Prove that `[ -> a , -> b , -> c+ -> d]=[ -> a , -> b , -> c]+[ -> a , -> b , -> d]`

Prove that [a xx b,a xx c,d]=(a*d)[a,b,c]

A B C D IS A PARALLELOGRAM AND O is any point in its interior. Prove that: a r\ ( A O B)+\ a r\ ( C O D)=\ a r\ ( B O C)+\ a r( A O D) a r\ ( A O B)+a r\ (C O D)=1/2\ a r(^(gm)A B C D) Given: A parallelogram A B C D\ a n d\ O is a point in its interior. To Prove: a r\ (\ A O B)+\ a r( C O D)=a r\ ( B O C)+a r( A O D)

During the transformation of ._(c )X^(a) to ._(d)Y^(b) the number of beta -particles emitted are a. d + ((a - b)/(2)) - c b. (a - b)/(c ) c. d + ((a - b)/(2)) + c d. 2c - d + a = b

Prove that the points (a, b), (c, d) and (a-c, b-d) are collinear, if ad = bc.

In Fig. 4.101, A D and C E are two altitudes of A B C . Prove that A E F ~ C D F (ii) A B D ~ C B E (iii) A E F ~ A D B (iv) F D C ~ B E C (FIGURE)

A B C D E is pentagon, prove that vec A B + vec B C + vec C D + vec D E+ vec E A = vec0 vec A B+ vec A E+ vec B C+ vec D C+ vec E D+ vec A C=3 vec A C

If A B C D is a parallelogram, then prove that a r\ (\ A B D)=\ a r\ (\ B C D)=\ a r\ (\ A B C)=\ a r\ (\ A C D)=1/2\ a r\ (|""|^(gm)A B C D)

If A=[[a, b],[ c ,d]] , then a d j\ A is [[-d,-b],[-c, a]] (b) [[d,-b],[-c ,a]] (c) [[d, b],[ c, a]] (d) [[d, c],[ b ,a]]

If a, b, c, d, e be 5 numbers such that a, b, c are in AP, b, c, d, are in GP & c, d, e are in HP then (i) Prove that a,c,e are in GP. (ii) Prove that e=(2b-a)^2//a . (iii) If a=2& e=18 , find all possible vlaues of b,c,d.

In Figure, A B C D is a parallelogram and X\ a n d\ Y are points on the diagonal B D such that D X=B Y . Prove that A X C Y is a parallelogram (ii) A X=C Y ,\ A Y=C X A Y B\ ~= C X D (iv) A X D~=\ C Y B