MPTET Varg 2 Math Mock Test - 5 - MPTET MCQ

Concept:

Energy

• The capacity of doing work is called energy.
• SI unit of energy is the joule.
• Various forms of energy are heat energy, light energy, nuclear energy, electrical energy, mechanical energy, etc.
• Energy can be converted from one form to another.
• This happens through various processes or using certain appliances.
• For example, when a bulb glows by electricity, electrical energy is converted into light energy.

Explanation:

Windmill

• A windmill has large blades just as a fan.
• It wind rotates the blades attached to the turbine.
• The turbine rotates and electrical energy is produced.
• Energy to the motion is kinetic energy.
• So, the windmill converts the kinetic energy of blades into electrical energy.

Electric cell

• Chemical reactions take place in the cell.
• This makes a potential difference between the two terminals.
• When connected with the conductor, current flows.
• In this way, chemical energy is converted into electrical energy.

Electric bulb

• An electric bulb converts the electrical energy into light and heat energy.
• A filament present in the bulb burns and light energy is produced out of it.

Microphone

• A microphone receives the sound signal and converts it into electrical signals.
• In the microphone, there is a diaphragm.
• It vibrates and the diaphragm makes the coil attached to it vibrate.
• A permanent magnet near the coil induces a current in the coil by the process of electromagnetic induction.
• So, sound energy is converted into electrical energy.

Conclusion:

The correct sequence is hence a) - (ii), (b) - (iv), (c) - (i), (d) - (iii)

Concept:

Logarithms:

• If ab = x, then we say that loga x = b.
• log aⁿ = n log a.

Calculation:

We know that 27 = 3³ and 9 = 3².

Changing the given logarithms to log 3, we get:

log 27 = 1.432

⇒ log 3³ = 1.432

Using log an = n log a, we get:

⇒ 3(log 3) = 1.432

⇒ log 3 = 0.4773

Now, log 9 = log 3² = 2(log 3) = 2 × 0.4773 = 0.9546.

Given:

The total surface area of a solid hemisphere is 1848 cm².

Concept used:

Total surface area of a hemisphere = 3πR²

where R is the radius of the hemisphere

Diameter = 2 × Radius

Calculation:

Let the radius of the hemisphere be R.

According to the concept,

3πR² = 1848

⇒ R² = 1848/3π

⇒ R² = 1848/(3 × 22/7)

⇒ R² = (1848 × 7)/66

⇒ R² = 196

⇒ R² = 14²

⇒ R = 14 (∵ Radius can't be negative)

Now, the diameter of the flat surface of the hemisphere

⇒ 2 × 14

⇒ 28 cm

∴ The diameter of the flat surface of the hemisphere is 28 cm.

Given:

Area of base (circle) of the cylinder = 1386 cm²

Height (h) of the cylinder = 30 cm

Given:

The angles of a triangle are in Arithmetic Progression.

The greatest angle is three times at least angle.

Concept Used:

In a triangle,

∠A + ∠B + ∠C = 180°

Where,

∠A, ∠B and ∠C = Angle of the triangle

Calculation:

Let the angle ∠A, ∠B and ∠C be an arithmetic progression

Let the greatest and least angle be ∠C and ∠A

According to the question, we have

∠C = 3∠A

The ∠A, ∠B and ∠C are an arithmetic progression

⇒ ∠B = 2∠A

Now,

∠A + ∠B + ∠C = 180°

⇒ ∠A + 2∠A + 3∠A = 180°

⇒ 6∠A = 180°

⇒ ∠A = 30°

The greatest angle = ∠C

⇒ ∠C = 3 × 30°

⇒ ∠C = 90°

∴ The greatest angle of the triangle is 90°.

