UP TGT Mathematics Mock Test - 3 - UPTET MCQ

Professor Meena Charanda, Principal of Kalindi College, Delhi University, received the ‘International Culture Award’ in 2024 for her exceptional contributions to education and social service. The honor was bestowed upon her at the India International Center on March 30, 2024.

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”.

G. Sathiyan made history by winning the men's singles trophy at the WTT Feeder Beirut 2024 event, becoming the first Indian to achieve this feat in a World Table Tennis Feeder Series tournament. This victory showcased his exceptional skills and talent in table tennis on the international stage.

Concept:

Parallelogram Law of Forces:

If two forces, acting at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point. 

Resultant force Fr, of any two forces F1 and F2 with an angle θ between them, can be given by vector addition as

'

CONCEPT:

Centre of gravity:

• The centre of gravity is a theoretical point in the body where the total weight of the body is thought to be concentrated.
• The centre of gravity of a body is a point where the total gravitational torque on the body is zero.
• The centre of gravity of a body may lie inside or outside the body.
• The centre of gravity of the body coincides with the centre of mass in uniform gravity or gravity-free space.
• If the body is so extended that gravitational acceleration varies from part to part of the body, then the centre of gravity and centre of mass will not coincide.
• Let a body consists of n number of particles and the gravitational acceleration is uniform for all the particles of the body.
• Then the position of the centre of gravity is given as,

Where w1, w2,..., and wn = weight of the particles

EXPLANATION:

• We know that the centre of gravity of a solid hemisphere of radius r lies at a distance of  from its base. Hence, option 1 is correct.

Concept:

The number of words from n letters where a letter is repeated m times

= n!/m!

where n! = n(n - 1)(n - 2)....3.2.1

Calculation:

If all the vowels come together, let us consider them as a single letter,

let α = (IEEEE)  {Vowels in the word INDEPENDENCE}

So, the number of words from the letters of the word INDEPENDENCE, in which vowels always come together

⇒ the number of words from the letters of the word NDPNDNC(IEEEE)

⇒ the number of words from the letters of the word NDPNDNCα =  = 3360

{N is repeated 3 times and D is repeated two times}

And the total number of words that can be made by α 

{changing the vowels together}

= the number of words from letters IEEEE = 5!/4! = 5

So, the number of words that can be made out from the letters of the word INDEPENDENCE,

in which vowels always come together = 3360 × 5 = 16800.

∴ The correct answer is option (1).

Given:  

Green balls = 20

Red balls = 15

Formula Used:

Probability of any event, P(E) = 

Where, P(E) = Probability of an event

n(E) = Total number of required outcomes

n(S) = Total number of possible outcomes

Where, n = number of object and r = selecting object at a time

 = n

Calculation:

According to the question,

Total number of balls = 20 green balls + 15 red balls = 35 balls

n(S) = 

And, n(E) = 

Probability to choose a green ball = 

∴ Probability  to choose a green ball is 4/7.

Concept:

Calculation:

⇒ x₁ + x₂ + x₃ +..........+ xn - 2n = 110

⇒ x1 + x2 + x3 +.........+xn = 110 + 2n      ------(1)

⇒ x1 + x2 + x3 +........+ xn - 2n = 20

⇒ x1 + x2 + x3 +........+ xn = 20 + 5n      ------(2)

Equating Equation (1) and (2),we get

⇒ 110 + 2n = 20 + 5n

⇒ 110 - 20 = 5n - 2n

⇒ 90 = 3n

⇒ n = 30

CONCEPT:

• Equation of motion: The following equations of motion are used in kinematics.

v = u + at

$\mathrm{s}=\mathrm{u}\mathrm{t}+\frac{1}{2}\mathrm{a}{\mathrm{t}}^2$

v2 = u2 + 2as

$S_n=u+\frac{a}{2}(2n-1)$

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

CALCULATION:

Given - initial velocity (u) = 0 m/s, n = 2, and acceleration of body under motion = a

• The distance (in meters) travelled in the 2^nd second is given by

$\Rightarrow S_n=u+\frac{a}{2}(2n-1)$

$\Rightarrow S_n=\frac{a}{2}(2\times 2-1)$

$\Rightarrow S_n=\frac{a}{2}(4-1)$

Given:

cos 9α = sin α

Concept:

sin(90° - θ) = cosθ 

tan 45° = 1

Calculation:

Let us assume the value of tan 5α be X

⇒ cos 9α = sin α     ----(1)

⇒ sin (90° - 9α) = cos 9α     ----(2)

⇒ By equation (1) and (2)

⇒ sin (90° - 9α) = sin α

⇒ 90° = 10α

⇒ α = 9° 

⇒ X = tan 5α = tan (5 × 9°) = tan 45° = 1

∴ The required result will be 1.

Calculation:

Given, Curves are y² = 2y - x and y-axis.

⇒ x = 2y - y² ...(1)     and x = 0 ....(2)

From (1) and (2), we get,

2y - y² = 0

⇒ y(2 - y) = 0

⇒ y = 0 or y = 2

Thus the curve meets the y-axis at (0,0) and (0,2)

So, the required area = 

= 

Concept:

Property of definite integral:

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 

Let ϕ = x + y + z - 5

⇒ 

⇒ 

Given:

Calculation:

⇒ x2/(x + 1)2 - 5x/(x + 1) + 6 = 0

multiplying the above by (x + 1)², we get 

⇒ x2 - 5x(x + 1) + 6(x + 1)2

⇒ x2 - 5x2 - 5x + 6x2 + 6 + 12x = 0

⇒ 2x2 + 7x + 6 = 0

⇒ 2x2 + 4x + 3x + 6 = 0

⇒ 2x(x + 2) + 3(x + 2) = 0

⇒ x = -3/2, -2

∴ (1 + 1/x) = 1 + 1/(-3/2) = 1 - 2/3 = 1/3

(1 + 1/x) = 1 - 1/2 = 1/2

∴  or 

Concept:

Subgroup:- A non-empty subset H of a group G is called a subgroup of G if H is itself a group with respect to the same operation that defined for G.

