MPTET Varg 2 Math Mock Test - 3 - MPTET MCQ

Problem-solving abilities can be developed when we encourage children in solving problems independently or in groups without providing any direct support. Besides promoting problem-solving abilities in children, they should be encouraged to pose problems.

Key Points

• Problem-posing in mathematics means generating problems from the content. Posing relevant problems indicates the level of understanding of the concepts, processes, and procedures of mathematics.
• Problem posing involves generating new problems and questions to explore a given situation, as well as reformulating a problem during the course of solving the problem related to it. Teachers should encourage such practices in the classroom as much and as frequently as possible.

Hence, it can be concluded that problem-posing in mathematics means generating problems from the content.

The logarithm of 1/9 in a system whose base is 3:

We know the value of log₃ 1 = 0 and log₃9 = 2

Given

Distance = same

Speed of A and B = 3 ∶ 4

Formula used

Distance = speed × time

When distance is the same, Speed and time become inversely proportional.

Solution

Since distance is the same, the time ratio will be the reverse of the speed ratio.

Time taken by A and B = 4: 3

Let the time taken by A and B = 4x and 3x

⇒ 4x - 3x = 20

x = 20 minutes

A's time = 4x

⇒ 4x = 4 × 20 = 80 minutes

⇒

The correct option is 2.

Given:

Concept used:

(a - b)² = a² + b² - 2ab

Calculation:

Now, squaring both sides

∴ The required value is .

Given:

log32 x = 0.8

Concept Used:

If log_ax = n is given

Then, x = aⁿ

Calculation:

In log32 x = 0.8

The value of a = 32

The value of n = 0.8

Then, x = (32)^0.8

⇒ x = (2⁵)^0.8

⇒ x = 2^{5 × 0.8}

⇒ x = 2⁴ = 16

∴ The value of x is 16.

Given:

A cuboid of length 36 m, breadth 18 m and height 9 m is melted and recast into a cube.

Concept used:

Volume of cuboid = L × H × B

Volume of cube = a³

Here,

L = Lenght

B = breadth

H = height

a = side of the cube

Calculation:

Let the side of the cubes to be formed be a

According to the question,

Volume of cuboid = volume of the cube

So,

36 × 18 × 9 = a³

⇒ 5832 = a3

⇒ a³ = 18³

⇒ a = 18

So, the diagonal of the cube is 18√3

∴ The length of the diagonal of the cube is 18√3.

Given:

Formula used:

Mean = (∑f_ix_i) / N

where f_i = frequency of i_th term

_Xi = i^th term

N = total observations

Calculations:

Mean = (∑f_ixi)/N

⇒ Mean = 661/27

⇒ Mean = 24.48

∴ The mean age of all patients is 24.48 years.

Given:

Formula used:

If base is same

Log a - Log b = Log(a/b)

Log(aⁿ) = n Log a

Calculation:

According to the question,

(Log 128 - Log 8)/Log 4

Using the log formula

⇒ Log(128/8)/Log 4

⇒ Log 16/Log 4

⇒ Log(4²)/Log 4

Using the formula Log(an) = n Log a

⇒ 2 Log 4/Log 4

⇒ 2

∴ The value of  is 2.

Given:

The radius of the hemisphere = 7a

Formula used:

T.S.A of hemisphere = 3πr²

Calculation:

T.S.A of hemisphere = 3πr2

∴ The answer is 462 a².

Mathematics though it occupies a significant place in our school curriculum. Therefore, mathematics has become a part of universal education and has been made a compulsory subject of study. It is impossible to think of activity without the use of mathematics. The following values justify the importance of the mathematics curriculum.

Key Points

Mathematical Aspects:

• Mathematics teaches us how to analyse a situation, how to come to a decision, to check thinking and its results, to perceive relationships, to concentrate, to be accurate and to be systematic in our work habits.
• Mathematics develops the ability to perform necessary computations with accuracy and reasonable speed. It also develops an understanding of the processes of measurement and of the skill needed in the use of instruments of precision.

Social Aspects of mathematics:

• The routine activities of daily life demand a mastery of a number of facts and a number of processes. To read with understanding much of the materials in newspapers requires considerable mathematical vocabulary. A few such terms are per cent, discount, commission, dividend, invoice, profit and loss, wholesale and retail, taxation, etc. As civilization is becoming more complex, many terms from the electronic media and computers are being added.
• Mathematical operations like addition, subtraction, multiplication, division and so on, are used in our daily activities. From poor to rich, all have to use Mathematics in their real lives in one or another way.
• Certain decisions require sufficient skill and understanding of quantitative relations. The ability to sense problems, formulate them specifically and solve them accurately requires systematic thinking.
• To understand many institutions and their management problem, a quantitative viewpoint (modelling) is necessary. It is illuminating to hear from an economist, an architect, an engineer, an aviator, or a scientist what in mathematics is helpful to them as workers.
• Many vocations need mathematical skills.

