UP TGT Mathematics Mock Test - 3 - UPTET MCQ

The Correct Answer  is A: 1 Only

Explanation for the Correct Answer:
The Bhimbetka rock shelters are an archaeological site in central India that spans the prehistoric Paleolithic and Mesolithic periods, as well as the historic period. It exhibits the earliest traces of human life on the Indian subcontinent and evidence of Stone Age habitation starting at the site in Acheulian times.
1. Evidence of Use of Tools and Implements:

• Bhimbetka caves indeed demonstrate the continuity of human evolution from the Lower Paleolithic period to the Mesolithic period to the Chalcolithic period. This is primarily because these caves contain a rich collection of archaeological findings.
• The site has yielded tools and implements from all these periods, providing concrete evidence of human evolution and progress in tool-making techniques and survival strategies.
• The artifacts found here include tools from the Old Stone Age (Lower Palaeolithic), tools from the Middle Stone Age (Mesolithic), as well as tools from the New Stone Age (Chalcolithic).
• This shows a clear progression and continuity of human activities and evolution over a long period of time, which is why statement 1 is correct.

2. Presence of Shankhalipi Inscriptions:

• While the Bhimbetka caves do contain ancient inscriptions, the presence of "Shankhalipi" inscriptions specifically is not a significant factor in demonstrating the continuity of human evolution from the Lower Paleolithic period to the Mesolithic period to the Chalcolithic period.
• Shankhalipi, also known as shell-script, is a unique and lesser-known script found in India, and it is not associated with the Paleolithic, Mesolithic, or Chalcolithic periods.
• Therefore, the presence of Shankhalipi inscriptions in the Bhimbetka caves does not directly support the evidence of continuity of human evolution over these periods. This is why statement 2 is not correct.

So, the correct answer to the question is option A: 1 only.

The correct option is B.
When we talk about the Vijayanagara Empire, we all remember the popular and great king, Sri Krishnadevaraya and the capital city of his kingdom, Hampi. Many people do not know about the second capital and summer capital of Vijayanagara Empire, Penukonda. Penukonda means “Big Hill”.

The correct answer is '1920'.

Key Points

• The Shiromani Gurdwara Parbandhak Committee (SGPC) is an organization in India responsible for the management of Gurdwaras, Sikh places of worship in the states of Punjab and Himachal Pradesh and the union territory of Chandigarh.
• SGPC also administers Darbar Sahib in Amritsar.
• SGPC was established in 1920 in Punjab when Gurdwara Reform Movement was started.
• The SGPC is governed by the president of SGPC.
• This post got legal status after the passing of the Sikh Gurdwaras Act, 1925 by legislation in British India.

The correct answer is 'Rock salt'.

Key Points

• Punjab is rich in rock salt.
• It has extensive deposits of rock salt.
• The Salt Range is a series of hills and low mountains between the valleys of the Indus and Jhelum rivers, located in the northern part of the Punjab region of Pakistan.
• It forms one of the richest salt fields in the world.

Additional Information

• Gold
  • ​Karnataka is the largest producer of gold in India.
  • Karnataka produces 80% of the gold in India.
  • It is known as the 'Land of Gold'.
• Coal
  • ​Jharkhand has the highest coal reserves in India.
  • India is the fifth largest coal-producing country in the world and the second fastest mined in the world.
• Oil
  • ​The three largest producers of mineral oil or crude oil in India are Rajasthan (23.7%), Gujarat (12.5%), and Assam (12.1%).
  • Rajasthan is the largest producer of petroleum in India followed by Gujarat and Assam.

The correct answer is 'An ornament worn in front of turban'.

Key Points

• The Sarpech also known as an aigrette is an ornament worn in front of a turban.
• The turban ornament was worn by significant Hindu, Sikh, and Muslim princes.
• Sar means "head" or "front" and pech means "screw", giving the word "sarpech".
• The literal meaning is "that which is screwed onto the front (of the turban)".

Important Points

Concept:

Let y = f(x) be a function defined on an interval I.

Let x₁, x₂ be any two points in I, where x₁, x₂ are not the end points of the interval. Then

• f(x) is a strictly increasing function if f(x₁) < f(x₂) whenever x₁ ≤ x₂ ∀ x₁, x₂ ∈ I.

