To determine the relationship between three given straight lines, we need to analyze their slopes. Lines are parallel if they have the same slope. Lines are perpendicular if the product of their slopes is -1. Lines are intersecting if their slopes are different. Lines are concurrent if they intersect at a single common point.
The first line is \(x + y - 3 = 0\).
Here, \(A = 1\) and \(B = 1\).
So, the slope \(m_1 = -(1)/(1) = -1\).
The second line is \(x + y + 2 = 0\).
Here, \(A = 1\) and \(B = 1\).
So, the slope \(m_2 = -(1)/(1) = -1\).
The third line is \(3x + 3y - 7 = 0\).
Here, \(A = 3\) and \(B = 3\).
So, the slope \(m_3 = -(3)/(3) = -1\).
We have \(m_1 = -1\), \(m_2 = -1\), and \(m_3 = -1\).
Since \(m_1 = m_2 = m_3\), all three lines have the same slope.
Correct Option: D) parallel
To solve this problem, we need to follow the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). In this expression, we have addition and division. According to the order of operations, division must be performed before addition.
Correct Option: D) 7525
The question asks to identify the Indian independence movement during which Mahatma Gandhi gave the famous slogan 'Do or Die'. To answer this, we need to recall the key slogans and their associated movements led by Mahatma Gandhi.
Correct Option: C) Quit India Movement — Mahatma Gandhi gave the 'Do or Die' slogan during the Quit India Movement in 1942, urging Indians to make every effort to achieve independence.
To solve this blood relation problem, we need to construct a family tree or a set of relationships based on the given statements. We will identify the gender of each person and their connections to others to determine the relationship between O and F.
Thus, O is F's son's daughter.
Correct Option: D) Son's daughter
This question is based on blood relationships involving maternal uncles across two generations. The nurse states that her maternal uncle and the maternal uncle of the boy's maternal uncle are the same person. We need to trace the family relationships to determine how the boy is related to the nurse.
A maternal uncle is the brother of one's mother.
Therefore, the nurse's maternal uncle is her mother's brother.
Let this person be U.
The boy's maternal uncle is the brother of the boy's mother.
Let this person be M.
Therefore, M is the brother of the boy's mother.
The nurse states that her maternal uncle and the maternal uncle of M are the same person.
Therefore, U is also the maternal uncle of M.
This means that U is the brother of M's mother.
U is the brother of the nurse's mother and also the brother of M's mother.
In the direct family relationship used to solve the question, M's mother is the nurse's mother.
Therefore, M is the nurse's brother.
M is the maternal uncle of the boy.
Therefore, M is the brother of the boy's mother.
Since M is the nurse's brother, the nurse can be the boy's mother.
Thus, the boy is the nurse's son.
Among the given options, Son is the relationship that satisfies the family connections.
Therefore, option D is correct.
Correct Option: D) Son
The nurse's maternal uncle is also the maternal uncle of the boy's maternal uncle. Following the direct family relationship, the boy's maternal uncle is the nurse's brother. Since the boy's maternal uncle is his mother's brother, the nurse is the boy's mother. Therefore, the boy is her son.
This is a blood relation problem. We need to deduce the relationships between the family members based on the given statements and then determine the relationship between E and D. We can use a family tree diagram or logical deduction to solve this.
Now let's combine the information to find the relationship between E and D:
Therefore, E is the father of D.
D) Father — Based on the deductions, E is the father of D.
Who has taken over as the new CEO of the Railway Board on 1st Jan 2021?
Piyush Goyal
V K Yadav
Ashutosh Gangal
Suneet Sharma
This question tests knowledge of recent appointments in key government bodies, specifically the Indian Railway Board. Candidates should be aware of significant personnel changes in public sector undertakings and ministries.
Correct Option: D) Suneet Sharma was appointed as the new Chairman & CEO of the Railway Board on January 1, 2021.
To determine the position of a point \((x_1, y_1)\) with respect to a circle given by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), we substitute the coordinates of the point into the equation. Let \(S(x, y) = x^2 + y^2 + 2gx + 2fy + c\). Then:
The given equation of the circle is:
\[ x^{2} + y^{2} - 3x - 4y + 1 = 0 \]Let \(S(x, y) = x^{2} + y^{2} - 3x - 4y + 1\). The given point is \((1, 2)\).
Substitute the coordinates of the point \((1, 2)\) into the expression \(S(x, y)\):
\[ S(1, 2) = (1)^{2} + (2)^{2} - 3(1) - 4(2) + 1 \]Calculate the value:
\[ S(1, 2) = 1 + 4 - 3 - 8 + 1 \]\[ S(1, 2) = 5 - 3 - 8 + 1 \]\[ S(1, 2) = 2 - 8 + 1 \]\[ S(1, 2) = -6 + 1 \]\[ S(1, 2) = -5 \]Since \(S(1, 2) = -5\), which is less than 0, the point \((1, 2)\) lies inside the circle.
Correct Option: D) lies inside the circle
This question tests knowledge of the collective responsibility of the Council of Ministers in an Indian state. Understanding the parliamentary system and the specific bodies to which the executive is accountable is key.
Correct Option: D) The Council of Ministers of a State is collectively responsible to the Legislative Assembly of the State.
Article 164(2) of the Indian Constitution clearly states that "The Council of Ministers shall be collectively responsible to the Legislative Assembly of the State." This is a cornerstone of the parliamentary system at the state level, ensuring that the executive remains accountable to the directly elected representatives of the people.
The question asks to identify India's first national academy dedicated to music, dance, and drama. This requires knowledge of the history and establishment of major cultural institutions in India.
Correct Option: A) Sangeet Natak Akademi