Thus from the above-mentioned points, it is clear that the given statement reflects mathematical aspects.

Given:

70% of the students in a school are boys

Number of girls is 345

Calculation:

Total number of students = Number of girls / (1 - Percentage of boys)

Plugging in the given values, we get:

Total number of students = 345 / (1 - 0.7) = 345 / 0.3 = 1150.

Given:

Speed of the train = 60 kmph

Length of train = 500 meters

Formula Used:

Time taken by train to cross a pole = Length of train/Speed of train

Calculation:

Speed of train = 60kmph = 60 x 5/18 = 50/3 m/sec

Time taken by train to cross a pole = Length of train/Speed of train

⇒ 500/(50/3)

⇒ (500 x 3)/50

⇒ 10 x 3

⇒ 30 sec

∴ Time taken by the train to cross a pole is 30 seconds.

Given:

The ratio of the areas of two squares = 9 : 1

Concept used:

Area of square = a²

Perimeter of square = 4a

Calculation:

Let the area of square be A₁ and A₂ respectively

Now, according to the question

Now, Perimeter of square is

∴ The required ratio of their perimeters is 3 : 1.

Concept used:

1. Two triangles are said to be similar if they have corresponding

angles equal & sides in the same proportion.

2. Condition for similarity - SAS

If two corresponding sides are in proportion with one included angle equal.

3. Two triangles are said to be congruent if their corresponding

sides & angles are equal.

Explanation:

option 1) Two congruent triangles are necessarily similar

⇒ True because congruent triangles will be having equal angles.

option 2) All equiangular triangles are similar

⇒ True, as corresponding angles will be equal.

option 3) Two isosceles right triangles are similar

⇒ True, they will be similar by the SAS rule

option 4) All isosceles triangles are similar

⇒ False, All isosceles triangles may not be having

corresponding angles equal.

∴ The correct answer is 4)

Concept

Let us suppose a be the first term and d be the common difference of an AP. Then the nth term of an AP is given by: an = a + (n - 1) × d

Let us suppose there is an AP consisting of m terms, then the nth term from the end of the AP is (m – n + 1)th term from the beginning.

CALCULATION:

Given:

p, 9, 3p - q. 3p + q are in AP

Let d be a common difference, then

d = 9 - p ----(I)

d = (3p - q) - 9 --- (II)

d = (3p + q) - (3p - q) ---(III)

From (I) and (II) we get

9 - p = 3p - q - 9

⇒ 4p - q = 18 ....(1)

From II and III we get

3p - q - 9 = 3p + q - 3p + q

⇒ 3p - q - 9 = 2q

⇒ 3p - 3q = 9

⇒ p - q = 3 ....(2)

Solving equations (1) and (2), we get

p = 5, q = 2

Hence common difference d = 9 - p = 9 - 5 = 4 and first term a = 5.

So, T₂₀₁₀ = a + (2010 - 1) × d

= 5 + 2009 × 4

= 5 + 8036

T₂₀₁₀ = 8041

Given:

Radius of cylinder(r) = 7 cm

Height of cylinder(h) = 12 cm

Formula used:

T.S.A of cylinder = 2πr(r + h)

Calculation:

T.S.A of cylinder = 2 × 22/7 × 7 × (7 + 12)

⇒ 44 × 19

⇒ 836 cm²

∴The answer is 836 cm² .

Shortcut TrickA person invests Rs. P for two years in such a scheme that it will provide a% p.a. for the first year and b% p.a. for the second year.

The compound interest earned, C.I = P × (100% + a%) × (100% + b%) - P

According to the question,

The compound interest earned = 10000 × (100% + 10%) × (100% + 20%) - 10000

C.I = (10000 × 110% × 120%) - 10000

C.I = 10000 × 110/100 × 120/100 - 10000 = Rs. 3200

Therefore, 'Rs. 3200' is the required answer.

Alternate MethodGiven:

Principal = Rs. 10,000

Rate for first year = 10% p.a.

Rate for second year = 20% p.a.

