UP TGT Mathematics Mock Test - 4 - UPTET MCQ

The executive board of the International Monetary Fund (IMF) approved a proposal to make a 50 per cent increase in the quota allocated to members in proportion to their current quotas. The proposal will now be considered and made effective by the board of governors as part of concluding the 16th general review of quotas.
Currently, India has a quota of special drawing rights (SDR) 13,114.4 million which denotes a share of 2.75 per cent, making it the eighth-largest quota-holding country in the IMF.

• Statement 1 is correct: The Gupta era saw a more developed judicial system, with civil and criminal laws clearly demarcated, indicating advancements in legal frameworks.
• Statement 2 is incorrect: Guilds of artisans and merchants were actually governed by their own laws rather than state laws, allowing them autonomy in managing internal affairs and maintaining discipline.
• Statement 3 is correct: The involvement of Brahmanas in the judicial process alongside kings underscores the influence of the varna system on governance and legal proceedings.

The correct answer is '1577'.

Key Points

• Amritsar was founded by Shri Guru Ramdas ji, the fourth guru of the Sikhs in about 1577 A.D.
• Before the city was founded, the area was covered with thick forests and had several lakes.
• The construction of Amrit Sarovar from which the city gets its present name was also started by Shri Guru Ramdas.
• His successor, Shri Guru Arjan Dev, completed the project and located the Harmandir Sahib in its midst.
• Maharaja Ranjit Singh took over full control of Amritsar by 1802 A.D.

The correct answer is Rocky soil.

Key Points

• The land of Haryana can be divided into the following three parts:
  • Hill:
    • The soil of this region is rocky and this type of soil can be seen on the hills of Morni.
    • In the southern part of the state, due to the presence of hills of the Aravalli Mountains, rocky and sandy soils are found here.
    • The soil in the hilly region of the state is thin and hard.
  • Plains:
    • The soil of the plains of the state is fertile.
    • It is brownish-yellow in color.
    • This soil has come with the flow of rivers like Yamuna, Saraswati etc.
    • Many crops are grown in this region and the most fertile region is also considered to be the plains of Haryana.
  • Sandy:
    • Sandy soil is found in the southwestern part of Haryana.
    • The color of this soil is light brown.
    • This soil comes with the winds blowing from the neighboring state of Rajasthan.

The Correct Answer is Chandulal Madhavlal Trivedi.

Key Points

• Sir Chandulal Madhavlal Trivedi was an Indian administrator and civil servant who served as the first Indian governor of the state of Punjab (then East Punjab) after Independence in 1947.
• He subsequently served as the first Governor of Andhra Pradesh from its creation in 1953 until 1957.
• Born- 2 July 1893
• Died- 15 March 1980
• The Governor is appointed by the president for a term of 5 years.
• Power and functions of Governor:-
  • Executive Power related to administration, appointments, and removals.
  • Legislative powers related to lawmaking and the state legislature, that is Vidhan Sabha or Vidhan Parishad.
  • Discretionary powers to be carried out according to the discretion of the Governor.

Additional Information

Chakravarthi Rajagopalachari:-

• Chakravarthi Rajagopalachari was the first Governor of West Bengal.
• Tenure:- 15 August 1947 to 21 June 1948
• Current Governor Of West Bengal:- Jagdeep Dhankhar

Maharaj Man Singh II ( Rajpramukh):-

• Maharaj Man Singh II ( Rajpramukh) was the first Governor of Rajasthan.
• Tenure:- 30 March 1949 to 31 October 1956
• Current Governor of Rajasthan:- Kalraj Mishra

Dharma Vira:-

• Dharma Vira was the first Governor of Haryana.
• Tenure:- 1 November 1966 to 14 September 1967
• Current Governor of Haryana:- Satyadev Narayan Arya

The correct answer is IDA.

Key Points

• International Development Association (IDA) provides low- or no-interest loans to low-income countries.
• It is one of the five development institutions of the World Bank.
• About IDA (International Development Association):
  • The founding of the International Development Association (IDA) in 1960 put greater stress on the poorest countries, part of a constant shift toward the eradication of poverty becoming the Bank Group’s primary goal.
  • With 173 shareholder nations, IDA aims to reduce poverty by providing loans (called “credits”) and grants for programs that boost economic growth, reduce inequalities, and improve people’s living conditions.
  • IDA is one of the largest sources of assistance for the world’s 75 poorest countries, 39 of which are in Africa, and is the single largest source of donor funds for necessary social services in these countries.

