DSSSB TGT Mathematics Mock Test - 1 - DSSSB TGT/PGT/PRT MCQ

• Logical: Mathematics is based on logic and reasoning. It involves deductive reasoning to reach conclusions based on given premises.


• Precise: Mathematics requires exactness and precision in its language and notation. It does not tolerate ambiguity.


• Universal: Mathematical truths are not subjective; they hold true regardless of cultural or personal biases.


• Abstract: Mathematics deals with abstract concepts that may not have a physical representation in the real world.


• Problem-solving: Mathematics is a tool for problem-solving in various fields such as science, engineering, and finance.


• Challenging: While mathematics can be challenging, it is also rewarding and fulfilling when one grasps its concepts and applications.

• Developing New Theories: Fundamental research aims to develop new theories by discovering broad generalizations in various fields of study.


• Understanding the Basics: It focuses on understanding the fundamental principles underlying a phenomenon or a concept.


• Advancing Knowledge: Fundamental research contributes to the advancement of knowledge in a particular area by exploring new ideas and concepts.


• Creating a Foundation: It helps in creating a solid foundation for further research and innovation in the respective field.


• Long-term Impact: The outcomes of fundamental research often have long-term impacts on society, technology, and various industries.

• Action Research: Action research is a research methodology that involves active participation in a situation to solve a specific problem. It focuses on practical solutions and involves a cyclical process of planning, acting, observing, and reflecting.

• S. M. Corey: S. M. Corey is credited with applying the concept of action research for the first time. She is known for her work in the field of education and psychology, particularly in the development of action research methodologies.

• Pioneering Role: S. M. Corey's contribution to the field of action research has been significant as she was one of the first researchers to apply this methodology in educational settings.


• Impact on Research Practices: Her work has influenced the way researchers approach problems and seek solutions through active engagement and reflection.


• Continued Relevance: The principles and methods of action research developed by S. M. Corey continue to be used in various fields to address real-world issues and bring about positive change.

• Legacy of S. M. Corey: S. M. Corey's pioneering work in action research has left a lasting impact on research practices and continues to inspire researchers to engage in participatory and reflective processes to effect change.

• Definition of Incidental Correlation: Incidental correlation refers to the relationship between two variables that are not causally related but appear to be correlated due to some external factor.


• Role of the Teacher: In incidental correlation, the teacher plays a crucial role in identifying and addressing any misconceptions or misinterpretations that students may have regarding the correlation between variables.


• Guidance and Support: Teachers can provide guidance and support to students in understanding the difference between correlation and causation, helping them to critically analyze the data and draw appropriate conclusions.


• Encouraging Critical Thinking: Teachers can encourage students to think critically about the factors that may be influencing the correlation between variables and to consider alternative explanations.


• Importance of Education: Education plays a key role in helping individuals to distinguish between incidental correlation and true causation, and teachers are instrumental in imparting this knowledge to their students.

• Definition: Correlation in mathematics indicates a joint relationship between two variables.


• Explanation: When two variables have a correlation, it means that they tend to vary together. If one variable increases, the other variable also tends to increase, or if one variable decreases, the other variable also tends to decrease.


• Types of Correlation: There are different types of correlation such as positive correlation, negative correlation, and no correlation. Positive correlation means that as one variable increases, the other variable also increases. Negative correlation means that as one variable increases, the other variable decreases. No correlation means that there is no relationship between the two variables.


• Correlation Coefficient: The strength and direction of the correlation between two variables can be measured using a correlation coefficient. The correlation coefficient ranges from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation.

• Correlation in mathematics is important for understanding the relationship between different variables and making predictions based on data analysis.


• It helps in identifying patterns and trends in data, which can be useful for decision-making in various fields such as finance, economics, and science.

• Backward children: Text-books provide a structured way of learning that can help backward children catch up with their peers. They present information in a clear and organized manner, making it easier for these students to understand and retain the material.



• Gifted children: Even gifted children can benefit from text-books as they offer a comprehensive overview of a subject, helping them deepen their understanding and explore new concepts. Text-books can also serve as a reference guide for these students to delve into advanced topics.



• Average students: Text-books are designed to cater to the needs of average students by presenting information in a way that is easy to comprehend. These students can use text-books to reinforce their learning, clarify doubts, and improve their academic performance.



• All the above: Text-books are versatile tools that can benefit students of all abilities. They serve as a foundation for learning, providing essential information, explanations, and examples that cater to a wide range of learning styles and abilities.

• Induction: Induction is a method used in mathematics to prove a formula or statement for all natural numbers. It involves proving a base case, typically when n=1, and then showing that if the formula holds for a specific value n, it also holds for n+1. This method is widely accepted and used in mathematical proofs.