Formula used:

(1.) Interest = Effective rate × Principal

(2.) Effective rate = R₁ + R₂ +

Calculation:

According to the question,

Effective rate = 10 + 20 +  = 32%

Interest = 10000 × 32% = Rs. 3200

Therefore, 'Rs. 3200' is the required answer.

Benjamin Bloom's taxonomy is a set of three hierarchical models that refers to the classification of educational learning objectives.

Key Points

B. S. Bloom has divided the cognitive objectives into six categories (under Bloom's taxonomy) arranged from the lowest to the highest level of functioning.

Their classification is given as under:

Hence, it could be concluded that knowledge skill of bloom’s taxonomy will be evaluated here.

Logarithms were invented by Scottish Mathematician John Napier. Logarithms refers to the quantity representing the power, to which a fixed number (the base) must be raised to produce a given number.

Concepts:

• If a, b, c are in A.P ⇔ b =
• If a, b, c are in G.P ⇔ b² = ac
• If a, b, c are in H.P ⇔

Calculation:

If log_ℓ x, log_m x, log_n x are in A.P.

Then, 2log_mx = log_ℓ x + log_nx

⇒ 2(log x / log m) = (log x / log l) + (log x / log n)

⇒ 2(1 / log m) = (1 / log l) + (1 / log n)

⇒ (2 / logm) = (log n + log l) / (log l . log n)

⇒ log n² = log_ℓ^m [log ℓn]

⇒ log n² = log(ℓn)^log_ℓ^m

∴ n² = (ℓn)loglm

Van Heile's theory provides an insight to the teacher about how the students learn geometry at different levels. It helps in describing how the students learn at each level and pass to another level and shapes their learning of geometry at each level of learning.Key PointsThe Van Hiele levels are described below:

• Level 0: Visualization-The students can recognize shapes by their whole appearance that should just like the exact shape. They can also compare the figures with their prototypes (exemplar) or everyday things but can not identify the properties of geometric figures. For example, they can compare the shape of a circle with bangles, coins, wheels, etc. but are unable to identify and describe the properties of a circle.
• Level 1: Analysis -They will learn the functions and parts of a figure. They can describe the properties of a figure and recognize the figures with the same properties. For example, they can identify the shapes and describe their properties such as a circle is a closed rounded figure with no corners.
• Level 2: Abstraction or informal deduction -The students will be able to understand the relationships between the properties of a figure. They can take part in informal deductive discussions and can discuss the different characteristics of figures. For example, the opposite sides of a parallelogram are parallel. The opposite sides of a square and rectangle are also parallel which means the square and rectangle are also a parallelogram.
• Level 3: Deduction or formal deduction -At this level, the students become aware of the more complex geometrical concepts. They can prove an abstract statement on geometric properties to conclude. For example, they can prove that the square is a rectangle but a rectangle can not be a square.
• Level 4:Rigor -The last level of geometrical learning belongs to the senior secondary and university-level classes. The students can compare different geometrical results. For example, the sum of all three angles of a triangle is 180 degrees and is compared to the other properties or other results (to find exterior or interior angles of a triangle) related to the triangle to solve geometrical problems.

Therefore, the correct sequence is Recognition of figures, Analysis of components of figures, Relationships among figures, Deductive Reasoning & formal proofs, and Axiomatic Reasoning.

Calculation:

Total time from 7:00 a.m to 4:30 pm = 9 hrs 30 minutes

From the above-given data,

The parking rates for more than 8 hrs and up to 12 hrs is Rs. 150.

Concept:

Histogram: A histogram is a display of statistical information that

uses rectangles to show the frequency of data items in successive

numerical intervals of equal size.

Explanation:

This is the image of the histogram.

The height of the rectangle in the histogram represents the frequency

Concept:

Distance formula between (x1, y1) and (x2, y2) = √(x2 - x1)2 + (y2 - y1)2

Calculation:

A (7, 6) and B (-3, 4)

Any point on the y-axis has coordinates of the form (0, y).

Since the point is equidistant from both points A (7, 6) and B (-3, 4)

Let the point be P (x, y)

As per the distance formula

32 + (y − 4)2 = 72 + (y − 6)2

16 - 8y = 40 + 36 -12y

4y = 60, y = 15

Therefore the correct answer is (0, 15).

The correct option is 9k_x = k_y.

CONCEPT:

Kinetic energy is a scalar quantity. Kinetic energy is the energy posses by the object when it is in motion.

