Numerical Ability Test

Result

Numerical Ability Test - 13

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Weekly Quiz Competition

Question 1
5 / -1

ABCD is a cyclic quadrilateral and BC is a diameter of the circle. If ∠DBC = 29°, then ∠BAD = ?

A
122°

B
119°

C
129°

D
111°

Solution

Given:
∠DBC = 29°
Concept used:
Angle in a semicircle is a right angle.
We know that sum of all the sides of triangles is 180°
The opposite angles in a cyclic quadrilateral is 180°
Calculation:
According to the question
∠DBC = 29°
∠BDC = 90° [Angle in a semicircle is 90°]
We know that sum of all the sides of triangles is 180°
⇒ ∠DBC + ∠BDC + ∠BCD = 180°
⇒ 29° + 90° + ∠BCD = 180°
⇒ 119° + ∠BCD = 180°
⇒ ∠BCD = (180° – 119°)
⇒ ∠BCD = 61°
Now,
We know that, the opposite angles in a cyclic quadrilateral is 180°
So,
⇒ ∠BAD + ∠BCD = 180°
⇒ ∠BAD + 61° = 180°
⇒ ∠BAD = (180° – 61°)
⇒ ∠BAD = 119°
∴ The required value of ∠BAD is 119°
Question 2
5 / -1

ABCD is a cyclic quadrilateral in which AB = 21 cm, BC = 16.8 cm and CD = 14 cm. If AC bisects BD, then the length of AD is:

A
25.2 cm

B
24.8 cm

C
18.2 cm

D
22.8 cm

Solution

Given:
ABCD is a cyclic quadrilateral
AB = 21 cm
BC = 16.8 cm
CD = 14 cm
Concept Used:
Cyclic quadrilateral: A quadrilateral whose all the four vertices lie on the circle.
If a diagonal of the cyclic quadrilateral is bisected, then the products of the adjacent sides touching it is equal.
Hence, AB × BC = AD × DC
Calculation:
According to the question,
In the figure, ABCD is a cyclic quadrilateral and AC bisects BD
Now, according to the concept:
⇒ 21 × 16.8 = AD × 14
⇒ AD = (21 × 16.8)/14 = 25.2 cm
∴ The length of AD is 25.2cm.
Important Points
The sum of pair of opposite angles of a cyclic quadrilateral is\(180^\circ\)
⇒\(\angle DAB~+~\angle BCD~=~180^\circ\)⇒ \(\angle ABC~+~\angle CDA~=~180^\circ\)
Note: Inverse of this theorem is also true.
Question 3
5 / -1

Find the area of the quadrilateral shown in the following figure (underlined not measured)

A
144

B
54

C
63

D
123

Solution

Formula used:
Area of triangle = 1/2 × base × height
Calculation:
According to the question
Area of quadrilateral = Area of ΔABC + Area of ΔADC
⇒ (1/2 × 15 × 15) + (1/2 × 3 × 21)
⇒ (225/2) + (63/2)
⇒ (225 + 63)/2
⇒ 288/2
⇒ 144
∴ The required area of quadrilateral is 144
Question 4
5 / -1

Two angles of a quadrilateral are 120° each. If the other two angles are also equal then what is the measurement of each of the other two angles?

A
80°

B
70°

C
90°

D
60°

Solution

Given:
Two angles of a quadrilateral are 120° each.
Other two angles are also equal
Formula used:
Sum of angles of a quadrilateral = 360°
Calculation:
Let the other two angles be x° each
⇒ 120° + 120° + x° + x° = 360°
⇒ 2x° = 120°
⇒ x° = 60°
∴ Measurement of each of the other two angles is 60°
Question 5
5 / -1

Looking at the given parallelogram, where angle B and D are (7x - 30)° and 5x° respectively, we can say that the value of x is:

A
105°

B
65°

C
15°

D
75°

Solution

Concept:
The opposite angles of a parallelogram are equal
Calculation:
According to the question
⇒ (7x – 30)° = 5x°
⇒ (7x – 5x) = 30
⇒ 2x = 30
⇒ x = 15°
∴ The value of x is 15°
Question 6
5 / -1

The number of diagonals in a 19 - gon is:

A
161

B
133

C
171

D
152

Solution

Formula used:
No. of diagonals in polygon = {n(n – 3)/2}
Calculation:
According to the question
⇒ {19(19 – 3)/2}
⇒ {19(16/2)}
⇒ (19 × 8)
⇒ 152
∴ The required number of diagonals in a 19 - gon is 152
Question 7
5 / -1

If the sum of interior angles of a polygon is 3420°, find the number of sides of the polygon.

A
19

B
20

C
21

D
22

Solution

Given:
The Sum of interior angles of the polygon is 3420°
Formula used:
Sum of interior angles of polygon = (n - 2) × 180
Where n = number of sides
Calculation:
According to the question:
(n - 2) × 180 = 3420
⇒ n - 2 = 19
⇒ n = 21
∴ The number of sides of the polygon is 21
Question 8
5 / -1

In the given figure, ABCD is a parallelogram in which ∠DCB = 85° and ∠ADB = 55° then, ∠ABD is equal to:

A
35°

B
40°

C
45°

D
50°

Solution

Given:
∠DCB = 85°
∠ADB = 55°
Concept used:
In a parallelogram, the opposite angles are equal and the opposite sides are parallel
The sum of all the angles of a triangle is 180°
Calculations:
Opposite angles of the parallelogram
⇒ ∠DCB = ∠DAB
⇒ ∠DAB = 85°
Sum of angles of triangle = ∠ABD + ∠ADB + ∠DAB
⇒ ∠ABD + 55° + 85° = 180°
⇒ ∠ABD = 180° - 140°
⇒ ∠ABD = 40°
∴ The value of ∠ABD is 40°
Question 9
5 / -1

BD is a diagonal of a rhombus ABCD. If ∠ADB = 50°, then the measure of ∠DCB is

A
60°

B
75°

C
80°

D
100°

Solution

Given:
BD is a diagonal of a rhombus ABCD
∠ADB = 50°
Formula used:
Sum of all interior angles of a triangle = 180º
Opposite angles are equal for a rhombus.
Angles opposite to equal sides are equal.
Calculation:
ABCD is a rhombus.
We know that all sides are equal for a rhombus.
Therefore, AB = BC = CD = DA
For △ABD, AB = DA
∴ ∠ADB = ∠ABD = 50°
Now, ∠BAD = 180º - (50° + 50°) = 80°
For ABCD rhombus, ∠BAD = ∠DCB
Because, opposite angles are equal for a rhombus.
∴ ∠DCB = 80°
∴ The measure of ∠DCB is 80°.
Question 10
5 / -1

In the given figure, AB is the diameter of the circle. ∠ADC = 100°. Find ∠CAB

A
30°

B
10°

C
40°

D
50°

Solution

Concept used:
The sum of opposite angles of a cyclic quadrilateral is 180°.
The angle made by the diameter of a circle at any point of the circle is 90°.
Calculation:
In the given figure, ABCD is a cyclic quadrilateral.
⇒ ∠D + ∠B = 180°
⇒ 100° + ∠B = 180°
⇒ ∠B = 180° - 100° = 80°
Now, in ΔABC,
∠ACB is made by diameter AB
⇒ ∠ACB = 90°
So, in ΔABC,
∠CAB + ∠B + ∠ACB = 180°
⇒ ∠CAB + 80° + 90° = 180°
⇒ ∠CAB = 180° - 170° = 10°
∴ The value of ∠CAB is 10°.