Here, 

• f(x) is a non–decreasing function if f(x₁) ≤ f(x₂) whenever x₁ ≤ x₂ ∀ x₁, x₂ ∈ I.

Here, 

• f(x) is strictly decreasing function if f(x₁) > f(x₂ ) whenever x₁ ≤ x₂ ∀ x₁, x₂ ∈ I.

Here, 

• f(x) is a non-increasing function if f(x₁) ≥ f(x₂) whenever x₁ ≤ x₂ ∀ x₁, x₂ ∈ I.

Here, 

Solution:

Given: f(x) = 2x³ - 24x + 27

First find f'(x).

 ⇒ f'(x) = 6x² - 24

Now if f’(x) = 0

 6x² - 24 = 0

⇒ 6(x² - 4) = 0

⇒ 6(x - 2)(x + 2) = 0

⇒ (x - 2)(x + 2) = 0

⇒ x = 2, -2 

As we can see that, f’(x) < 0 for all 

Hence, the given function is strictly decreasing (-2, 2)

∴ The correct option is (4)

Given:

18 binders bind 900 books in 10 days. 

Concept used:

Total work = Efficiency (Work done per hour) × Total time taken

Calculation:

Efficiency of each binders = 900 ÷ (18 × 10) = 5 books per hour

Now, the number of required binders to bind 600 books in 12 days

⇒ 600 ÷ (12 × 5)

⇒ 10

∴ 10 binders will be required to bind 600 books in 12 days.

Let “a” be the acceleration:

Then 

Velocity at the end of 10 sec

$v^2=u^2+2a\left(s\right)=2a\left(50a\right)=100a^2$

v = 10 a

For the next 10 seconds

$s_2=ut+\frac{1}{2}a{t}^2=10a\left(10\right)+\frac{1}{2}\left(a\right)\left(100\right)=150a$

Given:  

350 workers can do work in 60 days.

Concept used :

Total work = total worker × total number of days

Calculation:

Let the number of workers should be added = x

The work has to complete 4 days before, So the work has to be completed in (60 - 4) = 56 days

As per the question,

350 × 60 = 56 ( 350 + x)

⇒ 350 + x = 375

⇒ x = 375 - 350

⇒ x = 25

∴ 25 more workers should be added

Concept:

Integral of the product of two functions

Calculation:

Given, I = ∫ex{1 + ln x + x ln x}dx

⇒ I = ∫ex dx + ∫ e^x ln x dx + ∫ e^x x ln x dx

⇒ I = ex + ∫ ex ln x dx + ∫ ex x ln x dx

Computing the last integral  ∫ ex x ln x dx by taking f(x) = x ln x and g(x) = e^x

⇒ I = ex + ∫ ex ln x dx + [x ln x ∫ ex dx - ∫(x ln x)' ∫ ex dx dx]

⇒ I = ex + ∫ ex ln x dx + x ln x ex - ∫(1 + ln x)ex dx

⇒ I = ex + ∫ ex ln x dx + x ln x ex - ∫ex dx - ∫ex ln x dx

⇒ I = ex + x ln x ex - ex 

⇒ I = x ln x ex + c

∴ The correct option is (1).

Concept:

• The cartesian product of a set A = {a,b} and B = {e,f} is either A × B = {(a,e),(a,f),(b,e),(b,f)} or B × A = {(e,a),(e,b),(f,a),(f,b)}
• If two sets have m and n number of elements respectively, then their cartesian product will have m × n number of elements.
• (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)
• n(A) gives the number of elements in set A.

Calculation:

Given, set A and B have 99 elements in common.

Hence, n(A ∩ B) = n(B ∩ A) = 99

Now, according to the formula above, 

n((A × B) ∩ (B × A)) = n((A ∩ B) × (B ∩ A)) = n((A ∩ B)) × n((B ∩ A)) = 99²

Hence, there are 99² common elements in the sets A × B and B × A 

Concept:

AM-GM inequality:

If x1.  . . Xn ≥ 0, then

Calculation:

Let a²x and b²y are two numbers.

