{P}-\overrightarrow{Q}\right)$$ where $$\overrightarrow{P}\perp \overrightarrow{Q}$$, when act at an angle $$\theta$$1 to each other, the magnitude of their resultant is $$\sqrt{3(P^2+Q^2)}$$, when they act at an angle $$\theta$$2, the magnitude of their resultant becomes $$\sqrt{2(P^2+Q^2)}$$. This is possible only when $${\theta }_1<{\theta }_2$$. Statement II : In the situation given above. $$\theta$$1 = 60$${}^{\circ }$$ and $$\theta$$2 = 90$${}^{\circ }$$ In the light of the above statements, choose the most appropriate answer from the options given below :-

Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is :