Union:- Union of two or more sets is the set containing all the elements of the given sets.

Calculation:

Let G = (Z, +) be a group

⇒ H₁ = {0, ±2, ±4, ±6, ....}

⇒ H₂ = {0, ±3, ±6, ±9, ....}

⇒ H₁ U H₂ = {0, ±2, ±3, ±4, ±6, ±8, ±9,....}

⇒ 2, 3 ∈ H₁ U H₂

⇒ But, 2 + 3 = 5 ∉ H₁ U H₂

⇒ Closure property does not hold in H₁ U H₂.

∴ (H1 ∪ H2, +) is not a subgroup of (Z, +).

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:

• When any two consecutive numbers in a series have the same difference, they are in Arithmetic Progression or AP. An example of AP is 1 , 3 , 5 ...
• The difference between these two consecutive numbers of the series is called the common difference and it is often denoted by d.
• The sum of the first n integers is 

Now, let us assume that the sum of n terms of this series is 

Similarly, the sum of (n-1) terms is 

Therefore, the n^th term of the series is 

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 + 

⇒  = 

Calculation

When a die is thrown there are 6 outcomes

s = {1,2,3,4,5,6}

Prime numbers = 2,3,5

P (prime number) = 3/6 = 1/2

Formula: 

Average speed = Total distance covered ÷ Total time 

Shortcut Trick

Concept:

We know that the domain of a function f(x) is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes. 

Given:

Analysis:

For domain,

f(x) ≥ 0

(x - 2) (x - 1) ≥ 0

Case: 1

(x - 2) ≥ 0 and (x – 1) ≥ 0

x ≥ 2 and x ≥ 1

overall: x ≥ 2

{x|2 ≤ x < ∞} = [2, ∞)

Case: 2

(x - 2) ≤ 0 and (x - 1) ≤ 0

x ≤ 2 and x ≤ 1

overall : x ≤ 1

{x|-∞ < x ≤ 1} = (-∞, 1]

∴ The domain of the function is:

(-∞, 1] ∪ [2, ∞)

Given:

ⁿC₁₀ = ⁿC₁₅    -------(1)

Concept:
ⁿC_r = 

Calculation:

We can write,
nC₁₀ =   ------(2)
ⁿC₁₅ =   ------(3)

By equation (1) comparing equations (2) and (3), we get

 ------(4)

Comparing both sides of equation (4), we get

10 = n - 15, and

⇒ 15 = n - 10

⇒ n = 25

Now, 27Cn for n = 25, will be

²⁷C₂₅ =  = 

Concept:

Let the two equations be:

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

Then,

1. For unique solutions, 

2. For infinitely many solutions, a1a2=b1b2=c1c2

3. For no solution, 

Concept:

Binomial theorem

(1 + x)n = 1 + nC1 x + nC2 x2 + .....+ nCn-1 xn-1 + nCn xn

(1 - x)n = 1 - nC1 x + nC2 x2 + ..... nCn-1 xn-1 + (-1)n nCn xn

Calculation:

(1 + x)m (1 - x)n 

⇒  [1 + mx + m(m - 1)/2 x2 +.....+ xm].[1 - nx + n(n - 1)/2 x2 +.....+ xn​]
⇒ (1 + x)m (1 - x)n = [1 + (m - n)x + []x2 +......{terms of degree ≥ 3}

⇒ Coefficient of x = m - n = 3 -----(i)

And Coefficient of x2 = = - 6 ----(ii)

⇒ m(m - 1) + n(n - 1) - 2mn = -12

⇒ m2 - m + n2 - n - 2mn = - 12

⇒ (m - n)2 - (m + n) = - 12

From (i),

⇒ 9 - (m + n) = -12

⇒ m + n = 21      ----(iii)

Now from (i) and (iii),

2m = 24 ⇒ m = 12

∴ The correct answer is option (3).

Concept

The equation of motions are 

v = u + at 

v² = u² + 2as                

Calculation:

Given:

Mass of the block, m = 5 kg, Inclination angle of the plane, θ = 30°, Initial velocity of the block, u = 0 m/s, Distance travelled by the block, s = 3.6 m

Force acting on the block along with the inclination of plane =  

Acceleration of the block along the inclined plane, a =  

Applying equation of motion along the inclined plane.

v² = u² + 2as

v² = 0 + 2 × 5 × 3.6

∴ v = 6 m/s.

Given:

α and β are the roots for the quadratic equation ax2 + bx + c = 0

Formula used:

(1.) (a + b)² = a² + b² + 2ab

(2.) For quadratic equation,

α and β are the roots for the quadratic equation ax2 + bx + c = 0

(2.1.) α + β = 

(2.2.) αβ =  

Calculation:

The above expression can be written as:

Concept:

Calculation:

Given,



∴ The correct answer is option (2).

Concept:  

• To find the maximum or minimum values for a function f(x) of a single variable, the following is done.
• Partial derivative of the function with respect tp x (f_x) calculated.
• Values of x is calculated by putting the partial derivative equal to zero.
• When the values of x are obtained, we will do the following:
• For values very slightly less than x, we need to find out whether f_x is positive or negative.
• For values very slightly more than x, we need to find out whether f_x is positive or negative.
• Based on the result, it is possible to establish whether x is a point of maxima or minima.

Calculation: