Simplify: [(26 ÷ \( \frac{2}{2} \)) × {6 ÷ 2 × (\( \frac{12-2}{2} \))} ÷ {5 × 6 ÷ 3}]
To simplify the given expression, we must follow the BODMAS/PEMDAS rule (Brackets, Orders/Exponents, Division, Multiplication, Addition, Subtraction). We will solve the innermost brackets first and proceed outwards, performing operations in the specified order.
First, let's write down the expression:
\[ \left[ \left( 26 \div \frac{2}{2} \right) \times \left\{ 6 \div 2 \times \left( \frac{12-2}{2} \right) \right\} \div \left\{ 5 \times 6 \div 3 \right\} \right] \]Simplify the innermost parentheses:
In the first term: \( \frac{2}{2} = 1 \)
In the second term: \[\frac{12-2}{2} = \frac{10}{2} = 5\]
Substitute these values back into the expression:
\[ \left[ \left( 26 \div 1 \right) \times \left\{ 6 \div 2 \times 5 \right\} \div \left\{ 5 \times 6 \div 3 \right\} \right] \]Now, simplify the terms inside the curly braces and the first parenthesis:
First parenthesis: \( 26 \div 1 = 26 \)
First curly brace: \( 6 \div 2 \times 5 \)
Following BODMAS, perform division first, then multiplication:
\[ 6 \div 2 = 3 \]\[ 3 \times 5 = 15 \]Second curly brace: \( 5 \times 6 \div 3 \)
Following BODMAS, perform multiplication first, then division (or left to right for multiplication and division):
\[ 5 \times 6 = 30 \]\[ 30 \div 3 = 10 \]Substitute these simplified values back into the expression:
\[ \left[ 26 \times 15 \div 10 \right] \]Finally, perform the operations inside the square bracket from left to right (multiplication and division have equal precedence):
\[ 26 \times 15 = 390 \]\[ 390 \div 10 = 39 \]A) The simplified value of the expression is 39.
The question asks about the advantages of decentralised planning within India's Panchayati Raj system. We need to identify the option that best describes a benefit of empowering local bodies in planning and development.
Correct Option: C) It allows local communities to tailor need-based development initiatives.
Decentralised planning, especially through the Panchayati Raj system, empowers local communities to identify their specific needs and priorities. This enables them to design and implement development initiatives that are directly relevant and beneficial to their unique local context, rather than relying on uniform, top-down plans that might not address local realities effectively. This direct involvement ensures that resources are allocated to address the most pressing local issues, leading to more effective and sustainable development.
The problem requires us to modify a given number based on specific rules for its even and odd digits, and then find the sum of all odd digits in the newly formed number. We will process each digit of the original number, apply the given rules, and then identify and sum the odd digits in the resulting number.
The sum of all the odd digits in the new number is 24.
D) 24 — As calculated, the sum of the odd digits in the new number (3, 5, 9, 7) is \(3 + 5 + 9 + 7 = 24\).
The question asks for the location of a specific event, the STREE Summit 2025, on a particular date. This requires knowledge of current affairs and event locations. Such questions often test awareness of major conferences and summits held in India.
Correct Option: D) Hyderabad
The STREE Summit 2025 on Women's Safety and Empowerment was held in Hyderabad on 15 April 2025. Hyderabad has been a host city for various national and international conferences and summits, making it a suitable venue for such an event.
The given series is an alternating series, meaning the pattern involves two different operations applied alternately to consecutive terms.
The pattern observed is an alternating subtraction of 2 and addition of 10.
The last operation was \(+10\) (from 918 to 928). Therefore, the next operation should be \(-2\).
Missing term \( = 928 - 2 = 926 \)
If the missing term is 926, the next operation should be \(+10\).
\(926 + 10 = 936\)
This matches the last given term in the series, confirming our pattern and the missing term.
C) 926 — The series follows an alternating pattern of subtracting 2 and adding 10. Following this pattern, \(928 - 2 = 926\).
