MPTET Varg 2 Math Mock Test - 7 - MPTET MCQ

Formula used:

Average speed =

Calculation:

Total distance = 40 × (33/60) + 62 × (33/60)

Average speed = Total distance/Time taken

Average speed =

Average speed = 102/2 = 51 km/hr

∴ The correct answer is 51 km/hr.

The Van Hiele's levels originated in 1957 given by Pierre Van Hiele and his wife from the Utrecht University in the Netherlands. It helped in shaping the curriculum throughout the world that especially influenced the learning of geometry at a large level.

• It provides an insight to the teacher about how the students learn geometry at different levels. It describes how the students learn at each level and pass to another level.

Key Points

Van Hiele levels:

• In early schooling, the learners began to learn about shapes and try to differentiate various shapes from each other.
• The students learn according to their individual differences, the age can be different in each stage as they learn at their own pace.
• The Van Hiele levels are described below:

So, it is concluded that according to Van Hiele's stages, the child is on the analysis stage.

Given:

Production of mobile in the company over the years.

Calculations:

Percentage increase in 2002 (production is 45) compared to the previous year 2001 (production is 30) =

Percentage increase in 2003 (production is 65) compared to the previous year 2002 (production is 45) =

Percentage increase in 2005 (production is 70) compared to the previous year 2004 (production is 50) =

Percentage increase in 2007 (production is 80) compared to the previous year 2006 (production is 55) =

From the above calculation, in the year 2002, the percentage increase is maximum

∴ The answer is 2002

The correct answer is Force of friction.

Key Points

• A boat comes to rest if we stop rowing. The force responsible for changing the state of motion of objects is Force of friction.
  • ​Friction between water and the boat brings it to a stop once you stop rowing.
• Force of friction
  • It is the opposing force when two surfaces come in contact with each other to move either in the same direction or in opposite directions.​
  • The frictional forces are majorly affected by the texture of the surface of the objects that came in contact with each other and the amount of force exerted on them.
  • The formula for frictional force is given by F = μN
  • Frictional Forces are primarily classified into four different types-
    • rolling friction
    • sliding friction
    • static friction
    • kinetic friction

Important Points

• Force of Gravity
  • It is a force that attracts any objects with mass.
  • It is the force by which a planet or other body draws objects toward its center.
  • Objects with more mass have more gravity.
  • Gravity also gets weaker with distance. Hence, the closer objects are to each other, the stronger their gravitational pull is.
  • Gravitational Force = Gm¹m²/r² is the Gravitational force between two objects measured in Newton (N). G is the Universal Gravitational Constant with a value of 6.674 × 10^-11 Nm²kg^-2.
• Muscular force
  • It is any force applied by using our muscles like arms or legs.​
• Magnetic force
  • ​It is the attraction or repulsion that arises between electrically charged particles because of their motion.
  • The force is perpendicular to both the velocity v of the charge q and the magnetic field.
  • Magnetic Force F = qvB sinθ
    • F = magnetic force
    • q = charge of moving particle
    • v = particle velocity
    • B = magnetic field
    • θ = angle between velocity and magnetic field vectors

Formula used:

1. (a^m)ⁿ = a^mn
2. a^m × aⁿ = a^m+n
3. a^m/aⁿ = a^m-n

Calculation:

[(216)4 ÷ 364] × 65 = 6m

⇒ [(6³)⁴ ÷ (6²)⁴] × 6⁵ = 6^m

By using the above properties

⇒ [(6^12-8] × 6⁵ = 6^m

⇒ 6⁴ × 6⁵ = 6^m

⇒ 6⁹ = 6^m

∴ m = 9

Given:

The average of 8 numbers is 25.

each number is increased by 15 %

Calculations:

Let's call the original 8 numbers as x.

The average of x is 25,

So the sum of x is 25x

Now, we need to increase each number by 15%.

So, the new number will be x + 0.15x = 1.15x.

The sum of the new numbers will = 25 × 1.15x

So, the new average of the numbers will = 28.75.

Therefore, the new average of the 8 numbers after increasing by 15% is 28.75.

Concept used:

The opposite angles of a rhombus are equal.

The diagonal of a rhombus bisects the angle through which it passes.

Solution:

∠OCB = 50° = ∠OCD

∠COD = ∠BOC = 90°

In an ΔOCD, we know that

∠OCD + ∠DOC + ∠ODC = 180°

50° + 90° + ∠ODC = 180°

∠ODC = 180° - 90° - 50°

∠ODC = 40° = ∠BDC

Hence, the correct option is 40°.

