RPSC Senior Grade II (Mathematics) Mock Test - 2 - REET MCQ

Aims in mathematics education refer to the overarching goals and purposes of teaching and learning mathematics.

Key Points

• They reflect the broader aspirations of what educators hope to achieve in terms of students' mathematical knowledge, understanding, and application.
• For example, some common aims in mathematics education include developing critical thinking skills, fostering mathematical reasoning, promoting problem-solving abilities, and cultivating a deep understanding of mathematical concepts.
• Mathematics objectives typically focus on specific mathematical concepts, skills, or problem-solving strategies.
• While aims in mathematics education are broad and general, providing the overarching direction for teaching and learning mathematics, objectives are more specific and actionable, guiding the instructional focus and assessment of student progress.

Hence, it can be concluded that objectives are very broad and general is not true for the objectives of mathematics education when compared with its aims.

• Estimating the whole course content/set of competencies for the class during the year.
• Tentative and temporal distribution of the content and relevant pedagogic process.
• Arranging the given course content /set of competencies in a teaching-learning
  sequence.
• Planning and identification of relevant pedagogic resources for a given set of conceptual areas.
• For each lesson within the unit decide about the appropriate teaching methods, teaching aids, student activities, and the evaluation procedure.
• Scope for adaptation and modification of teaching strategies according to the learners' requirements.

• On the other hand planning of units strictly in the order as given in the textbook is not an aspect of unit planning in mathematics because many situations arise where there will be a need to adapt or change the plan according to the student's needs, so the unit plan needs to be flexible, not strict.

  Hence option 2 is the correct one

Analytic and synthetic are methods that use reasoning and arguments to discover relationships.

Key Points
Analysis method:- In this method, we proceed from unknown to known. It is derived from the word analysis which means breaking up or resolving a thing into its constituent elements.

• In this method, we break up the unknown problem into simpler parts and then see how these can be recombined to find the solution.
• So, we start with what is to be found out and then think of further steps or possibilities that may connect the unknown built the known and find out the desired result. In this method, the teacher begins with what is required to be probed and arrives at what is given.
  • It leads to the conclusion of the hypothesis
  • It leads to unknown to known
  • It leads to abstract to concrete
• Analytic statements are not considered statements of proof for the problem. Rather analysis is considered the means of discovering the proof.
• This type of proof is the reverse way of the proof given in the textbooks.

Synthetic method:- In this method, we proceed from known to unknown. Synthetic is derived from the word synthesis. It is the complement of analysis.

• Synthetic Euclidean geometry is a good model of a deductive structure and is favorable to the learning of reasoning and to the development of precision of thought. In any proposition, we have a hypothesis, which may be the information given in the proposition or a set of axioms, definitions, principles, or relationships which have been proved earlier, and a conclusion, that is, the result to be proved or arrived at.
• Synthesis is combining elements to produce something new. So in it, we combine several facts, perform certain mathematical operations and arrive at a solution. That is we start with the known data and connect it with the unknown part. In this method, the teacher starts from the given data and arrives at what is required to be proved.
  • It leads to the hypothesis of the conclusion
  • It leads to known to unknown
  • It leads to concrete to abstract
• The synthetic method provides proof of the problem.
• This is the type of proof we come across in nearly all mathematics textbooks and literature.

Hence, In the Analytic method, the proof of a geometric theorem proceeds just the reverse way of the proof given in the textbooks.

In elementary school years, the learning outcomes are focused on developing useful capabilities and also on developing the ability to think and reason mathematically. Useful capabilities include conceptual understanding and ability to understand and solve problems in the areas of numbers, number operations, fractions, shapes and spatial thinking, measurement, problem-solving, patterns and data handling (NCERT, 2008). Including the above capabilities, the process of assessment in mathematics includes five major dimensions of mathematical learning for which the probable tools and techniques to be used are suggested in the following table:

Hence, all are the tools and techniques of assessment for learning mathematics.

Concept:
In cyclic group, if O(G) = n, Then the element a^k for k=0,1,2....(n-1) are generator of G if GCD(k,n)=1

Explanation:
For G= {a, a2, a3, a4, a5, a6 = e} if a^k is generator
⇒ GCD(k,6)=1 for k=1,2,3,4,5,6
Let's check
GCD(1,6)=1(valid)
GCD(2,6)=2(Invalid)
GCD(3,6)=3(Invalid)
GCD(4,6)=2(Invalid)
GCD(5,6)=1(valid)
GCD(6,6)=6(Invalid)
So, the generator of G are a and a⁵

Concept:
The following are the properties of a parabola of the form: x² = - 4ay where a > 0

• Focus is given by (0, - a)
• Vertex is given by (0, 0)
• Equation of directrix is given by: y = a
• Equation of axis is given by: x = 0
• Length of latus rectum is given by: 4a
• Equation of latus rectum is given by: y = - a

Calculation:
Given: Equation of the parabola is x² = - 8y
The given equation of parabola can be re-written as x² = - 4 ⋅ 2y ---(1)
Now by comparing the equation (1) with x² = - 4ay we get
⇒ a = 2
As we know, equation directrix of the parabola of the form x² = - 4ay is given by: y = a
So, equation of directrix of given parabola is: y = 2
Hence, option B is the correct answer.

