UP TGT Mathematics Mock Test - 2 - UPTET MCQ

The word monsoon comes from the Arabic word mausim, which means weather. Owing to the yearly appearance of torrential rain, indicating a marked shift in weather, mausim gradually became monsoon.

The development of a country is generally determined by its per capita income, its average literacy level and health status of the people in the country. 

Development of the country is a broad term that develops the per capita income, and living standard of the people. 

It also decreases the poverty level, crime rate and illiteracy of among the people.

Key Points

• The energy crisis is the concern that the world’s demands on limited natural resources are exhausting.
• Resources, like fossil fuels that are used to power industrial society, are diminishing as the demand rises.
• These natural resources are in limited supply.
• While they do occur naturally, it can take hundreds of thousands of years to replenish the stores.
• It has nothing to do with demand and supply but highlights the fact that there are limited resources for use and consumption.

Thus, the energy crisis means the danger of extinction of fossil fuels like coal and petrol.

Important Points

There are basically two sources of energy:

The Union Ministry of road transport and highways is planning to extend its Incident Management Services (IMS) to all national highways.
The purpose of this step is to minimize road fatalities and improve safety for road users. Incident Management Services (IMS) on national roads will be provided, run, and maintained by businesses.

Concept:

Calculations:

Given,   and c→=i^−j^−k^.

From the above, we can find the magnitude of both the given vectors.

 
and 

Explanations:

Group Homorphisms :

Let (G,o) & (G’,o’) be 2 groups, a mapping “f ” from a group (G,o) to a group (G’,o’) is said to be a homomorphism

if –f(a o b) = f(a) o' f(b) for all a, b ∈ G

∴ option 1 is correct 

Explanation:

The centre of gravity of solid hemisphere lies on the central radius at a distance of (3r/8) from the plane base.

Important Points

Similarly, the centre of gravity for different shapes are:

Concept:

Identities:

• i² = -1
• i³ = -i
• i⁴ = 1
• i⁴ⁿ = 1
• i⁴ⁿ⁺¹ = i
• i⁴ⁿ⁺² = -1
• i⁴ⁿ⁺³ = -i

Alternate Method

Concept:

Euler's formula states that for any real number x,

Where i = √-1

Given:

an  and bn  are bounded sequences

Concept used: 

A sequence is bounded if all the terms of the sequence are ≤ N s.t there does not exist any term in the sequence which is > N.

Calculations:

an  and bn  are bounded sequences

⇒ a_n < M and b_n < M' 

⇒ a_n  + b_n < M + M'

∴ all the term of the sequence of a_n + b_n < M + M' 

∴ a_n + b_n is also bounded. 

Given:

Group G is defined as a * b = ab, where a, b ∈ R and ab ≠ 0

Concept used: 

In a Group 

Identity of an element: a*e = e*a = a

Inverse of an element: a^-1a = e = a^-1a

Calculations:

ab *(c) = ab for find identity

⇒ ab * 1 = ab 

⇒ (ab)1 = ab

⇒ 1 is the identity 

Let d is the inverse of ab 

⇒ (ab)*d = 1 = d*(ab)

⇒ (ab)d = 1

⇒ d = 1/ab 

∴ 1/ab is the inverse of ab = (ab)^-1

Given:

In the given figure, A tangent AC touches the circle (with centre at O) at point B. If ∠DBC = 30∘, then find the value of ∠BOD.

Concept used:

Alternate segment theorem:

As per the above figure, the angle (θ) between a tangent and a chord is equal to the angle (θ) in the alternate segment.

The angle subtended by an arc at the centre is twice to that of the angle subtended by it at any point on the remaining part of the circumference of the circle.

Calculation:

According to the question, the required figure is:

By using the Alternate segment theorem:

∠DEB = ∠DBC = 30^∘ 

Since the angle subtended by an arc at the centre is twice to that of the angle subtended by it at any point on the remaining part of the circumference of the circle,

Therefore, 

⇒ ∠DOB = 2∠DEB

⇒ ∠DOB = 2 × 30∘ 

⇒ ∠DOB = 60∘ 

∴ The required angle is 60^∘,

Concept:

Using substitution method in indefinite integral and using formula 

Solution:

Substitute u = e^x → 

$=-\cos u+C$

Undo substitution u = e^x

∴ The correct option is (2)

Concept: 

We know that the general equation for a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

The general equation for a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center and r is the radius.

Hence, The center and radius of equation (1) is 

Given: 

A right circular cone with vertex as (2, -3, 5), axis making equal angle with Coordinate axis and semi-vertical angle θ = 30°.