Additional Information

• About World Bank:
  • The Bretton Woods Conference, formally known as the United Nations Monetary and Financial Conference, was a meeting of delegates from 44 nations that met from July 1 to 22, 1944 in Bretton Woods, New Hampshire (USA), to agree upon a series of new rules for international financial and monetary order after the conclusion of World War II.
  • The two major attainments of the conference were the creation of the International Bank for Reconstruction and Development (IBRD) and the International Monetary Fund (IMF).
  • Founded in 1944, the International Bank for Reconstruction and Development (IBRD) — presently called the World Bank — has grown to a closely associated group of five development institutions.
  • With a total of 189 member countries, the World Bank Group is a unique global partnership.

The three main indicators of the Human Development Index (HDI) are life expectancy at birth, gross enrollment ratio, and per capita income. These indicators provide a comprehensive measure of a country's development, encompassing health, education, and economic well-being.

Concept:

• Equation of motion: The mathematical equations used to find the final velocity, displacements, time, etc. of a moving object without considering force acting on it are called equations of motion.
• These equations are only valid when the acceleration of the body is constant and they move on a straight line.

There are three equations of motion:

v = u + at

s = ut + ½ at²

v² = u² + 2as

Where, v = final velocity, u = initial velocity, s = distance travelled by the body under motion, a = acceleration of body under motion, and t = time taken by the body under motion.

Calculation:

A body falling from rest ⇒ u = 0

Distance travelled = h

Attain velocity (final velocity) = v

Applying third equation of motion

v² – u² = 2as

v² – 0 = 2gh [here a = g free fall under gravity]

⇒ v² = 2gh ⇒ h = v²/2g

To attain final velocity = 2v

Let it travel h^’ from starting

v² – u² = 2as

(2v)² – 0 = 2gh’ [here a = g free fall under gravity]

⇒ h’ = 4 (v²/2g)

Total distance to travel to achieve v to 2v is-

⇒ h’ = 4h

⇒ 4h - h = 3h  (more to fall)

Hence the correct answer is 3h.

Concept:

• Projectile motion: Projectile motion is the motion of an object projected into the air, under only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.
• Initial Velocity: The initial velocity can be given as x components and y components.

• Component of initial velocity in x-direction, (ux) = ucosθ
• Component of initial velocity in the y-direction, (uy) = usinθ
• In the case of projectile motion, we can see a free-fall motion of a body on a parabolic path with constant velocity.
• If a body is thrown at a certain angle then during its movement, we get two components of velocity as given below.

• And thus, the range of a projectile is the displacement of a particle along the x-axis and can be given as:

The range of the projectile,

• Whereas the time of flight is the total time for which projectile stayed in the air.

, 

The angle of projection = θ

Initial velocity = u

Gravitational acceleration = g

Time of flight = t

Range of projectile = R

Explanation:

As given above,

v = initial velocity of projectile 

θ = angle with x-axis

Time of flight for the projectile, 

And the range of the projectile, 

Thus, by comparing the above two equation range of projectile can also be modified as

But  (By using trigonometric relation)

Concept:

Formula:

• log mⁿ =  n.log m
• d(uv)dx=vdudx+udvdx
• dxdx=1

Solution:

Let y = x^x

Taking log both sides, we get

⇒ log y = log(x)^x

⇒ log y = x.log(x)     [∵ log mⁿ = n.log m ]

Differentiating with respect to x, we get

∴ The correct option is (1)

CONCEPT:

• Centre of mass: Centre of Mass of a body is a point at which the whole of the mass of the body appeared to be concentrated.

The Centre of masses of different bodies is given below in the table: 

EXPLANATION:

• From the above table, it is clear that the center of mass of a uniform circular half ring is at a distance of   from the center.
• So the correct answer is option 2.

Concept used: 

A group is cyclic iff there exists an element of order whose order is equal to the order of the group 

Calculations:

o(S₃) = 3!

contains 6 elements of the Symmetric group S(3), of degree 3, written in cycles are : {(1), (12), (13), (23), (123), (321)}.

o(1) = 1

o(12), o(23) and o(13) = 2

o(123),  and  o(321) = 3 

no element having order equal to order of group 

∴ option 1 is incorrect.

o(D4) = 8 

 D₄ = {1, r, r², r³ , s, sr, sr², sr³}

o(s), o(sr), o(sr2), o(sr3) = 2

o(1) = 1, o(r) = 2 , o(r2) = 3, o(r3) = 4

no element having order equal to order of the group 

∴ option 3 is incorrect.

o(K₄) = 4 

K4  = {e, a, b, ab}

o(e) = 1 , o(a) = 2, o(b) = 2, o(ab) = 2 

No element having order equal to the order of group 

∴ option 4 is incorrect.

o(Z₅) = 5 

Z5 = {0, 1, 2, 3, 4}

o(0) = 1 o(1) = 5, o(2) = 5 , o(3) = 5 , o(4) = 5 

all elements having order = order of the group 

all are the generator of the group.

