Solved Examples

1. LCM of 27,314and53is${\displaystyle \frac{2}{7},{\displaystyle \frac{3}{14}\mathrm{a}\mathrm{n}\mathrm{d}{\displaystyle \frac{5}{3}\mathrm{i}\mathrm{s}}}}$

a. 45

b. 35

c. 30

d. 25

Correct Option: C

Explanation:

LCM of numeratorsHCF of denominators=LCM of 2,3,5HCF of 7,14,3=301=30${\displaystyle \frac{\text{LCM of numerators}}{\text{HCF of denominators}}=\frac{\text{LCM of 2,3,5}}{\text{HCF of 7,14,3}}=\frac{30}{1}=30}$

2. About the number of pairs which have 16 as their HCF and 136 as their LCM, the conclusion can be

a. only one such pair exists

b. only two such pairs exist

c. many such pairs exist

d. no such pair exists

Correct Option: D

Explanation:

HCF is always a factor of LCM. ie., HCF always divides LCM perfectly.

3. The HCF of two numbers is 12 and their difference is also 12. The numbers are

a. 66, 78

b. 94, 106

c. 70, 82

d. 84, 96

Correct Option: D

Explanation:

The difference of required numbers must be 12 and every number must be divisible by 12. Therefore, they are 84, 96.

4. The HCF of two numbers is 16 and their LCM is 160.  If one of the numbers is 32, then the other number is 

a. 48

b. 80

c. 96

d. 112

Correct Option:b

Explanation:

The number = HCF×LCMGiven number=16×16032=80

5. HCF of three numbers is 12. If they are in the ratio 1:2:3, then the numbers are

a. 12,24,36

b. 10,20,30

c. 5,10,15

d. 4,8,12

Correct Option: A

Explanation:

Let the numbers be a, 2a and 3a.

Then, their HCF = a  so a=12

The numbers are 12,24,36

Geometry - Lines

Geometry is basically the study of different properties of point, line and plane.

Angle
Right angle: An angle of 90° is called a right angle.
Acute angle: An angle of less than 90° is called an acute angle.
Obtuse angle: An angle of greater than 90° is called an obtuse angle.
Reflex angle: An angle of greater than 180° is called reflex angle.
Complementary angles: If the sum of the two angles is 90°, then they are called complementary angles.
Supplementary angles:  If the sum of the two angles is 180°, then they are called supplementary angles.

Angle Relations: 

Vertically opposite angles: If two lines intersect each other, then the vertically opposite angles are equal.
∠$\mathrm{\angle }$a = ∠$\mathrm{\angle }$c and ∠$\mathrm{\angle }$b = ∠$\mathrm{\angle }$d
a, c and b, d are vertically opposite angles.

Angles in parallel lines: If a transversal intersects two parallel lines:

a. each pair of consecutive interior angles are supplementary.
∠$\mathrm{\angle }$4 + ∠$\mathrm{\angle }$5 = 180°
∠$\mathrm{\angle }$3 + ∠$\mathrm{\angle }$6 = 180°

b. each pair of alternate interior angles are equal.
∠$\mathrm{\angle }$5 =∠$\mathrm{\angle }$3, ∠$\mathrm{\angle }$6 = ∠$\mathrm{\angle }$4
∠$\mathrm{\angle }$5 = ∠$\mathrm{\angle }$3 =∠$\mathrm{\angle }$1 =∠$\mathrm{\angle }$7
∠$\mathrm{\angle }$6 = ∠$\mathrm{\angle }$4 = ∠$\mathrm{\angle }$2 = ∠$\mathrm{\angle }$8

Collinear: Three or more than three points are said to be collinear, if there is a line which contains them
all.
Concurrent: Three or more than three lines are said to be concurrent, if there is a point which lies on them
all.

Proportionality Theorem:

In the above diagram, ABBC=DEEF${\displaystyle \frac{AB}{BC}=\frac{DE}{EF}}$ or ACBC=DFEF${\displaystyle \frac{AC}{BC}=\frac{DF}{EF}}$