To solve this problem, we need to understand the concepts of fourth proportion and third proportion. We will first calculate the fourth proportion for the given numbers. Then, we will use this value to find the third proportion, which will allow us to determine the value of A.
If \(a, b, c, d\) are in proportion, then \(a:b = c:d\), which means \[\frac{a}{b} = \frac{c}{d}\]. Here, \(a=3\), \(b=9\), \(c=4\), and \(d\) is the fourth proportion.
\[ \frac{3}{9} = \frac{4}{d} \]Cross-multiplying gives:
\[ 3 \times d = 9 \times 4 \] \[ 3d = 36 \] \[ d = \frac{36}{3} \] \[ d = 12 \]So, the fourth proportion is 12.
If \(a, b, c\) are in continued proportion, then \(a:b = b:c\), which means \[\frac{a}{b} = \frac{b}{c}\]. Here, \(a=A\), \(b=12\), and \(c\) is the third proportion.
We are given that the third proportion to A and 12 is the same as the fourth proportion calculated in step 1, which is 12.
So, \(c = 12\).
\[ \frac{A}{12} = \frac{12}{12} \]From this equation:
\[ \frac{A}{12} = 1 \] \[ A = 12 \times 1 \] \[ A = 12 \]Correct Option: A) 12
This problem involves calculating the combined work rate of two pipes, one filling and one emptying. We will determine the individual rates of filling and emptying, then find the net rate when both pipes operate simultaneously. Finally, we will use the net rate to find the total time required to fill the tank.
The first pipe fills the tank in 12 hours. So, in one hour, it fills \(\frac{1}{12}\) of the tank.
\[ \text{Filling rate} = \frac{1}{12} \text{ tank/hour} \]The second pipe empties the tank in 14 hours. So, in one hour, it empties \(\frac{1}{14}\) of the tank.
\[ \text{Emptying rate} = \frac{1}{14} \text{ tank/hour} \]Since one pipe is filling and the other is emptying, their effects are opposite. The net rate is the filling rate minus the emptying rate.
\[ \text{Net rate} = \text{Filling rate} - \text{Emptying rate} \]\[ \text{Net rate} = \frac{1}{12} - \frac{1}{14} \]To subtract the fractions, find a common denominator, which is the Least Common Multiple (LCM) of 12 and 14. The LCM of 12 and 14 is 84.
\[ \text{Net rate} = \frac{7}{84} - \frac{6}{84} \]\[ \text{Net rate} = \frac{7 - 6}{84} \]\[ \text{Net rate} = \frac{1}{84} \text{ tank/hour} \]If the net rate is \(\frac{1}{84}\) of the tank filled per hour, then the time taken to fill the entire tank (1 unit) is the reciprocal of this rate.
\[ \text{Time} = \frac{1}{\text{Net rate}} \]\[ \text{Time} = \frac{1}{\frac{1}{84}} \]\[ \text{Time} = 84 \text{ hours} \]Correct Option: D) 84
The tank will be filled in 84 hours when both pipes are opened simultaneously.
The question asks to identify the city chosen to host the Asia-Pacific Economic Cooperation (APEC) Summit in November 2025. This requires knowledge of recent international event hosting decisions.
Correct Option: C) Gyeongju — Gyeongju was officially selected on June 27, 2024, to host the APEC Summit in November 2025, beating out other major South Korean cities.
The question asks about the primary purpose of ISRO's semi-cryogenic engine, expected by 2025. This requires knowledge of ISRO's current and future rocket propulsion systems and their applications.
Correct Option: B) To power the semi-cryogenic booster stage of the Launch Vehicle Mark-3 (LVM3)
ISRO's semi-cryogenic engine (SCE-200) is being developed primarily to replace the solid strap-on boosters and the L110 liquid core stage of the LVM3, thereby forming a more powerful and efficient semi-cryogenic booster stage. This will significantly enhance the LVM3's payload capacity and reduce launch costs.