To expand the given expression \((6a + 2b + 7c)^2\), we will use the algebraic identity for the square of a trinomial, which is \((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\).
Correct Option: D) 36a² + 4b² + 49c² + 24ab + 28bc + 84ac
The problem requires us to evaluate a mathematical expression after interchanging certain operators. We need to replace '+' with '−' and '−' with '+', and similarly '×' with '÷' and '÷' with '×'. After the operator interchange, we will solve the new expression following the order of operations (BODMAS/PEMDAS).
So, the value of the expression after interchanging the operators is 72.
Correct Option: B) 72
Two sets of numbers are given below. In each set of numbers, certain mathematical operation(s) on the first number result(s) in the second number. Similarly, certain mathematical operation(s) on the second number result(s) in the third number and so on. Which of the given options follows the same set of operations as in the given sets?
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into their constituent digits. E.g. 13 – Operations on 13 such as adding to/subtracting from/multiplying with 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
19-57-114-134; 21-63-126-146
18-54-108-118
17-51-102-112
20-60-120-180
16-48-96-116
The problem asks us to identify a sequence of numbers that follows the same mathematical operations as two given example sequences. We need to find the pattern of operations (addition, subtraction, multiplication, division) between consecutive numbers in the given sets and then apply that pattern to the options.
Analyze the first given set: 19-57-114-134
From 19 to 57:
\(57 \div 19 = 3\). So, the operation is multiplication by 3. \[ 19 \times 3 = 57 \]
From 57 to 114:
\(114 \div 57 = 2\). So, the operation is multiplication by 2. \[ 57 \times 2 = 114 \]
From 114 to 134:
\(134 - 114 = 20\). So, the operation is addition of 20. \[ 114 + 20 = 134 \]
The pattern for the first set is: \(\times 3\), \(\times 2\), \(+ 20\).
Analyze the second given set: 21-63-126-146
From 21 to 63:
\(63 \div 21 = 3\). So, the operation is multiplication by 3. \[ 21 \times 3 = 63 \]
From 63 to 126:
\(126 \div 63 = 2\). So, the operation is multiplication by 2. \[ 63 \times 2 = 126 \]
From 126 to 146:
\(146 - 126 = 20\). So, the operation is addition of 20. \[ 126 + 20 = 146 \]
The pattern for the second set is also: \(\times 3\), \(\times 2\), \(+ 20\).
Conclusion about the pattern: The consistent pattern of operations is: multiply by 3, then multiply by 2, then add 20.
Apply the pattern to the options:
A) 18-54-108-118
B) 17-51-102-112
C) 20-60-120-180
D) 16-48-96-116
Correct Option: D) 16-48-96-116
This option perfectly follows the identified pattern of operations: multiply by 3, then multiply by 2, then add 20.
To solve this problem, we need to track Tapan's movements step-by-step, determining his final position relative to his starting point (Point A). We can use a coordinate system or simply visualize the movements in terms of North/South and East/West displacements. After finding his final position (Point P), we calculate the shortest distance and direction needed to return to Point A.
Point A is at \((0,0)\) and Point P is at \((4,0)\).
To reach Point A from Point P, Tapan needs to move from \((4,0)\) to \((0,0)\).
The change in the x-coordinate is \(0 - 4 = -4\).
The change in the y-coordinate is \(0 - 0 = 0\).
This means he needs to move 4 km in the negative x-direction, which corresponds to the west direction.
The shortest distance is 4 km, and the direction is west.
A) 4 km to the west - This matches our calculation. Tapan is 4 km east of Point A, so he needs to drive 4 km west to return to Point A.
Which of the following statements is/are correct?
1) Nari Shakti Vandan Adhiniyam was introduced in 2023.
2) Nari Shakti Vandan Adhiniyam ensures 33% reserved seats for women in the Parliament, Legislative Assemblies and Delhi Assembly.
3) Nari Shakti Vandan Adhiniyam will be in force for 10 years only.
The question asks us to identify the correct statements regarding the Nari Shakti Vandan Adhiniyam. We need to examine the year of its introduction, the legislative bodies covered by the women's reservation provisions, and the duration of the reservation. The key distinction is that the reservation applies to the Lok Sabha and specified Legislative Assemblies, not to both Houses of Parliament.
Understand the Nari Shakti Vandan Adhiniyam:
The Nari Shakti Vandan Adhiniyam, officially known as the Constitution (One Hundred and Sixth Amendment) Act, 2023, provides for approximately one-third (33%) reservation of seats for women in specified legislative bodies.
Its objective is to increase women's representation in directly elected legislative institutions.
Analyse Statement 1:
Statement: "Nari Shakti Vandan Adhiniyam was introduced in 2023."
The Women's Reservation Bill was introduced in the Lok Sabha on September 19, 2023. It was subsequently passed by both Houses of Parliament and received Presidential assent on September 28, 2023.
Therefore, Statement 1 is correct.
Analyse Statement 2:
Statement: "Nari Shakti Vandan Adhiniyam ensures 33% reserved seats for women in the Parliament, Legislative Assemblies and Delhi Assembly."
The Act provides for approximately one-third reservation for women in the following legislative bodies:
However, the Parliament of India consists of the President, Lok Sabha and Rajya Sabha. The women's reservation provisions apply to the Lok Sabha but do not extend to the Rajya Sabha.
