Free Student aptitude test, this topic is mainly used to preparation for campus recruitment program. It involves several topics. But here we can revise the few topics. The topics are arithmetic and geometric progression and analytical geometry. Analytical geometry involves the main topics such as find the equation of the line and slope of the line. These are revised problems with solutions.

**Free student aptitude test Example 1:**

** ** Prove that the sum of n arithmetic means between two numbers is n times the single
A.M between them.

**Solution:**

Let A_{1}, A_{2}… A_{n} be the n A.M’s between a and b.

From the example

A1+ A2 + A3 + … + An = (n(a+b))/2

= n × (A.M between a and b)

= n (single A.M between a and b)

**Free student aptitude test Example 2:**

** ** If A(-2,5) and (4,-5). Then solve the equation of the locus of a point. Given
Statement PA_{2} – PB_{2} = 20.

**Solution:**

A(− 2, 5) and B(4, − 5) are the two given points. Let P(x_{1}, y) is any point on the locus.
Given that PA_{2} – PB_{2} = 20.

(x_{1} + 2)^{2} + (y_{1} − 5)^{2} − [(x_{1}− 4)^{2} +
(y_{1} +5)^{2}] = 20

x_{1} ^{2} + 4 x_{1} + 4 + y_{1} ^{2} − 10 y_{1} + 25
− [x_{1} ^{2} −8 x_{1} + 16 + y_{1} ^{2} + 10 y_{1} + 25] = 20

Simplifying the equation we can get,

12 x_{1} − 16 y_{1} − 32 = 0

Divided by2 we can get,

I.e. 6 x_{1} − 8y_{1} − 16 = 0

Again divided by 2 we get,

3 x_{1}-8 y_{1}-16 =0

The locus of (x_{1}, y_{1}) is 3x − 8y − 16 = 0

**ree student aptitude test Example 3:**

** ** Find the equation of the straight line parallel to 3x + 2y = 9 and which passes
through the point (4, − 3).

**Solution:**

The straight line parallel to 3x + 2y − 9 = 0 is of the form

3x + 2y + k = 0 … (1)

The point (4, − 3) satisfies the equation (1)

Hence 12 − 6 + k = 0 i.e. k = − 6

∴ 3x + 2y − 3 = 0.

This is the equation of the given line **3x + 2y = 9** which passes through the given point