The ratio of the lengths of two corresponding sides of two similar triangles is 17 : 13. The ratio of the areas of these two triangles, in the order mentioned, is:
289 : 169
17 : 13
17\(\sqrt{17}\) : 13
290 : 170
This problem involves the properties of similar triangles. A key theorem states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Correct Option: A) 289 : 169 is the correct ratio of the areas, as it is the square of the ratio of the corresponding sides.
The question provides the number of children in 10 families and asks us to calculate the number of children per family. This requires finding the arithmetic mean (average) by dividing the total number of children by the total number of families.
Identify the given information:
The numbers of children in the 10 families are:
2, 1, 3, 4, 2, 3, 5, 1, 1, 3
Total number of families = 10
Calculate the total number of children:
Add the number of children in all 10 families.
\[ 2+1+3+4+2+3+5+1+1+3=25 \]Therefore, the total number of children is 25.
Apply the arithmetic mean formula:
The average number of children per family is calculated as:
\[ \text{Average}=\frac{\text{Total number of children}}{\text{Total number of families}} \]Substituting the values:
\[ \text{Average}=\frac{25}{10} \] \[ \boxed{\text{Average}=2.5} \]Therefore, the average number of children per family is 2.5.
Correct Option: B) 2.5 — The total number of children in the 10 families is 25. Dividing 25 by 10 gives an average of 2.5 children per family. Hence, Option B is the correct answer.
A) 2.8: This is incorrect because an average of 2.8 would require 28 children across 10 families, whereas the given families have 25 children.
C) 2.4: This is incorrect because an average of 2.4 would require 24 children across 10 families, whereas the total is 25.
D) 3: This is incorrect because an average of 3 would require 30 children across 10 families, whereas the given total is 25.
To find the combined ratio A:B:C:D from given individual ratios A:B, B:C, and C:D, we need to make the common terms in consecutive ratios equal. This is achieved by finding the Least Common Multiple (LCM) of the common terms and multiplying the ratios by appropriate factors.
Given:
The common term is B. The values for B are 3 and 4. The LCM of 3 and 4 is 12.
To make B equal to 12 in \(A:B\):
\[ A:B = (1 \times 4) : (3 \times 4) = 4:12 \]To make B equal to 12 in \(B:C\):
\[ B:C = (4 \times 3) : (3 \times 3) = 12:9 \]Now, combine these two to get \(A:B:C\):
\[ A:B:C = 4:12:9 \]Given:
The common term is C. The values for C are 9 and 6. The LCM of 9 and 6 is 18.
To make C equal to 18 in \(A:B:C\):
Multiply each part of \(A:B:C\) by \(\frac{18}{9} = 2\):
\[ A:B:C = (4 \times 2) : (12 \times 2) : (9 \times 2) = 8:24:18 \]To make C equal to 18 in \(C:D\):
Multiply each part of \(C:D\) by \(\frac{18}{6} = 3\):
\[ C:D = (6 \times 3) : (7 \times 3) = 18:21 \]Now, combine these two to get \(A:B:C:D\):
\[ A:B:C:D = 8:24:18:21 \]A) 8:24:18:21 is the correct combined ratio.
The question asks to identify the correct statements regarding the Quad Leaders' Summit. We need to evaluate each statement based on recent developments and historical facts related to the Quad grouping.
Based on the evaluation, statements 1 and 3 are correct, while statement 2 is incorrect.
Correct Option: A) 1 and 3 only
The question asks to identify the sport in which Kapil Parmar won India's first-ever medal at the Paralympic Games 2024. This is a factual recall question related to recent sporting achievements.
Kapil Parmar indeed made history by winning India's first-ever medal in Judo at the Paralympic Games 2024. This was a significant milestone for Indian para-sports.
A) Judo — Kapil Parmar secured India's first-ever medal in Judo at the Paralympic Games 2024, making this the correct answer.
In a row of 25 students facing north, Manoj is 15th from the right end. If Harish is 10th to the left of Manoj, what is Harish's position from the left end of the row?
The problem involves finding the position of a person from one end of a row, given their position from the other end and the position of another person relative to them. We will first find Manoj's position from the left end, then use that to find Harish's position from the left end.
A) 1st is the correct answer. As calculated, Harish's position from the left end is 1st.
If x is inversely proportional to y, and y = 6 when x = 7, find the value of x, when y = 64.
\( \frac{21}{32} \)
\( \frac{23}{35} \)
\( \frac{22}{33} \)
\( \frac{21}{33} \)
The problem states that \(x\) is inversely proportional to \(y\). This means that their product is a constant. We can use the given values to find this constant and then use it to find the new value of \(x\) when \(y\) changes.
Understand Inverse Proportionality:
If \(x\) is inversely proportional to \(y\), it can be written as:
This implies that \(xy = k\), where \(k\) is the constant of proportionality.