Key Points

• Soda bottles are made of thick glass so that they can withstand the pressure in summer.
• Soda bottles are generally made of 'Polyethylene Terephthalate', commonly known as PET plastic, or glass using the blow molding process.
• Usually, the plastic bottles used to hold potable water, soda, and other drinks are made from PET material because that is both strong and light so that they can withstand pressure in summer.
• *To be able to withstand internal gas pressure, the bottle is made from thick, heavy glass

It is known that language can either help or hinder learning. If language is used correctly and with clarity, it helps in thinking but if it reveals imperfect meanings it creates a misunderstanding. Since mathematics deals in abstractions and itself is a way of thinking, it creates a dependent relationship between the notions and the language used to describe them.

• Mathematical language facilitates thinking by complementing ordinary language. Consider how a child gets the notion of a circle. A child handles, manipulates, and observes the shapes of objects like a wheel, bangles, the ring, etc. He may experiment with a model or may stand in a circle while playing.

Important Points

• Also, mathematics is itself a language; it has its own symbols and rules for correct usage. In spoken language, a usage indicates what words mean, in mathematics, careful defining sharpens word meanings. Mathematical language is clear, concise, consistent, and cogent. Pupils who get the idea and describe it incorrect language are less confused than pupils who memorize terms representing ideas that remain as strange as the terms themselves.

Thus from the above-mentioned points, it is clear that if the language of mathematics reveals imperfect meanings then it creates a misunderstanding.

Given:

Length (l) = 4.2 m = 42 dm

Breadth (b) = 30 dm

Height (h) = 180 cm = 18 dm

Formula used:

Surface Area (SA) of Cuboid = 2 [(l × b) + (b × h) + (h × l)]

Calculations:

According to the question,

SA = 2 [(42 × 30) + (30 × 18) + (18 × 42)]

⇒ SA = 2 [(1260) + (540) + (756)]

⇒ SA = 2 [2256]

⇒ SA = 2 × 2556 = 5112 dm²

∴ The surface area (in dm2) of a cuboid is 5112 dm².

Given:

Area of rectangle = 180 cm²

Ratio of length (l) and breadth (b) = 9/5

Formula used:

Perimeter of a Rectangle (P) = 2 × (l + b)

Area of a Rectangle = l × b.

Calculations:

According to the question,

l/b = 9/5

⇒ l = 9b/5

Area of rectangle = l × b

⇒ 180 = (9b/5) × b

⇒ 900 = 9b²

⇒ b = √100 = 10 cm

Then, length = (9 × 10)/5 = 18 cm

The perimeter of the rectangle:

⇒ P = 2 × (l + b) = 2 × (18 + 10)

⇒ P = 2 × (28) = 56 cm

∴The Perimeter of the Rectangle is 56 cm.

Given:

Volume of cuboid = 864 cu. cm

Area of ​​the base is 108 sq. cm

Formula Used:
Volume of Cuboid = Length × Breadth × Height

Since, Length and Breadth makes the base of cuboid,

Area of Base = Length × Breadth

Volume of Cuboid = Area of Base × Height

Calculation:

Area of Base × Height = 864

⇒ 108 × Height = 864

⇒ Height = 864 ÷ 108 = 8 cm

Hence, Height of cuboid is 8 cm.

Concept:

Supplementary Angles: Pair of angles that always sum up to 180°.

Calculation:

According to the concept discussed above

⇒ x + 2x = 180°

⇒ 3x = 180°

⇒ x = 180°/3 = 60°

∴ 2x = 2 × 60° = 120°

Given:

The total surface area of the right circular cone is 360π m2.

The radius is 10 m.

Formula used:

(1) A = πrl + πr2

Where,

A, is the total surface area of the right circular cone

l, is the slant height of the right circular cone

r, is the radius of the right circular cone

(2) l² = r2 + h2

Where,

l, is the slant height of the right circular cone

r, is the radius of the right circular cone

h, is the height of the right circular cone

Calculation:

According to the question, the required figure is:

Let 'l' be the slant height of the right circular cone.

According to the question,

⇒ A = πrl + πr²

⇒ 360π = [(π × 10 × l) + (π × 10²)]

⇒ 360 = [10l + 100]

⇒ 36 = l + 10

⇒ l = 26 m

Now,

⇒ l² = r² + h²

⇒ 262 = 102 + h2

⇒ 676 = 100 + h2

⇒ h2 = 576

⇒ h = 24 m

∴ The height of the right-circular cone is 24 m.

