MPTET Varg 2 Math Mock Test - 8 - MPTET MCQ

Given:

Some triangles and quadrilaterals are given.

Find the line of symmetry.

Concept used:

The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shapes and divide into two identical halves.

Calculation:

In isosceles triangle,

Only one line of symmetry.

In Kite,

Only one line of symmetry.

In rhombus,

Two lines of symmetry.

In rectangle, Two line of symmetry.

In scalene triangle,

No any line of symmetry.

In parallelogram,

No any line of symmetry.

∴ Option 4 both have no same line of symmetry.

∴ Option 4 is correct.

Given:

log10 (x - 9) + log10 x = 1

Concept Used:

log_b (M) + log_b (N) = log_b(MN)

Calculations:

log10 (x - 9) + log10 x = 1

⇒ log10 (x - 9)(x) = 1

⇒ log10 (x - 9)(x) = log₁₀10

⇒ x² - 9x - 10 = 0

⇒ x² - 10x + x - 10 = 0

⇒ x(x - 10) + 1(x - 10) = 0

⇒ (x + 1)(x - 10) = 0

⇒ x = -1 and x = 10

∴ option 4 is correct

Given:

The altitude drawn to the base of an isosceles triangle is 8 cm and the perimeter of the triangle is 32 cm.

Concept used:

Area of a triangle = 1/2 × base × altitude

Calculation:

Let the equal sides of the isosceles triangle be x

So, base = 32 - 2x

As we in an isosceles triangle the altitude drawn to the base bisects the base

So, according to that,

BD = DC = BC/2 = 16 - x

Now,

8² + (16 - x)² = x²

⇒ 64 + 256 + x² - 32x = x²

⇒ 32x = 320

⇒ x = 10

So, base is 32 - 20

⇒ 12 cm

Now,

Area = 1/2 × 8 × 12

⇒ 48 cm²

∴ The required answer is Option 4.

Concept:

Formula of Logarithms:

 here a ≠ 1 and a > 0 and x be any number.

Properties of Logarithms:

Calculation:

Given: log 2 = 0.301 and log 5 = 0.699.

As we know that, 

⇒ log (5 / 2) = log (5) - log (2)

By substituting the values of log 5 and log 2 in the above equation we get,

⇒ log (5 / 2) = 0.699 - 0.301 = 0.398

Hence, log (5 / 2) ≈ 0.4.

A mathematical game is an activity that engages children for amusement or fun. It should be a part of the mathematics classroom as it will be a child-centered and happy classroom where children get opportunities to observe, explore and share experiences.

Key Points

Characteristics of mathematical games-

• Mathematical games must not be confused with open-ended play and there is a set of rules that governs decision making. The rules should be based on mathematical ideas and the game should end when the goal is reached.
• Mathematical games provide non- didactic feedback to the children be the minimum amount of teacher intervention.
• Mathematical games promote processes of anticipation and planning in children which help them stimulate interest in a topic and develop mathematical skills.

Hence, we conclude that the correct statements are (b) and (c).

Given:

The angle of elevation of the top of a building at a distance of 70 m from its foot on a horizontal plane is found to be 60°.

Concept used:

tanθ = P/B

Here,

P = Perpendicular

B = Base

Calculation:

Let AB be the building and BC be the distance

Now,

In ΔABC

tan60° = AB/BC

⇒ √3 = AB/70

⇒ AB = 70√3

So, the height = 70√3 m

∴ The height of the building is 70√3 m.

Given:

The area of the rhombus = 240 cm2

Length of the one diagonal = 16 cm

Formula used:

(1.) Area of rhombus = 1/2 × d₁ × d₂

Where,

d₁ = length of the first diagonal

d₂ = length of the second diagonal

a = side length of the rhombus

(3.) Perimeter of rhombus = 4a

Calculation:

According to the question,

Now,

⇒ 15² + 8² = a2

⇒ a² = 225 + 64

⇒ a2 = 289

⇒ a = 17 cm

Perimeter of rhombus = 4 × 17 = 68 cm

Therefore, '68 cm' is the required answer.

Given:

47 + {7 + [61 − (21 ÷ 3)] ÷ 9}

Concept Used:

Follow the BODMAS rule according to the table given below:

Calculation:

47 + {7 + [61 − (21 ÷ 3)] ÷ 9}

= 47 + {7 + [61 − 7] ÷ 9}

= 47 + {7 + 54 ÷ 9}

= 47 + {7 + 6}

= 47 + 13

= 60

Hence, the required value is 60.

Mathematics is interesting by using games, quizzes, riddles, tricks, etc. as learning activities. Every child deserves to be successful and confident in school life. Mathematical puzzles, riddles, and tricks develop many life skills in learners required to lead a successful and happy life. Children love games and puzzles, so learning through games makes the learning of Mathematics joyful and interesting.

