The problem requires simplifying an algebraic expression. We will use the algebraic identity \((x-y)(x+y) = x^2 - y^2\) to simplify the product of the first two terms, and then combine it with the third term.
Identify the algebraic identity to be used. The first part of the expression is \((1.5a^2 - 2.5b^2)(1.5a^2 + 2.5b^2)\). This matches the form \((x-y)(x+y)\), where \(x = 1.5a^2\) and \(y = 2.5b^2\).
Apply the identity \((x-y)(x+y) = x^2 - y^2\):
\[ (1.5a^2 - 2.5b^2)(1.5a^2 + 2.5b^2) = (1.5a^2)^2 - (2.5b^2)^2 \]Calculate the squares:
So, the product simplifies to \(2.25a^4 - 6.25b^4\).
Now, substitute this back into the original expression:
\[ (1.5a^2 - 2.5b^2)(1.5a^2 + 2.5b^2) + 6.25ab \]\[ = 2.25a^4 - 6.25b^4 + 6.25ab \]Since there are no like terms to combine, this is the simplified form of the expression.
Correct Option: B) \(2.25a^4 - 6.25b^4 + 6.25ab\)
To find the average earning per vehicle, we need to calculate the total earnings from all vehicles and then divide it by the total number of vehicles. The problem states that the average occupancy is 100%, meaning all seats in all vehicles are filled.
Correct Option: C) Rs 115
The question asks about the number of groups and periods in the Modern Periodic Table. This is a fundamental concept in chemistry, and the structure of the periodic table is standardized globally.
Correct Option: B) The Modern Periodic Table has 18 groups and 7 periods.
The problem requires us to first modify the given series by removing all numbers. Then, we need to identify two specific letters based on their positions from the left and right ends of the modified series. Finally, we count the number of letters in the English alphabet that fall between these two identified letters.
Correct Option: A) One
Consider the below statements and identify the correct answer.
Statement I: The chemical formula for baking soda is CaSO₄ \( \frac{1}{2} \)H₂O.
Statement II: Baking soda is used for making toys, materials for decoration, and for making surfaces smooth
Both the statements are true.
Statement II is true, and Statement I is false.
Both the statements are false
Statement I is true, and Statement II is false.
The question asks us to identify the correct statement(s) regarding baking soda. We need to evaluate the chemical formula given in Statement I and the uses mentioned in Statement II for baking soda.
C) Both the statements are false. Both statements contain incorrect information regarding baking soda. Statement I incorrectly identifies the chemical formula of baking soda as that of Plaster of Paris, and Statement II incorrectly attributes the uses of Plaster of Paris to baking soda.
To determine the nature and size of the image formed by a convex mirror, we will use the mirror formula and the magnification formula. We need to correctly apply the sign conventions for the given values.
The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]Substitute the known values:
\[ \frac{1}{+6.0} = \frac{1}{v} + \frac{1}{-20.0} \]\[ \frac{1}{v} = \frac{1}{6.0} - \frac{1}{-20.0} \]\[ \frac{1}{v} = \frac{1}{6.0} + \frac{1}{20.0} \]Find a common denominator (60):
\[ \frac{1}{v} = \frac{10}{60} + \frac{3}{60} \]\[ \frac{1}{v} = \frac{13}{60} \]\[ v = \frac{60}{13} \approx +4.62 \text{ cm} \]Since \(v\) is positive, the image is formed behind the mirror, which means it is a virtual image.
The magnification formula is:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]Substitute the values of \(v\), \(u\), and \(h_o\):
\[ \frac{h_i}{3.0} = -\frac{+4.62}{-20.0} \]\[ \frac{h_i}{3.0} = +\frac{4.62}{20.0} \]\[ h_i = 3.0 \times \frac{4.62}{20.0} \]\[ h_i = 3.0 \times 0.231 \]\[ h_i = +0.693 \text{ cm} \]Since \(h_i\) is positive, the image is erect. The magnitude of \(h_i\) is \(0.693 \text{ cm}\), which is less than the object height of \(3.0 \text{ cm}\). This means the image is diminished.
Based on the calculations:
Therefore, the image is less than 3.0 cm in height and erect.
A) Less than 3.0 cm, erect. This matches our findings that the image height is approximately 0.693 cm (less than 3.0 cm) and the image is erect.
The problem asks us to find a set of three numbers that follow the same logical rule as the given sets: (5, 21, 2) and (7, 33, 4). We need to identify the mathematical relationship between the first, second, and third numbers in the given sets and then apply this rule to the options.
Analyze the first given set: (5, 21, 2)
Let the numbers be \(A\), \(B\), and \(C\). So, \(A=5\), \(B=21\), \(C=2\).
We need to find a relationship between \(A\), \(B\), and \(C\). Let's try operations involving multiplication, addition, and subtraction.
Consider the operation: \((A \times C) + C^2\)
\[ (5 \times 2) + 2^2 = 10 + 4 = 14 \]This does not equal 21.
Consider the operation: \((A \times C) + A\)
\[ (5 \times 2) + 5 = 10 + 5 = 15 \]This does not equal 21.
Consider the operation: \((A \times C) + \text{some constant}\)
\[ (5 \times 2) + K = 21 \] \[ 10 + K = 21 \] \[ K = 11 \]This is a possibility, but the constant \(K\) should ideally be related to \(A\) or \(C\) for a general rule.
Let's try a different approach. What if the middle number is derived from the first and third numbers using multiplication and addition/subtraction?
Consider the relationship: \((A \times \text{some number}) + C\)
Let's try to relate \(B\) to \(A\) and \(C\) using a pattern like \((A \times X) + Y = B\).
What if the rule is \((A \times C) + (\text{something related to } A \text{ or } C)\)?
Let's try: \(A \times (C + X) + Y = B\)
Consider the rule: \((A \times C) + (A + C)\)
\[ (5 \times 2) + (5 + 2) = 10 + 7 = 17 \]This is not 21.
Consider the rule: \((A \times \text{some multiplier}) + C\)
Let's try to find a multiplier for \(A\). If we multiply \(A\) by 3, we get \(5 \times 3 = 15\). Then \(15 + 6 = 21\). The 6 could be \(3 \times C\).
So, rule could be: \((A \times 3) + (C \times 3)\)
\[ (5 \times 3) + (2 \times 3) = 15 + 6 = 21 \]This works for the first set! So, the rule is \(B = (A \times 3) + (C \times 3)\), which can be simplified to \(B = 3 \times (A + C)\).
Verify the rule with the second given set: (7, 33, 4)
Here, \(A=7\), \(B=33\), \(C=4\).
Apply the rule \(B = 3 \times (A + C)\):
\[ 3 \times (7 + 4) = 3 \times 11 = 33 \]The rule holds true for the second set as well.
Apply the rule to the options:
We are looking for an option \((A, B, C)\) where \(B = 3 \times (A + C)\).
C) (8, 51, 9)
A) (6, 40, 8)
B) (10, 93, 3)
D) (11, 24, 3)
The question asks to identify the incorrect statement about magnetism. We need to evaluate each statement based on the fundamental properties of magnets and magnetic fields.
Correct Option: C) A number of field lines intersect at a given point inside a magnetic field.
This statement is incorrect. Magnetic field lines never intersect each other. If they did, it would imply two directions for the magnetic field at the point of intersection, which is not possible.
This problem involves tracking a person's direction after a series of turns. We need to visualize each turn and the resulting direction. The key is to determine the direction Sujlam was facing just before the final turn at point P.
Therefore, just before turning left at point P, Sujlam was facing South.
C) South — As per the step-by-step analysis, Sujlam was facing South just before making the final left turn at point P.
The following table shows the break-up of the costs (in Rs) incurred by a company in the production of five products (A, B, C, D, E). Study the table and answer the question.
| Product | A | B | C | D | E |
| Raw Material | 400 | 200 | 700 | 300 | 400 |
| Manufacturing Cost | 80 | 40 | 150 | 80 | 100 |
| Advertisement | 180 | 70 | 200 | 170 | 200 |
| Labour | 200 | 180 | 400 | 140 | 250 |
| Others | 50 | 10 | 50 | 10 | 50 |
| Total Cost | 900 | 500 | 1500 | 700 | 1000 |
Which of the following products has the maximum manufacturing cost in comparison to its total cost of production?
To find the product with the maximum manufacturing cost in comparison to its total cost, we need to calculate the ratio of Manufacturing Cost to Total Cost for each product. The product with the highest ratio will be the answer.
Correct Option: C) D