UP TGT Mathematics Mock Test - 4 - UPTET MCQ

Answer: B. Yerragudi Minor Rock Edict

The Ashokan Rock Edict found written partly in the Boustrophedon style is the Yerragudi Minor Rock Edict. The reasons for this are:

• Boustrophedon style refers to a writing style where alternate lines of text are reversed, with the direction of writing and reading alternating from left to right and then right to left. This style of writing was used in some ancient scripts, such as Greek and Etruscan.
• Yerragudi Minor Rock Edict, also known as the Gavimath inscription, is located in the Koppal district of Karnataka, India. It is one of the earliest inscriptions of Emperor Ashoka, dating back to the 3rd century BCE.
• The inscription is written in the Prakrit language using the Brahmi script, and it contains the early teachings of Buddhism, as well as the principles of Dhamma (righteousness).
• The Boustrophedon style can be observed in some parts of the Yerragudi Minor Rock Edict, making it the correct answer for this question.

November 25 is recognized as the International Day for the Elimination of Violence Against Women, a significant date designated by the United Nations to spotlight and address the global issue of violence faced by women.

The correct answer is the Blackbuck.

Important Points

• The Blackbuck is the state animal of Haryana. In India hunting of Blackbuck is ban under the Wildlife Protection Act of 1972, due to excessive hunting and habitat degradation of Blackbucks during the 20th century.
• The Blackbuck or an Indian antelope are found in India, Nepal, and Pakistan. They have a lifespan of approximately 10 to 15 years.

Additional Information

• Black Francolin of the pheasant family is the State bird of Haryana.
• There are 2 National parks located in Haryana viz, Sultanpur National Park located in Gurugram and Kalesar National Park located in Yamunanagar.
• Chinkara also known as Indian Gazelle is the state animal of Rajasthan.

SURYA KIRAN's 17th edition is being held in Pithoragarh, Uttarakhand, emphasizing the enduring collaboration between India and Nepal in military cooperation.

• Along with Buddha's images, other Buddhist images of Bodhisattvas like Avalokiteshvara, Padmapani, Vajrapani, Amitabha, and Maitreya Buddha started getting sculpted.

• However, with Vajrayana Buddhism's rise, many Bodhisattva images were added as a part of the personified representations of certain virtues or qualities, as propagated by the Buddhist religious principles for the masses' welfare.

Answer:
The correct combination regarding the ideals of our Constitution is option C: Sovereign, Socialist, Secular, Democratic Republic. Let's break down each ideal:
Sovereign:
- The Constitution declares India as a sovereign nation, which means it is independent and not under the control of any external authority.
- India has the authority to govern itself and make decisions without interference from other countries.
Socialist:
- The Constitution aims to establish a socialist society in India.
- This means promoting social and economic equality, reducing inequalities in income and wealth, and ensuring equal opportunities for all citizens.
Secular:
- India is a secular nation, which means it does not promote any specific religion or give preferential treatment to any particular religion.
- The State maintains an equal distance from all religions and treats all citizens equally, irrespective of their religion.
Democratic Republic:
- India is a democratic republic, where the power to govern is vested in the hands of the people.
- The citizens have the right to elect their representatives through free and fair elections, and these representatives make decisions on their behalf.
- The President is the head of state, and the Prime Minister is the head of government.
In summary, the correct combination of ideals of our Constitution is C: Sovereign, Socialist, Secular, Democratic Republic.

The correct answer is option (1) i.e. 1 November 1966.

• Punjab Province in British India was divided into East Punjab and West Punjab.
• East Punjab is in India and West Punjab is in Pakistan.
• Chandigarh is the capital of both the states Haryana and Punjab, earlier Haryana was a part of Punjab till 1 November 1966.
• Amarinder Singh is the Chief Minister and V. P. Singh Badnore is the present governor of the state. (May 2020)
• Punjab is known as 'The land of five rivers'.
• Vaisakhi is the main festival of the state.

India is set to chair the International Sugar Organisation (ISO) in 2024, underlining its prominent position as the world's largest consumer and second-largest producer of sugar.

Concept:

Hyperbola:

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Concept:

For n people to be seated in a circular manner, the total arrangements will be (n - 1)!.

Calculation:

Here, 10 people are seated at the round table.

∴ Total number of ways in which 10 persons can sit along with a round table = (10 - 1)! = 9!∴  

So, the total number of elementary events = 9!

Now, if we regard two particular persons as one person, then we will be left with only 9 persons. These 9 persons can be seated along with a round table in (9 - 1)! = 8!

Also, those two particular persons can be arranged among themselves in 2! ways.

∴ Number of favorable events = 8! × 2!

⇒ Required probability = (8! × 2!)/9! = 2/9

Hence, the probability that two particular persons sit together is 2/9.

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°.

Calculation:    

Total letters = 8

Letter 'E' is repeating  3 times  

Letter 'N' is repeating 2 times

⇒ Number of words can be formed = 

⇒ Number of words can be formed = 

⇒ Number of words can be formed = 3360

∴ 3360 words can be formed.

