Let $$A(1,1),B(-4,3),C(-2,-5)$$ be vertices of a triangle $$ABC,P$$ be a point on side $$BC$$, and $${\Delta }_1$$ and $${\Delta }_2$$ be the areas of triangles $$APB$$ and $$ABC$$, respectively. If $${\Delta }_1:{\Delta }_2=4:7$$, then the area enclosed by the lines $$AP,AC$$ and the $$x$$-axis is :

The equations of the sides $$\mathrm{AB},\mathrm{BC}$$ and CA of a triangle ABC are $$2x+y=0,x+\mathrm{p}y=39$$ and $$x-y=3$$ respectively and $$\mathrm{P}(2,3)$$ is its circumcentre. Then which of the following is NOT true?

Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q$$\left(k_1,k_2\right)$$, then $$k_1+k_2$$ is equal to :

Let $$m_1,m_2$$ be the slopes of two adjacent sides of a square of side a such that $$a^2+11a+3\left(m_1^2+m_2^2\right)=220$$. If one vertex of the square is $$(10(\cos \alpha -\sin \alpha ),10(\sin \alpha +\cos \alpha ))$$, where $$\alpha \in \left(0,\frac{\pi }{2}\right)$$ and the equation of one diagonal is $$(\cos \alpha -\sin \alpha )x+(\sin \alpha +\cos \alpha )y=10$$, then $$72\left({\sin }^4\alpha +{\cos }^4\alpha \right)+a^2-3a+13$$ is equal to :

Let $$\mathrm{A}(\alpha ,-2),\mathrm{B}(\alpha ,6)$$ and $$\mathrm{C}\left(\frac{\alpha }{4},-2\right)$$ be vertices of a $$△\mathrm{ABC}$$. If $$\left(5,\frac{\alpha }{4}\right)$$ is the circumcentre of $$△\mathrm{ABC}$$, then which of the following is NOT correct about $$△\mathrm{ABC}$$?

Two vectors $$\overrightarrow{A}$$ and $$\overrightarrow{B}$$ have equal magnitudes. If magnitude of $$\overrightarrow{A}$$ + $$\overrightarrow{B}$$ is equal to two times the magnitude of $$\overrightarrow{A}$$ $$-$$ $$\overrightarrow{B}$$, then the angle between $$\overrightarrow{A}$$ and $$\overrightarrow{B}$$ will be :

$$\overrightarrow{A}$$ is a vector quantity such that $$|\overrightarrow{A}|$$ = non-zero constant. Which of the following expression is true for $$\overrightarrow{A}$$ ?

Which of the following relations is true for two unit vector $$\hat{A}$$ and $$\hat{B}$$ making an angle $$\theta$$ to each other?

From the base of a pole of height 20 meter, the angle of elevation of the top of a tower is 60$${}^{\circ }$$. The pole subtends an angle 30$${}^{\circ }$$ at the top of the tower. Then the height of the tower is :

Let AB and PQ be two vertical poles, 160 m apart from each other. Let C be the middle point of B and Q, which are feet of these two poles. Let {\pi \over 8} and $$\theta$$ be the angles of elevation from C to P and A, respectively. If the height of pole PQ is twice the height of pole AB, then tan2$$\theta$$ is equal to

A tower PQ stands on a horizontal ground with base $$Q$$ on the ground. The point $$R$$ divides the tower in two parts such that $$QR=15\mathrm{m}$$. If from a point $$A$$ on the ground the angle of elevation of $$R$$ is $$60^{\circ }$$ and the part $$PR$$ of the tower subtends an angle of $$15^{\circ }$$ at $$A$$, then the height of the tower is :

Let a vertical tower $$AB$$ of height $$2h$$ stands on a horizontal ground. Let from a point $$P$$ on the ground a man can see upto height $$h$$ of the tower with an angle of elevation $$2\alpha$$. When from $$P$$, he moves a distance $$d$$ in the direction of $$\overrightarrow{AP}$$, he can see the top $$B$$ of the tower with an angle of elevation $$\alpha$$. If $$d=\sqrt{7}h$$, then $$\tan \alpha$$ is equal to

The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $$45^{\circ }$$. Let R be a point on AQ and from a point B, vertically above $$\mathrm{R}$$, the angle of elevation of $$\mathrm{P}$$ is $$60^{\circ }$$. If $$\angle \mathrm{BAQ}=30^{\circ },\mathrm{AB}=\mathrm{d}$$ and the area of the trapezium $$\mathrm{PQRB}$$ is $$\alpha$$, then the ordered pair $$(\mathrm{d},\alpha )$$ is :

A horizontal park is in the shape of a triangle $$\mathrm{OAB}$$ with $$\mathrm{AB}=16$$. A vertical lamp post $$\mathrm{OP}$$ is erected at the point $$\mathrm{O}$$ such that $$\angle \mathrm{PAO}=\angle \mathrm{PBO}=15^{\circ }$$ and $$\angle \mathrm{PCO}=45^{\circ }$$, where $$\mathrm{C}$$ is the midpoint of $$\mathrm{AB}$$. Then $$(\mathrm{OP}{)}^2$$ is equal to :

The angle of elevation of the top of a tower from a point A due north of it is $$\alpha$$ and from a point B at a distance of 9 units due west of A is $${\cos }^{-1}\left(\frac{3}{\sqrt{13}}\right)$$. If the distance of the point B from the tower is 15 units, then $$\cot \alpha$$ is equal to :

If the constant term in the expansion of {\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}} is 2k.l, where l is an odd integer, then the value of k is equal to:

Let n $$\ge$$ 5 be an integer. If 9n $$-$$ 8n $$-$$ 1 = 64$$\alpha$$ and 6n $$-$$ 5n $$-$$ 1 = 25$$\beta$$, then $$\alpha$$ $$-$$ $$\beta$$ is equal to

The term independent of x in the expansion of (1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0 is :

If \sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} }, where $$\alpha$$ $$\in$$ R, then the value of 16$$\alpha$$ is equal to

The remainder when (2021)2023 is divided by 7 is :