In ` A B C ,s i d e sb , c` and angle `B` are given such that `a` has two valus `a_1a n da_2dot` Then prove that `|a_1-a_2|=2sqrt(b^2-c^2sin^2B)`

In A B C ,a , ca n dA are given and b_1,b_2 are two values of the third side b such that b_2=2b_1dot Then prove that sinA=sqrt((9a^2-c^2)/(8c^2))

If in triangle ABC, a, c and angle A are given and c sin A lt a lt c , then ( b_(1) and b_(2) are values of b)

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec b . If the angle between aa n db is pi/6, then prove that |a_1a_2a_3b_1b_2b_3c_1c_2c_3|=1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)

If A,B,C are the angles of a triangle and tan A=1,tan B=2, prove that tan C=3 If a,b,c are the corresponding sides,then prove that (a)/(sqrt(5))=(b)/(2sqrt(2))=(c)/(3)

If the line a_1 x + b_1 y+ c_1 = 0 and a_2 x + b_2 y + c_2 = 0 cut the coordinate axes in concyclic points, prove that : a_1 a_2 = b_1 b_2 .

In a delta ABC, a,c, A are given and b_(1) , b_(2) are two values of third side b such that b_(2)=2b_(1). Then, the value of sin A.

If b,c B are given, and if b lt c , show that (a_1 -a_2)^2 +(a_1 + a_2)^2 tan^2 B= 4b^2 , where a_1,a_2 are the two values of the third side.

If the tangent and normal to xy=c^2 at a given point on it cut off intercepts a_1, a_2 on one axis and b_1, b_2 on the other axis, then a_1 a_2 + b_1 b_2 = (A) -1 (B) 1 (C) 0 (D) a_1 a_2 b_1 b_2

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three nonzero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec bdot If the angle between vec aa n d vec b is pi//6 , then the value of |a_1b_1c_1a_2b_2c_2a_3b_3c_3| is a.0 b. 1 c. 1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2) d. 3/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)

The distributive law from algebra states that for all real numbers c,a1 and a2,we have c(a_1+a_2)=c a_1+c a_2 Use this law and mathematical induction to prove that,for all natural numbers,ngeq2,if c,a_1,a_2,....,an are any real numbers,then c(a_1+a_2+.....+a_n)=c a_1+c a_2+....+c a_n