Given:

Shadow length = 31m

Angle of elevation = 60°

Concept used:

Tan θ = P/B

Calculation:

In ΔPQR

⇒ tan 60° = PQ/QR

⇒ tan 60° = PQ/31

⇒ √3 = PQ/31

⇒ PQ = 31√3 m

∴ The height of the skyscraper is 31√3 m.

Calculations :

log 10 - log 5 + log 25

log 10 - log 5 + log 5² (log x2​ = 2logx)

⇒ log 10 - log 5 + 2 log 5

⇒ log 10 + log 5 [log a + log b = log (a.b)]

⇒ log 50

∴ Option 1 is the correct answer.

Given:

The number is 8x5215.

Concept used:

A number is divisible by 9 if the sum of all digit is divisible by 9.

Calculation:

Number = 8x5215

The sum of all digit = 8 + x + 5 + 2 + 1 + 5

⇒ 21 + x

Nearest multiple 9 = 27

To make the no. divisible by 9 then we have to replace the x by 6

∴ The least value of x is 6.

The correct answer is Ductility
Key Points

• Ductility is the ability of a material to be drawn or plastically deformed without fracture.
• It is therefore an indication of how ‘soft’ or malleable the material is.
• The ductility of steels varies depending on the types and levels of alloying elements present.
• An increase in carbon, for example, will increase the strength but decrease the ductility.

Additional Information

Production of mobile in a company over the years.

Calculations:

Production of mobile in 2003 and 2004 is 65 and 50

Required percentage decline = 
⇒ 1500/65  = 23.07% ~ 23%

∴ The answer is 23%

Learning Mathematics is concerned with both comprehending the facts, theories, rules, and laws, and learning much broader mathematics. As a teacher, one should make his/her students understand that mathematics has various applications in daily life.

Key Points

Mathematical reasoning and the process of reasoning especially in the context of mathematics. In mathematical reasoning, the students make use of various processes to conclude the problems in mathematics that they encounter.

In mathematical language, there are two kinds of reasoning.

• Inductive reasoning: Induction is the form of reasoning in which a general law or principle is derived from a study of particular objects or specific processes. For example, on adding 2 and 4, we get 6. Similarly, on adding two even number result is also even. It is based on the logic that if something is true for a particular case and is further true for a reasonably adequate number of cases, then it is true for all such cases.
• Deductive reasoning: In this reasoning first you give the relevant formula, principles, and ideas to students and explain further its application of the formula to problems. The students in your class come to understand how the formula can be used or applied. For example – when you are going to teach about profit and loss, directly you announced the formula of interest i.e. I = PTR/100, and solve different related problems by using this formula. The students just observe your method of solving and then they memorize this formula for further use.

Hence, we can conclude that the students make use of various processes to conclude the problems in mathematics that they encounter, this process is known as reasoning.

Given:

Aman drinks glasses of water daily = 14

The capacity of the glass = 300 ml

1 week = 7 days

Calculation:

Aman drinks water in a week = 14 × 7 = 98 glasses

⇒ Total quantity = 98 × 300 = 29400 ml

1 L = 1000 ml

⇒ 29400 ml = 29400 ÷ 1000 = 29.4L

∴ 29.4 L of water he drinks in 1 week.

Given :

Index of year 1997 = 200

Index of year 1996 = 100

Calculation :

Required ratio :

⇒ 200 : 100

⇒ 2 : 1

Hence the correct answer is "2 : 1".

Given:

Angle of depression = 30°

After flying 60 meters, the angle of depression becomes 60°

Calculation:

In both conditions, perpendicular is same.

So, tan 30° = 1/√ 3 = Perpendicular/base and

tan 60° = √ 3/1 = Perpendicular/base

For same height given in the question, we multiply √ 3

tan 30° = (1/√ 3) × (√3/√ 3) = √ 3/3

tan 60° = √ 3/1

Difference of distance covered = 3 - 1 = 2 units

⇒ 2 units = 60

⇒ √ 3 unit = (60/2) × √ 3 = 30√ 3

∴ The Required answer is 30√ 3 meters.