Then apply AM-GM inequality

Concept:

Mean value theorem for integrals:

Let f be continuous on [a, b]. Then there is a point x₁ in (a, b) such that,

f' (x1) = [f (b) - f (a)]/(b - a) 

Calculation:

Given: f (x) = $\frac{1}{x}$

Concept:

The dot product of two vectors is defined as:

Additional Information
Cross/Vector product of two vectors is defined as:

Concept:

• A ⋃ B denotes the elements which are in set A or set B.
• A ∩ B denotes the elements which are in both set A and set B.

Calculation:

In the plot, the red line depicts y = x and the blue curve depicts y = e^x.

It can be seen from the plot that the two lines do not intersect anywhere. Hence, there is no common y for a given x in sets A and B. 

Therefore, A ∩ B = ϕ 

Given:

Equation (1) = 3x - 2y - 11 = 0

Equation (2) = x + y = 7

Intersect point = P(α, β)

Concept used:

We will solve the linear equations.

The value of x and y will be the coordinates of intersecting point P.

Calculation:

3x - 2y = 11      ------(i)

x + y = 7      ------(ii)

Multiplying eq (ii) we get,

3x + 3y = 21      ----(iii)

By solving equation (i) and (ii) we get

5y = 10

⇒ y = 2

Now putting the value of y in eq (ii) we get,

x = 5,

P (α, β) = P (5, 2)

So, α = 5, β = 2 

Now,

(3α + 5β)

⇒ 3 × 5 + 5 × 2

⇒ 15 + 10

⇒ 25

∴ The value of (3α + 5β) is 25.

Let ϕ = x + y + z - 5

⇒ 

⇒ 

Given:

A is a square matrix and A² = A.

Calculations:

Consider (I - A)³ + A, where I is the identity matrix.

(I - A)(I - A)(I - A) + A 

⇒ (I - A - A + A²)(I - A) + A 

⇒ (I - A - A + A)(I - A) + A 

⇒ (I - A)(I -A) + A 

⇒ (I - A - A + A²) + A 

⇒ (I - A - A + A) + A  (As A² = A)

⇒ I

Hence, option 1 is correct .

Concept:

Mean: Mean (average) is the sum of all quantities divided by no of quantities of the data set.

Mode: It is the most repeated value in the data set.

Median: Median is the middlemost value of the data set.

Skewness:

• The skewness is a measure of the asymmetry of the probability distribution assuming a unimodal
• distribution (assuming there is only one peak in the distribution of values). We can say that the
• skewness indicates how much our underlying distribution deviates from the  normal distribution
• since the normal distribution has a skewness of 0. Generally, we have three types of skewness.

Symmetrical: When the skewness is close to 0 and the mean is almost the same as the median

Negative skew:

• When the left tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the right tail.
• In this case, we can use also the term "left-skewed" or "left-tailed" and the median is greater than the mean.

Positive skew:

• When the right tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the left tail.
• In this case, we can use also the term "right-skewed" or "right-tailed" and the median is less than the mean.

Explanation:

• We know that when the left tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the right tail. In this case, we can use also the term "left-skewed" or "left-tailed". and the median is greater than the mean.
• So, the frequency curve corresponding to the data obtained is skewed to the left and the graph obtained is as below:

For the above left-skewed curve, Mode > Median > Mean.

∴ Mode > Median > Mean for left-skewed curve.

Given: 

The number of elements of order 5 in Z15 ⊕ Z5

Concept used:  number of elements of order d in Zn is ϕ(d)

ϕ(d) = dm - dm-1 where m is the power of d 

The number of elements of order d in Zn ⊕ Zm

The elements are of the form (a, b) where L.C.M (o(a), o(b)) = d 

Calculation:

The number of elements of order 5 in Z15 ⊕ Z5

The number of elements of order 5 in Z₁₅ = 4 because ϕ(5¹) = 5¹ - 5¹-1 = 4

The number of elements of order 5 in Z₅ = 4

Now, 

The number of elements of order 5 in Z15 ⊕ Z5

L.C.M (o(a), o(b)) = 5 

possible LCM = (1, 5) = 5,  LCM=(5, 1)= 5, LCM= (5, 5) = 5 

LCM = (1, 5) = no of element of order 1 in  Z15 × no of element of order 5 in  Z₅ = 1 × 4 = 4

LCM = (5, 1) = no of element of order 5 in  Z₁₅ ×  no of element of order 1 in  Z₅ = 4 ×​ 1 = 4 

LCM = (5, 5) = no of element of order 5 in  Z15 ×  no of element of order 5 in  Z₅ = 4 ×​ 4 = 16

Total number of elements = 4 + 4 + 16 = 24  

∴ The number of elements of order 4 in Z15 ⊕ Z₅ = 24

Given:

A sum of money becomes 114% of the principal in one year when it is invested at R% p.a. (at simple interest).

