ABCD is a trapezium such that `A B||C Da n dC B` is perpendicular to them. If `/_A D B=theta,B C=p ,a n dC D=q` , show that `A B=((p^2+q^2)sintheta)/(pcostheta+qsintheta)`

Let A B C D be a tetrahedron such that the edges A B ,A Ca n dA D are mutually perpendicular. Let the area of triangles A B C ,A C Da n dA D B be 3, 4 and 5sq. units, respectively. Then the area of triangle B C D is 5sqrt(2) b. 5 c. (sqrt(5))/2 d. 5/2

In a triangle, if the angles A , B ,a n dC are in A.P. show that 2cos1/2(A-C)=(a+c)/(sqrt(a^2-a c+c^2))

a cot theta+b cosec theta=p and b cot theta+ a cosec theta=q then p^(2)-q^(2) is equal to

Let A B C D be a quadrilateral with area 18 , side A B parallel to the side C D ,a n dA B=2C D . Let A D be perpendicular to A Ba n dC D . If a circle is drawn inside the quadrilateral A B C D touching all the sides, then its radius is 3 (b) 2 (c) 3/2 (d) 1

Let A B C be a triangle with incentre at Idot Also, let Pa n dQ be the feet of perpendiculars from AtoB Ia n dC I , respectively. Then which of the following results are correct? (A P)/(B I)=(sinB/2cosC/2)/(sinA/2) (b) (A Q)/(C I)=(sinC/2cosB/2)/(sinA/2) (A P)/(B I)=(sinC/2cosB/2)/(sinA/2) (d) (A P)/(B I)+(a Q)/(C I)=sqrt(3)if/_A=60^0

In A B C , a point P is taken on A B such that A P//B P=1//3 and point Q is taken on B C such that C Q//B Q=3//1 . If R is the point of intersection of the lines A Qa n dC P , ising vedctor method, find the are of A B C if the area of B R C is 1 unit

P_1n dP_2 are planes passing through origin L_1a n dL_2 are two lines on P_1a n dP_2, respectively, such that their intersection is the origin. Show that there exist points A ,Ba n dC , whose permutation A^(prime),B^(prime)a n dC^(prime), respectively, can be chosen such that A is on L_1,BonP_1 but not on L_1a n dC not on P_1; A ' is on L_2,B 'onP_2 but not on L_2a n dC ' not on P_2dot

If a, b, c, d are in G.P., then (a + b + c + d)^(2) is equal to

A B C D is a tetrahedron and O is any point. If the lines joining O to the vrticfes meet the opposite faces at P ,Q ,Ra n dS , prove that (O P)/(A P)+(O Q)/(B Q)+(O R)/(C R)+(O S)/(D S)=1.