Therefore, Statement 2 is incorrect because it refers to Parliament as a whole instead of specifically mentioning the Lok Sabha.
Analyse Statement 3:
Statement: "Nari Shakti Vandan Adhiniyam will be in force for 10 years only."
Article 334A, inserted by the Constitution (One Hundred and Sixth Amendment) Act, 2023, specifies a reservation period of 15 years, with provision for continuation as determined by Parliament through legislation.
Therefore, Statement 3 is incorrect because it mentions 10 years instead of 15 years.
Identify the correct statements:
Therefore, only Statement 1 is correct.
Correct Option: A) 1 only — The Nari Shakti Vandan Adhiniyam was introduced in 2023, making Statement 1 correct. Statement 2 is incorrect because the reservation applies to the Lok Sabha and specified Legislative Assemblies, not to Parliament as a whole, since the Rajya Sabha is excluded. Statement 3 is incorrect because the reservation period is 15 years rather than 10 years. Hence, Option A is the correct answer.
B) 1 and 3: This is incorrect because Statement 3 wrongly specifies a reservation period of only 10 years. The Act provides for a period of 15 years, with provision for continuation through parliamentary legislation.
C) 2 and 3: This is incorrect because both Statements 2 and 3 are incorrect. Statement 2 wrongly refers to Parliament as a whole, whereas the reservation applies specifically to the Lok Sabha and specified Legislative Assemblies. Statement 3 incorrectly mentions 10 years instead of 15 years.
D) 1 and 2: This is incorrect because Statement 2 refers to Parliament as a whole. The reservation provisions cover the Lok Sabha, State Legislative Assemblies and Delhi Legislative Assembly, but not the Rajya Sabha. Therefore, Statement 2 is not factually accurate as written.
The question asks to identify the synthetic chemicals primarily responsible for ozone depletion in the atmosphere. We need to recall the major substances known to cause this environmental problem.
Correct Option: A) Chlorofluorocarbons (CFCs) are the primary synthetic chemicals responsible for the depletion of ozone in the atmosphere. Their widespread use led to significant thinning of the ozone layer, particularly over the Antarctic, leading to the Montreal Protocol to phase out their production.
To identify the substance reduced in a chemical reaction, we need to understand the concept of reduction. Reduction can be defined in several ways: gain of hydrogen, loss of oxygen, gain of electrons, or decrease in oxidation state. In the given reaction, we will look for the species that loses oxygen or gains hydrogen.
Correct Option: A) CuO is the substance reduced because it loses oxygen to form copper (\(\text{Cu}\)).
To find the median of a given set of data, the first step is to arrange the data in ascending or descending order. Once ordered, the median is the middle value if the number of data points is odd. If the number of data points is even, the median is the average of the two middle values.
List the given data:
\(78, 33, 59, 56, 61, 36, 48, 42, 97, 57, 62\)
Count the number of data points (n):
There are \(11\) data points.
Arrange the data in ascending order:
\(33, 36, 42, 48, 56, 57, 59, 61, 62, 78, 97\)
Determine the position of the median:
Since the number of data points \(n = 11\) is odd, the median is the value at the position given by the formula \(\frac{n+1}{2}\).
Median position \[= \frac{11+1}{2} = \frac{12}{2} = 6^{th}\] position.
Identify the median value:
Looking at the sorted list, the \(6^{th}\) value is \(57\).
\(33, 36, 42, 48, 56, \textbf{57}, 59, 61, 62, 78, 97\)
Therefore, the median of the given data is \(57\).
Correct Option: C) 57
The question asks which type of pollution can be minimized by implementing proper sewage treatment. We need to understand what sewage is and how its treatment affects different environmental compartments.
Correct Option: D) Water pollution. Proper sewage treatment is specifically designed to clean wastewater, preventing the discharge of harmful contaminants into water bodies. This directly minimizes water pollution by reducing pathogens, organic matter, and nutrients that would otherwise degrade aquatic ecosystems and human health.
The problem involves two numbers whose initial ratio is given. We are then given conditions for how the numbers change and their new ratio. We need to set up algebraic equations based on these ratios and solve for the original numbers, finally calculating their sum.
Represent the original numbers: Let the two original numbers be \(5x\) and \(8x\), since their ratio is \(5:8\).
Apply the given changes: The first number is increased by 13, so it becomes \(5x + 13\). The second number is decreased by 6, so it becomes \(8x - 6\).
Formulate the equation based on the new ratio: After the changes, the new ratio is \(4:6\), which simplifies to \(2:3\). So, we can write the equation:
\[ \frac{5x + 13}{8x - 6} = \frac{4}{6} \]Simplify the ratio on the right side:
\[ \frac{5x + 13}{8x - 6} = \frac{2}{3} \]Solve the equation for \(x\): Cross-multiply to solve for \(x\):
\[ 3(5x + 13) = 2(8x - 6) \] \[ 15x + 39 = 16x - 12 \]Subtract \(15x\) from both sides:
\[ 39 = 16x - 15x - 12 \] \[ 39 = x - 12 \]Add 12 to both sides:
\[ x = 39 + 12 \] \[ x = 51 \]Calculate the original numbers: First number \( = 5x = 5 \times 51 = 255 \). Second number \( = 8x = 8 \times 51 = 408 \).
Find the sum of the original numbers: Sum \( = 255 + 408 = 663 \).
Correct Option: D) 663