Find the Constant of Proportionality (\(k\)):
We are given that \(y = 6\) when \(x = 7\). Substitute these values into the equation \(xy = k\):
Use the Constant to Find the New Value of \(x\):
We need to find the value of \(x\) when \(y = 64\). Using the same constant \(k = 42\):
Solve for \(x\):
Divide both sides by 64:
Simplify the Fraction:
Both 42 and 64 are divisible by 2:
A) \( \frac{21}{32} \)
The calculated value of \(x\) is \( \frac{21}{32} \), which matches option A.
The question asks about the specific structure that needs to be blocked in males to prevent sperm transfer during contraceptive surgical procedures. This requires knowledge of the male reproductive system and the path of sperm.
C) Vas deferens — The vas deferens is the tube that transports sperm from the epididymis to the ejaculatory duct. Blocking or cutting this tube (as in a vasectomy) effectively prevents sperm from being part of the ejaculate, thus achieving contraception.
The solution of the system of linear equations is:
x + y + z = 6, 2x − y + 3z = 14, −x + 2y − z = −2
x = \( \frac{-4}{3} \), y = \( \frac{4}{3} \), z = 6
x = \( \frac{4}{3} \), y = \( \frac{-4}{3} \), z = 5
x = \( \frac{5}{3} \), y = \( \frac{-5}{3} \), z = 6
x = \( \frac{-5}{3} \), y = \( \frac{5}{3} \), z = −6
To solve a system of linear equations, we can use methods like substitution, elimination, or matrix methods (Cramer's rule, inverse matrix). The elimination method is often efficient for three variables.
Given the system of linear equations:
Step 1: Eliminate \(y\) from Equation 1 and Equation 2.
Add Equation 1 and Equation 2:
\[\begin{array}{rcl} (x + y + z) + (2x - y + 3z) &=& 6 + 14 \ 3x + 4z &=& 20 \quad \text{(Equation 4)} \end{array}\]Step 2: Eliminate \(y\) from Equation 1 and Equation 3.
Multiply Equation 1 by 2: \(2(x + y + z) = 2(6) \implies 2x + 2y + 2z = 12\).
Add this modified Equation 1 to Equation 3:
\[\begin{array}{rcl} (2x + 2y + 2z) + (-x + 2y - z) &=& 12 + (-2) \ (2x - x) + (2y + 2y) + (2z - z) &=& 10 \ x + 4y + z &=& 10 \end{array}\]Wait, this is not correct. We need to eliminate \(y\). Let's re-do this step.
Multiply Equation 1 by 2: \(2x + 2y + 2z = 12\).
Add this to Equation 3: \(-x + 2y - z = -2\).
This would give \(x + 4y + z = 10\), which still has \(y\). We need to eliminate \(y\).
Let's use Equation 1 and Equation 3 to eliminate \(y\).
Multiply Equation 1 by 2: \(2x + 2y + 2z = 12\).
Subtract Equation 3 from this modified Equation 1:
\[\begin{array}{rcl} (2x + 2y + 2z) - (-x + 2y - z) &=& 12 - (-2) \ 2x + 2y + 2z + x - 2y + z &=& 12 + 2 \ 3x + 3z &=& 14 \quad \text{(Equation 5)} \end{array}\]Step 3: Solve the system of Equation 4 and Equation 5.
Equation 4: \(3x + 4z = 20\)
Equation 5: \(3x + 3z = 14\)
Subtract Equation 5 from Equation 4:
\[\begin{array}{rcl} (3x + 4z) - (3x + 3z) &=& 20 - 14 \ z &=& 6 \end{array}\]Step 4: Substitute \(z = 6\) into Equation 5 to find \(x\).
\[\begin{array}{rcl} 3x + 3(6) &=& 14 \ 3x + 18 &=& 14 \ 3x &=& 14 - 18 \ 3x &=& -4 \ x &=& \frac{-4}{3} \end{array}\]Step 5: Substitute \(x = \frac{-4}{3}\) and \(z = 6\) into Equation 1 to find \(y\).
\[\begin{array}{rcl} x + y + z &=& 6 \ \frac{-4}{3} + y + 6 &=& 6 \ \frac{-4}{3} + y &=& 0 \ y &=& \frac{4}{3} \end{array}\]So, the solution is \(x = \frac{-4}{3}\), \(y = \frac{4}{3}\), \(z = 6\).
Correct Option: A) \(x = \frac{-4}{3}\), \(y = \frac{4}{3}\), \(z = 6\)
The question asks to identify the IIT whose students and professors developed an innovative fire rescue drone in June 2024. This is a factual recall question related to recent technological advancements from Indian educational institutions.
D) IIT Dharwad — In June 2024, students and professors of IIT Dharwad developed an innovative fire rescue drone. This drone is designed to assist in firefighting operations, particularly in difficult-to-reach areas, by providing real-time data and potentially delivering small payloads. The project was led by Dr. P. Seshu, a professor at IIT Dharwad, along with a team of students.