## Saturday, June 7, 2008

### Solutions to June 1 2008 practice set

1. State whether the statement is true or false

A projectile fired form the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.

At the top of the path, the vertical component of velocity is zero and the particle has only horizontal component. Hence the speed (the magnitude of the velocity) is minimum at the top of the path.

2. The number of vectors of unit length perpendicular to vectors a = (1,1,0) and b = (0,1,1) is

a. three
b. two
c. one
d. infinite
e. none of these

If vector (x,y,z) is a unit vector perpendicular to (1,1,0) their scalar product is zero
=> x +y = 0
=> y = -x

If vector (x,y,z) is a unit vector perpendicular to (0,1,1) their scalar product is zero
=> y+z = 0
=> y = -z

As it is a unit vector x²+y²+z² = 1
=> 3y² = 1
=> y = ±1/SQRT(3)

This gives two value of y and hence there are two vectors

3. Let z1 and z2 be complex numbers such that z1≠z2 and |z1| = |z2|. If z1 has positive real part and z2 has negative imaginary part, then (z1+z2)/(z1-z2) may be

a. real and positive
b. zero
c. real and negative
d. purely imaginary
e. none of these

Assume z1 = a+ib, a is positive means a>0,
z2 = c+id, d<0

|z1| = |z2|
=> a²+b² = c²+d²

(z1+z2)/(z1-z2) = [(a+c) + i(b+d)]/ [(a-c) + i(b-d)]

Taking the multiplicative inverse of denominator and multiplying the numerator

= [(a+c) + i(b+d)]* [(a-c) - i(b-d)]/[(a-c) ² + (b-d) ²]

= [(a²-c²)+(b²-d²)]+i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]

=[{(a²+b²)-(c²+d²)}+i{(b+d)(a-c)-(a+c)(b-d)}]/ [(a-c) ² + (b-d) ²]

As a²+b² = c²+d² the first term of the numerator is zero.

Hence
(z1+z2)/(z1-z2)
= i[(b+d)(a-c)-(a+c)(b-d)]/ [(a-c) ² + (b-d) ²]

So this is a pure imaginary number.
If b+d as well as a+c are equal zero, the expression can be zero also.

4. If two compounds have the same empirical formula but different molecular formulae they must have ------

a. different percentage composition
b. different molecular weights
c. same viscosity
d. same vapour density

Different molecular formulae but same empirical formula imply same empirical mass but different molecular masses