According to the question,

∠BCA = 2 ∠BAC

so

∠BCA = 60°

∠BAC = 30°

So from sine rule,

AC/sin90° = BC/sin30°

AC/1 = BC/(1/2)

AC = 2 BC

Given:

C.P of a table = Rs. 800

Loss % = 40%

Formula Used:

S.P = {(100 - Loss %)/100} × C.P

Where,

S.P = Selling price

C.P = Cost price

Calculation:

According to the question, we have

S.P = {(100 - 40)/100) × 800

⇒ S.P = (60/100) × 800

⇒ S.P = Rs. 480

∴ The selling price of the table is Rs. 480.

Given:

Radius of hemisphere = 51 cm.

Formula used:

T.S.A of hemisphere = 3π r²

Calculation:

∴ The answer is "24501.42".

Given:

The weight of a bag of tea was 350 kg, 280 kg, 270 kg, 360 kg, 310 kg, 300 kg

Concept used:

Range = Highest term - lowest term

Calculation:

Highest term = 360 kg

Lowest term = 270 kg

Range = 360 - 270 = 90 kg

∴ Required range is 90

Given:

The difference between the Compound (CI) and the Simple interests (SI) is ₹432.

Time (T) = 3 years

The Rate (R) of interest = 20% per annum

Concept used:

Compound Interest = P(1 + R%)^T

Simple Interest = (P × R × T)/100

Calculations:

According to the question,

CI - SI = Rs. 432

⇒ 432 = [P(1 + 20%)3 - 1] - [(P × 20 × 3)/100]

⇒ 432 = P[(1 + 20%)3 - 1] - P[(20 × 3)/100]

⇒ 432 = P[(1 + 0.20)3] - P[6/10]

⇒ 432 = P[(1.20)3 - 1] - P[3/5]

⇒ 432 = P[1.728 - 1 - 3/5]

⇒ 432 = P[(3.64 - 3)/5]

⇒ 432 = P[0.64/5]

⇒ P = 432/0.128

⇒ P = Rs. 3375

∴ The principal lent was Rs. 3375.

Shortcut Trick
20% = 20/100 = 1/5

Time = 3 years

Hence assume that the Principle is 5³ = 125 Rs.

Simple Interest for 3 years = 75 Rs.

Compound Interest for 3 years = 91 Rs.

⇒ 91 - 75 = 16

Given that,

⇒ 16 = 432

⇒ 1 = 432/16

The principle, 125

⇒ 125 × 432/16 = 3375 Rs.

∴ The principal lent was Rs. 3375.

Given

Altitude of triangle = 12 cm

Concept

Altitude of an equilateral triangle is a median of an equilateral triangle.

Length of altitude of equilateral triangle = √3/2 × side of the triangle

Calculation

Let the side of an equilateral triangle is x cm.

12 = √3/2 × x

⇒ 24/√3 cm

Perimeter of triangle = 3(side)

⇒ 3 × (24/√3)

⇒ 24√3 cm.

Alternate Method
Given:

Height of the equilateral triangle = 12 cm

Formula used:

The perimeter (P) of a triangle is = Sum of all the sides

Concept used:

Pythagorean Theorem = (Hypotenuse)² = (Perpendicular)² + (Base)²

⇒ sin 60° = √3/2

Calculations:

Considering the height of the triangle as perpendicular then,

⇒ sinθ = (Perpendicular)/(Hypotenuse)

⇒ sin 60° = 12/H

⇒ H = (12 × 2)/√3 = 24/√3

Now the perimeter of the triangle,

⇒ P = 3 × Side

⇒ P = 3 × 24/√3 = 24√ 3

∴ The perimeter of the triangle is 24√3.

