MPTET Varg 2 Math Mock Test - 2 - MPTET MCQ

Given:

The least common multiple of a and b is 42

Calculation:

According to the question,

LCM(a, b) = 42

LCM(5a, 11b) = LCM{LCM(5,11)(a,b)}

LCM(5a, 11b) = 5 × 11 × 42 = 2310

∴ The LCM of 5a and 11b is 2310.

Given:

The diagonal of the square is 20 cm.

Concept used:

Perimter = 2√2 x diagonal

Calculation:

⇒ Perimeter = 2√2 x 20 = 40√2 cm

∴ The diagonal of square is 20 cm, then perimeter of the square must be 40 √2 cm.

Calculation:

LCM of (25, 40 and 56) = 1400

So,the smallest number which when divided by 25, 40 and 56 leaves 13 as remainder is 1400 + 13 = 1413

∴ The correct answer is 1413.

Given, the quadratic polynomial is x² + 99x + 127

We have to find the zeros of the polynomial.

Using the quadratic formula,

x = [-b ± √b² - 4ac]/ 2a

Here, a = 1, b = 99 and c = 127.

x = [-99 ± √99² - 4(1) (127)]/ 2(1)

x = [-99 ± √(9801 - 508)]/ 2

x = [-99 ± √9293]/2

x = [-99 ± 96.4]/2

Now, x = (-99 + 96.4)/ 2 = -2.6/2 = -1.3

x = (-99 - 96.4)/ 2 = -195.4/ 2 = -97.7

The roots are -1.3 and -97.7.

Therefore, the zeros of the polynomial are both negative.

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 mathematical puzzles, riddles and tricks develop all of the above skills.

Mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems that makes it a universal subject.

Key Points Nature of Mathematics:

• Mathematics reveals hidden patterns that help us understand the world around us.
• The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty. Thus, it has a wide scope of generalization.
• Mathematicians see Mathematics as a language, they believe that just like any other language mathematics has its own vocabulary, grammar, and rules which are known as terms, definitions, operational signs, formulas, etc.
• Mathematics is deductive and inductive in nature, so it can be claimed that due to its nature of deduction and induction it can be predicted after getting the conclusion based on the reasoning.

​Hence, from the above-explained points, we can say that "the scope of Generalisation in Mathematics is very limited" is not related to Nature of mathematics.

Concept used:

Calculation:

∴ The square root of 0.9 is equal to 0.9487.

Number of females patient in city A = 3/8 × 64 = 24

Number of females patient in city B = 4/9 × 54 = 24

Number of females patient in city C = 1/4 × 16 = 4

Number of females patient in city D = 5/12 × 48 = 20

Number of females patient in city E = 1/3 × 27 = 9

The total females = 24 + 24 + 4 + 20 + 9 = 81

Since the number given is in thousands = 81000

Average = 81000/5

⇒ 16200

The correct answer is 2 and 3 only.

Key Points

Pressure

• It is the force per unit area.
• The SI Unit of Pressure is the pascal (Pa).
• A pascal can be defined as a force of one newton applied over a surface area of a one-meter square.
• The pressure is dependent on the area over which the force is acting, without any change in the force, the pressure can be increased and decreased. Hence, Statement 1 is not correct.
• The Force applied is to be constant if the surface becomes smaller the pressure increases and vice versa.
• The drinking straw is a very thin pipe that is used to drink Soft drinks.
• The drinking straw works on the existence of atmospheric pressure. Hence, Statement 2 is correct.
• The lower end of the drinking straw is dipped in the soft drink. When we suck at the upper end of the straw with our mouth, the pressure of the air inside the straw and in our inner mouth is reduced. But the pressure acting on the surface of soft drinks is equal to the atmospheric pressure.
• Greater atmospheric pressure acting on the surface of the soft drink pushes the soft drink up the straw into our mouth.
• The gas pressure inside an inflated balloon is always greater than the atmospheric pressure outside. Hence, Statement 3 is correct.
• Pressure is the force acting per unit area of a surface. When the air is filled inside the balloon the pressure inside the balloon increases and the balloon becomes spherical and exerts equal pressure in all directions in order to balance the atmospheric pressure.

