To find the tallest person, we need to establish the height relationships between all the individuals (P, Q, R, S) based on the given statements. We will represent "taller than" with the symbol \(>\) and combine the inequalities to form a complete order.
Correct Option: D) R is the tallest person as per the derived order of heights.
To find the Highest Common Factor (HCF) of two numbers expressed in their prime factorization form, we need to identify the common prime factors and take the lowest power of each common prime factor present in both numbers.
First number: \( N_1 = 2^{4} \times 3^{4} \times 5^{3} \times 7^{2} \)
Second number: \( N_2 = 2^{2} \times 3^{6} \times 5^{5} \)
The prime factors common to both numbers are 2, 3, and 5. The prime factor 7 is only present in \(N_1\), so it is not a common factor.
The HCF is the product of these lowest powers:
\[ \text{HCF} = 2^{2} \times 3^{4} \times 5^{3} \]D) \( 2^{2} \times 3^{4} \times 5^{3} \)
If sinθ − cosθ = 0, (angle in first quadrant) then the value of \( \sin ^{3}\theta + 3\cos ^{3}\theta \) is:
\(\frac{1}{\sqrt{2}}\)
2
\(\sqrt{2}\)
2\(\sqrt{2}\)
The problem provides an equation involving sine and cosine functions and asks for the value of an expression involving their cubes. We will first solve the given equation to find the value of \(\theta\), and then substitute this value into the expression to find its result.
Given the equation: \(\sin \theta - \cos \theta = 0\)
Rearrange the equation:
\[ \sin \theta = \cos \theta \]Divide both sides by \(\cos \theta\) (assuming \(\cos \theta \neq 0\), which is true for the first quadrant):
\[ \frac{\sin \theta}{\cos \theta} = 1 \] \[ \tan \theta = 1 \]Since \(\theta\) is in the first quadrant, the value of \(\theta\) for which \(\tan \theta = 1\) is \(\theta = 45^{\circ}\) or \(\theta = \frac{\pi}{4}\) radians.
Now, we need to find the value of the expression \(\sin^3 \theta + 3\cos^3 \theta\).
Substitute \(\theta = 45^{\circ}\) into the expression:
We know that \(\sin 45^{\circ} = \frac{1}{\sqrt{2}}\) and \(\cos 45^{\circ} = \frac{1}{\sqrt{2}}\).
Calculate \(\sin^3 \theta\):
\[ \sin^3 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^3 = \frac{1^3}{(\sqrt{2})^3} = \frac{1}{2\sqrt{2}} \]Calculate \(\cos^3 \theta\):
\[ \cos^3 45^{\circ} = \left(\frac{1}{\sqrt{2}}\right)^3 = \frac{1^3}{(\sqrt{2})^3} = \frac{1}{2\sqrt{2}} \]Substitute these values back into the expression:
\[ \sin^3 \theta + 3\cos^3 \theta = \frac{1}{2\sqrt{2}} + 3\left(\frac{1}{2\sqrt{2}}\right) \] \[ = \frac{1}{2\sqrt{2}} + \frac{3}{2\sqrt{2}} \] \[ = \frac{1+3}{2\sqrt{2}} \] \[ = \frac{4}{2\sqrt{2}} \] \[ = \frac{2}{\sqrt{2}} \]Rationalize the denominator:
\[ = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \] \[ = \frac{2\sqrt{2}}{2} \] \[ = \sqrt{2} \]Correct Option: C) \(\sqrt{2}\)
The question asks to find the relationship between the first pair of words and apply the same relationship to the third word to find the fourth word. The relationship is based on classification or category.
A) Vegetable — Kiwi is a fruit. Similarly, Bottle Gourd is commonly classified and consumed as a vegetable.
The question asks to identify the Indian state that has the giant squirrel as its state animal. This is a factual question related to the state symbols of various Indian states, a common topic in general knowledge sections of competitive exams.
Correct Option: B) Maharashtra
The Indian Giant Squirrel (Ratufa indica) is the state animal of Maharashtra. It is locally known as "Shekru" and is found in deciduous and evergreen forests of the state.
The question asks to identify the first reserve forest of India. This is a factual question related to Indian environmental history and protected areas. Recalling key historical facts about India's forest conservation efforts will lead to the correct answer.
C) Satpura National Park is indeed the first reserve forest of India, declared in 1865. It is located in the Hoshangabad district of Madhya Pradesh.
Simplify.
\(\sqrt{56+\sqrt{185+88\sqrt{\sqrt{36}+19}}}\)
To simplify the given expression, we will work from the innermost square root outwards. We will evaluate each square root step by step until the entire expression is simplified to a single value.
First, simplify the innermost part of the expression: \(\sqrt{36}\).
\[ \sqrt{36} = 6 \]Substitute this value back into the expression:
\[ \sqrt{56+\sqrt{185+88\sqrt{6+19}}} \]Simplify the term inside the next square root: \(6+19\).
\[ 6+19 = 25 \]Substitute this value back:
\[ \sqrt{56+\sqrt{185+88\sqrt{25}}} \]Evaluate \(\sqrt{25}\):
\[ \sqrt{25} = 5 \]Substitute this value back:
\[ \sqrt{56+\sqrt{185+88 \times 5}} \]Perform the multiplication: \(88 \times 5\).
\[ 88 \times 5 = 440 \]Substitute this value back:
\[ \sqrt{56+\sqrt{185+440}} \]Perform the addition inside the square root: \(185+440\).
\[ 185+440 = 625 \]Substitute this value back:
\[ \sqrt{56+\sqrt{625}} \]Evaluate \(\sqrt{625}\). We know that \(25^2 = 625\).
\[ \sqrt{625} = 25 \]Substitute this value back:
\[ \sqrt{56+25} \]Perform the final addition: \(56+25\).
\[ 56+25 = 81 \]Evaluate the final square root: \(\sqrt{81}\).
\[ \sqrt{81} = 9 \]Thus, the simplified value of the expression is 9.
C) 9 is the correct answer as derived from the step-by-step simplification.
To solve this problem, we need to divide the total mass of the cake by the number of boys. The total mass is given in kilograms (kg), and the options are in grams (g). Therefore, the first step is to convert the total mass of the cake from kilograms to grams. Then, divide this total mass in grams by the number of boys to find out how much cake each boy receives.
The total mass of the cake is 1.5 kg.
We know that \(1 \text{ kg} = 1000 \text{ g}\).
So, to convert 1.5 kg to grams:
\[ 1.5 \text{ kg} = 1.5 \times 1000 \text{ g} = 1500 \text{ g} \]The total mass of the cake in grams is 1500 g.
The cake is divided equally among 10 boys.
Amount of cake each boy gets is:
\[ \text{Amount per boy} = \frac{\text{Total mass of cake}}{\text{Number of boys}} \]\[ \text{Amount per boy} = \frac{1500 \text{ g}}{10} \]\[ \text{Amount per boy} = 150 \text{ g} \]Therefore, each boy will get 150 g of cake.
Correct Option: B) 150 g
The question asks for the year in which the Pulwama terror attack occurred. This is a factual recall question related to significant events in recent Indian history, often covered in current affairs sections of competitive exams.
Correct Option: B) 2019
The Pulwama terror attack occurred on February 14, 2019, when a suicide bomber of the Jaish-e-Mohammed (JeM) terrorist group attacked a convoy of security personnel on the Jammu-Srinagar National Highway in Pulwama district, Jammu and Kashmir. This tragic incident resulted in the deaths of 40 Central Reserve Police Force (CRPF) personnel.
The question gives the ratio of two numbers as 3:2. After subtracting 8 from the first number and 6 from the second number, their ratio becomes 5:4. We need to represent the original numbers using a common multiplier and form an equation to determine their values.
Identify the given information:
Represent the original numbers:
Let the two numbers be 3x and 2x, since their ratio is 3:2.
\[ \text{First Number}=3x \] \[ \text{Second Number}=2x \]Form an equation using the new ratio:
After subtracting 8 from the first number and 6 from the second number, the ratio becomes 5:4.
\[ \frac{3x-8}{2x-6}=\frac{5}{4} \]Cross-multiplying:
\[ 4(3x-8)=5(2x-6) \]Solve the equation:
Expanding both sides:
\[ 12x-32=10x-30 \]Rearranging:
\[ 12x-10x=32-30 \] \[ 2x=2 \] \[ x=1 \]Calculate the original numbers:
Substituting x = 1:
\[ \text{First Number}=3\times1=3 \] \[ \text{Second Number}=2\times1=2 \]Therefore, the original numbers are 3 and 2.
Check the resulting ratio:
After subtracting the given values:
\[ 3-8=-5 \] \[ 2-6=-4 \]The resulting ratio is:
\[ -5:-4=5:4 \]Therefore, the original numbers are 3 and 2.
Correct Option: C) 3, 2 — Let the original numbers be 3x and 2x. Using the given condition:
\[ \frac{3x-8}{2x-6}=\frac{5}{4} \] \[ 12x-32=10x-30 \] \[ x=1 \]
Hence, the original numbers are 3 and 2. Option C is the correct answer.
A) 2, 3: This is incorrect because the original ratio of 2 and 3 is 2:3, not 3:2.
B) 24, 16: This is incorrect because subtracting 8 and 6 gives 16 and 10, whose ratio is 8:5, not 5:4.
D) 16, 24: This is incorrect because the original ratio of 16 and 24 is 2:3, not 3:2.