We need to find the measure of the angle formed by the bisectors of two adjacent angles (∠C and ∠D) in a quadrilateral. We will use the property that the sum of angles in a quadrilateral is 360° and the property that the sum of angles in a triangle is 180°.
In quadrilateral ABCD, the sum of all interior angles is 360°.
\[ \angle A + \angle B + \angle C + \angle D = 360^\circ \]We are given \(\angle A = 86^\circ\) and \(\angle B = 72^\circ\). Substitute these values into the equation:
\[ 86^\circ + 72^\circ + \angle C + \angle D = 360^\circ \]Calculate the sum of \(\angle A\) and \(\angle B\):
\[ 158^\circ + \angle C + \angle D = 360^\circ \]Solve for the sum of \(\angle C\) and \(\angle D\):
\[ \angle C + \angle D = 360^\circ - 158^\circ \]\[ \angle C + \angle D = 202^\circ \]Let CO and DO be the bisectors of \(\angle C\) and \(\angle D\) respectively. They meet at point O. In \(\triangle DOC\), we have:
\[ \angle ODC = \frac{1}{2} \angle D \]\[ \angle OCD = \frac{1}{2} \angle C \]The sum of angles in \(\triangle DOC\) is 180°:
\[ \angle DOC + \angle ODC + \angle OCD = 180^\circ \]Substitute the bisected angles:
\[ \angle DOC + \frac{1}{2} \angle D + \frac{1}{2} \angle C = 180^\circ \]Factor out \(\frac{1}{2}\):
\[ \angle DOC + \frac{1}{2} (\angle D + \angle C) = 180^\circ \]Substitute the value of \((\angle C + \angle D)\) from step 4:
\[ \angle DOC + \frac{1}{2} (202^\circ) = 180^\circ \]Calculate \(\frac{1}{2} (202^\circ)\):
\[ \angle DOC + 101^\circ = 180^\circ \]Solve for \(\angle DOC\):
\[ \angle DOC = 180^\circ - 101^\circ \]\[ \angle DOC = 79^\circ \]Correct Option: B) 79°
Six friends, A, B, C, D, E and F, are sitting in a circle facing the centre of the circle. A is 2nd to the right of E. Only one person sits between F and D. B and D are immediate neighbours. F is 3rd to the right of B. C is second to the right of D. Who sits between D and F?
This problem involves arranging six friends around a circular table based on given positional clues. We will build the arrangement step-by-step using the clues, starting with the most definitive ones, and then deduce the positions of all friends to answer the question.
Let's combine the clues to form a coherent arrangement.
Let's try to fix a position and build from there. Let's place E at the bottom (position 6 for reference).
E _ _ _ _ _
From Clue 1: A is 2nd to the right of E. So, if E is at position 6, A is at position 2.
E (6) _ (1) A (2) _ (3) _ (4) _ (5)
Now consider Clue 2: F is 3rd to the right of B. And Clue 3: B and D are immediate neighbours. And Clue 4: Only one person sits between F and D.
Let's try placing B. If B is at position 1, then F is at position 4. (B _ _ F)
If B is at position 1, D must be at position 6 (next to B). But E is already at position 6. So B cannot be at position 1.
Let's try placing B at position 3.
E (6) _ (1) A (2) B (3) _ (4) _ (5)
If B is at position 3, F is 3rd to the right of B. Counting clockwise from B: position 4, position 5, position 6. So F is at position 6. But E is at position 6. So B cannot be at position 3.
Let's try placing B at position 4.
E (6) _ (1) A (2) _ (3) B (4) _ (5)
If B is at position 4, F is 3rd to the right of B. Counting clockwise from B: position 5, position 6, position 1. So F is at position 1.
E (6) F (1) A (2) _ (3) B (4) _ (5)
Now, let's use Clue 3: B and D are immediate neighbours. Since B is at position 4, D can be at position 3 or position 5.
Case 1: D is at position 3.
E (6) F (1) A (2) D (3) B (4) _ (5)
Check Clue 4: Only one person sits between F and D. F is at 1, D is at 3. A is at 2, so A sits between F and D. This is consistent.
Check Clue 5: C is second to the right of D. D is at 3. Second to the right of D would be position 5. So C is at position 5.
E (6) F (1) A (2) D (3) B (4) C (5)
This arrangement uses all 6 friends and satisfies all conditions:
The final arrangement (clockwise from F) is F, A, D, B, C, E.
The question asks: Who sits between D and F?
Looking at our arrangement (F, A, D, B, C, E), between D and F (going clockwise from D to F) we have B, C, E. Going anti-clockwise from D to F, we have A.
In a circular arrangement, "between X and Y" usually refers to the shorter arc. Here, F-A-D is the shorter arc with A between F and D. The longer arc would be F-E-C-B-D.
Therefore, A sits between D and F.
D) A is the person who sits between D and F in the final arrangement.
The question asks about the primary function of the Reserve Bank of India (RBI). To answer this, we need to recall the core responsibilities and roles assigned to the RBI as the central bank of India.
A) It conducts consolidated supervision of the financial sector in India. This is a key function of the Reserve Bank of India. The RBI is responsible for regulating and supervising commercial banks, cooperative banks, non-banking financial companies (NBFCs), and other financial institutions to ensure the stability and health of the Indian financial system. This includes setting prudential norms, conducting inspections, and taking corrective actions.
The question asks to identify the indigenously developed automatic train protection system that was tested in India in March 2022. This requires knowledge of recent developments in Indian Railways technology.
Correct Option: A) KAVACH is the indigenously developed Automatic Train Protection (ATP) system, also known as the Train Collision Avoidance System (TCAS). It was successfully tested in March 2022 by Indian Railways to enhance safety by preventing collisions, controlling train speed, and ensuring adherence to signal aspects.
Match the following:
| Compound | Name |
| i. C6H6 | a. Ethane |
| ii. H2C=CH2 | b. Benzene |
| iii. CH3-CHO | c. Acetaldehyde |
| iv. CH3COCH3 | d. Ethylene |
| v. CH3CH3 | e. Acetone |
i-b, ii-d, iii-c, iv-e, v-a
i-d, ii-c, iii-e, iv-b, v-a
i-d, ii-c, iii-a, iv-b, v-e
i-a, ii-b, iii-c, iv-e, v-d
To solve this matching question, we need to correctly identify the common names for the given chemical formulas or structures. This requires basic knowledge of organic chemistry nomenclature.
Correct Option: A) i-b, ii-d, iii-c, iv-e, v-a is the correct combination based on our step-by-step matching.
The problem presents a number series and asks us to find the next term. We need to identify the pattern or rule governing the sequence by examining the differences between consecutive terms.
The differences are \(-1, -4, -9, -16\). These numbers are the negative squares of consecutive natural numbers:
So, the pattern is that each term is obtained by subtracting the square of the next natural number from the previous term.
Following this pattern, the next difference should be \(-5^2 = -25\).
Therefore, the next term in the series will be \(158 - 25 = 133\).
A) 133 is the correct answer because it follows the established pattern of subtracting consecutive squares (\(1^2, 2^2, 3^2, 4^2, 5^2\)) from the preceding term.
The question asks about the retirement pattern of members of the Rajya Sabha (Council of States) in India. The Rajya Sabha is a permanent body and is not subject to dissolution. Its members serve for a fixed term, and a specific fraction of them retire periodically to ensure continuity.
B) One-third of the members retire every second year. This statement accurately describes the retirement process for members of the Rajya Sabha as per the Indian Constitution.
The question asks about the location where the World Health Organization (WHO) and India agreed to establish a Global Centre for Traditional Medicine in March 2022. This is a factual question related to current affairs and international collaborations in the health sector.
C) Gujarat is the correct answer. The WHO Global Centre for Traditional Medicine (GCTM) was established in Jamnagar, Gujarat, through an agreement signed in March 2022.
The question asks to identify the scientist credited with formulating the laws of inheritance. This is a direct knowledge-based question from the field of biology, specifically genetics.
Correct Option: C) Gregor Mendel is known for his experiments with pea plants in the mid-19th century. Through careful observation and statistical analysis of trait inheritance, he formulated the fundamental principles of heredity, which are now known as Mendel's Laws of Inheritance (Law of Segregation, Law of Independent Assortment, and Law of Dominance).
To find the volume of a cube, we need to know the length of its side. The formula for the volume of a cube is the cube of its side length (\(V = s^3\)). We will substitute the given side length into this formula and calculate the result.
The side length of the cube (\(s\)) is given as 12 cm.
The volume \(V\) of a cube with side length \(s\) is given by:
\[ V = s^3 \]Substitute \(s = 12 \text{ cm}\) into the volume formula:
\[ V = (12 \text{ cm})^3 \]Calculate \(12 \times 12 \times 12\):
So, the volume \(V\) is:
\[ V = 1728 \text{ cm}^3 \]Correct Option: B) 1728 cm³