∴ option 2 is correct.

Given:

If Mohan speaks the truth 70% of the times, Sohan speaks the truth 50% of the times.

Calculation:

Probability of Mohan speaking truthP(Mohan) = 70% = 70/100 = 7/10

Probability of Sohan speaking truthP(Sohan) = 50% = 50/100 = 1/2

Probability of two mutually exclusive events to occur simultaneously = P(Mohan) × P(Sohan)

⇒ 7/10 × 1/2

⇒ 7/20

⇒ 0.35

∴ Required probability is 0.35

Given:

Three sides of the traingle are a = 5, b = 7 and c = 3

Concept used:

Cosine rule = the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

Example:

As per the diagram, Cosine rules to find the length of the sides a, b & c of the triangle ABC is given by

cos x = (b2 + c2 – a2)/2bc

cos y = (a2 + c2 – b2)/2ac

cos z = (a2 + b2 – c2)/2ab

Calculation:

Applying cosine formula in ΔABC

CosB = [(5² + 3² - 7²)/(2 × 5 × 3)]

⇒ CosB = [(25 + 9 - 49)/30]

⇒ CosB = [(- 15)/30]

⇒ CosB = -(1/2)

⇒ CosB = Cos120°

So, B = 120°

∴ Angle B is 120°.

The correct answer is 270 J.

• Given.
• mass, m= 15 kg.
• Velocity, v= 6 ms^-1.
• 

                                    = 0.5 × 15 × 6²

                                    = 270 J

Kinetic Energy

• If a body of mass m is moving with velocity v, then kinetic energy KE:
  • 
  • where p is the linear momentum.
  • When momentum is doubled, kinetic energy becomes four times.
  • If a body is moving in a horizontal circle then its kinetic energy is the same at all points, but if it is moving in a vertical circle, then the kinetic energy is different at a different point

Concept:

Let  , if , then x = p, y = q and z = r

Solution:

Given that,  and 

If , then

x - y = 3 ....(1)

and x + y = 2 ....(2)

Concept:

• Idempotent Matrix: Idempotent matrix is a square matrix that when multiplied by itself, gives back the same matrix. A matrix M is said to be an idempotent matrix if M² = M.
• Further every identity matrix can be termed as an idempotent matrix.
• The idempotent matrix is a singular matrix and can have non-zero elements.

Solution:

• A matrix that, when multiplied by itself, yields itself is called an idempotent matrix.

∴ The correct option is (1)

Additional Information Involutory matrix: A matrix A is said to be involutory if A² = I

Symmetric matrix: A matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself (A^T = A)

Given: 

The tangent at a point P, on a circle with centre O, intersects the diameter BA produced at Q.

∠QPB = 103o

Concept used:

The angle subtended by diameter on any point on the circumference of a circle is 90°.

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Calculation:

According to the question, the required figure is:

As we know,

The angle subtended by diameter on any point on the circumference of a circle is 90°.

So, 

∠BPA = 90°

Now, 

∠BPQ = ∠BPA + ∠APQ

⇒ 103° = 90° + ∠APQ

⇒ ∠APQ = 13°

As per the alternate segment theorem.

∠PBQ = ∠APQ = 13°

In ΔBPA,

∠PBA + ∠BAP + ∠APB = 180°

⇒ 13° + ∠BAP + 90° = 180°

⇒ ∠BAP + 103° = 180°

⇒ ∠BAP = 77°

Now,

∠BAP + ∠PAQ = 180°

⇒ 77° + ∠PAQ = 180°

⇒ ∠PAQ = 103°

∴ The measure of ∠PAQ is 103°.

Concept:

The general equation of sphere is given as:

x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0  ----(1)

centre at (-u, -v, -w) and

radius  ---(2)

Calculation:

Given:

x2 + y2 + z2 + x + y - 2 = 0

Compare given equation with equation (1)

x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0

u = -0.5, v = -0.5, w = 0, d = -2

Radius can be calculated from equation (2)

= √5/2

Given:

S2: x² +y² +z² + 2ux + 2vy + 2wz + d = 0

S1: x² +y² +z² +2u₁x + 2v₁y + 2w₁z + d₁=0

Calculations:

The conditions such that two spheres are orthogonal to each other is

2uu1 + 2vv1 + 2ww1 = d + d₁

Concept:

In a G.P., 

nth term is calculated by: a_n = ar^n-1, where a = first term and r = common ratio.