Given:

The length of each a pair of opposite sides of a parallelogram is 16 cm

The perpendicular distance between these two sides is 10 cm

Concept used:

Area of a parallelogram = Base × Height (perpendicular distance between parallel sides)

Calculation:

Area of the parallelogram

⇒ 16 × 10

⇒ 160 cm²

∴ The area of the parallelogram is 160 cm².

Given:

The radius of the base of a right circular cone is increased by 15% keeping the height fixed.

Concept used:

Volume of cone = 1/3πr²h

Calculation:

Let the radius be r

After an increase in radius, it would be 1.15r.

Volume of new cone = 1/3π(1.15r)2h

And, Volume of old cone = 1/3πr²h

⇒ % Change in volume = [Increase in volume/old volume] × 100%

⇒ 32.25%

∴ The volume of the cone will be increased by 32.25%.

Shortcut Trick

If the radius of the cone is increased by x% and the height of the cone is increased by y%,

The percentage increase in the volume of the cone =

According to the question,

x = 10

y = 20

Therefore,

The percentage increase in the volume of the cone,

⇒ ΔV = (20 + 20 + 5 + 0.2) × 100

⇒ ΔV = 45.2%

∴ The required answer is 45.2%

According to NCF 2005, the main goal of mathematics education at the primary level is the development of children's ability in mathematization. Basically, it means that children should learn to think about any situation using the language of mathematics. They can use the tools and techniques of mathematics in real life.

Important Point

• In Mathematics there is a certain way of thinking, people connect the mathematical concepts with real-life which help them think more logically and practically.
• The National Curriculum Framework 2005 is among the four National Curriculum Frameworks approved by the National Council of Educational Research and Training NCERT in India.
• At the Primary level, children learn from concrete objects and visualization processes.
• It must be ambitious, coherent (logical and consistent) and should focus on principles of mathematics.

Hence, we conclude that a 'certain way of thinking and reasoning' means adopting the exploratory approach, use of manipulatives, connecting concepts to real life, involving students in discussions.

Shortcut Trick
The cubes of side lengths x cm, y cm and z cm three melted as cast into a single cube.

The side length of the casted cube =

According to the question,

The side length of the first cube, x = 10 cm

The side length of the second cube, y = 12 cm

The side length of the third cube, z = 15 cm

Therefore,

The side length of the casted cube =

∴ The required answer is 18.27 cm

Alternate Method

Given:

The side length of the first cube = 10 cm

The side length of the second cube = 12 cm

The side length of the third cube = 15 cm

Formula used:

Volume of cube = a³

where,

a = side length of the cube

Calculation:

The volume of the first cube = 10³ = 1000 cm³

The volume of the second cube = 12³ = 1728 cm³

The volume of the third cube = 15³ = 3375 cm³

Since all three cubes are melted and cast into a single cube,

Therefore,

The volume of casted cube = 1000 + 1728 + 3375 = 6103 cm³

Let A be the side length of the casted cube.

⇒ A³ = 6103

⇒ A ≈ 18.27 cm

∴ The required answer is 18.27 cm.

Alternate Method 

The volume of the first cube = 103 = 1000 cm3

The volume of the second cube = 123 = 1728 cm3

The volume of the third cube = 153 = 3375 cm3

Since all three cubes are melted and cast into a single cube,

Therefore,

The volume of casted cube = 1000 + 1728 + 3375 = 6103 cm3

Now, we can also solve through option.

Apply the formula for the volume of cube in each value of side length of the cube.

Option 1: 22³ = 10648 cm³

Option 2: 403 = 64000 cm3

Option 3: 183 = 5832 cm3

Option 4: 103 = 1000 cm3

In comparing we get, option B gives the nearest answer.

∴ The required answer is 18.27 cm.

The correct answer is Joule/second.

