Start the test Question no. 1 Suppose, ω and ω 2 are two distinct cube roots of unity different from 1. Then, what is (ω 1 - ω 2 ) 2 equal to? 3   1   -1   -3 Question no. 2 What is the argument of the complex number (−1 − i) , where i = √(−1)?   5π/4   -5π/4   3π/4 None of these Question no. 3 What is one of the square roots of 3 + 4i, where i = √(−1)? 2+i   2-i   -2 + i   -3 - i Question no. 4 Consider the following statements I. ( ω 10 +1) 7 + ω = 0 II. ( ω 105 +1) 10 = p 10 for some prime number p, where, ω ≠ 1 is a cubic root of unity. Which of the above statement(s) is/are correct? Only I   Only II   Both I and II   Neither I nor II Question no. 5 The origin and the roots of the equation z 2 + pz + q = 0 form an equilateral triangle, if p 2 = q ...

Biochemical oxygen demand Biochemical Oxygen Demand (BOD) is a way to measure how much oxygen is needed by microorganisms to break down organic material in water. Think of it as a test to see how "dirty" the water is with organic waste. The higher the BOD, the more polluted the water is, because it means there's a lot of organic matter that needs to be broken down. How does BOD affect aquatic life? High Biochemical Oxygen Demand (BOD) levels can significantly impact aquatic life. The following are: Oxygen Depletion: When BOD is high, it means there’s a lot of organic matter in the water that microorganisms need to break down. This process consumes a lot of oxygen, reducing the amount available for fish and other aquatic organisms. Stress on Aquatic Life: Low oxygen levels can stress aquatic organisms, making it difficult for them to survive. Fish, in particular, need a certain amount of dissolved oxygen to breathe. Death of Aquatic Organisms: If ...

Probability ⚫What is Probability ? ⚪ Probability is a measure of chance that tells us how likely an event is to happen. It’s always represented as a number between 0 and 1, inclusive. ⚪ In probability , Experiment is refered as action and event can be termed as the required action. ⚪ Suppose Event = E and Sample Space = S , the probability is : ⚪ If P(E) = 0 , it means it is null , happening of this event is impossible. Similarly, If P(E) = 1 , it is sure that event will happen (100% chance). ⚪ ⚪                     ⚪ Independent Probability: Probability of A when A event is independent of B (doesn't depend on B) ⚪ Conditional Probability : The probability of B event , when A events already happens (Event A and B are dependent) ⚪ Probability of happening at least one of independent event : Let P 1 , P 2 ....... P n are the probability ...

Straight Lines Question no. 31 Let A be the non-void set of the children in a family. The relation ' x is a brother of y ' on A is looks_one Reflexive looks_two Symmetric looks_3 Transitive looks_4 None of these Answer Option looks_3 0 Solution very soon 16 Question no. 32 Let A = {1,2,3,4} and let R = { (2,2) , (3,3) , (4,4) , (1,2) } be a relation on A. Then R is looks_one Reflexive looks_two Symmetric looks_3 Transitive looks_4 None of these Answer Option looks_two q Solution ver soon 2 Question no. 33 The void relation on a set A is looks_one Reflexive looks_two Symmetric and Transitive looks_3 Reflexive and Symmetric looks_4 Reflexive and Transitive Answer Option looks_3 5: 7 Solution very soon 3 Question no. 34 Let R 1 be arelation defined by R 1 = {(a,b) | a ≥ b , a,b ∊ R }, then looks_one An equivalence relation on R looks_two Reflexive ,...

Set, Relation and Function Question no. 16 In a class of 55 students , the number of students studying different subjects are 23 in Mathematics, 24 in physics, 19 in chemistry , 12 in mathematics and physics , 9 in mathematics and chemistry , 7 in physics and chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is looks_one 6 looks_two 9 looks_3 7 looks_4 All of these Answer Option 1. 0 Solution very soon 16 Question no. 17 If A,B and C are any three sets , then A ✕( B ⋃ C ) looks_one (A ✕ B) ⋃ (A ✕ C) looks_two (A ⋃ B) ✕ (A ⋃C) looks_3 (A ✕B) ⋂ (A ✕C) looks_4 None of these Answer Option 1 is correct answer solution ver soon 2 Question no. 18 If A, B and C are any three sets , then A - (B ⋃ C) is equal to looks_one (A-B) ⋃ (A-C) looks_two (A-B) ⋂ (A-C) looks_3 ...

Basic Structure in Indian Constitution The concept of the "basic structure" of the Indian Constitution is a doctrine established by the Supreme Court of India that implies certain features of the Constitution are so fundamental to its identity that they cannot be altered or destroyed through amendments by the Parliament. Here are some key points regarding basic structure of Indian Constitution: Doctrine of Basic Structure: This doctrine was first introduced in the landmark case of Kesavananda Bharati vs. State of Kerala in 1973. The Supreme Court ruled that while the Parliament has wide powers to amend the Constitution, it cannot change the basic structure. Essential Features: The essential features of the basic structure include the supremacy of the Constitution, republican and democratic form of government, secular character, federal character, separation of powers, unity and integrity of the nation, and individual freedom. Landmark Cases: Several...

Study Material For UPSC In this page, you will get free study material of important topic of UPSC Examination with previous year question These are some important topics which must be covered : 1. Preamble in Indian constitution 2. Fundamental Duties in Indian Constitution 3. Fundamental Rights in Indian Constitution 4. DPSP in Indian Constitution 5. Comptroller and Auditor General of India 6. Attorney General Of India 7. Central Vigilance Commission 8. Central Information Commission 9. Basic Structure of Indian COnstitution