Important Points

Benefits of recreational activities to develop following skills

• Encourage mathematical reasoning and logical thinking;
• Develop strategic thinking;
• Guide in concept development;
• Develop a positive attitude towards Mathematics;
• Help to develop the construction of ideas;
• Develop a variety of connections with the content that can form positive memories of learning;
• Improve fundamental operations of Mathematics;
• Help teachers to cater to the diverse needs of children;
• Help in the development of problem-solving, critical thinking skills, decision-making skills, etc.

Hence, we have concluded that It is important to conduct mathematical recreational activities and challenging geometrical puzzles in the class as they are helpful to enhance the spatial and analytical ability of every learner.

Explanation:

Screw gauge: A screw gauge is an instrument that is used to find the diameter of the wire or thin metal sheet.

The least count of the Screw gauge is equal to 0.01 mm.

The least count of the screw gauge

The number of divisions on a circular scale = 50

0.01 mm 

⇒ Pitch = 0.5 mm

Hence, option 1) is the correct answer.

Given:

Ratio of radius = 2 : 5

Ratio of Heights = 5 : 7

Formula Used:

Volume of cylinder = πr²h

(where r = radius, h = height)

Calculation:

Let, the radii be 2x and 5x and the heights be 5y and 7y.

For 1st cylinder, radius = 2x, height = 5y

For 2nd cylinder, radius = 5x, height = 7y

According to question,

Ratio of their volume(1st : 2nd) = π × (2x)² × 5y : π × (5x)² × 7y

⇒ 4x² × 5y : 25x² × 7y

⇒ 4 : 35

∴ The ratio of their volume is 4 : 35.

Concept used:

If the speed of a man in still water is u km/hr. and the speed of the current is v km/hr., then we can formulate the below relations:
The speed with current = (u+v) km/hr. or downstream
Speed against current = (u-v) km/hr. or upstream

Calculations:
We have been given a man's speed with the current is 12 km/hr. and the speed of the current is 1.5 km/hr.
Therefore, we will substitute the known values from the question in relation as follows,
Man's speed (with current) = u+v = 12 km/hr. and v = 1.5 km/hr.

u = 12 - v = 12 - 1.5 = 10.5 km/hr

we actually need to find the speed of man against the current of water. We have both values u and v, so we can put these values in above stated formula for the speed of man against the current which is
Speed against the current of water = u-v
Now we know u = 10.5 km/hr and v = 1.5 km/hr
Putting these values in the formula u-v = 10.5 - 1.5 = 9 km/hr

Hence, the correct option is 1.

Given:

PT is a tangent to a circle, whose centre is O.

OP = 17 cm

OT = 8 cm

Formula used:

Pythagoras Theorem: In a right-angle triangle, the hypotenuse is equal to -

OP² = OT² + TP²

Calculation:

This, the length of the tangent TP is equal to:

TP2 = OP2 - OT2

TP2 = (17)2 - (8)2

TP2 = 289 - 64

TP2 = 225

TP = √225 = 15

∴ The length of the tangent segment PT is equal to 15 cm.

Given:

The circumference of the two circles is 110 cm and 330 cm respectively.

Concept used:

Circumference of a circle = 2πR (R = Radius)

Calculation:

Let the radii of the bigger and smaller circles be R and r respectively.

Difference between their radii = (R - r)

According to the concept,

2πR - 2πr = 330 - 110

⇒ 2π(R - r) = 220

⇒ 44/7 × (R - r) = 220

⇒ (R - r) = (220 × 7)/44

⇒ (R - r) = 35

∴ The difference between their radii is 35 cm.

Given:

LCM = 4104 , HCF = 9, one of the number = 171

Concept used:

LCM × HCF = Product of the numbers

Calculations:

LCM × HCF = Product of the numbers

⇒ 4104 × 9 = 171 × other number

⇒ other number = 4104 × 9/171 = 216

Hence, The Required value is 216.

Calculation:

Here we can see that when DE || BC

⇒ AD/AB = AE/AC

And

⇒ AD/DB = AE/EC

And

⇒ DB/AB = CE/AC

∴ We can see that all options are satisfied except for option 3.

∴ The correct answer is option 3.

Given:

The numbers are 12/5, 14/15 and 16/17

Concept used:

HCF = HCF of numerator/LCM of the denominator

Calculation:

LCM of 5, 15, 17 = 255

HCF of 12, 14, 16 = 2

So, HCF = 2/255

∴ The required answer is 2/255.

Concept:

• Radiation: The process of the emission of heat energy by hot body in form of waves or particles is called radiation.
• Sand radiates heat more quickly than the earth.
• Further, deserts have very little water vapor in the air. Trapping of heat is thus difficult.