Concept used:

The Square has all congruent sides and angles are the right angle.

The opposite sides are parallel.

Solution:

The properties of the square, match the given properties.

R represents a square shape.

Hence, the correct option is 3.

CONCEPT:

Methods of Heat Transfer:

Conduction:

• The heat transfer through direct physical contact is known as conduction.
• Heat transferred from the electric burner of a stove to the bottom of a pan is transferred by conduction.
• This method of heat transfer is applicable in

Radiation:

• When electromagnetic radiation is emitted or absorbed to transfer the heat or When heat is transferred in the form of electromagnetic waves is known as radiation.
• Electromagnetic radiation includes microwaves, radio waves, infrared radiation, visible light, ultraviolet radiation, gamma rays, and X-rays.
• The method does not require any medium so heat from the sun reaches us by this method.

Convection

• It is heat transfer by the movement of its molecules.
• It occurs only in fluids (liquid and gas) not in solids because solid particles can not move from one place to another.
• For example, in a pot boiling water on the stove, or in thunderstorms, where hot air rises up to the base of the clouds.
• The sea breeze and land breeze occur due to convection.

EXPLANATION:

Sea breeze:

• This process takes place for the duration of the day. Both the sea and the land surface are heated up by the sun.

• The sea heats up slower than the land because it has a much higher heat capacity.

• Thus, the temperature over the land surface increases, in turn, heating up the surrounding air.

• Expansion occurs in the less dense warm air and an area over the land having low pressure is developed. At the same time on the top of the sea, a high-pressure area develops.

• Due to the difference in pressure and the air flows from the high pressure over the sea to the low pressure over the land.

• Hence, option 3 is correct.

Given:

In a triangle ΔABC, 2∠A = 3∠B = 6∠C

Concept Used:

The sum of angles of a triangle = 180°

Calculation:

According to the question,

⇒ 2∠A = 3∠B = 6∠C

⇒ ∠A = 3∠C and ∠B = 2∠C

According to the concept,

⇒ ∠A + ∠B + ∠C = 180°

⇒ 3∠C + 2∠C + ∠C = 180°

⇒ 6∠C = 180°

⇒ ∠C = 30°

∴ ∠A = 3∠C = 90° and ∠B = 2∠C = 60° and ∠C = 30°

Given:

PQ ∥ RS

∠P = 85°

Concept used:

The sum of the interior angles on the same side of a transversal line is always 180 degrees.

The sum of opposite angles in a cyclic quadrilateral is 180°.

Calculation:

In a cyclic quadrilateral PQRS.

∠P = 85°

The sum of opposite angles in a cyclic quadrilateral is 180°.

∠P + ∠R = 180°

⇒ ∠R = 180° - ∠P

⇒ ∠R = 180° - 85°

⇒ ∠R = 95°

Now, ∠Q = 180° - ∠R

⇒ ∠Q = 180° - 95°

⇒ ∠Q = 85°

∴ The correct answer is 85°.

The word "mathematics" has been used in two distinct and different senses, i.e., one as a method used to solve the problems of quantity, space, order, etc., and the second as a set of laws or generalizations of truths that are, discovered. According to National Curriculum Framework-2005, the main goal of Mathematics education in school is the mathematization of the child’s thought process. Mathematics relies on logic, reasoning, problem-solving, creativity, and mathematical way of thinking. These skills can be useful in many other subjects.

Key Points

• Mathematics is the science of quantity and space.
• Mathematics is the science of calculations and measurement.
• Mathematics is the method of the progress of various subjects.
• Mathematics is the abstract form of science.
• Mathematics is the systematized, organized, and exact branch of science.
• Mathematics is an important means of generalization.
• Mathematics is the means of drawing conclusion and judgement.
• Mathematics is the logical subject and science of logical reasoning.