Given:
A glass jar contains 6 black, 8 blue, 4 white, and 3 red marbles.

Concept used:
1. Probability of an event = Occurance of Favourable Outcomes ÷ Occurance of Total Outcomes
2. nCr =

Calculation:
Total number of blue and red marbles = 8 + 3 = 11
Ways of drawing one blue or red marble = ¹¹C₁ = 11
Ways of drawing amongst all marbles = 21C1 = 21
Now, if a single marble is chosen at random from the jar, the probability that it is blue or red
⇒ 11/21
∴ If a single marble is chosen at random from the jar, the probability that it is blue or red is 11/21.

Given:
Two unbiased dice are thrown.

Formula Used:
Probability of an event = Number of favorable outcomes / Total number of possible outcomes

Calculation:
Total number of possible outcomes when two dice are thrown = 6 × 6 = 36
Favorable outcomes for getting a sum of 10:
(4,6), (5,5), (6,4)
Number of favorable outcomes = 3
Probability of getting a sum of 10 = Number of favorable outcomes / Total number of possible outcomes
Probability = 3 / 36
Probability = 1 / 12
The correct answer is option 2 (5/36).

Given:
Length of diagonal 1 (d₁) = 10 cm
Length of diagonal 2 (d₂) = 8.2 cm

Formula Used:
Area of rhombus = (1/2) × d₁ × d₂

Calculation:
Area = (1/2) × 10 × 8.2
⇒ Area = 5 × 8.2
⇒ Area = 41 cm²
The area of the rhombus is 41 cm².

Concept:

Order: The order of a differential equation is the order of the highest derivative appearing in it.

Degree: The degree of a differential equation is the power of the highest derivative occurring in it, after the Equation has been expressed in a form free from radicals as far as the derivatives are concerned.

Calculation:

Given,

⇒

⇒

⇒ Degree = 2 and Order = 5.

∴ The order and degree of the differential equation are 2, 5 repectively.

The correct answer is Option 4.

Formula Used:
Sin C + sin D = 2 sin(C + D)/2 × cos(C - D)/2
∫ tanx. secx dx = secx + c
∫ sinx dx = - cosx + c

Calculation:



⇒ (tany × secy) dy = 2 sinx dx
⇒ ∫ (tany × secy) dy = ∫ 2 sinx dx
⇒ secy = - 2cosx + c
∴ sec y + 2 cos x = c

Concept:

Derivatives of Trigonometric Functions:

∫ ex [f'(x) ​+ f(x)] dx = ex f(x) + C.

Calculation:

= ∫ ex [f'(x)​ + f(x)] dx, where f(x) = -cot x.
= e^x f(x) + C
= -ex cot x + C.
Hence, the correct option is: 2) e^x cot x + C

Given,
Integrating both sides,
⇒ dy/dx = a, where a is an arbitrary constant
Again integrating,
⇒ y = ax + b, where a,b are arbitrary constant
∴ The correct answer is option (1).

Let any analytical function be

f(x + iy) = (x + iy)²

= (x² – y²) + (2 ixy)

Now ϕ (x, y) = (x² – y²)

and ψ(x, y) = 2xy

Now by first derivation, we can see,

Which shows,

And

Concept:
The equation of the form |A - λI| = 0 is called the characteristic equation where,
A is a square matrix
λ is an eigenvalue and,
I is an identity matrix, and the non-trivial solution of this equation is the eigenvalues.
The eigenspace is the space generated by the eigenvectors corresponding to the same eigenvalue - that is, the space of all vectors that can be written as a linear combination of those eigenvectors i.e, If λ is any eigen value ,A is any square matrix, I is an identity matrix and X is any eigen vector then we have | A - λI |.X = 0.

