The question asks us to identify the FALSE statement among the given options regarding properties of a triangle. We will analyze each statement based on fundamental triangle inequalities and angle properties.
Since statement C is false, it is the correct answer.
C) In any triangle, the sum of any two angles of the triangle is greater than the measure of the third angle. This statement is FALSE. As shown in the step-by-step analysis, in an obtuse triangle (e.g., angles \(30^\circ, 20^\circ, 130^\circ\)), the sum of the two smaller angles (\(30^\circ + 20^\circ = 50^\circ\)) is less than the largest angle (\(130^\circ\)).
The question asks why the innermost part of a flame appears black. This relates to the combustion process and the availability of oxygen in different regions of the flame.
B) Because of the presence of un-burnt carbon particles. This is the direct reason for the black appearance of the innermost part of the flame, resulting from incomplete or no combustion due to insufficient oxygen.
If Raju's earning is 40% less than Venkat's earning, by how much percentage is Venkat's earning more than Raju's earning?
200%
100%
\( \frac{200}{3} \)%
\( \frac{400}{3} \)%
This problem involves percentage calculations. We are given that Raju's earning is 40% less than Venkat's earning. We need to find by what percentage Venkat's earning is more than Raju's earning. We will assume a base value for Venkat's earning to make calculations easier.
Correct Option: C) \( \frac{200}{3} \)%
The problem asks us to find what percentage of 'a' is 'b', given a relationship between 60% of \((a-b)\) and 40% of \((a+b)\). We will set up an equation based on the given information, simplify it to find a relationship between 'a' and 'b', and then express 'b' as a percentage of 'a'.
Formulate the equation from the given information:
The problem states that 60% of \((a - b)\) is the same as 40% of \((a + b)\). We can write this mathematically as:
Simplify the equation:
We can multiply both sides by 100 to remove the decimals, or simply work with the decimals:
Rearrange the terms to group 'a' and 'b':
Subtract \(0.4a\) from both sides and add \(0.6b\) to both sides:
Express 'b' as a percentage of 'a':
We have the relationship \(b = 0.2a\). To express 'b' as a percentage of 'a', we multiply the decimal by 100:
Thus, b is 20% of a.
A) 20 — As calculated, b is 20% of a.
The question asks to identify a famous Kathak dancer from the given options. This requires knowledge of prominent personalities in Indian classical dance forms.
Correct Option: D) Birju Maharaj is a legendary Kathak dancer, choreographer, and guru. He was a leading exponent of the Lucknow Kalka-Bindadin gharana of Kathak. His contributions to Kathak are immense, and he is widely recognized as one of the greatest Kathak dancers of all time.
The question gives the speeds at which Vinod travels from his home to his office and returns home, along with the total journey time. Since the distance travelled in both directions is equal, we can use the relationship between speed, distance and time to calculate the distance between his home and office.
Identify the given information:
Recall the relationship between speed, distance and time:
\[ \text{Time}=\frac{\text{Distance}}{\text{Speed}} \]Let the distance between Vinod's home and office be x km.
Since he travels to the office and returns along the same distance, the distance for each part of the journey is x km.
Calculate the time taken for each journey:
Time taken from home to office:
\[ t_1=\frac{x}{20} \]Time taken from office to home:
\[ t_2=\frac{x}{5} \]The total journey takes 4 hours.
Therefore:
\[ \frac{x}{20}+\frac{x}{5}=4 \]Solve the equation:
Taking the LCM of 20 and 5:
\[ \frac{x+4x}{20}=4 \] \[ \frac{5x}{20}=4 \]Simplifying:
\[ \frac{x}{4}=4 \]Multiplying both sides by 4:
\[ x=16 \]Determine the required distance:
The distance between Vinod's home and office is 16 km.
Correct Option: C) 16 km — Vinod travels to his office at 20 km/h and returns at 5 km/h. Since the total journey takes 4 hours, the distance x satisfies:
\[ \frac{x}{20}+\frac{x}{5}=4 \] \[ \frac{5x}{20}=4 \] \[ x=16\text{ km} \]
Hence, Option C is the correct answer.