Calculations:

Let P (x, y, z) be any point on the surface of the cone and v(2, - 3, 5) be a vertex of the cone D.R.'s of VP are x - 2, y + 3, z - 5,

Since the axis of the cone makes an equal angle with the coordinate axis and hence D. R's of the axis is 

Concept:

Calculation:

Given:

y = e3x + e-5x 

At x = 0

we have

Concept:

The equation of parabola having their vertices at the origin and foci on the x-axis is 

Calculations:

The equation of parabola having their vertices at the origin and foci on the x-axis is

               ....(1)

Differentiating on both side, we get

⇒ 

⇒ 

Put this value in equation (1), we get

⇒ 

⇒ y(y - 2xy') = 0, Here y' = 

∴ y-2xy' = 0

Hence,  the differential equation to the family of parabolas having their vertices at the origin and foci on the x-axis is y - 2xy' = 0

Concept:

Steps for obtaining adjoint of the matrix 

We need to calculate cofactors of all elements

⇒ Cij  or Aij=(-1)(i+j)  Mij
Here, Mij  = Minor of matrix A

Transpose of the cofactor is called adjoint of the matrix

 Adj (A) = CT

Calculation:

Given that,

The cofactor α = A₁₁ = δ 

The cofactor β = A₁₂ = -γ 

The cofactor γ = A₂₁ = -β 

The cofactor δ = A₂₂ = -α  

⇒  adj 

Shortcut Trick
 

Given:

Formula used:

sinθ/cosθ = tanθ 

sec2θ – tan2θ = 1

Calculation:

⇒ 

⇒ (1 + secθ – tanθ)/(1 + secθ + tanθ)(secθ –tanθ)

⇒ (1 + secθ – tanθ)/(secθ + sec²θ + secθ tanθ – tanθ – tanθ secθ – tan²θ)

⇒ (1 + secθ – tanθ)/(secθ + sec2θ – tan²θ – tanθ)

⇒ (1 + secθ – tanθ)/(secθ + 1 – tanθ)     [sec2θ – tan2θ = 1]

⇒ 1

∴ The required value is 1.

CONCEPT:

CALCULATION:

Here, we have to find the area of the region bounded by the curves y = x3 + 4x + 2, the line x = 2, x  = 5 and the x - axis

So, the area enclosed by the given curves is given by 

As we know that, 

Concept:

A square matrix B which of size n × n is considered to be symmetric if and only if  BT = B.

A skew-symmetric matrix is a square matrix that is equal to the negative of its transpose matrix, A square matrix B which of size n × n is considered to be symmetric if and only if B^T = - B.

Calculation:

A is symmetric A^T = A

B is skew-symmetric B^T = -B

      ------(i)

Transpose

  --- [We know (A + B)^T = A^T + B^T]

As A is symmetric & B is skew-symmetric 

A^T = A & B^T = -B

      ------(ii)

From (i) + (ii)

A = 

From (i) - (ii)

B = 

then AB = 

Given:

S = 1/n  

Concept used:

Bolzano Weierstrass Theorem:

The theorem says that there is a limit point for a bounded infinite set S.

Let Ø ≠ S ⊆ R and S be bounded and infinite. Then there exists a limit point of S (in R ). 

Limit point:

L is the limit point of the sequence if all the terms of the sequence lie in the neighborhood of L except finite terms 

Calculations:

S = 1/n = 1, 1/2, 1/3 , 1/4,  .......

as S is the infinite and bounded set 

all the terms are > 0 and ≤  1

∴ by Bolzano Weierstrass Theorem it must have limit point 

as all the terms of the sequence lie in the neighborhood of the 0 

∴ 0 is the limit point of the sequence 

Shortcut Trick
Given Expression is 

We can write this expression like

∴ The value of n is 99.

Alternate Method
Given:

The give expression 1/ (1×2) + 1/ (2×3) + 1/ (3×4) …………. 1/n(n+1) = 99/100

Calculation       

1/ (1×2) + 1/ (2×3) + 1/ (3×4) …………. 1/n(n+1) = 99/100

Is also 1/ (1×2) + 1/ (2×3) + 1/ (3×4) …………. 1/99(99+1)

Comparing both sides,  

1/n(n+1) = 1/99(99+1)

Here n = 99

Hence, the value of n = 99

Given:

Concept used:

Trace of the Matrix: Sum of the diagonal Elements of the Matrix.

Calculations:

$A=\left[\begin{array}{cc}2& 4\\ 3& 2\end{array}\right]$

The diagonal elements are 2, 2 

⇒ Sum of the diagonal elements is 2 + 2 = 4 

∴ Trace = 4 

Concept:

• Here we have to use the multiplication and addition properties of the matrixes.

Calculation:

Given: 
 

∵ AI = A

⇒ A2 + 2A = 3A

The equation for the angle bisector for ax2 + 2hxy + by2 = 0 is given by

Given:

x2 + 2pxy - y2 = 0 and x2 - 2qxy - y2 = 0 bisect angles between each other.

Calculation: