MPTET Varg 1 Mathematics Mock Test - 2 - MPTET MCQ

हमेशा बहुवचन में प्रयुक्त होने वाला शब्द है: 'अमीर'।
स्पष्टीकरण

• उदाहरण- वह अमीर है, वे अमीर हैं।
• दोनों में प्रयुक्त हो सकता है।

अन्य विकल्प
सामग्री - एकवचन शब्द का वाक्य प्रयोग – हवन के लिए सामग्री कम पड़ेगी।
बहुवचन शब्द का वाक्य प्रयोग – पूजा के लिए सभी आवश्यक सामग्री आ गई है।
नारी - एकवचन शब्द का वाक्य प्रयोग – नारी की अवहेलना भगवान का अपमान है।
बहुवचन शब्द का वाक्य प्रयोग – आज की नारियाँ पुरुषों से किसी मामले में कम नही।
अन्यसंबंधित बिंदु

"लक्ष्मी + छाया" का संधि-पद का सही उदाहरण होगा- 'व्यंजन संधि'

• लक्ष्मी + छाया = लक्ष्मीच्छाया (ई + छ = च्छ)
• (स्वर के बाद अगर छ् वर्ण आ जाए तो छ् से पहले च् वर्ण बढ़ा दिया जाता है।)

व्यंजन संधि:-

• जब संधि करते समय व्यंजन के साथ स्वर या कोई व्यंजन के मिलने से जो रूप में परिवर्तन होता है, उसे ही व्यंजन संधि कहते हैं।

उदाहरण-

• आ + छादन = आच्छादन
• संधि + छेद = संधिच्छेद

स्पष्टीकरण
व्यंजन संधि के अन्य उदाहरण:-

• उत् + नति = उन्नति
• सत् + जन = सज्जन
• सम् + योग = संयोग
• दिक् + अम्बर = दिगम्बर
• अभी + सेक = अभिषेक

महत्वपूर्णबिंदु

The correct answer is - "a"
Explanation:

• From the given options, the most appropriate answer is "a".
• Indefinite Article "A" & "An" uses before singular countable nouns.
  • Example - A person, A woman, An MLA, etc.
• Article "A" can be used before collective noun (Countable).
• Here, Books is countable so Article "A" is used.
• Thus, Option 1 is correct option.

Other Related Points

• Definite Article "The" is used before things which are particular or specific plural nouns.
  • Example - The sun, The fort, etc.

The correct answer is Indore.
Explanation:

• Madhya Pradesh's first bone bank is being set up at MGM Medical College, Indore.​
  • The bone bank is like an eye bank.
  • In this, the bones donated by the donor or removed during the operation are kept in a deep freezer at a temperature of -40 to -80 degrees Celsius.
  • Before this, the bones to be preserved are thoroughly tested for antigens, and infections.
  • And they are used after keeping them in a deep freezer for six months.
  • Bones broken in an accident or accident can also be replaced from a 'bone bank'.​

The correct answer is 26 January 1950.

Explanation:

• The Indian Constitution, which came into effect on 26 January 1950, has the dubious distinction of being the longest in the world.
• The Constitution of India was framed between December 1946 and November 1949. During this time its drafts were discussed clause by clause in the Constituent Assembly of India.
• In all, the Assembly held eleven sessions, with sittings spread over 165 days.
• The Constituent Assembly had 300 members. Of these, six members played particularly important roles. Three were representatives of the Congress, namely, Jawaharlal Nehru, Vallabh Bhai Patel, and Rajendra Prasad.
• Besides this Congress trio, a very important member of the Assembly was the lawyer and economist B.R. Ambedkar. Serving with him were two other
  lawyers, K.M. Munshi from Gujarat and Alladi Krishnaswamy Aiyar from Madras.

• These six members were given vital assistance by two civil servants. One was B. N. Rau, Constitutional Advisor to the Government of India, and the other was the Chief Draughtsman, S. N. Mukherjee.

Hence, we conclude that the Indian Constitution came into effect on 26 January 1950.

Important PointsTimeline of Formation of the Constitution of India

By checking options, we get

As we observed the first three series (i, iii, and iv) contain 10 characters in the same sequence, but ii (858plo7op07) has 12 characters, and also one '0' is more.

Therefore, the 'ii' is the INCORRECT series.

Hence, "Option (2)" is the correct answer.