• Planning: Planning is an important step in problem-solving, but it is not the method used for establishing formulae in mathematics. Planning involves organizing thoughts and steps to solve a problem effectively.


• Synthesis: Synthesis is the act of combining different ideas or information to form a new whole. While synthesis may be used in the process of establishing formulae, it is not the primary method for doing so.


• None of these: This option is incorrect as induction is a widely accepted method for establishing formulae in mathematics.

• Non-projected: Non-projected teaching aids include materials that do not require a screen or projector, such as flashcards, models, and real-life objects. These aids can engage students in a hands-on way, making learning more interactive and memorable.


• Projected: Projected teaching aids, like slideshows or videos, can be helpful in presenting information in a visually appealing manner. However, they may not always be as effective as direct experience in engaging students and facilitating deep understanding.


• Direct experience: Direct experience, such as experiments, field trips, or hands-on activities, allows students to actively participate in the learning process. This type of teaching aid promotes critical thinking, problem-solving skills, and a deeper understanding of the subject matter.


• None of these: This option is incorrect because direct experience is often considered the most effective teaching aid due to its ability to immerse students in the learning process and enhance their understanding through firsthand experiences.

Therefore, the most effective teaching aid among the options provided is direct experience. It allows students to actively engage with the material, leading to better retention and comprehension of the content.

• In the order of exercise: This approach involves organizing the content based on the sequence of exercises provided in the textbook. Each chapter or section may focus on specific types of problems or concepts, with exercises following a logical progression.



• In problematic order: This method involves structuring the content based on the level of difficulty of the problems. Starting with simpler concepts and gradually progressing to more complex problems can help students build their skills and confidence.



• In logical order: Organizing the content in a logical order ensures that students can easily follow the flow of concepts and build upon their understanding step by step. This approach helps students see the connections between different topics and enhances their overall comprehension.



• In all the above order: Incorporating a combination of the above approaches can provide students with a well-rounded learning experience. By including exercises in different orders, students can develop a deep understanding of mathematical concepts, improve problem-solving skills, and enhance critical thinking abilities.

By developing content in a mathematics textbook using a variety of orders, educators can create a comprehensive learning resource that meets the diverse needs of students and supports their academic growth.

• Science of Logical Reasoning: Mathematics is the science of logical reasoning, where principles and rules are applied to solve problems systematically and accurately.



• Process of Problem Solving: Mathematics involves the process of problem solving, where various mathematical concepts are used to analyze, interpret, and solve real-world problems.



• Important means of Generalisation: Mathematics is an important means of generalization, where patterns and relationships are identified and generalized to make predictions and draw conclusions.



• An Applied Science: Mathematics is also considered as an applied science, as it is used in various fields such as engineering, physics, economics, and many more to solve practical problems and make advancements in technology.

Therefore, the most suitable definition of mathematics is that it is the science of logical reasoning, where principles and rules are applied to solve problems systematically and accurately. Mathematics also involves the process of problem solving, is an important means of generalization, and is considered an applied science in various fields.

Given:
AB = 12 cm and BC = 9 cm
AD is the bisector of ∠A
By using Pythagoras theorem,


By using the angle bisector theorem,

∴ The length of CD is 5 cm.

If the given integral of the form  then replace ax² +bx + x by

So replace y² + 4y + 7  by

So I = 

Given 
Thus, when t → ∞, the given expression assumes the indeterminate form ∞/∞

Series is 30, 27, 24. 21,...
Here, a = 3o, d = -3 
The maximum value is obtained when all the terms are positive. 
T_n ≥ 0
30 + (n - 1) (-3) > 0
n - 1 < 10
n < 11
So, n = 10, i.e. 11^th term is zero.
So, the sum is (10/2)[2(30) + 9(-3)]
= 5[60 - 27]
= 5(33)
= 165

Given 
f(y) is defined only when
(i) y² + 4 ≠ 0 (denominator of a function can never be equal to zero)
(ii) y - 1 ≥ 0   ( function under the square root is always non-negative)
So y² + 4 ≠ 0is true for all real values of y
And y-1 ≥ 0
⇒ y≥1
Hence domain of function 

We have to use Leibnitz formula in order to differentiate given definite integral.
According to Leibnitz formula

For increasing function

Given 
Take common x , y and z from R₁,R₂ and R₃  respectively

Now Take common x , y and z from C₁, C₂ and C₃ respectively

Expand along R₂

Let the number be n.
Now, according to the question, if n is divided by 3 it leaves 2 as the remainder.
Then, by Euclid’s division lemma,
n = 3q+2, where, 0≤2≤3
⇒ 3n = 3(3q+2)
⇒ 3n = 9q+6
⇒ 3n+6 = 9q+6+6 = 9q+12
⇒ 3n+6 = 3(3q+4)
⇒ 3n+6 = 3m, where m = 3q+4
Therefore, if 3n+6 is divided by 3 , then the remainder will be 0.

