GATEWAY TO CGL MAINS 2015 : Quiz on Advanced Maths

Quadrilateral: A quadrilateral is any closed shape that has four sides. The sum of the measures of the angles is 360⁰. Some of the known quadrilaterals are square, rectangle, trapezium, parallelogram and rhombus.

Square: A square is regular quadrilateral that has four right angles and parallel sides. The sides of a

square meet at right angles. The diagonals also bisect each other perpendicularly.

If the side of the square is a, then its perimeter = 4a, area = a^2 and the length of the diagonal = √2a

Rectangle: A rectangle is a parallelogram with all its angles equal to right angles.

Properties of a rectangle:

Sides of rectangle are its heights simultaneously.

Diagonals of a rectangle are equal: AC = BD.

A square of a diagonal length is equal to a sum of squares of its sides’ lengths, i.e. AC² = AD² + DC².

Area of a rectangle = length × breadth

Parallelogram: A parallelogram is a quadrangle in which opposite sides are equal and parallel.

Any two opposite sides of a parallelogram are called bases, a distance between them is called a height.

Area of a parallelogram = base × height

Perimeter = 2(sum of two consecutive sides)

Properties of a parallelogram:

1. Opposite sides of a parallelogram are equal(AB = CD, AD = BC ).

2. Opposite angles of a parallelogram are equal ( A = C, B = D ).

3. Diagonals of a parallelogram are divided in their intersection point into two :

( AO = OC, BO = OD ).

4. A sum of squares of diagonals is equal to a sum of squares of four sides:

AC² + BD² = AB² + BC² + CD² + AD².

Rhombus: If all sides of parallelogram are equal, then this parallelogram is called a rhombus.

Diagonals of a rhombus are mutually perpendicular ( AC perpendicular BD ) and divide its angles into two ( DCA = BCA, ABD = CBD ) etc. 

Area of a rhombus = 1/2 x product of diagonals 

= 1/2 x AC x  BD

REGULAR POLYGONS : A regular polygon is a polygon with all its sides equal and all its interior angles equal. All vertices of a regular lie on a circle whose center is the center of the polygon.

•Each interior angle of a regular polygon = 180(n-2)/n

•Sum of all the angles of a regular polygon = n x180(n- 2)/n = 180(n-2).

Some Questions based on above Concept:

1.How many diagonals are there in a octagon ? 

A.10 

B.14 

C.18 

D.20 

2.The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4,  the largest angle is : 

A.120°

B.134°

C.144°

D.150°

3.The ratio of the measure of an angle of a regular octagon to the measure of its exterior angle is: 

A.1 : 3

B.2 : 3

C.3 : 1 

D.3 : 2

4.The sum of the interior angles of polygon is 1440°. The number of sides of the polygon is: 

A.9 

B.10 

C.8 

D.12 

5.In the adjoining figure, the value of x and y are (here ABCD is a parallelogram) :

A.6,4  

B.5,4 

C.4, 5 

D.None of these

6.The diagonals of rectangle ABCD meet at O. If ∠BOC = 44°, then ∠OAD is equal to: 

A.90°

B.60°

C.100°

D.68°

7.ABCD is cyclic trapezium whose sides AD and BC are parallel to each other. If ∠ABC = 72°, then the measure of the ∠BCD is: 

A.162°

B.18°

C.108°

D.72°

8.A regular polygon is inscribed in a circle. If a  side subtends an angle of 36° at the centre, then the number of sides of the polygon is: 

A.5

B.10 

C.12 

D.9

9.ABCD is a ∥ gm, E is the mid-point of AB and CE bisects ∠BCD. Then ∠DEC is :  

A.60°

B.90°

C.100°

D.120°

10.Each interior angle of a regular polygon is 18° more than eight times an exterior angle. The number of sides of the polygon is: 

A.10 

B.15 

C.20 

D.25 
SOLUTION 
1.(d) No. of diagonals of a polygon of n sides 
= n(n-3)/2=(8(8-3))/2=20

2.(c) angles be x, 2x, 3x, 4x 
so x + 2x + 3x + 4x = 360°
⇒10x=360° ⇒x=36°
so  Largest angle = 4x = 144°

3.(c)Exteteror angle = 360/8 = 45 
Interior angle = 180 – 45 = 135
Interior angle : exterior angle s
= 135 : 45 = 3 : 1

4.(b) Sum of all interior angles = 
(n -2) * 180°
1440 = (n -2) * 180 
N = 10(n ----- no. of sides)

5.(a) 7x = 42
X = 6  and 8y = 32 
Y = 4 

6.(d) Since, the diagonals of a rectangle bisect each other. 
so OA = OD ⇒ ∠ODA= ∠OAD
But, ∠AOD = 44 (vertically opposite angle to ∠BOC) 
so OAD = ½ (180° – 44°)
= ½ (136°) = 68°

7.(d)

 ∠ABC + ∠CDA = 180°
∠CDA = 180° – 72°
= 108°
so ∠ADE = 180° – 108°
= 72°
AD∥ BC 
∠BCD = ∠ADE = 72° (corresponding angles)

8.(b) Let no. of sides = n 
Each equal side subtends equal angle at the centre. 
n * 36 = 360°
n = 360/36 = 10

9.(b) 
10.(c) (n-2)/n * 180 
= 8 * 360°/n + 18 
=>(n -2) * 10 = 160 + n 
=>10n – 20 = 160 + n 
=>9n = 180 
=>n = 20