Explanation:

• Since sand radiates heat more quickly than plain, heat gets radiated much quickly than plain which makes the desert cooler.
• So, Nights are cooler in the deserts than in plains because sand radiates heat more quickly than earth.

Important Points

• Sand absorbs heat energy from the sun more quickly than air, thus days are hotter than plain.
• This factor is responsible for hotter days in deserts.
• Further this absorbed heat is radiated quickly when sunsets and makes night cooler.

Given:

Base Area = 154 sq cm

CSA = 550 sq cm

Formula used:

CSA(cone) = π x r x l

r = radius of the base

l = slant height of the cone

Base area = 22/7 x r x r

Volume(Cone) = 22/7 x r x r x h

Calculation:

Base area = 22/7 x r x r = 154

r = 7 cm

CSA(cone) = 22/7 x r x l = 550

l = 25 cm

h² = 25² - 7²

h = 24 cm

Volume(Cone) = 1/3x 22/7 x r x r x h = 1/3 x 22/7 x 7 x 7 x 24

Volume = 1232 cm³

Hence, the correct answer is option 1).

Given:

The summation of n terms for the given series =

Formula used:

Calculation:

According to the question,

The sum of two terms = 13

Therefore,

Average = 13/2 = 6.5

Therefore, '6.5' is the required answer.

Concept:

Physical quantity:

• The quantities which are measurable are known as a physical quantities.
• Types of quantity are:
  • Scalar
  • Vector

Explanation:

Momentum:

• Momentum is the quantity of motion of a moving body.
• It is also measured as the product of its mass, and velocity.
• Its SI unit is Kgms^-1.

Impulse:

• Impulse is a term that quantifies the overall effect of a force acting over time.
• Its SI unit is Ns or Kgms-1.

Thus, the correct answer is momentum, and impulse.

Additional Information

Velocity:

• Velocity is a vector quantity that refers to the rate at which an object changes its position.
• Its SI unit is ms^-1.

Acceleration:

• Acceleration is a vector quantity that refers to the rate at which an object changes its velocity.
• Its SI unit is ms-2.

Given:

Amount = Rs. 8800

Time = 4 years

Rate = 25%

Formula Used:

Amount = Principal + SI

P = principal

R = Rate of interest

T = Time

Calculation:

⇒ 8800 - P = P

⇒ 8800 = 2P

⇒ P = Rs. 4400.

Given:

The exterior angle of regular octagon is x degrees.

Formula used:

Each exterior angle of the polygon = 360^o/n

Where, n = number of sides of a polygon

Calculation:

A regular octagon will have all its sides equal in length.

Total sides of octagon = 8.

So, the exterior angle (x) = 360/8 = 45°

∴ The answer is 45°

Calculation:

From the given diagram,

2x = 120°

x = 60°

2z = 60°

z = 30°

So, a = 180 - 60 - 30 = 90°

2t = 50°

t = 25°

So,

b = 180 - 30 - 25 = 125°

So, a + b = 215°

∴ Value of a + b is 215°

Given:

Dimensions = 12 cm × 6 cm × 4 cm

Concept used:

Diagonal of the cuboid =

Calculation:

The length of the longest rod that can be kept in a rectangular box = Diagonal of the cuboid

Diagonal of the cuboid =

∴ The correct answer is 14 cm.

Given:

sides = 10, 6.5 and a

Concept used:

The third side of the triangle is always, less than the sum of two other sides of the triangle.

Solution:

10 < a="" +="" />

a >= 4

The least value a can have is 4 as a whole number.

Hence, the correct option is 4.

Each number is perfect square except 181.

169 = 13²

121 = 11²

Given:

sin² 45° + cos2 45° - tan2 45°

Concept used:

Calculation:

sin² 45° + cos² 45° - tan² 45°

∴ sin² 45° + cos² 45° - tan² 45° is equal to 0.

Concept:

The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is obtained by using the Pythagoras' Theorem: d² = (x₁ - x₂)² + (y₁ - y₂)².

Calculation:

Using the formula for the distance between two points:

AP² = (x - 4)2 + (y - 1)2

BP² = (x + 1)2 + (y - 4)2

Since the distances are equal, we have:

AP² = BP²

⇒ (x - 4)2 + (y - 1)2 = (x + 1)2 + (y - 4)2

⇒ x² - 8x + 16 + y² - 2y + 1 = x² + 2x + 1 + y² - 8y + 16

⇒ 10x = 6y

⇒ 5x = 3y.

Given:

The cost price of 10 articles equals the selling price of 12 articles.

Concept:

Solution:

As per the question,

The cost price of 10 articles equals the selling price of 12 articles.

10 CP = 12 SP

Loss % = 16.66 %

Hence, option 3 is correct.