A right angle is divided into three positive parts `alpha,betaa n dgammadot` Prove that for all possible divisions `t a nalpha+tanbeta+tangamma>1+tanalphatanbetatangammadot`

Let alpha,betaa n dgamma be some angles in the first quadrant satisfying tan(alpha+beta)=(15)/8a n dcos e cgamma=(17)/8, then which of the following hold(s) good? (a) alpha+beta+gamma=pi (b) cotalpha+cotbeta+cotgamma=cotalphacotbetacotgamma (c) tanalpha+tanbeta+tangamma=tanalphatanbetatangamma (d) tanalphatanbeta+tanbetatangamma+tangammatanalpha=1

If alpha and beta are positive acute angles and cos2alpha=(3cos2beta-1)/(3-cos2beta) ,then prove that tan alpha=sqrt2tanbeta

The base of a triangle is divided into three equal parts. If t_1, t_2,t_3 are the tangents of the angles subtended by these parts at the opposite vertex, prove that (1/(t_1)+1/(t_2))(1/(t_2)+1/(t_3))=4(1+1/(t2 2))dot

Prove that , 2^(2n)-3n-1 is divisible by 9 for all positive intergral values of n greater than 1 .

By the principle of mathematical induction prove that (5^(2n)-1) is always divisible by 24 for all positive integral values of n.

vec u , vec v and vec w are three non-coplanar unit vecrtors and alpha,beta and gamma are the angles between vec u and vec v , vec v and vec w ,a n d vec w and vec u , respectively, and vec x , vec y and vec z are unit vectors along the bisectors of the angles alpha,betaa n dgamma , respectively. Prove that [ vecx xx vec y vec yxx vec z vec zxx vec x]=1/(16)[ vec u vec v vec w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2) .

If hat a , hat b ,a n d hat c are three unit vectors, such that hat a+ hat b+ hat c is also a unit vector and theta_1,theta_2 and theta_3 are angles between the vectors hat a , hat b ; hat b , hat ca n d hat c , hat a respectively, then among theta_1,theta_2 and theta_3 . a. all are acute angles b. all are right angles c. at least one is obtuse angle d. none of these

Match the statements/expressions given in column I with the values given in Column II. Column I, Column II In R^2, if the magnitude of the projection vector of the vector alpha hat i+beta hat j on sqrt(3) hat i+ hat ji ssqrt(3) and if |alpha| is /are, (p) 1 Let aa n db be real numbers such that the function f(x)={-3a x^2-2,x<1 bx+a^2, xgeq1 Differentiable for all x in Rdot Then possible value (s) of a is/are, (q) 2 Let omega!=1 be a complex cube root of unity. If (3-3omega+2omega^2)^(4n+3)+(2+3omega-3omega^2)^(4n+3)+(-3-2omega+3omega^2)^(4n+3)=0 , then possible values (s) of n is /are, (r) 3 Let the harmonic mean of two positive real numbers aa n db be 4. If q is a positive real number such that a ,5,q ,b is an arithmetic progressin, then the values (s)of|q-a| is /are, (s) 4 , (t) 5

If vector vec b=(t a nalpha,-1, 2sqrt(sinalpha//2))a n d vec c=(t a nalpha, t a nalpha, -3/(sqrt(sinalpha//2))) are orthogonal and vector vec a=(1,3,sin2alpha) makes an obtuse angle with the z-axis, then the value of alpha is

ABC is a right angled Delta in which angleB=90^@ and BC =a. If n points L_1,L_2,L_3,...,L_n on AB are such that AB is divided into (n+1) equal parts and L_1M_1,L_2M_2,...,L_nM_n are line segments parallel to BC and points M_1,M_2,...,M_n are on AC, then the sum of the lengths of L_1M_1,L_2M_2,...,L_nM_n is: