The problem states that the area of the rectangle remains constant even when its dimensions change. We will first calculate the initial area of the rectangle using the given length and breadth. Then, using this constant area and the new length, we will find the new breadth.
The formula for the area of a rectangle is given by:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]Given initial length \((L_1 = 20 \text{ cm})\) and initial breadth \((B_1 = 8 \text{ cm})\).
\[ \text{Initial Area} = 20 \text{ cm} \times 8 \text{ cm} = 160 \text{ cm}^2 \]The problem states that the area remains the same. So, the new area \((A_2)\) is equal to the initial area \((A_1)\).
\[ A_2 = A_1 = 160 \text{ cm}^2 \]The new length \((L_2)\) is given as 25 cm. Let the new breadth be \((B_2)\).
Using the area formula again:
\[ A_2 = L_2 \times B_2 \]Substitute the known values:
\[ 160 \text{ cm}^2 = 25 \text{ cm} \times B_2 \]Now, solve for \((B_2)\):
\[ B_2 = \frac{160 \text{ cm}^2}{25 \text{ cm}} \]\[ B_2 = 6.4 \text{ cm} \]C) 6.4 cm — The new breadth of the rectangle, when its length is increased to 25 cm and the area remains constant at 160 cm², is 6.4 cm.
We need to determine the total number of people in a row based on their relative seating positions. We will build the arrangement step-by-step using the given clues, starting from the most restrictive conditions.
"nobody sits to the left of A": This means A is at the extreme left end of the row.
Arrangement so far: A _ _ _ _ _ _ _
"M sits third to the right of A": If A is at position 1, M is at position 4.
Arrangement so far: A _ _ M _ _ _ _
"M is second to the left of P": This means P is to the right of M, with one person between them. If M is at position 4, P is at position 6.
Arrangement so far: A _ _ M _ P _ _
"Only one person sits between P and S": Since P is at position 6, S can be at position 4 (which is M, so not possible as people are distinct) or position 8. Thus, S must be at position 8.
Arrangement so far: A _ _ M _ P _ S
"Q sits to the immediate right of S": If S is at position 8, Q is at position 9.
Arrangement so far: A _ _ M _ P _ S Q
"nobody sits to the right of Q": This means Q is at the extreme right end of the row.
The complete arrangement is: A _ _ M _ P _ S Q
Now, let's count the positions. The arrangement shows 9 distinct positions occupied by people. The blanks represent other people whose identities are not specified but who occupy a seat.
The final arrangement is: A (person) (person) M (person) P (person) S Q
Counting the positions from A to Q, we have 9 positions.
D) 9. Based on the step-by-step construction, the final arrangement shows 9 people in the row.
The question asks to identify the Prime Minister of India during the period when the Third Five-Year Plan was formulated and implemented. This requires knowledge of the timelines of India's Five-Year Plans and the tenures of its Prime Ministers.
Correct Option: C) Jawaharlal Nehru
Which of the following ionic compounds has the highest boiling point?
The boiling point of an ionic compound is primarily determined by the strength of the ionic bonds, which is related to the lattice energy. Lattice energy depends on the charge of the ions and their size. Higher charge and smaller ionic radii lead to stronger electrostatic attraction and thus higher lattice energy and boiling point.
B) CaCl\(_2\)
Calcium chloride (\( \text{CaCl}_2 \)) has a boiling point of \( 1935^\circ \text{C} \). It has a \( +2 \) cation, leading to strong electrostatic forces. While \( \text{Mg}^{2+} \) is smaller, \( \text{CaCl}_2 \) exhibits a higher boiling point than \( \text{MgCl}_2 \).
This problem involves calculating the annual equal payment required to discharge a debt over a specified period with compound interest. This is a common application of annuities, specifically finding the installment amount when the present value of the debt is known. We will use the formula for the present value of an ordinary annuity or calculate the present value of each installment separately and sum them up.
B) 15,435 is the calculated annual equal payment that will discharge the debt.
The question asks for a key reason behind Mendel's success in discovering the laws of inheritance. We need to evaluate each option based on Mendel's experimental methodology and contributions to genetics.