Formula used:

Final amount = P + 

Where, 

P = Principal

R = Rate of interest

T = Time of investment

Calculation:

Let A and P be the final amount and principal.

According to the question,

⇒ 114% of P = P(1 + )

⇒  × 100 P = P(1 + )

⇒ P = P(1 + )

⇒  = 1 + 

⇒  = 

Concept:

 where a < c < b

Calculation:

Given function y = [|x - 3|], in 1 < x < 5,

where |x - 3| = 3 - x  when 1 < x < 3

and |x - 3| = x - 3 when 3 < x < 5

So function will be defined as, 

 y = 1 when 1 < x < 2

 y = 0 when 2 < x < 3

 y = 0 when 3 < x < 4

 y = 1 when 4 < x < 5

So, 

⇒  = (2 - 1) + 0 + (5 - 4) = 2

∴ The correct answer is option (2).

Concept:

Following steps to finding maxima and minima using derivatives.

• Find the first derivative of the function.
• Set the derivative equal to 0 and solve. This gives the values of the maximum and minimum points.
• Now we have to find the second derivative.
  • If f"(x) is less than 0 then the given function is said to be maxima
  • If f"(x) Is greater than 0 then the function is said to be minima

Calculation

Given:

 where a, b and x > 0

Now 

,f''(x) = + ve

∴  f(x) has the least value at 

Sum of the volumes of the individual spheres = Volume of the hollow sphere

Volume of a sphere $=\frac{4}{3}\pi r^3$

⇒ Volume of the first sphere of radius 1 cm = $\frac{4}{3}\pi r^3=\frac{4}{3}\pi$ cm³

Volume of the second sphere =  cm³

Volume of the hollow sphere =

where R is the external radius and r is the internal radius

Given, R - r = 1

r = R -1

Hence, we have

⇒ 217 = R² + R² – R + R² + 1 – 2R

⇒ 217 = 3R² – 3R + 1

⇒ R² - R - 72 = 0

⇒ (R - 9)(R + 8) = 0

Concept Used:

For any vector , there are two parts one is the direction and the other is magnitude.

For a constant vector the direction as well as magnitude both remains constant.

And the direction of the zero vector is arbitrary.  

Calculation:

| p̅ (s)| is a non-zero constant,

Hence, Its derivation must be zero.

, i.e., zero vector.

Hence, it must have an arbitrary direction. i.e., it can have any direction.

Given:

For a quadratic equation a and b are the roots for the equations, If a + b = 2 and a2 + b2 = 20, 

Formula used:

(1.) The general form of writing quadratic equations:

x² - (Sum of roots)x + (Product of roots) = 0

(2.) For a quadratic equation, a and b are the roots for the equations

Sum of roots = a + b

Product of roots = ab

(3.) (a + b)² = a² + b² + 2ab

Calculation:

According to the question,

⇒ (a + b)² = a² + b² + 2ab

⇒ 22 = 20 + 2ab

⇒ 4 = 20 + 2ab

⇒ 2ab = -16

⇒ ab = -8

Now, the required quadratic equation.

⇒ x2 - 2x - 8 = 0

Therefore, 'x2 - 2x - 8 = 0' is the required answer.

Since A is a scalar matrix, we have 

Concept:

• tan θ = perpendicular/base
• tan 30° = 1/√3

• tan 60° = √3 

Calculation:

Let the pole be CD and the tower be AB.

Suppose the height of the pole be h. 

In ΔABD,

tan 60° = AB/BD

⇒ √3 = 20/BD

⇒ BD = 20/√3 m

Now, In ΔAEC,

tan 30° = AE/EC

⇒ 20 = 60 - 3h

⇒ 3h = 40

⇒ h = 40/3 = 

Hence, the height of the pole is .

Given:

Solution:

I =