A) 10 km: This is incorrect because travelling 10 km at 20 km/h and returning at 5 km/h would take 2.5 hours, not 4 hours.
B) 6 km: This is incorrect because travelling 6 km in each direction at the given speeds would take only 1.5 hours.
D) 8 km: This is incorrect because travelling 8 km at 20 km/h and returning at 5 km/h would take 2 hours, not 4 hours.
The question asks to identify the option that is different from the rest. This is a classification question where we need to find a common characteristic among three options and identify the one that does not share that characteristic.
C) American football — This ball is oval-shaped (prolate spheroid), unlike the other options which are spherical.
The question asks for the year in which The Right of Children to Free and Compulsory Education Act was introduced in India. This requires knowledge of key legislative milestones related to education in India.
A) 2009 is the correct answer. The Right of Children to Free and Compulsory Education Act was introduced (passed by Parliament) in 2009. It subsequently came into force on April 1, 2010.
The problem can be solved by translating the given word problem into a mathematical equation. We need to find the value of 'x' based on the relationship described: "60% of x added to 60, then the result is x + 20."
Translate the percentage into a decimal or fraction:
60% can be written as \(\frac{60}{100}\) or \(0.6\).
Formulate the expression for "60% of x":
This is \(\frac{60}{100} \times x\) or \(0.6x\).
Formulate the expression for "60% of x added to 60":
This means \(0.6x + 60\).
Set up the equation based on the given condition:
The problem states that "the result is x + 20". So, we set the expression from step 3 equal to \(x + 20\):
\[ 0.6x + 60 = x + 20 \]Solve the equation for x:
To solve for \(x\), we need to gather all terms involving \(x\) on one side and constant terms on the other side. Subtract \(0.6x\) from both sides:
\[ 60 = x - 0.6x + 20 \]\[ 60 = 0.4x + 20 \]Subtract \(20\) from both sides:
\[ 60 - 20 = 0.4x \]\[ 40 = 0.4x \]Divide both sides by \(0.4\):
\[ x = \frac{40}{0.4} \]\[ x = \frac{40}{\frac{4}{10}} \]\[ x = 40 \times \frac{10}{4} \]\[ x = 10 \times 10 \]\[ x = 100 \]A) 100 is the correct value for x.
We are given the perimeter and area of a right-angled triangle. We need to find the length of its hypotenuse. We can use the formulas for the perimeter and area of a right-angled triangle, along with the Pythagorean theorem, to set up equations and solve for the sides of the triangle.
Let the two perpendicular sides of the right-angled triangle be \(a\) and \(b\), and the hypotenuse be \(c\).
The perimeter of the triangle is given by \(P = a + b + c\). We are given \(P = 56\) cm.
\[ a + b + c = 56 \quad \text{(Equation 1)} \]The area of the triangle is given by \(A = \frac{1}{2}ab\). We are given \(A = 84\) cm².
\[ \frac{1}{2}ab = 84 \]\[ ab = 168 \quad \text{(Equation 2)} \]According to the Pythagorean theorem, for a right-angled triangle:
\[ a^2 + b^2 = c^2 \quad \text{(Equation 3)} \]From Equation 1, we can express \(a+b\) in terms of \(c\):
\[ a + b = 56 - c \]Square both sides of this equation:
\[ (a + b)^2 = (56 - c)^2 \]\[ a^2 + b^2 + 2ab = (56 - c)^2 \]Substitute \(a^2 + b^2 = c^2\) (from Equation 3) and \(ab = 168\) (from Equation 2) into the squared equation:
\[ c^2 + 2(168) = (56 - c)^2 \]\[ c^2 + 336 = 56^2 - 2(56)c + c^2 \]\[ c^2 + 336 = 3136 - 112c + c^2 \]Subtract \(c^2\) from both sides:
\[ 336 = 3136 - 112c \]Rearrange the equation to solve for \(c\):
\[ 112c = 3136 - 336 \]\[ 112c = 2800 \]\[ c = \frac{2800}{112} \]\[ c = 25 \]Thus, the length of the hypotenuse is 25 cm.
A) 25 cm — The calculations show that the length of the hypotenuse is 25 cm.