The question asks to identify the organ responsible for secreting bile. This requires knowledge of the functions of various organs in the digestive system.
Correct Option: B) The liver is the organ responsible for secreting bile. It produces bile continuously, which is then transported to the gall bladder for storage and concentration, or directly to the duodenum.
The question asks for the chemical formula of lime water. To answer this, we need to recall the common names and chemical formulas of calcium compounds.
Correct Option: C) Ca(OH)₂ is the chemical formula for calcium hydroxide, which is commonly known as lime water.
Evaluate: cos 20° cos 40° cos 80°
\( \frac{1}{2} \)
\( \frac{1}{32} \)
\( \frac{1}{8} \)
\( \frac{1}{4} \)
To evaluate the given trigonometric expression, we will use the trigonometric identity for the product of cosines: \( \cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)] \). Alternatively, we can use a specific identity for products of the form \( \cos \theta \cos 2\theta \cos 4\theta \dots \cos (2^{n-1}\theta) \).
The given expression is: \( \cos 20^\circ \cos 40^\circ \cos 80^\circ \).
We can use the general identity for the product of cosines: \[\cos \theta \cos 2\theta \cos 4\theta = \frac{\sin (2^3 \theta)}{2^3 \sin \theta}\]. In this case, \( \theta = 20^\circ \) and \( n=3 \) (since we have \( 2^0\theta, 2^1\theta, 2^2\theta \)).
Substitute \( \theta = 20^\circ \) into the identity:
\[ \cos 20^\circ \cos (2 \cdot 20^\circ) \cos (4 \cdot 20^\circ) = \frac{\sin (2^3 \cdot 20^\circ)}{2^3 \sin 20^\circ} \]\[ = \frac{\sin (8 \cdot 20^\circ)}{8 \sin 20^\circ} \]\[ = \frac{\sin 160^\circ}{8 \sin 20^\circ} \]We know that \( \sin (180^\circ - x) = \sin x \). Therefore, \( \sin 160^\circ = \sin (180^\circ - 20^\circ) = \sin 20^\circ \).
Substitute this back into the expression:
\[ = \frac{\sin 20^\circ}{8 \sin 20^\circ} \]Cancel out \( \sin 20^\circ \) from the numerator and denominator (since \( \sin 20^\circ \neq 0 \)):
\[ = \frac{1}{8} \]C) The value of the expression \( \cos 20^\circ \cos 40^\circ \cos 80^\circ \) is \( \frac{1}{8} \).
On which date is the Constitution Day of India celebrated?
The question asks for the date on which Constitution Day of India is celebrated. This requires knowledge of important dates in Indian constitutional history.
Correct Option: D) 26 th November
The problem provides the sum of squares of the first 14 natural numbers and asks for the sum of squares of a different series. We can observe a pattern in the second series that relates it to the first series. By factoring out common terms, we can express the second sum in terms of the given sum.
We are given the sum of squares of the first 14 natural numbers:
\[ 1^{2} + 2^{2} + 3^{2} + \dots + 14^{2} = 1015 \]We need to find the value of:
\[ 3^{2} + 6^{2} + 9^{2} + \dots + 42^{2} \]Notice that each term in the second series is a multiple of 3. We can rewrite each term as follows:
So, the series can be written as:
\[ (3 \times 1)^{2} + (3 \times 2)^{2} + (3 \times 3)^{2} + \dots + (3 \times 14)^{2} \]Using this rule, we can write each term as:
\[ 3^{2} \times 1^{2} + 3^{2} \times 2^{2} + 3^{2} \times 3^{2} + \dots + 3^{2} \times 14^{2} \]We can take \(3^2\) common from all terms:
\[ 3^{2} (1^{2} + 2^{2} + 3^{2} + \dots + 14^{2}) \]We know that \( 1^{2} + 2^{2} + 3^{2} + \dots + 14^{2} = 1015 \). Substitute this into the expression:
\[ 3^{2} \times (1015) \]First, calculate \(3^2 = 9\). Then multiply by 1015:
\[ 9 \times 1015 = 9135 \]Thus, the value of the given series is 9135.
A) 9135 is the correct answer, as calculated above.
A, B and C can complete a job in 5 days. B, C and D can complete the same job in 10 days. C, D and A can complete the same job in 15 days and D, A and B can complete the same job in 30 days. How long A, B, C and D together will take to finish the job?
8 days
8 \( \frac{1}{2} \) days
7 \( \frac{1}{2} \) days
7 days
This problem involves work and time. We can solve it by calculating the individual work rates of the groups and then summing them up to find the combined work rate of A, B, C, and D. The total work is considered as 1 unit. The work rate is the reciprocal of the time taken to complete the job.
Adding the work done by all groups in 1 day:
\[ \left( \frac{1}{5} + \frac{1}{10} + \frac{1}{15} + \frac{1}{30} \right) \]The least common multiple (LCM) of 5, 10, 15, and 30 is 30.
\[ \frac{6}{30} + \frac{3}{30} + \frac{2}{30} + \frac{1}{30} = \frac{6+3+2+1}{30} = \frac{12}{30} = \frac{2}{5} \]The sum \(\frac{2}{5}\) represents the work done in 1 day by (A+B+C) + (B+C+D) + (C+D+A) + (D+A+B). Notice that each person (A, B, C, D) appears 3 times in this sum.
So, \[3 \times (\text{Work done by A+B+C+D in 1 day}) = \frac{2}{5}\]
Divide the total sum by 3:
\[ \text{Work done by (A+B+C+D) in 1 day} = \frac{1}{3} \times \frac{2}{5} = \frac{2}{15} \]Time is the reciprocal of the work rate:
\[ \text{Time} = \frac{1}{\text{Work done in 1 day}} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ days} \]C) 7 \( \frac{1}{2} \) days. This matches our calculated time of 7.5 days.
The question asks about the focus theme of World Environment Day in 2018. To answer this, one needs to recall the specific theme chosen for that year's observance.
Correct Option: B) Plastic pollution
The question asks to identify the African country that first launched the Malaria vaccine for children on World Malaria Day. This requires knowledge of recent global health initiatives and specific country-level actions regarding malaria prevention.
Correct Option: D) Malawi
To simplify the given expression, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This stands for Brackets/Parentheses, Orders/Exponents, Division, Multiplication, Addition, and Subtraction. Operations are performed from left to right for Division and Multiplication, and then for Addition and Subtraction.
The simplified value of the expression is \(23\).
Correct Option: A) 23 is the correct simplification of the given expression following the order of operations.
The question asks about the founder of the Indian National Congress. This requires knowledge of key figures involved in the formation of this important political organization in Indian history.
Correct Option: B) A.O Hume was a retired British Indian Civil Service officer who played a pivotal role in the formation of the Indian National Congress in 1885. He is widely regarded as its founder.