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? alpha+beta+gamma=pi cotalpha+tanbeta+tangamma=tanalphatanbetatangamma tanalpha+tanbeta+tangamma=tanalphatanbetatangamma tanalphatanbeta+tanbetatangamma+tangammatanalpha=1

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/t_2^2)dot

Using Binomial theorem, prove that 6^(n)-5n always leaves remainder 1 when divided by 25 for all positive integer n.

Using Binomial theorem, prove that 6^(n)-5n always leaves remainder 1 when divided by 25 for all positive interger n .

If the normals to the parabola y^2=4a x at P meets the curve again at Q and if P Q and the normal at Q make angle alpha and beta , respectively, with the x-axis, then t a nalpha(tanalpha+tanbeta) has the value equal to 0 (b) -2 (c) -1/2 (d) -1

In given figure, an adiabatic cylindrical tube of volume 2V_(0) is divided in two equal parts by a frictionless adiabatic separator. An ideal gas in left side of a tube having pressure P_(1) and temperature T_(1) where as in the right side having pressure P_(2) and temperature T_(2).C_(p)//C_(v) =gamma is the same for both the gases. The separator is slid slowly and is released at a position where it can stay in equilibrium. Find (a) the final volumes of the two parts (b) the heat given to the gas in the left part and (c) the final common pressure of the gases,

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

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_2a n dtheta_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,a n dtheta_3dot a. all are acute angles b. all are right angles c. at least one is obtuse angle d. none of these

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 a alpha=tan^(-1)2 b. alpha=-tan^(-1)2 c. alpha=tan^(-1)2 d. alpha=tan^(-1)2