Calculation:
First ,we shall find all eigen values using the equation | A - λI | = 0.
we have T(x, y, z) = (x - y, 2x + 3y + 2z, x + y + 2z)

then the matrix form of this can be written as = A (Say)

Consider, |A - λI| = 0

So on solving the above quadratic equation we get the following eigen values
⇒ λ₁ = 1, λ₂ = 2, λ₃.
So the corresponding eigen space of the three eigen values can be found by the equation |A - λI | .X = 0.
Let X = be an eigen vector.
Consider,
(A - λI)X= 0
corresponding to λ₁ eigenvalue we have,

On doing matrix multiplication we get the following set of equations,
⇒ - x₂ = 0
2x₁ + 2x₂ + 2x₃ = 0
Therefore x₂ = 0 so we have x₁+x₃ = 0 i.e. and are dependent on each other.
so let x₁ = 1 then, x₃ = -1
∴ eigenspace corresponding to the eigen value λ₁ = 1 is ( 1, 0, -1 )
Similarly, corresponding to λ₂ eigen value we have,

On doing matrix multiplication we get the following set of equations,
⇒ -x1 - x2 = 0
2x1 + x2 + 2x3 = 0 and
x1 + x2 = 0
Therefore x₂ = - x₁ i.e. x₁ and x₂ are dependent on each other.
so, let x₁ = 2 then, x₂ = - 2
and 2x1 + x2 + 2x3 = 0
⇒ x₃ = -1
∴ eigenspace corresponding to the eigen value λ₂ = 2 is ( 2, -2, -1).
Finally, corresponding to λ3 eigenvalue we have,

On doing matrix multiplication we get the following set of equations,
⇒ 2x₁ - x₂ = 0
2x₁ + 2x3 = 0
x₁ + x₂ -x₃ = 0
Therefore x₃ = - x₁ i.e. x₁ and x₃ are dependent on each other.
Let x₁ = 1 then x₃ = -1
and x1 + x2 - x3 = 0
⇒ x₂ = -2
∴ eigenspace corresponding to the eigen value λ₃ = 3 is (1,-2,-1) .
Hence, the correct answer is option 2)

Concept:
Properties of eigenvalues:
1) Sum of eigenvalues = Sum of the diagonal elements of the matrix
2) Product of eigenvalues = Determinant of the matrix

Calculation:
The eigenvalues are in the ratio of 3:1, then the eigenvalues are 3λ and λ.
x + 2 = 4λ

............(i)
2x - 1 = 3λ² .........(ii)

Putting the value of (i) in (ii), we get:

3(x+2)² = 32x - 16
3x² - 20x + 28 = 0
x (3x - 14) -2 (3x - 14) = 0
(3x - 14)(x - 2) = 0
x = 2,14/3

To express a group G as an internal direct product of subgroups a, b, c, ... z, you typically mean that each element of G can be represented uniquely as a product of elements from each of these subgroups.

This implies that every element g in G can be expressed uniquely as g = a . b . c . ... . z, where a is from subgroup a, b is from subgroup b, and so forth.

So the correct option is:
G = a . b . c ... z
This expression represents the unique product from each subgroup. Note that the operation here isn't necessarily numerical multiplication— it's the group operation, whatever that might be in the specific context of the group in question.

A check board matrix is one in which elements are 1 if (i + j) is even and elements are zero if (i + j) is odd. Therefore for 2 x 2 matrix A checkboard matrix is of the form:

here its Rank is 2 because the number of non-zero rows in the echelon form of the given matrix is 2. Hence, the rank is 2.

If f is a continuous function from one metric space X to another metric space Y, then the property being referred to is typically called the "Continuity of f". In more precise terms, a function f: X → Y is continuous if for any open set O in Y, the preimage f^-1(O) is an open set in X. This use of open sets is a standard way to define continuity in topology and metric spaces then the property being referred the "Continuous invariant of f".
Hence option (iii) is correct.

Concept:

Newton’s divided difference polynomial method:
Second order polynomial interpolation using Newton’s divided difference polynomial method is as follows,
Given (x0,y0), (x1,y1), (x2,y2) be the data points and f(x) be the quadratic interpolant, then f(x) is given by
f(x) = b0 + b1(x – x0) + b2 (x – x0)(x – x1);
Where
b0 = f(x0);

Calculation:
Given f(0) = 1, f(1) = 3, f(3) = 55;
⇒ (0,1), (1,3), (3,55) are the data points;
The polynomial will be f(x) = b0 + b1(x) + b2 (x)(x – 1);

Substituting the constant b0, b1, b2 in the quadratic interpolant,
⇒ f(x) = 1 + 2x + 8 (x)(x – 1) = 1 + 2x + 8x2 – 8x = 1 - 6x + 8x2;
The unique polynomial of degree 2 will be f(x) = 1 - 6x + 8x2;
Now f(2) = 21

Calculation:
Given curve is y2 = 4a(x + a) ⇒ y2 = 4ax + 4a²
Differentiating with respect to x:
⇒ 2y = 4a
⇒ a =
Substituting the value of a in the given equation.

