PROBLEMS ON PERMUTATIONS AND COMBINATIONS, NUMBER THEORY, CIRCLES, SIMILARITY

1. GRE QUANT
PERMUTATIONS AND COMBINATIONS, ALGEBRA,NUMBER
THEORY,CIRCLES, SIMILARITY

2. Question 1
A manager has the choose 6 persons. From the 11 candidates available, he has already chosen 3.
How many different combinations of the remaining candidates does the manager choose from
Since the manager has already chosen 3 candidates, he now has to choose 3 out of 8.
This can be done in 8𝐶3 =56 ways

3. Question 2
2 − 5𝑥 ≤
6𝑥 − 5
3
, 𝑥 ∈ 𝑅
Represent on a number line
6 − 15𝑥 ≤ 6𝑥 − 5
11≤ 21𝑥
11
21
≤ 𝑥
0.5 ≤ 𝑥
0. 0.5. 1

4. Question 3
S consists of all the positive integers <81 that are not equal to the square of an intager
Quantity A Quantity B
Number of integers in S 72
Squares of integers <81 are 1, 4, 9, 16, 25, 36, 49, 64
These are 8 in number.
Number of integers in S = 80 – 8 = 72
The 2 quantities are equal

5. Question 4
What arclength is intercepted by an inscribed angle of 210 on a circle with radius = 12
B
21
A B
42
Required to find arclength AB
Angle at the centre = 420
𝑎𝑟𝑐𝑙𝑒𝑛𝑔𝑡ℎ 𝐴𝐵 = 𝑟𝜃
= 12 × 42 ×
𝜋
180
=17.6

6. A triangle is inscribed in a circle with shorter sides 3 cm and 4 cm. If the longer side is a diameter,
Find its length
If the longer side is a diameter, we use the
property, angle in a semicircle is a
right angle
3
4
By Pythagoras’ theorem, longer side is the hypotenuse which is 5 cm
QUESTION 5

7. Triangles PQR and XYZ are similar. If PQ= 6cm, PR = 4 cm, XY = 9 cm, find XZ
QUESTION 6
We draw a rough sketch
R
p
Q R
X
Y Z
6 4
9
x

8. We use the following property
Corresponding sides of similar triangles are proportional
𝑃𝑄
𝑋𝑌
=
𝑃𝑅
𝑋𝑍
6
9
=
4
𝑥 x= 6 cm
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