`E= 1/2 mv^2` give dimensions of E.

Find out the dimension of (GIM^(2))/(E^(2)) where E = energy G = gravitational constant I =impulse of force and M = mass.

If e_1 and e_2 be the eccentricities of a hyperbola and its conjugate, show that 1/(e_1^2)+1/(e_2^2)=1 .

If vec e_1, vec e_2, vec e_3a n d vec E_1, vec E_2, vec E_3 are two sets of vectors such that vec e_idot vec E_j=1,ifi=j and vec e_idot vec E_j=0a n difi!=j , then prove that [ (vec e_1, vec e_2 ,vec e_3)][ (vec E_1, vec E_2, vec E_3) ]=1.

In the following circuit fig, a 10m long potentiometer wire with resistance 1,2 ohm/m, a resistance R_1 and an accumulator of e.m.f. 2V are connected in series. When the e.m.f. of thermocouple is 2.4 mV the deflection in galvanometer is zero. The current supplied by the accumulator will be

int_(-pi/2)^(pi/2)(e^(|sinx|)cosx)/(1+e^(tanx))dx is equal to (a) e+1 (b) 1-e (c) e-1 (d) none of these

If E_(1) ,E_(2),E_(3)…E_(n) be n independent events such that P(Ei)=(1)/(1+i) for 1 le I le n then the chance that none of E_(1),E_(2),E_(3) … E_(n) occur is

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x) and f(1/2)=1 . Then the value of int f(x)dx lies in the interval for x:[1/2,1] (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)