Thus, it can be concluded that Mathematics is the science of measurement, logical subject, science of number and space.

Given:

A cuboid having dimension:

Length = 4 cm, breadth = 6 cm, height = 6 cm

Formula used:

The radius of hemisphere = 1/2 × (√4h² + l² + b²)

Calculation:

2D image of the hemisphere:-

The radius of the hemisphere =

The radius of hemisphere = 1/2 × (√4h² + l² + b²)

⇒ 1/2 × √{4 × (6)²} + (4)² + (6)²

⇒ 1/2 × √144 + 16 + 36

⇒ 1/2 × √196 = 1/2 × 14 = 7 cm

∴ The radius of hemisphere = 7 cm

Given:

{(5 × 5 × 5 × 5 × 5 × 5)5 × (5 × 5 × 5 × 5 × 5)8} ÷ (5 × 5) = (625)a

Concept Used:

(x^m)ⁿ = x^mn

x^m × xⁿ = x^m+n

Calculation:

⇒ According to the question,

⇒ {(5 × 5 × 5 × 5 × 5 × 5)5 × (5 × 5 × 5 × 5 × 5)8} ÷ (5 × 5) = (625)a

⇒ {(5⁶)5 × (5⁵)8} ÷ (5²) = (625)a

⇒ {5³⁰ × 5⁴⁰} ÷ (5²) = (5⁴)a

⇒ {5⁷⁰} ÷ (5²) = 5^4a

⇒ 5⁷⁰/5² = 54a

⇒ 5⁶⁸ = 54a

⇒ Comparing the powers in the above equation

⇒ 68 = 4a

⇒ a = 17

Therefore, the value of a is 17.

Given:

BA || ED and BC || EF.

Concept used:

Corresponding Angles Theorem: It states that when a transversal line intersects the parallel lines, then the corresponding angles formed are congruent.

Explanation:

Produce DE to intersect BC at P

Since AB and ED are parallel and when DE is produced to intersect BC, it acts as a transversal line.

Hence, following the above-given theorem ∠ABC = ∠DEF.

CONCEPT:

• Thermal Conduction: It is the ability of a material to conduct heat and it is measured by thermal conductivity.
• Thermal conductivity is a property of a material. It depends only on the material.
• It is denoted by K and the SI unit is watt per (meter.kelvin)(W/m-K).

EXPLANATION:

• The thermal conductivity of any material is a property of that material. So it is independent of length, mass, and area of cross-section. So options 1, 2, and 3 are wrong.
• The thermal conductivity only depends on the material of the rod. So option 4 is correct.

Important Point

• Some values of the thermal conductivity of common materials are.

1. Copper – 385 W/m-K
2. Aluminium – 209 W/m-K
3. Brass – 109 W/m-K
4. Air – 0.0238 W/m-K

Given:

CI @ 11% is 6,963

Formula:

A = P (1 + r/100)^t

CI = P (1 + r/100)^t - P

Calculation:

P (1 + 11/100)² - P = 6963

⇒ P × 1.11² - P = 6963

⇒ 1.2321P - P = 6963

⇒ 0.2321P = 6963

⇒ P = 6963/0.2321

⇒ P = 30,000

∴ The sum is ₹ 30,000.

Shortcut TrickCalculation:

Effective rate for 2 years at rate 11% = 23.21%

So,

⇒ P × 23.21% = 6963

⇒ P = (6963/23.21) × 100

⇒ P = Rs. 30,000

∴ The correct answer is 30,000.

Formula used:

cos θ = Base / Hypotenuse

Calculation:

Let, the length of the ladder be L

By using the above formula;

cos θ = Base / Hypotenuse

cos 60° = 2.5/L

⇒ 1/2 = 2.5/L (∵ cos 60° = 1/2)

⇒ L = 2 × 2.5 = 5 m

∴ The length of the ladder is 5 meters.

