Given `sinalpha+sinbeta=l`, `cosalpha+cosbeta=m`, `tan(alpha/2)tan(beta/2)=n`, (A) `(l^2+m^2)(1-n)=2l(1+n)` (B) `(l^2+m^2)(1+n)=2l(1-n)` (C) `(l^2+m^2)(1-n)=2m(1+n)` (D) `(l^2+m^2)(1+n)=2m(1-n)`

|[(m+n)^(2), l^(2), mn], [(n+l)^(2), m^(2), ln], [(l+m)^(2), n^(2), lm]| =(l^(2) +m^(2) +n^(2))(l-m)(m-n)(n-l)(l+m+n)

If the direction cosines of two lines are l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2) , then find the direction cosine of a line perpendicular to these lines. a) l_(1)+l_(2),m_(1)+m_(2),n_(1)+n_(2) b) l_(1)-l_(2),m_(1)-m_(2),n_(1)-n_(2) c) m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1) d) l_(1)+2l_(2),m_(1)+2m_(2),n_(1)+2n_(2)

If l , m, n are the direction cosines of a line, then: a) l^(2)+m^(2)+n^(2)=1 b) l^(2)+m^(2)+n^(2)=0 c) l+m+n=1 d) l+m+n=0

If (alpha, beta) is the foot of perpendicular from (x_1, y_1) to line lx+my+n=0 , then (A) (x_1 - alpha)/l = (y_1 - beta)/m (B) (x-1 - alpha)/l) = (lx_1 + my_1 + n)/(l^2+m^2) (C) (y_1 - beta)/m = (lx_1 + my_1 +n)/(l^2 +m^2) (D) (x-alpha)/l = (lalpha+mbeta+n)/(l^2+m^2)

If (alpha, beta) is the foot of perpendicular from (x_1, y_1) to line lx+my+n=0 , then (A) (x_1 - alpha)/l = (y_1 - beta)/m (B) (x-1 - alpha)/l) = (lx_1 + my_1 + n)/(l^2+m^2) (C) (y_1 - beta)/m = (lx_1 + my_1 +n)/(l^2 +m^2) (D) (x-alpha)/l = (lalpha+mbeta+n)/(l^2+m^2)

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If L(m,n)=int_(0)^(1)t^(m)(1+t)^(n),dt , then prove that L(m,n)=(2^(n))/(m+1)-n/(m+1)L(m+1,n-1)

If l : m = 2 (1)/(2) : 1(2)/(3) and m : n = 1 (1)/(4) : 3 (1)/(2) , find l : m : n

Two lines with direction cosines l_1,m_1,n_1 and l_2,m_2,n_2 are at righat angles iff (A) l_1l_2+m_1m_2+n_1n_2=0 (B) l_1=l_2,m_1=m_2,n_1=n_2 (C) l_1/l_2=m_1/m_2=n_1/n_2 (D) l_1l_2=m_1m_2=n_1n_2