Memory refers to the ability to retain information and reproduce it over some time when required to perform a cognitive task.
It has been conceptualized as a process comprised of three stages;
(i) encoding, (ii) storage, and (iii) retrieval.
Explanation:
Memory in a system is the electronic holding place (storage). Memory is used to store not only the data but also the programs. The memory of a computer is divided into many similar cells or locations, each of which is individually addressable.
Long-Term Memory:

• It is long-term memory (LTM) through which you perform many activities in life and adapt to different situations. It is LTM to which you refer when you speak, read, recognize faces, play football and suddenly remember where you put the key to the room that you could not find before.
• The information is stored for a short period and then gets deleted. Therefore, it is also called inactive memory.

Other Related Points
Sensory Memory:
This receives information from the various sensory receptors in the environment. Here, the information is held for a very brief period, perhaps a few seconds.

Short-Term Memory:
Some memory holds information for fairly short intervals – say up to a minute and this is known as short-term memory.

Hence, we can conclude that Long-term memory is the correct answer.

Planning is an important tool in the teaching-learning process as proper planning helps teacher in setting and achieving learning objectives.
Explanation:
A plan prepared keeping in view of the lessons to be taught month-wise in an academic year is called 'year plan' as it is a systematic annual teaching plan which helps teachers in:

• establishing proper correlation between different lessons.
• engaging students effectively in teaching-learning process.
• outlining learning objectives and creating realistic timeline.
• maintaining a standard teaching pattern throughout the year.
• making teaching-learning process more regular and organised.

Hence, it could be concluded that a plan prepared keeping in view of the lessons to be taught month-wise in an academic year is called 'year plan'.
Other Related Points

Learning Disability is a neurological disorder that affects the way one receives and processes information. There are various types of learning disabilities in children.
Explanation:

• Dyspraxia is a developmental disorder of the brain in childhood causing difficulty in muscle control as well as activities requiring coordination and movement.
• This motor skill development is a type of learning disability specifically related to motor skills.
• It is also known as developmental coordination disorder or motor learning disability.
• Children with dyspraxia struggle with coordination and movement.

Hence, it can be inferred that a disorder characterized by problems with movement and motor skill coordination is Dyspraxia.

Given:
A flagstaff of height h is surmounted on a verticle tower.
Angles of elevation of the bottom and top of the flagstaff are θ and 2θ respectively.
Formula used:
1. tan θ = Perpendicular/Base

2. tan 2θ =

3. cos 2θ =

Calculation:
Let the height of the tower be x.

In right triangle ABC
tan θ = AC/AB = x/AB
⇒ AB = x/tan θ ------(1)
In right triangle ABD
tan 2θ = AD/AB = (x + h)/AB

⇒ = (x + h)/AB [Using formula (2)]

⇒ = [From equation (1)]

⇒ =

⇒

⇒

⇒

⇒ = cos2θ [Using formula (3)]

⇒ x = hcos 2θ
∴ The height of the tower is h cos 2θ

Explanation:
An injective function (injection) or one-to-one function is a function that maps distinct elements of its domain to distinct elements of its codomain.
A total number of injective functions from a set having 'n' elements to a set having 'm' elements is given by ^mP_r.
We have set A has 3 elements and set B has 4 elements.
Mapping is from A to B.
∴ m = 4 and n = 3
∴ Number of injective mappings

=

=

=

= 4! = 24

 Concept:

• If a polynomial y = f(x) cuts the x-axis at x = a, then f(a) = 0.
• If the polynomial y = f(x) touches the x-axis, then its maxima/minima is 0, or it has repeated roots, or its discriminant is 0.
• For a polynomial y = f(x), at the maxima/minima.

Calculation:
The given polynomial is y = f(x) = x2 + kx + 16.
.⇒ dy/dx = 2x + k
At minima/maxima, dy/dx = 0.
⇒ dy/dx = 2x + k = 0
⇒ x = (- k/2)
Maxima/minima should be 0, for it to touch the x-axis.

⇒ k²/4 = 16

.

Concept:
Let V be a vector space of dimension n. Let W be a subspace of V. Then
dim(W) ≤ n
If dim(H) = n, then H = V.
Calculation:
Given that, W is the subspace of vector space V.
As discussed above, we know that dimension of subspace W should be less than or equal to the dimension of vector space V. Mathematically,
dim(W) ≤ dim(V)
Other Related Points
The subspace of a vector space: A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V.
A vector space W is said to be a subspace of a vector space V if it satisfies the following conditions: 

Given:

x2 - 11x + 18 = 0

Concept Used:

For the standard form of quadratic equation,

ax² + bx + c = 0

Sum of roots = -b/a

Product of roots = c/a

Calculation:

Let the roots of the quadratic equation be α and β

In the given equation,

a = 1, b = –11, c = 18

Sum of roots =

Product of roots =

Squaring both sides of the sum of squares,

⇒

⇒

⇒

⇒

Therefore, the required value of α2 + β² is 85.