• Work is defined as the component of force along the direction of motion times the displacement of the object acting under a certain motion.
• Joule - The SI unit of work is the joule (J).
• Erg - It is the unit of work in the CGS system.
  • Erg is a unit of work equal to 10⁻⁷ joules.
• Newton metre (Nm) - Newton meter is the unit of work in terms of force and displacement.
• Units of work done is
  

Shortcut Trick
The perimeter of the semi-circle with radius r is given by

πr + 2r = 36r/7

According to the question,

⇒ 72 = 36r/7

⇒ r = 14 cm

Therefore, '14 cm' is the required answer.

Alternate Method
Given:

Perimeter of a semi-circle = 72 cm

Formula used:

The perimeter of semicircle = πr + 2r

Calculation:

According to the question,

⇒ 72 = πr + 2r

⇒ 72 = (π + 2) × r

⇒ 72 = (22/7 + 2) × r

⇒ 72 = 36r/7 × r

⇒ r = 14 cm

Therefore, '14 cm' is the required answer.

Mistake Points
The perimeter of a semicircle is the sum of half of the

circumference of the circle and its diameter i.e. (2r + r).

Don't take it equal to πr.

Mean = (21 + 11 + 27 + 8 + 5 + 12 + 7 + 23 + 3 + 14 + 9 + 19) / 12

= 159/12 = 13.25

Arranging the observations in ascending order,

3, 5, 7, 8, 9, 11, 12, 14, 19, 21, 23, 27

Since the number of observations is even,

Median = (6^th term + 7^th term) / 2 = (11 + 12)/2 = 11.5

Difference between mean and median of the given data = 13.25 - 11.5 = 1.75

Given:

Area of the floor of a cubical room = 192 m²

Formula used:

Diagonal of the cube = √3 × side of cube

Floor area = side² (floor of the cube is in the shape of a square)

Calculation:

According to the question,

The floor area of the cube = side²

⇒ 192 = side²

⇒ side of cube = √192 = 8√ 3 m

Diagonal = √3 × side of the cube

⇒ √3 × 8 × √3

⇒ 8 × 3 = 24 m

∴ The length of the longest rod that can be kept in that room is 24 m.

Concept:

The amount of electrical energy transferred to an appliance depends on its power and the duration of time it is used.

Electrical energy is measured in kilowatt-hours, kWh. One unit is 1 kWh.

Electrical energy transferred is given as,

E = P × t

Where, P is the power in kilowatts (kW), T is the time in hours (h)

Calculation:

Given,

Power, P = 100 kW

Time, t = 1 hour

∴ E = P × t = 100 × 1 = 100 kWh

E = 100 × 3.6 × 10⁶ J = 36 × 10^{7 J (∵ 1 kWh = 3.6 × 106 J)}

Hence, option (2) is the correct answer.

The correct answer is Insulation.

Key Points

• Insulation is a general term used to describe material that creates barriers to the transmission of electricity, heat, moisture, shock, or sound between insulated surfaces of adjacent bodies. These materials could be insulating heat, cold, electricity, sound, or radiation.
• To reduce heat loss or gain, insulation is utilised in buildings and manufacturing processes.
• Although its major goal is to save money, it also allows for more precise temperature management and worker protection.
• It reduces rusting caused by moisture on cold surfaces. These materials are porous and have a lot of latent air cells.
• The building industry uses thermal insulation to decrease heat loss or gain through the home envelope (external walls, windows, roofs, foundation, etc.).
• Thermal insulation provides thermal comfort inside homes by maintaining a comfortable temperature.

Important Points

• The flow of a fluid (liquid or gas) between locations of differing temperatures is known as convection.
• Heat energy is transferred through collisions between surrounding atoms or molecules in the process of conduction.
• Solidification, commonly known as freezing, is a phase transition in which a liquid becomes a solid when its temperature is decreased below its freezing point.

Given:

Dimension of matchboxes = 4 cm × 3 cm × 1.5 cm

Dimension of cardboard box = 30 cm × 30 cm × 20 cm

Formula:

Volume of cuboid = l × b × h

Calculation:

Let number of matchboxes be x.

According to the question

4 × 3 × 1.5 × x = 30 × 30 × 20

⇒ × = (30 × 30 × 20)/(4 × 3 × 1.5)