Given:

PQRS is a cyclic quadrilateral

PQ is extended up to T and U on both sides

∠SPU = 68° , ∠RQT = 57°

Concept used:

Sum of the opposite angles of a cyclic quadrilateral is 180°

Calculation:

As per the question,

∠SPU = 68°,

so, ∠SPQ = 180° - 68°

⇒ 112°

So, ∠SRQ = 180 - 112°

⇒ 68°

Again,

∠RQT = 57°,

So, ∠RQP = 180° - 57°

⇒ 123°

Now,

∠PSR = 180° - 123°

⇒ 57°

So the value of ∠SRQ - ∠PSR = 68° - 57°

⇒ 11°

The value of ∠SRQ - ∠PSR is 11°

Given:

The top of two poles of height 24 m and 36 m are connected by a wire.

The wire makes an angle of 60° with the horizontal.

Concept used:

sinθ = perpendicular/Hypotenuse

sin 60° = √3/2

Calculation:

From the above figure,

The length of the wire is AD,

In ΔADE,

On multiply and divide by √3

∴ The required value is 8√3 m.

Given:

(tan57° + cot37°)/ (tan33° + cot53°)

We know cot(A) = tan(90-A)​ So, [tan 33° = cot 57° and cot37° = tan 53° ]

⇒ (tan57° + tan 53°)/ (cot 57° + cot 53°)

⇒ (tan57° + tan 53°)/ (1/tan 57° + 1/tan 53°)

⇒ (tan57° + cot37°)/ {(tan53° + tan57°)/tan53°× tan57°}

tan53° × tan57°

Concept Used:

The median is the middle number in a data set when the numbers are listed in either ascending or descending order.

If total number of observation (n) is even then, Median = [(n/2)th + (n/2 + 1)th]/2

Calculation:

Factors of 12 - 1, 2, 3, 4, 6, 12

The middle value is 3 and 4

⇒ Median = (3 + 4)/2 = 3.5

∴ The median of factors of 12 is 3.5

The correct option is 3 i.e. 3.5

Calculation:

The capacity of 1 bottle is y litre

Now, A/Q

(4/5) - 6y + 4y = 3/4

⇒ 2y = (4/5) - (3/4)

⇒ 2y = 1/20

⇒ y = 1/40

One bottle can fill 1/40 part of the oil tank.

∴ 40 bottles of oil can the contain in the whole tank.

Concept Used:-

Euclid has given some geometrical axioms based on the general assumption which can be assumed without proving them. In this problem, we will use the 4th axiom of Euclid which state that, “the things which coincide with one another will be equal to one another".

Explanation:-

In the given Figure, we have

AX = CY .........(1)

Since X is the mid-point of AC. Thus,

AX = CX .........(2)

From equations (1) and (2) we have,

CY = AX = CX .......(3)

Since Y is the mid-point of BC. Thus,

CY = BY ........(4)

From equations (3) and (4) we have,

AX = CX = CY = BY

Or,

⇒AX = BY

⇒2AX = 2BY

⇒AC = BC (Since X and Y are the mid-points of AC and BC respectively)

Thus, when X and Y are the mid-points of AC and BC and AX = CY. then AC = BC.

Hence, the correct option is 1.

Formula Used:

X% = X/100

Calculation:

(120% of 675) + 92 = (? % of 1240) + 716

⇒ 120/100 × 675 + 92 = ?/100 × 1240 + 716

⇒ 810 + 92 = 12.4 × ? + 716

⇒ 902 - 716 = 12.4 × ?

⇒ 186/12.4 = ? = 15

∴ The correct answer is 15

• The teacher needs to check the performance or the level of learning of students so that the learning can be facilitated in better ways.
• To monitor the learning of students, the teacher keeps evaluating the performance of learners by asking them questions about the learned concepts.

Important Points

The questions asked by the teacher in a natural classroom setting can be classified mainly into two types:

Open-ended questions

• The respondents are free to share, clarify and put their views. These questions may have more than one correct answer.
• The students will elaborate their point of view and the teacher will be able to see the things from a student's perspective.

​Close-ended questions

• These questions may generally have one answer and provides limited insight.
• The respondents respond in limited ways like responding in ‘yes’ or ‘no’, putting the sign ‘correct’ or ‘incorrect’, etc.

​Key Points

Let's understand in brief:

What is the area of a square having a side of 10 cm?

• This question has a one-limited response.
• The formula to find out the area of a square is side × side i.e., 10 × 10 = 100 cm2.
• So, it is a close-ended type question.