Concept:

Calculation:

Given that,

A particle having horizontal range (R) is twice the greatest height (H) attained by particle.

This mean that, R = 2H

∴ sin 2θ = sin² θ ⇒ 2sin θ.cos θ = sin² θ

tan θ = 2

also 

now using Pythagoras theorem as shown below we can find sin θ and cos θ

Therefore,  similarly 

Hence range of projectile is given as

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°.

Calculations:

Centre of the given sphere : (-1,-1,-1)

Since the required sphere is concentric with the given sphere, their centres will be the same.

We know that the equation of a sphere whose centre is (a,b,c) and radius r is, (x-?)² + (y-b)² + (z-?)²= r²

∴ Equation of required sphere with centre at (-1,-1,-1) and radius 3

= (x+1)² + (y+1)² + (z+1)²= 3² x² + 2x + 1 + y² + 2y + 1 + z² + 2z + 1 = 9

x² +y² +z² + 2x + 2y + 2z - 6 = 0, which is the required equation of the sphere.

Concept:

Derivatives of Trigonometric Functions:

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:

Coin tossed = 3 times

Formula used:

Probability = Desired Observations/Total Observations

Calculation:

According to questions,

Coin is tossed 3 times

∴ Total observations = (HHH, TTT, HHT, HTH, TTH, THT, HTT, THH)

From total observations, only 3 observations satisfy the condition,

So,

Probability of getting exactly 2 heads = 3/8

∴ The correct answer is 3/8.

Given:

Radius = 25 cm

Length of ​chord = 48 cm

Concept used:

The perpendicular from the center of a circle to a chord bisects the chord.

Pythagoras Theorem: Hypotenuse2 = Base2 + Perpendicular²

Calculation:

OA = OB = 25 cm  (Radius of circle)

AB = 48 cm

PB = AB/2 = 48/2

Using the above concept

PA = PB = 24 cm

Applying Pythagoras theorem in ΔOPB,

OB² = OP² + PB²

⇒ 25² = OP² + 24²

⇒ OP² = 625 - 576

⇒ OP2 = 49

⇒ OP = 7 cm

∴ The length of perpendicular drawn from the centre of the circle to the chord is 7 cm.

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:

Matrix Multiplication

If A is a matrix of order m×n and B is a matrix of order n×p, then the order of the product of matrices is m×p.

Steps:

• Take the elements of i^th row of the first matrix.
• Take the elements of j^th column of the second matrix.
• Multiply the corresponding elements.
• Add all the products.

Calculation:

Given:

 and 

Now, A² = 

Also given, A² = B

⇒ $\left[\begin{array}{cc}{\alpha }^2& 0\\ \alpha +1& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 5& 1\end{array}\right]$

⇒ α² = 1 and α + 1 = 5

Which is not possible at the same time.

∴ No real value of 'a' exists.

Concept:

Projectile motion: When a particle is projected obliquely near the earth surface, it moves simultaneously in horizontal and vertical directions. This type of motion is called projectile motion.

​

Where, u = projected speed, θ = Angle at which an object is thrown from the ground, g = Acceleration due to gravity = 9.8 m/s2

Maximum height is the maximum vertical distance travelled by the projectile from the horizontal plane.

For maximizing the vertical range, sin θ must be maximum, which is sin θ = 1

⇒ θ = 90° 

Concept:

• The family of circles through two circles C₁ and C₂ is C₁ + λ C₂

• If equation of circle is x2 + y2 + 2gx + 2fy + c = 0

           then centre of circle, (-g,-f)

• If equation of circle is x2 + y2 + 2gx + 2fy + c = 0

           then radius of circle, R = 

• The perpendicular distance of a point (x1,y1) from a line ax + by + c = 0 is

Calculation:

Family of circles through the points of intersection of x2 + y2 - 1 = 0, x2 + y2 - 2x - 4y + 1 = 0

is given by 

(x2 + y2 - 2x - 4y + 1) +λ(x2 + y2 - 1) = 0

⇒ (1 + λ)x2 + (1 + λ)y2 - 2x - 4y + (1 - λ) = 0

⇒ x2 + y2 - x - y +  = 0 __(i)

⇒ Centre of this circle ()

Concept:

Slope of tangent to a curve f(x), at a point (x_o​, y_o​) is given by f′(x),

Where f′(x) is derivative of f(x) at x = x_o​

Solution:

We have y = sin⁻¹(sin²x)

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.

Given:

Number are formed by using 6, 1, 2, 3

Calculation:

Number of ways to fill the first position = 4

Number of ways to fill second position = 3

Number of ways to fill third position = 2

Total number of ways = 4 × 3 × 2

⇒ Total number of ways = 24

∴ The total number of 3 digit number is 24.

Additional Information

If the repetition of number was allowed then we can take each digit in 4 different ways and so, the answer will be 4 × 4 × 4 = 64

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:

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: