A certain number of people are sitting in a straight row, facing north. B is fourth to the left of G. F is fourth to the right of C and second to the left of K, who is fifth to the right of A. D is not an immediate neighbour of either K or B. There are only three persons between J and A. J and K are sitting at extreme ends. G is second to the right of C. If all the positions between J and K are occupied, what is the total number of persons seated in the row?
The problem describes the relative positions of several people sitting in a straight row. We need to deduce the arrangement of all individuals based on the given clues and then count the total number of people. We will start by fixing the positions of individuals based on the most definitive clues and then fill in the gaps.
J _ _ _ A _ _ _ _ K
Let's count the positions: J (1) + 3 gaps + A (1) + 4 gaps + K (1) = 1 + 3 + 1 + 4 + 1 = 10 positions so far.
So the arrangement looks like:
J _ _ _ A _ _ _ _ K
From "F is second to the left of K", we have F _ K. This means F is immediately to the left of the position before K.
From our current arrangement: J _ _ _ A _ _ _ F K (since F is 2nd to the left of K, there is one person between F and K).
Now, "F is fourth to the right of C". This means C _ _ _ F.
Let's place C. If F is at position 8 (from the left, assuming J is 1), then C would be at position 4. Let's re-evaluate the numbering.
Current known positions:
1 2 3 4 5 6 7 8 9 10
J _ _ _ A _ _ _ F K
So, K is at position 10. F is at position 8 (second to the left of K).
Now, C is such that F is fourth to its right. So, C is at position 4 (8 - 4 = 4).
Revised arrangement:
1 2 3 4 5 6 7 8 9 10
J _ _ C A _ _ F _ K
C is at position 4. So G is at position 6 (4 + 2 = 6).
Revised arrangement:
1 2 3 4 5 6 7 8 9 10
J _ _ C A G _ F _ K
G is at position 6. So B is at position 2 (6 - 4 = 2).
Revised arrangement:
1 2 3 4 5 6 7 8 9 10
J B _ C A G _ F _ K
K is at position 10. B is at position 2.
The empty positions are 3, 7, 9.
D cannot be at position 9 (neighbour of K). D cannot be at position 3 (neighbour of B).
Therefore, D must be at position 7.
Final arrangement:
1 2 3 4 5 6 7 8 9 10
J B _ C A G D F _ K
Let's re-construct step-by-step with a visual representation of slots.
1. J and K are at extreme ends. J is at left end, K is at right end (from A_ _ _ _ K).
J _ _ _ _ _ _ _ _ K (10 positions minimum for A_ _ _ _ K and J _ _ _ A)
2. K is fifth to the right of A: A _ _ _ _ K
3. Three persons between J and A: J _ _ _ A
Combining these, we get: J _ _ _ A _ _ _ _ K. This accounts for 10 positions.
4. F is second to the left of K: F _ K. So, the arrangement becomes J _ _ _ A _ _ _ F K.
5. F is fourth to the right of C: C _ _ _ F. Placing C relative to F in the current arrangement: J _ _ C A _ _ F K.
6. G is second to the right of C: C _ G. Placing G relative to C: J _ _ C A G _ F K.
7. B is fourth to the left of G: B _ _ _ G. Placing B relative to G: J B _ C A G _ F K.
8. D is not an immediate neighbour of either K or B.
Current arrangement with known persons:
J B _ C A G _ F _ K
The empty slots are at position 3, 7, and 9 (if we assume J is 1st and K is 10th).
D cannot be at position 9 (neighbour of K). D cannot be at position 3 (neighbour of B).
So D must be at position 7.
Arrangement:
J B _ C A G D F _ K
The problem states: "If all the positions between J and K are occupied". This means there are no empty slots between J and K. The two remaining empty slots (position 3 and position 9) must be filled by two more people, let's call them P1 and P2.
So the complete arrangement is:
J B P1 C A G D F P2 K
Counting the total number of persons: 10.
A) Ten. Based on the step-by-step deduction, the final arrangement has 10 people.
We need to find Rajani's present age. We are given relationships between the ages of Rajani, her father, and her sister Lavanya, and the sum of their ages two years from now. We will set up equations based on these relationships and solve for Rajani's present age.
Let Rajani's present age be \(R\) years.
According to the problem, Rajani's father is three times older than Rajani. So, the father's present age is \(3R\) years.
Rajani is twice as old as her sister Lavanya. So, Lavanya's present age is \(\frac{R}{2}\) years.
Two years from now:
The sum of their ages two years from now will be 60 years. So, we can write the equation:
\[ (R + 2) + (3R + 2) + \left(\frac{R}{2} + 2\right) = 60 \]Now, let's solve this equation for \(R\):
\[ R + 3R + \frac{R}{2} + 2 + 2 + 2 = 60 \]\[ 4R + \frac{R}{2} + 6 = 60 \]\[ 4R + \frac{R}{2} = 60 - 6 \]\[ 4R + \frac{R}{2} = 54 \]To combine the terms with \(R\), find a common denominator:
\[ \frac{8R}{2} + \frac{R}{2} = 54 \]\[ \frac{9R}{2} = 54 \]Multiply both sides by 2:
\[ 9R = 54 \times 2 \]\[ 9R = 108 \]Divide by 9 to find \(R\):
\[ R = \frac{108}{9} \]\[ R = 12 \]Therefore, Rajani's present age is 12 years.
A) 12 years is the correct answer, as calculated above.
The marked price of a cooker is the same at four shops I, II, III and IV. Shop I allows two successive discounts of 20% and 15%; shop II allows successive discounts of 18% and 17%; shop III allows successive discounts of 25% and 10%; and shop IV allows successive discounts of 15%, 15%, and 5% on the marked price of the cooker. Which shop is selling the cooker at the lowest price?
To find which shop sells the cooker at the lowest price, we need to calculate the effective discount percentage for each shop. The shop offering the highest effective discount will have the lowest selling price. We can use the formula for successive discounts or calculate the final price by applying each discount sequentially.
D) III — Shop III offers the highest effective discount of \(32.5\%\), resulting in the lowest selling price.
The question asks to identify the economic concept that measures the responsiveness of the demand for one good (Tea) to a change in the price of another good (Coffee). This directly points to a specific type of elasticity of demand that deals with related goods.
B) Cross Elasticity of Demand — This measures how the quantity demanded of one good (Tea) responds to a change in the price of another good (Coffee). For substitute goods like Tea and Coffee, the cross elasticity of demand is positive, meaning an increase in the price of Coffee leads to an increase in the demand for Tea.
Consider the following statements about magnetic field of a bar magnet:
(a) It has magnitudes and directions both.
(b) Magnetic field lines are closed curves.
Which of the above statements is/are correct?
Only (b)
Both (a) and (b)
Only (a)
Neither (a) nor (b)
The question asks us to evaluate two statements regarding the properties of a magnetic field produced by a bar magnet. We need to recall the fundamental characteristics of magnetic fields and magnetic field lines.
Analyze statement (a): "It has magnitudes and directions both."
Analyze statement (b): "Magnetic field lines are closed curves."
Conclusion: Since both statements (a) and (b) are correct, the option that includes both is the right answer.
Correct Option: B) Both (a) and (b)
The question asks to identify the freedom fighter whose first definitive writing, 'Hind Swaraj', continues to generate critical interest globally. This requires knowledge of prominent Indian freedom fighters and their key literary works.
Correct Option: A) Mahatma Gandhi. 'Hind Swaraj' is Mahatma Gandhi's seminal work, written in 1909, which outlines his philosophy on self-rule, critique of modern civilization, and non-violence. It continues to be studied and debated worldwide.
The question asks to identify the thread-like structures that constitute the body (mycelium) of a fungus. This requires knowledge of basic fungal anatomy.
Correct Option: B) Hyphae are the thread-like structures that form the body, or mycelium, of a fungus. They are the fundamental structural units of most fungi.
Simplify the given expression.
\(\frac{3-5x^2}{1-x^3}+\frac{1}{1-x}-\frac{5x+4}{1+x+x^2}\)
\(\frac{x}{1-x^3}\)
\(\frac{2x^2}{1-x^3}\)
\(\frac{x^2}{1-x^3}\)
\(\frac{2x}{1-x^3}\)
To simplify the given expression, we need to combine the fractions. The key is to find a common denominator, which is \(1-x^3\). We will use the identity \(1-x^3 = (1-x)(1+x+x^2)\) to rewrite the terms with a common denominator and then combine them.
Identify the denominators in the given expression:
\[ \frac{3-5x^2}{1-x^3}+\frac{1}{1-x}-\frac{5x+4}{1+x+x^2} \]Recall the algebraic identity for the difference of cubes: \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\). Applying this, we have \(1-x^3 = (1-x)(1+x+x^2)\). This is the least common multiple (LCM) of the denominators.
Rewrite each term with the common denominator \(1-x^3\):
Now substitute these back into the original expression:
\[ \frac{3-5x^2}{1-x^3} + \frac{1+x+x^2}{1-x^3} - \frac{-5x^2+x+4}{1-x^3} \]Combine the numerators over the common denominator:
\[ \frac{(3-5x^2) + (1+x+x^2) - (-5x^2+x+4)}{1-x^3} \]Simplify the numerator:
\[ \frac{3-5x^2+1+x+x^2+5x^2-x-4}{1-x^3} \]Combine like terms in the numerator:
So the simplified numerator is \(x^2\).
The final simplified expression is:
\[ \frac{x^2}{1-x^3} \]C) \(\frac{x^2}{1-x^3}\) is the correct simplified expression obtained after combining the fractions with a common denominator and simplifying the numerator.
The question describes a compound X with three key properties: its use in glass and soap industries, the presence of ten molecules of water of crystallization, and a basic pH (>7) in solution. We need to identify the compound that matches all these characteristics.
Correct Option: D) Washing soda
Simplify the given expression.
\(\left(\frac{8}{125}\right)^{\frac{2}{3}}\times\left(\frac{27}{64}\right)^{-\frac{2}{3}}\div\left(\frac{5}{4}\right)^{-2}\)
\(\frac{4}{9}\)
\(\frac{2}{3}\)
\(\frac{3}{5}\)
\(\frac{3}{8}\)
To simplify the given expression, we will use the properties of exponents. Specifically, we will use the rules \((a^m)^n = a^{mn}\), \(a^{-n} = \frac{1}{a^n}\), and \[(\frac{a}{b})^n = \frac{a^n}{b^n}\]. We will first express the bases as powers of prime numbers, then apply the exponent rules, and finally perform the multiplication and division.
The given expression is:
\[ \left(\frac{8}{125}\right)^{\frac{2}{3}}\times\left(\frac{27}{64}\right)^{-\frac{2}{3}}\div\left(\frac{5}{4}\right)^{-2} \]We can rewrite the bases as powers:
Substitute these into the expression:
\[ \left(\frac{2^3}{5^3}\right)^{\frac{2}{3}}\times\left(\frac{3^3}{4^3}\right)^{-\frac{2}{3}}\div\left(\frac{5}{4}\right)^{-2} \] \[ \left(\left(\frac{2}{5}\right)^3\right)^{\frac{2}{3}}\times\left(\left(\frac{3}{4}\right)^3\right)^{-\frac{2}{3}}\div\left(\frac{5}{4}\right)^{-2} \]For the first term:
\[ \left(\left(\frac{2}{5}\right)^3\right)^{\frac{2}{3}} = \left(\frac{2}{5}\right)^{3 \times \frac{2}{3}} = \left(\frac{2}{5}\right)^2 = \frac{2^2}{5^2} = \frac{4}{25} \]For the second term:
\[ \left(\left(\frac{3}{4}\right)^3\right)^{-\frac{2}{3}} = \left(\frac{3}{4}\right)^{3 \times \left(-\frac{2}{3}\right)} = \left(\frac{3}{4}\right)^{-2} \]Using the rule \(a^{-n} = \frac{1}{a^n}\) or \[(\frac{a}{b})^{-n} = (\frac{b}{a})^n\]:
\[ \left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9} \]For the third term:
\[ \left(\frac{5}{4}\right)^{-2} = \left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25} \]Division by a fraction is equivalent to multiplication by its reciprocal:
\[ \frac{4}{25} \times \frac{16}{9} \times \frac{25}{16} \]Now, cancel out common terms:
\[ \frac{4}{\cancel{25}} \times \frac{\cancel{16}}{9} \times \frac{\cancel{25}}{\cancel{16}} = \frac{4}{9} \]Correct Option: A) \(\frac{4}{9}\)