Correct Option: A) he kept numerical count of the progeny
Mendel's success was largely due to his quantitative approach. By carefully counting and analyzing the numerical ratios of different traits in successive generations of pea plants, he was able to formulate precise laws of inheritance. This statistical rigor was a novel and critical aspect of his work that distinguished it from previous, less successful attempts to understand heredity.
To find the longest possible side length of square tiles that can perfectly pave a rectangular ground, we need to find the greatest common divisor (GCD) of its length and width. Since the dimensions are given in meters with decimals, it's best to convert them to centimeters first to work with whole numbers, and then find the GCD.
Given length \(L = 7.35\) meters
Given width \(W = 5.25\) meters
Since \(1 \text{ meter} = 100 \text{ centimeters}\):
\[ L = 7.35 \times 100 = 735 \text{ cm} \]\[ W = 5.25 \times 100 = 525 \text{ cm} \]We need to find GCD(735, 525). We can use the prime factorization method or the Euclidean algorithm.
Using prime factorization:
To find the GCD, we take the lowest power of the common prime factors:
\[ \text{GCD}(735, 525) = 3^1 \times 5^1 \times 7^1 = 3 \times 5 \times 7 = 105 \]The longest possible side length of the square tiles will be equal to the GCD of the length and width of the ground.
Side length \(s = 105 \text{ cm}\)
D) 105 cm — The greatest common divisor of 735 cm and 525 cm is 105 cm, which represents the longest possible side length of square tiles that can perfectly pave the ground without any cutting.
Which of the options below give the solution to the equation \(\frac{1}{x+3}+\frac{1}{x+5}\) = \( \frac{1}{6} \)?
x = 2 ± \(\sqrt{37}\)
x = 2 ± \(\sqrt{39}\)
x = 3 ± \(\sqrt{37}\)
x = 4 ± \(\sqrt{37}\)
To solve the given equation, we need to combine the fractions on the left-hand side, simplify the resulting expression, and then solve the quadratic equation that emerges. This typically involves finding a common denominator, cross-multiplication, and applying the quadratic formula.
Combine the fractions: The given equation is:
\[ \frac{1}{x+3} + \frac{1}{x+5} = \frac{1}{6} \]Find a common denominator for the left side, which is \((x+3)(x+5)\):
\[ \frac{(x+5) + (x+3)}{(x+3)(x+5)} = \frac{1}{6} \]Simplify the numerator:
\[ \frac{2x+8}{x^2+5x+3x+15} = \frac{1}{6} \] \[ \frac{2x+8}{x^2+8x+15} = \frac{1}{6} \]Cross-multiply:
\[ 6(2x+8) = 1(x^2+8x+15) \] \[ 12x+48 = x^2+8x+15 \]Rearrange into a standard quadratic equation form \(ax^2+bx+c=0\):
\[ x^2+8x-12x+15-48 = 0 \] \[ x^2-4x-33 = 0 \]Solve the quadratic equation using the quadratic formula: For an equation \(ax^2+bx+c=0\), the solutions are given by \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Here, \(a=1\), \(b=-4\), \(c=-33\).
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-33)}}{2(1)} \] \[ x = \frac{4 \pm \sqrt{16 + 132}}{2} \] \[ x = \frac{4 \pm \sqrt{148}}{2} \]Simplify the square root: We can write \(148\) as \(4 \times 37\).
\[ x = \frac{4 \pm \sqrt{4 \times 37}}{2} \] \[ x = \frac{4 \pm 2\sqrt{37}}{2} \]Final simplification:
\[ x = 2 \pm \sqrt{37} \]Correct Option: A) x = 2 ± \(\sqrt{37}\)
This problem involves calculating a future day of the week based on a given condition. The key is to understand the cyclical nature of days of the week, which repeats every 7 days. We will first determine "today's" day of the week and then find the day 121 days from "today" by using the concept of remainders when dividing by 7.
Therefore, 121 days from today (Saturday) will be Monday.
D) Monday is the correct day of the week, calculated by determining today's day and then counting forward the remainder of days after dividing 121 by 7.
This problem involves work and time, where individuals work at different rates and leave at different times. We will calculate the individual work rates, then determine the work done in different phases, and finally find the total time taken to complete the work.
C) 6.8