The correct answer is option 4.

Given:

length of a shadow cast by a pole is √3 times the length of the pole .

Calculation:

Let the length of shadow be '√3x' and the length of the pole be 'x'

So, tanθ
⇒ θ = 30°
∴ The answer is 30°.

Given

Radius: 4.2 m

Concept:

Volume of a hemisphere: πr³

Calculation:

Volume of hemisphere ⇒ π (4.2)³ = 155.736 π

1/21^th of volume ⇒ = 7.415 π ≈ 23.275

Therefore, 1/21^th of the volume is 7.3 m³

Concept:
Euclid’s Division Algorithm: If a and b are positive integers such that a = bq + r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.
HCF (Highest Common Factor), is the highest possible number that divides both numbers. (Remainder = 0)

Calculation:
Using Euclid’s Division Algorithm:
Step 1: H.C.F. of 29 and 24
29 = 24 × 1 + 5
5 ≠ 0

Step 2: H.C.F. of 24 and 5
24 = 5 × 4 + 4
24 = 5 × 4 + 1 + 3
1 and 3 ≠ 0

Step 3: H.C.F. of 5 and 1
5 = 1 × 5 + 0
Hence, correct order H.C.F. of 29 and 24 by using Euclid's division algorithm will be (b), (c), (a).

Concept:

Symmetric Matrix:
Any real square matrix A = (aij) is said to be symmetric matrix if and only if aij = aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = At then A is said to be a symmetric matrix.

Calculation:

A = At

On comparing
3 + x = 1 - x
⇒ x = - 1
And, y + 1 = 5 - y
⇒ y = 2
3x + y = 3(-1) + 2
∴ 3x + y = -1

Concept:
The distance between the directrices to the ellipse , a > b, is equal to .
The eccentricity of the above ellipse is given by e = .

Calculation:
Comparing the given equation of the ellipse with the standard form, we get:
a = 6 and b = 5.
Eccentricity, e = .
Distance between the directrices = .

If sinθ = x ⇒ θ = sin^-1x, for θ ∈ [-π/2, π/2]
sin (sin^-1 x) =x for -π/2 ≤ x ≤ π/2

Additional InformationPrincipal Values of Inverse Trigonometric Functions:

Formula used:
Formula of G.P. to find the number of terms
a_n = a₁(r)^{n - 1}
r = a₂/a₁
Where, a_n = nth term
a₁ = first term, a₂ = second term ,r = common ratio
n = number of terms

Calculation:
Here, we have 4,20,100 .............62500
As, these numbers are in G.P.
Where, a₁ = 4 , a₂ = 20 , a_n = 62500
⇒ r(common difference) = 20/4 = 5
⇒ a_n = a₁(r)^n-1
⇒ 62500 = 4(5)^n-1
⇒ 15625 = (5)^n-1
⇒ 5⁶ = 5^n-1
⇒ 6 = n -1
⇒ n = 7
Hence, the number of terms is 7.

Given:
A room is in the shape of a cube and the length of the longest rod placed in it is 40√3 cm

Calculation:
The length of the longest rod placed in the cubical room is equal to the length of the diagonal of the cubical room.
We know,
Length of the diagonal of a cube = a√3
Where a is the Length of each side of a cube.
⇒ ​a√3 = ​40√3 cm
⇒ a = 40 cm
Therefore, The Length of each side of a cube is 40 cm.
Now, The area of the floor = a3
⇒ The area of the floor = 403
⇒ The volume of the floor = 64000 cm³

Given:
A class has 175 students.
Number of students study mathematics = 100
Number of students Study physics = 70
Number of students study chemistry = 46
Number of students study math's and physics = 30
Number of students study math's and chemistry = 28
Number of students study physics and chemistry = 28
Number of students study math's, physics and chemistry = 18

Concept Used:
n(M_alone) = n(M) - n(P U M) - n(M U C) + n(M U P U C)
Here P = physics, C = chemistry, M = mathematics and union represent students studying both subjects.
Since we have to calculate students studying math's alone so the students studying both need to be subtracted and since common of all the three subjects is subtracted twice so we will add it for one time.

Explanation:
We have,
n(P U M) = 30
n(M U C) = 28
n(M U P U C) = 18
n(M) = 100
Using the above formula
n(M_alone) = 100 - 30 - 28 + 18
n(M_alone) = 60
∴ Option 3 is correct.