No. of ways to choose the 3 months = ¹²C₃ = 220
Now, we have 6 people and 3 months for the birthdays to fall in. 
Since none of these three months can have no birthday in it, the 3 possible ways of distributing the 6 people in 3 months would be (4, 1, 1), (3, 2, 1), (2, 2, 2) 
(4, 1, 1) 
No. of ways of rotating the distribution of the number of birthdays in these three months = 3!/2!
 No. of ways of rotating the distribution of the people = ⁶C₄ . ²C₁ . ¹C₁
Therefore, the total number of ways in (4, 1, 1) case 
= (3!/21).⁶C₄ . ²C₁ . ¹C₁ = 90 ways
(3, 2, 1) 
No. of ways of rotating the distribution of the number of birthdays in these three months = 3! 
No. of ways of rotating the distribution of the people = ⁶C₄ . ²C₁ . ¹C₁
Therefore, the total number of ways in (3, 2, 1) case 
3!.⁶C₄ . ²C₁ . ¹C₁ = 360 ways
(2, 2, 2) 
No. of ways of rotating the distribution of the number of birthdays in these three months
= 3!/3!
No. of ways of rotating the distribution of the people = ⁶C₄ . ²C₁ . ¹C₁
Therefore, total number of ways in (2, 2, 2) case =(3!/3!).⁶C₄ . ²C₁ . ¹C₁ = 90
The number of ways so that the birthdays of 6 people falls in exactly 3 calendar months = (90 + 360 + 90) • 220 = 118800

Given series is 2³ + 4³ + 6³ + ...

⇒ n = 6.

Point of intersection (1. 1)
The required line will be perpendicular to line joining (1, 1) and (2, 3).
The slope of line = m = -((2-1)/(3-1)) = -1/2 
Equation is y - 1 = -(1/2)(x-1)
2y - 2 = -x + 1 
2y + x = 3

Given 
Thus, at y =0, the given expression assumes the indeterminate form 0/0

{ Using binomial theorem)
So, 
Now,


= 5

As product of slope of these two lines is –1, hence these diagonals are perpendicular. So, PQRS must be a rhombus.

As f (y) is strictly increasing at y = 1
The substitute limiting value in the given functions

The component of  in the direction of the vector  projection of 
We know that the Projection of 
So projection of 

Let 
Let's substitute,

Now,

Substituting back xe^x + 1 = t, we get:

We have 
Taking all the integral values from π/2 to 3π/2 gives the integer.
 

Given f(y) = 
f(y) is discontinuous at y = 1 and y = 2
⇒ f(f(y)) is discontinuous when f(y) = 1 & 2
Now 1 - y = 1
⇒ y = 0 , where f(y) is continuous.
Also y + 2 = 1
⇒ y = -1 which does not belong to (1,2)
4-y = 1
⇒  y = 3 which belong to [2,4]
Now 
⇒ y = -1, which does not belong to [0,1]
Also y + 2 = 2
⇒ y = 0 which does not belong to (1,2)
4-y = 2
⇒ y = 2 which belong to [2,4]
Hence f(f(y)) is discontinuous at two points y = 2 and y = 3.

The r^th term of the series is given by:

Putting r = 1, 2,...., n-1, we get

Adding the above equations ,we get 

Obviously, the domain of the function is 0,3
Case -1:
f(t) = t when 0≤t≤1
Its range is [0,1] for 0≤t≤1
Case-2:
f(t) = t²+1; 1 < t ≤ 2
Consider h is very small positive number
When t = 1+h then f(t) = 
(1+ℎ)² +1 =1 +ℎ² +2ℎ +1 = ℎ² +2ℎ +2
As h is very small positive number so terms containing h can be neglected
So f(t) = 2 
When t = 2 then f(t) =5
Hence Its range is (2,5] for 1 < t ≤ 2 
Case-3
f(t) = t³; 2 < t ≤ 3
Consider h is a very small positive number
When t = 2+h then f(t) = 
(2+ℎ)³ = 8+ ℎ³ +12ℎ + 6ℎ²
As h is very small positive number so terms containing h can be neglected
So f(t) =8
When t = 3 then f(t) = 3³ = 27
Hence Its range is (8, 27] for 2 < t ≤ 3
Hence Range of f(t) over its domain = [0,1] ∪ [2,5] ∪ [8,27]