Hence, the correct answer is option 3).

Given :

The population of a village decreases each year by 20%.

And the present population of the village is 8,000

Formula used :

Old population × (decreased percent /100) ^time = present population

Calculation :

Let the population of village two years ago be x

According to the question,

⇒ x × 80/100 × 80/100 = 8,000

⇒ x = 8000 × 25/16

⇒ x = 12500

Concept:
Draw the constraints to find the feasible region:

• To draw the inequalities, first, draw the equation form of the inequalities.
• Convert all the constraints to equality and plot on the graph. Put the value of (x1, x2) obtained from the corner points of the feasible region and put it in the objective function.
• Now check the region which we have to choose depending on the sign of inequality.
• To check which region we need to choose put (0,0) in both the inequality. and check whether this inequality is satisfying or not.
• If it is satisfying the inequality then take the region containing (0,0) else the opposite side of (0,0).

Calculation:
Given:
7 x 1 + 12 x 2 ≥ 84

• Now, check all the given options by satisfying the points on this constraint.
• We have to choose the points which do not satisfy the constraints.
• The point (2,8) is lying in the lower plane of the given constraint so this is not satisfying the region of interest of the given constraints.

7 × 2 + 12 × 3 ≥ 84
⇒ 50 ≥ 84 (False)

• So, the correct answer is option 1

Concept Used:

For any function f(x) to be monotonically decreasing f'(x) < 0

Calculation:

f(x) = 2x3 + 3x2 − 12x − 4

Differentiating the above function w.r.t x

f'(x) = 6x2 + 6x − 12

for monotonically decreasing

⇒ f'(x) < 0

⇒ 6x2 + 6x − 12 < 0

⇒ x2 + x − 2 < 0

⇒ x2 + 2x − x − 2 < 0

⇒ x(x + 2) − 1(x+ 2) < 0

⇒ (x + 2) (x−1) < 0

⇒ x ∈ (-2,1)

Which is the same represents ]-2, 1[

Concept:

• Subspace of a vector space : A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V.

A vector space W is said to be a subspace of of a vector space V if it satisfies the following conditions: ​​

1. W
2. x, y ∈ W x + y ∈ W
3. α ∈ R ,x ∈ W α.x ∈ W

Calculation :
Let W = { x = (x₁, x₂, x₃) ∈ R³ | 2x1 - x2 + 3x3 = k}
Claim: Is to find the value of k.
So we shall check whether the three conditions to be a subspace is satisfied or not if we put those four k values given in the above options.
So to check W , for if k = 1,2,-1
Let us take a point (0,0,0) ∈ R³ ,then by definition of W if it were to satisfy the condition 2x₁ - x₂ + 3x₃ = k, we have no chance to get any of the k values 1,2,-1.
So, W fails to satisfy the first condition i.e. W to be a subspace .
Therefore, W will not be a subspace if k = 1,2,-1.
Next if k = 0 then we have (0,0,0) ∈ R³ such that 2x1 - x2 + 3x3 = 0 and so W
and if x = (x₁, x₂, x₃) ∈ R3 such that 2x1 - x2 + 3x3 = k and if y = (y₁, y₂, y₃) ∈ R3 such that 2y₁ - y₂ + 3y₃ = k.
Note, x + y = (x₁ + y₁, x₂ + y₂, x₃ + y₃)
Consider ,x + y = (2x1 - x2 + 3x₃) + (2y1 - y2 + 3y3)
⇒ x + y = 2(x₁ + y₁) - (x₂ + y₂) + 3(x₃ + y₃) = k
⇒ x + y ∈ W
Finally, let α ∈ R , x = (x₁, x₂, x₃) ∈ R ³ such that 2x1 - x2 + 3x3 = k
consider α.x = α.(x1, x2, x3)
⇒ α.x = α.(2x1 - x2 + 3x₃)
⇒ α.x = α.k ∈ W | k is an integer
Therefore, if k = 0 ,then W is a subspace.
Hence, the correct answer is option 2)

CONCEPT:

• Velocity (v): The rate of change of displacement of a body is called the velocity of that body.
  • Velocity is a vector quantity that has both magnitudes as well as direction.
• Acceleration (a): The rate of change of velocity is called the acceleration of the body.
  • Acceleration is also a vector quantity.
  • The slope of any velocity-time graph gives an acceleration of the body

a = dv/dt

Where t is time

CALCULATION:

Given that:

v = at + b

Acceleration (a) = dv/dt =

Given:
Two coins are tossed
Formula used:
P (E) = number of desired outcome/Total number of outcomes
Calculation:
Possible outcomes = {(H,H), (H,T), (T,H), (T,T)}
Desired outcomes = {(H,H), (H,T), (T,H)}
P(getting at least 1 tail) = 3/4
∴ The correct answer is 3/4.