Application:

In a G.P., If sixth term is 32

⇒ a₆ = ar^6-1 = 32

⇒  ar⁵ = 32......(equation 1)

And the eighth term is 128

⇒ a₈ = ar^8-1 = 128

⇒  ar⁷ = 128....(equation 2)

Dividing equation 1 by equation 2, we get,

r² = 4

⇒ r = 2

Concept:

ω is a cube root of unity.

Property of cube root of unity:

• ω3 = 1
• 1 + ω + ω2  = 0

Calculations:

Given, ω is a cube root of unity.

⇒ω³ = 1, ad 1 + ω + ω²  = 0

Now, consider the determinant

Taking ω and -ω common from C₂ and C₃ respectively,

Hence the required value is 

Concept:

If a cyclic group G is generated by an element 'a' of order 'n', then am is a generator of G if m and n are relatively prime.

Calculation:

Let a cyclic group G of order 50 generated by an element a, then

o(a) = o(G) = 50;

to determine the number of generator of G evidently,

G = {a, a2, a3, ........ , a⁵⁰ = e}

An element am € G is also a generator of G is H.C.F of m and 50 is 1.

H.C.F of (1, 50) is 1 silimarly, (3, 50) (7, 50), (9, 50) (11, 50), (13, 50), (17, 50), (19, 50), (21, 50), (23, 50), (27, 50), (29, 50), (31, 50), (33, 50), (37, 50), (39, 50), (41, 50), (43, 50), (47, 50), (49, 50)

Therefore, there are 20 generators of G.

Concept:

Symmetric Matrix:

• Square matrix A is said to be symmetric if the transpose of matrix A is equal to matrix A itself
• AT = A or A’ = A

Where,

AT or A’ denotes the transpose of matrix

• A square matrix A is said to be symmetric if aij = aji for all i and j

Where aij and aji is an elements present in the matrix.

Skew-Symmetric Matrix or Anti-symmetric:

• Square matrix A is said to be skew-symmetric if aij = −aji for all i and j.
• Square matrix A is said to be skew-symmetric if the transpose of matrix A is equal to the negative of matrix A ⇔ AT = −A
• Note that all the main diagonal elements in the skew-symmetric matrix are zero.

option(4)

CONCEPT:​

• Resolution of vectors into components: We have a vector (F) where the magnitude of the vector is F and the angle with horizontal is θ.

The vector has two components: 1. Vertical component and 2. Horizontal component

Vertical component (Fy) = F Sinθ

Horizontal component (Fx) = F Cosθ 

CALCULATION:

The horizontal component is responsible to move the stone along the ground.

F_x = 60cos60°  = 30 N

Hence, the effective value of the pull tending to move the stone on the ground is 30 N.

Equation of line segment:

-11(y + 1) = 10(x – 8)

-11y – 11 = 10x – 80

11y = -10x + 69       …… (1)

y = (-10x/11) + (69/11)

Equation of line perpendicular to above line:

y = (11x/10) + C

y = 1.1x + C

This line also passes from point (3, 10), then-

10 = 1.1 × 3 + C

C = 6.7

Equation of line passing through (3, 10) and perpendicular to line 11y = -10x + 69:

y = 1.1x + 6.7

10y = 11x + 67       ….. (2)

From (1) and (2):

x = -47/221 and y = 1429/221

Foot of perpendicular = (x, y)

= (-47/221, 1429/221)

Total people = 60

Hindi speaking people = 25

Both speaking people = 15

Only English speaking people = 60 - 25 = 35

So total English speaking people = 15 + 35 = 50

Below is the Venn diagram and shaded area represents the English speaking people.

Hence, correct answer is '50'.

Concept:

Chain Rule: If y is a function of u and u is a function of x

Calculation:

Concept:

1. 

2. Scalar Triple Product:

A scalar triple product is also called a box product.

It is evident that the scalar triple product of vectors means the product of three vectors. It means taking the dot product of one of the vectors with the cross product of the remaining two. It is denoted as

[a b c ] = a · [b × c]

3.The product is cyclic in nature ⇔ [ a b c ] = [ b c a ] = [ c a b ]

4. If any two of the a, b or c in [ a b c ] are the same then the value of the box product will be zero.

Calculation:

Given that,

Doing dot product on both sides with vector 

Additional Information
Volume: If ,and are the edges of a parallelepiped then the volume of the parallelepiped is given by: