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Given that \(\int\frac{{{\left.{d}{x}\right.}}}{{\sqrt{{{2}{a}{x}-{x}^{{2}}}}}}={a}^{{n}}{{\sin}^{{-{1}}}{\left(\frac{{{x}-{a}}}{{{a}}}\right)}}\) where a is a constant. Using dimensional analysis. The value of n is

(A) 1

(B) -1

(C) 0

(D) none of the above.

(A) 1

(B) -1

(C) 0

(D) none of the above.

Answer: (c )

The integral on LHS is in the from of log x, which is a number. Hence, \({a}^{{n}}\) must be a number, for which n = 0

The integral on LHS is in the from of log x, which is a number. Hence, \({a}^{{n}}\) must be a number, for which n = 0

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Statement -1 : Distance travelled in nth second has the dimensions of velocity.

Statement -2 : Because it is the distancce travelled in one (particular) second.

(A) Statement -1 is true, Statement -2 is true , and Statement -2 is correct explanation of Statement -1.

(B) Statement -1 is true , Statement -2 is true, but Statement -2 is not a correct explanation of Statement -1.

(C) Statement-1 is true, but Statement -2 is false.

(D) Statement-1 is false, but Statement -2 is true.

Statement -2 : Because it is the distancce travelled in one (particular) second.

(A) Statement -1 is true, Statement -2 is true , and Statement -2 is correct explanation of Statement -1.

(B) Statement -1 is true , Statement -2 is true, but Statement -2 is not a correct explanation of Statement -1.

(C) Statement-1 is true, but Statement -2 is false.

(D) Statement-1 is false, but Statement -2 is true.

What are the two angles of projection of a projectile projected with velocity of \({30}{m}/{s}\), so that the horizontal range is \({45}{m}\). Take, \({g}={10}{m}/{s}^{{{2}}}\).

Give an example of a physical wuantity (i) which has neigher unit nor direction (ii) has a dircetion but bot a vector (iii) can be either a vector or a scalar.

The potential energy function for a particle executing simple harmonic motion is given by \({V}{\left({x}\right)}=\frac{{{1}}}{{{2}}}{k}{x}^{{{2}}}\), where k is the force constant of the oscillatore. For \({k}=\frac{{{1}}}{{{2}}}{N}{m}^{{-{1}}}\), show that a particle of total energy 1 joule moving under this potential must turn back when it reaches \({x}=\pm{2}{m}.\)

A and B are two identical balls. A moving with a speed of \({6}{m}/{s}\), along the positive X-axis, undergoes a collision with B initially at rest. After collision, each ball moves along directions making angles of \(\pm{30}^{{\circ}}\) with the X-axis. What are the speeds of A and B after the collision ? I s this collision perfectly eleastic ?

A collision in which there is absolutely no loss of \({K}.{E}.\) is called an elastic collision. In such a collision, the linear momentum , total energy and kinetic energy, all are conserved.The coefficient of restitution \(/\) resilience of perfectly elastic collisions is unity.

Read the above passage and answer the following questions \(:\)

(i) When two bodies of equal masses undergo perfectly elastic collision in one dimension, what happens to their velocities ?

(ii) How is this fact applied in a nuclear reactor ?

Read the above passage and answer the following questions \(:\)

(i) When two bodies of equal masses undergo perfectly elastic collision in one dimension, what happens to their velocities ?

(ii) How is this fact applied in a nuclear reactor ?

The velocity of a particle at which the kinetic energy is eqyal to its rest mass energy is

(A) \({\left(\frac{{{3}{c}}}{{{2}}}\right)}\)

(B) \({3}\frac{{{c}}}{{\sqrt{{{2}}}}}\)

(C) \(\frac{{{\left({3}{c}\right)}^{{{1}/{2}}}}}{{{2}}}\)

(D) \(\frac{{{c}\sqrt{{{3}}}}}{{{2}}}\)

(A) \({\left(\frac{{{3}{c}}}{{{2}}}\right)}\)

(B) \({3}\frac{{{c}}}{{\sqrt{{{2}}}}}\)

(C) \(\frac{{{\left({3}{c}\right)}^{{{1}/{2}}}}}{{{2}}}\)

(D) \(\frac{{{c}\sqrt{{{3}}}}}{{{2}}}\)

Why spin angular velocity of a star is greatly enhanced when it collapses under gravitational pull and becomes a neutron star ?

How many of \({g}\) of \({S}\) are required to produce \({10}\text{moles}\) and \({10}{g}\) of \({H}_{{{2}}}{S}{O}_{{{4}}}\) respectively?

The pair of compounds which cannot exist in solution is:

(A) \({N}{a}{H}{C}{O}_{{{3}}}\) and \({N}{a}{O}{H}\)

(B) \({N}{a}_{{{2}}}{S}{O}_{{{3}}}\) and \({N}{a}{H}{C}{O}_{{{3}}}\)

(C) \({N}{a}_{{{2}}}{C}{O}_{{{3}}}\) and \({N}{a}{O}{H}\)

(D) \({N}{a}{H}{C}{O}_{{{3}}}\) and \({N}{a}{C}{l}\)

(A) \({N}{a}{H}{C}{O}_{{{3}}}\) and \({N}{a}{O}{H}\)

(B) \({N}{a}_{{{2}}}{S}{O}_{{{3}}}\) and \({N}{a}{H}{C}{O}_{{{3}}}\)

(C) \({N}{a}_{{{2}}}{C}{O}_{{{3}}}\) and \({N}{a}{O}{H}\)

(D) \({N}{a}{H}{C}{O}_{{{3}}}\) and \({N}{a}{C}{l}\)

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