K= 1/2mv²

CALCULATION:

Given:

The mass of both the objects is the same = m
The velocity of the first object = u
The velocity of the second object = 3u
K_x = kinetic energy of the first object
K_y = kinetic energy of the second object

If we take the ratio of the kinetic energy of two objects to set up a relation.

The value like m and u which is common get canceled out.

9k_x = k_y

Additional Information:

• Potential energy: The latent energy of an object at rest. If the object of mass m is placed at a height of h then the formula for potential energy is mgh. Where g is the acceleration due to gravity.

The correct answer is 'A, B, D and E only'.

Key Points

Application of 'FORCE'

• Force may bring a change in the shape of an object
  • When a force acts on an object, the object may change shape by bending, stretching or compressing - or a combination of all three shape changes.
• Force may make an object move ​
  • Force can cause an object to move or accelerate, to slow down or decelerate, to stop, or to change its direction.
• Force can change the direction of motion of an object
  • ​Force can make a stationary object move or bring a moving object to rest. Force can also change the direction of motion of the moving body.
• Force can change the speed of an object
  • ​Force can increase or decrease the speed of an object.
• Force can be push or pull
  • ​A force that changes the direction of an object towards you, would be a pull.
  • On the other hand, if it moves away, it is a push.

Explanation:

• Temperature is the degree of hotness or coldness of the substance.
• Heat is a form is of energy that is transferred from hotter (high temperature) to lower temperature.
• Heat transfer takes place until the equilibrium temperature is reached. That means the temperature of both bodies becomes the same.
• If the two bodies are at the same temperature, no heat transfer will take place.

So the correct option is 'not flow from iron ball to water or from water to iron ball'

Additional Information

• If there is a temperature difference, the heat flow is given as

H = m c Δ T

• m is mass of substance, c is specific heat capacity, Δ T changes in temperature.
• Heat is transferred through conduction, convection, and radiation.
• Conduction: The heat transfer through direct physical contact is known as conduction.
• Convection: Heat transfer takes place through the transfer of molecules.
• Radiation: Heat transfer takes place through Electromagnertic waves. This method does not require a medium.

Given:

The diameter of a sphere is 3.5 cm

Concept used:

Total surface area = 4πr2

Here r = radius

Calculation:

Total surface area = 4 × 22/7 × (3.5/2)²

⇒ 4 × 22/7 × 12.25/4

⇒ 22/7 × 12.25

⇒ 38.5

∴ The total surface area of the sphere is 38.5 cm2.

Mathematics: Mathematics is a systematized, organized, and exact branch of Science. It plays an important role in accelerating the social, economical, and technological growth of a nation. It helps in solving problems of life that need enumeration and calculation.

Key PointsMathematics is the gateway of all science. In school, those subject which is included in the curriculum must have certain aims and objectives based on which their nature is decided. The nature of Mathematics is enlisted in the following points:-

• Mathematics is based on understanding.
• Mathematics has its own language.
• Mathematics puts great emphasis on the child’s own methods of calculating and solving problems and rejects the previous practice of heavy emphasis on standard written algorithms.
• Mathematics is a science of discovery and logical reasoning.
• Mathematics is regarded as a powerful tool for interpreting the world and therefore should be rooted in real experience across the whole curriculum. Mathematics is brought out of the child’s everyday situations.
• Mathematics is an exact science.
• Mathematics with reason is rooted in action – learning through doing.
• Mathematics with reason puts less emphasis on representing numbers on paper as ‘sums’ and more emphasis on developing mental images in the child.

Important Points

The following are the characteristics of Mathematics:-

• Mathematics is an intellectual game.
• It deals with the art of concluding.
• It is a tool subject.
• It involves an intuitive method.
• It is the science of exactness, precision, and accuracy.
• It is the subject of a logical and specific sequence.
• It requires the application of rules and concepts to new situations.
• It is a logical study of structure and patterns.

Thus, it is concluded that Mathematics is changeable in the universe is incorrect about the nature of mathematics.

The correct answer is 27.8 km.

Key Points

• Rotation is the movement of the earth on its axis.
• The movement of the earth around the sun in a fixed path is called revolution.
• The circumference of the Earth at the equator is 40,075 km.
• Earth completes one rotation in 24 hours.
  • One hour is equal to 60 min so 24 hours =24 × 60=1440min.
  • Speed of rotation
    • Time = 40075/1440 = 27.8 km per min.
• So, the Earth rotates approximately 27.8 kilometers in one minute at the equator.

Confusion Points

• The rotational speed of Earth at the equator is 27.8 km/min.
• As we move closer to the poles, its speed decreases because the earth's circumference is decreasing.