Given:

Concept used:

We will take fully filled area is 1 unit, Half filled area is 1/2 unit, more than half filled area is 1 unit (approx 1), less than half filled area is 0 (approx 0)

Calculation:

In figure 1,

Area of square = 5 × 1 + 7 × 1/2 + 4 × 1 + 9 × 0 = 5 + 3.5 + 4 = 12.5 sq unit

In figure 2:

Area of square = 8 × 1 + 4 × 1 + 7 × 0 = 8 + 4 = 12 sq unit

In figure 3:

Area of square = 6 × 1 + 2 × 1/2 = 6 + 1 = 7 sq unit

In figure 4:

Area of square = 4 × 1 + 5 × 1/2 + 8 × 1 = 4 + 2.5 + 8 = 14.5 sq unit

∴ Area of figure 4 has larger area

Given:

Concept used:

Sin²θ + Cos²θ = 1

Calculation:

⇒ cosec x = 2

⇒ cosec x = cosec 30°

⇒ x = 30°

∴ The required answer is 30°.

Given:

Principal = Rs. 8000

Amount = Rs. 12600

Formula used :

A = P + SI

SI = (P × R × T)/100

Calculation:

According to the question,

SI = Rs.4600

⇒ 4600 = (8000 × R × 5)/100

⇒ 400R = 4600

⇒ R = 11.5%

∴ The correct answer is 11.5%.

Given

Average of 1 to 20 prime number

Concept

Prime number: Any number that is divisible by either itself or 1.

Formula Used:

Average = Total sum of all units/No. of units

The prime number between 1 to 20 are:-

⇒ 2, 3, 5, 7, 11, 13, 17, 19

Average = (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19)/8

⇒ 77/8

⇒ 9.625

The correct answer is 7.5 N.

Key Points

• Given,
  • mass = 5kg
  • time = 2s
  • Initial velocity = 4 ms-1
  • final velocity = 7 ms-1
  • Let the acceleration be 'a'.
  • Therefore,
    • 
    • a = 1.5 ms-2
  • We know that force is the product of mass and acceleration.
    • Therefore, force = 5 x 1.5 = 7.5 N

Given:

Calculation:

⇒ x = 1

∴ The answer is 1.

Concept:

If a, b, c are in A.P. then 2b = a + c.

Calculation:

Given, a, b, c are in arithmetic progression,

⇒ 2b = a + c

So, the equation ax2 - 2bx + c = 0

⇒ ax2 - (a + c)x + c = 0

⇒ ax(x - 1) - c(x - 1) = 0

⇒ (ax - c)(x - 1) = 0

⇒ x = 1, c/a

∴ The correct answer is option (1).

Given:

The equation x² – mx + 1 = 0 has two distinct roots.

Concept:

For any given quadratic equation,

ax² + bx + c = 0,

For two distinct roots,

D > 0

D = b² - 4ac

Solution:

From the given quadratic equation,

x² – mx + 1 = 0

a = 1

b = -m

c = 1

D = (-m)² - 4(1)(1)

D = m² - 4

For two distinct roots,

D > 0

m² - 4 > 0

m² > 4

|m| > 2

Hence, option 2 is correct.

Given:

cot θ + cosec θ = 5

Concept used:

cosec2θ - cot2θ = 1

Hypotenuse² = Perpendicular² + Base²

H² = P² + B²

Calculation:

cosec²θ - cot²θ = 1

⇒ (cosec θ + cot θ)(cosec θ - cot θ) = 1

⇒ 5(cosec θ - cot θ) = 1

⇒ cosec θ - cot θ = 1/5

Now,

cot θ + cosec θ + cosec θ - cot θ = 5 + 1/5

⇒ 2cosecθ = 26/5

⇒ cosecθ = 13/5

So, H/P = 13/5

So, B = √(13² - 5²)

⇒ √(169 - 25)

⇒ √144

⇒ 12

So, cos θ = B/H

⇒ 12/13

∴ The required answer is 12/13.

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.