Find 3 + 4 and 5 + 4 and 3 + 7 and 1 + 6, etc.

• If we add two particular numbers, then we will get their sum as an answer that is fixed.
• These types of questions have one response as an answer and a limited way of responding.
• 3 + 4 = 7, 5 + 4 = 9, 3 + 7 = 10, 1 + 6 = 7.
• Hence, it is also a close-ended type question.

If a dozen bananas cost 20 rupees, how much banana Shashank can buy for 50 rupees?

• One dozen bananas (12 bananas) cost 20 rupees.
• Two dozen bananas (24 bananas) will cost 40 rupees.
• And for 10 rupees, Shashank can buy half a dozen (6 bananas) of bananas.
• So, for 50 rupees Shashank can buy 30 bananas.
• This question has one unique answer. So, it is clearly a close-ended type question.

Madhu picked up a solid figure and described one of its characteristics by saying that it had a rectangular face. Identify the shape she may have picked up.

• Madhu picked a solid figure but not telling clearly which figure is that.
• She only described one of its characteristics that it has a rectangular face.
• The children can assume it can be a rectangle, a cuboid, or a square.
• So, the responses to this question vary depending on the knowledge of learners. Hence, it is an open-ended question.

Hence, It can be concluded that "identifying the shape Madhu may have picked up" is an open-ended type question.

Given:

37 - [19 - {6 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]

Concept used:

Follow the BODMAS rule according to the table given below:

Calculation:

37 - [19 - {6 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]

⇒ 37 - [19 - {2 - (6 - 3) ÷ 3}]

⇒ 37 - [19 - {2 - 3 ÷ 3}]

​⇒ 37 - [19 - {2 - 1}]

⇒ 37 - [19 - 1]

⇒ 37 - 18 = 19

∴ 37 - [19 - {6 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}] = 19

CONCEPT:

The median is that value of the given number of observations, which divides it into exactly two parts. So, when the data is arranged in ascending (or descending) order the median of ungrouped data is calculated as follows:

• When the number of observations (n) is odd, the median is the value of the observation.
• When the number of observations (n) is even, the median is the mean of  and the  observations.

CALCULATION:

Given set of numbers are 23, 34, 45, 67,-10, 78, 89 and 115.

Arranging numbers in ascending order -10, 23, 34, 45, 67, 78, 89 and 115.

As number of observation (n = 8) is even so median will be the mean of  and the  observations.

∴ Median = 

Given:

4/7 of total students are girls

Remaining are boys

there are 211 more girls than boys

Calculation:

Let the total number of students be x

∴ 4x/7 are girls

and 3x/7 are boys

Now,

Number of girls is 211 more than the boys

⇒ 4x/7 = 211 + 3x/7

⇒ x/7 = 211

⇒ x = 1477

∴ Number of boys are (3 × 1477)/7 = 633

Concept used:

Volume of a cube = a³

Volume of a cylinder = πr²h

Here,

a = edge of the cube

r = radius of the cylinder

h = height of the cylinder

Calculation:

Here,

Diameter of the base of the cylinder = edge of the cube

Again,

Height of the cylinder = edge of the cube

Now,

Let the edge of the cube be x

So, x = h = 2r

Now,

Ratio = x³ : πr2h

⇒ (2r)³ : πr²2r

⇒ 8r3 : πr32

⇒ 4 : π

∴ The required answer is 4 : π

Given:

The ratio of the areas of two circles is 16 ∶ 25.

Concept used:

Area of a circle = π × Radius²

Calculation:

Let the areas of the circles be 16a and 25a respectively and their radii be P and Q respectively.

According to the question,

π × P² = 16a ....(1)

π × Q² = 25a ....(2)

(1) ÷ (2)

P²/Q² = 16/25

⇒ P/Q = 4/5 (Considering the positive values only)

⇒ P : Q = 4 : 5

∴ The ratio of their radii is 4 : 5.

Given :

The ratio of the present ages of Rahim and Karim = 7 : 5

The ratio of ages of Rahim and Karim after two years = 3 : 2

Calculation :

Let the present ages of Rahim and Karim be '7a' and '5a' respectively.

According to the question,

⇒ 2 × (7a - 2) = 3 × (5a - 2)

⇒ 14a - 4 = 15a - 6

⇒ a = 2

So, Present age of Karim = 5a = 5 × 2 = 10 years

∴ The answer is 10 years.