Given:
A be a n×n matrix.
Concept used:
Rank of a matrix is n if and only if it is non singular.
Explanation:
We have given that A is n×n matrix.
then if it is non singular i.e. Det (A) ≠ 0 then rank of A is n. And converse is also true
If rank (A) = n then Det (A) ≠ 0
So, I ⇔ II
Det (A) ≠ 0

Concept:

If are any non-zero vector then

Calculation:

Given,

= 0

Given:

Edge of Cube = 12 m

Formula Used:

Volume of Cube = (Side of Cube)3

Calculation:

Volume of Cube = (12)³ = 1728 m³

1728 m³ Volume of air can be contained in the cubical box.

Concept Used:-

For a number "a", which can be written as,

The total number of positive divisor is given as,

T(a) = (a₁+1)(a2+1)

Here, p₁ and p₂ are the prime numbers.

Explanation:-

Given number is 252.

This number in its factored form can be written as,

252 = 2² × 3² × 7¹

Here, the 2, 3 and 7 are the prime numbers. So we have,

a₁ = 2, a2 = 3, a3 = 7

Power of 2 is 2, power of 3 is 2 and the power of 7 is 1.

Hence, the number of positive divisor of 252,

So, correct option is 4.

Given:

∠AOP = 2x + 4, ∠POQ = 3x

Calculation:

∠BOQ = ∠POQ = 3x

Now, ∠AOP + ∠BOQ + ∠POQ = 180

⇒ 2x + 4 + 3x + 3x = 180

⇒ 8x = 176

⇒ x = 22

So, ∠POB = 6x = 6 × 22 = 132°

∴The answer is 132° .

The given function can be represented as:

The given function f(x) is continuous at x = 0, if:

The above limit will become zero for all values of m > 0

For f(x) to be derivable at x = 0, should be continuous

i.e. the following evaluation must be zero:

will be zero for all values of m > 1

will be zero for all values of m > 2

From above all conditions, the given function is continuous and derivable at x = 0 for all values of m > 2

Given:

The angle of elevation of the top of a tower from a point on the ground is 30°.

After walking 30 meters towards the tower, the angle of elevation of the top becomes 60°.

Calculations:

Let AB be tower and C and D be the point of observation.

In ▵ABD, = tan60° = √3

=> AD =

=

In ▵ABC, = tan30° =

=> AC = AB × √3 = h√3

∴ CD = (AC - AD) = h√3 -

=> h√3 - = 30

=> = 30

=> 3h - h = 30√3

=> h =

=> h = 15√3

=> h = 15 × 1.73

=> h = 25.95~26

∴ The answer is 26 m.

Let,

By solving through integration by parts, we get

where C is constant

Concept Used:

A cyclic quadrilateral is a quadrilateral whose four vertices are on the circumference of a circle and the sum of opposite angles is 180°

A parallelogram is a quadrilateral whose opposite sides and angles are equal and both pairs of opposite sides are also parallel.

Calculation:

let ABCD be parallelogram,

∠ A=∠ C and ∠ B =∠ D

and for ABCD to be cyclic

∠ A +∠ C = 180° and ∠B +∠ D = 180°
⇒ 2∠A = 180° and ∠B = 180°
⇒ ∠A = 90° and ∠B = 90°
also opposite sides are equal.
⇒ ABCD is a rectangle.

Given:

The objective function z = 4x + 6y

The constraints are -

3x + 2y ≤ 12,

x + y ≥ 2,

and x, y ≥ 0

Concept:

The maximum or minimum value of a function occurs at one of the corner points of the feasible region.

Solution:

Making the graph of the constraints -

We see that the feasible region has corner points (0, 2), (2, 0), (4, 0) and (0, 6)

The values of the objective function z = 4x + 6y at these points respectively are 12, 8, 16 and 36

Hence maximum value occurs at (0, 6) and is equal to 36