In this type of question, we need to find a common pattern or rule that applies to three of the given pairs and identify the pair that does not follow that rule. We will analyze the positional values of the letters in the English alphabet.
Correct Option: B) BL
The question asks to identify the Indian batsman who achieved the unique feat of scoring three consecutive Test centuries in his first three International Test matches. This requires specific knowledge of cricket records.
Correct Option: A) Mohammad Azharuddin is the correct answer. He holds the unique record of scoring three consecutive centuries in his first three Test matches. He scored 110 on debut in Kolkata (1984-85), followed by 105 in Chennai and 122 in Kanpur, all against England.
The problem describes a relationship where the speed of a locomotive engine diminishes based on the square root of the number of wagons attached. We need to establish this relationship using the given data points and then use it to find the maximum number of wagons that can be attached without the speed falling below a certain threshold.
Correct Option: A) 169
The problem asks us to identify the option that is different from the rest, implying that three options share a common mathematical property while one does not. We need to examine each number to find a pattern or relationship.
D) 81 — This number is a perfect square (\(9^2\)) but not a perfect cube. The other three numbers are perfect cubes.
We are given two conditions involving two positive numbers: their sum and the difference of their squares. We can set up a system of two equations with two variables and solve for the numbers. Once we find both numbers, we can identify the greater one.
Let the two positive numbers be \(x\) and \(y\). Assume \(x\) is the greater number and \(y\) is the smaller number.
From the first condition, "The sum of two positive numbers is 27", we can write the equation:
\[ x + y = 27 \quad \text{(Equation 1)} \]From the second condition, "the difference of their squares is 81", we can write the equation:
Since we assumed \(x\) is the greater number, \(x^2\) will be greater than \(y^2\).
\[ x^2 - y^2 = 81 \quad \text{(Equation 2)} \]Recall the algebraic identity: \(a^2 - b^2 = (a - b)(a + b)\). Apply this to Equation 2:
\[ (x - y)(x + y) = 81 \quad \text{(Equation 3)} \]Substitute the value of \((x + y)\) from Equation 1 into Equation 3:
\[ (x - y)(27) = 81 \]Solve for \((x - y)\):
\[ x - y = \frac{81}{27} \] \[ x - y = 3 \quad \text{(Equation 4)} \]Now we have a system of two linear equations:
Equation 1: \(x + y = 27\)
Equation 4: \(x - y = 3\)
Add Equation 1 and Equation 4:
\[ (x + y) + (x - y) = 27 + 3 \] \[ 2x = 30 \] \[ x = \frac{30}{2} \] \[ x = 15 \]Substitute the value of \(x\) into Equation 1 to find \(y\):
\[ 15 + y = 27 \] \[ y = 27 - 15 \] \[ y = 12 \]The two numbers are 15 and 12. The greater of the two numbers is 15.
Correct Option: D) 15
The chord of contact of tangents drawn from a point on the circle \( x^{2} + y^{2} = a^{2} \) to the circle \( x^{2} + y^{2} = b^{2} \) touches the circle \( x^{2} + y^{2} = c^{2} \) such that bm = an cp, where m, n, p ∈ N. Find the value of m + n + p + 10.
We need to find the equation of the chord of contact from a point on the first circle to the second circle. Then, we will find the condition for this chord of contact to touch the third circle. By comparing the derived condition with the given relation, we can find the values of \(m\), \(n\), and \(p\), and subsequently calculate \(m+n+p+10\).
Let the point on the circle \(x^2 + y^2 = a^2\) be \(P(x_1, y_1)\). So, \(x_1^2 + y_1^2 = a^2\).
The equation of the chord of contact of tangents drawn from \(P(x_1, y_1)\) to the circle \(x^2 + y^2 = b^2\) is given by \(xx_1 + yy_1 = b^2\).
This chord of contact \(xx_1 + yy_1 = b^2\) touches the circle \(x^2 + y^2 = c^2\). The condition for a line \(Ax + By + C = 0\) to touch a circle \(x^2 + y^2 = r^2\) is that the perpendicular distance from the center \((0,0)\) to the line is equal to the radius \(r\).
Here, the line is \(x_1x + y_1y - b^2 = 0\), and the radius of the third circle is \(c\). So,
\[ \frac{|-b^2|}{\sqrt{x_1^2 + y_1^2}} = c \]We know that \(x_1^2 + y_1^2 = a^2\). Substituting this into the equation above:
\[ \frac{|-b^2|}{\sqrt{a^2}} = c \] \[ \frac{b^2}{a} = c \] \[ b^2 = ac \]The given relation is \(b^m = a^n c^p\). Comparing this with \(b^2 = a^1 c^1\), we get:
These values satisfy \(m, n, p \in N\).
We need to find \(m + n + p + 10\).
\[ m + n + p + 10 = 2 + 1 + 1 + 10 = 14 \]Correct Option: C) 14
The standard equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. We are given the radius and a point the circle passes through. The center lies on a given line. We will use these conditions to find the coordinates of the center and then form the equation of the circle.
Let the center of the circle be \((h, k)\). The radius is given as \(r = 3\).
The center \((h, k)\) lies on the line \(y = x - 1\). So, we can write \(k = h - 1\).
The equation of the circle is \((x - h)^2 + (y - k)^2 = r^2\). Substituting \(r = 3\) and \(k = h - 1\), we get:
\[ (x - h)^2 + (y - (h - 1))^2 = 3^2 \]\[ (x - h)^2 + (y - h + 1)^2 = 9 \]The circle passes through the point \((7, 3)\). Substitute \(x = 7\) and \(y = 3\) into the circle's equation:
\[ (7 - h)^2 + (3 - h + 1)^2 = 9 \]\[ (7 - h)^2 + (4 - h)^2 = 9 \]Expand and solve for \(h\):
\[ (49 - 14h + h^2) + (16 - 8h + h^2) = 9 \]\[ 2h^2 - 22h + 65 = 9 \]\[ 2h^2 - 22h + 56 = 0 \]\[ h^2 - 11h + 28 = 0 \]Factor the quadratic equation:
\[ (h - 4)(h - 7) = 0 \]So, \(h = 4\) or \(h = 7\).Find the corresponding values for \(k\) using \(k = h - 1\):
Now, we check which center gives the correct equation. We will use the general form \(x^2 + y^2 - 2hx - 2ky + h^2 + k^2 - r^2 = 0\).
Case 1: Center \((4, 3)\) and \(r = 3\)
\[ x^2 + y^2 - 2(4)x - 2(3)y + 4^2 + 3^2 - 3^2 = 0 \]\[ x^2 + y^2 - 8x - 6y + 16 + 9 - 9 = 0 \]\[ x^2 + y^2 - 8x - 6y + 16 = 0 \]This matches option B.Case 2: Center \((7, 6)\) and \(r = 3\)
\[ x^2 + y^2 - 2(7)x - 2(6)y + 7^2 + 6^2 - 3^2 = 0 \]\[ x^2 + y^2 - 14x - 12y + 49 + 36 - 9 = 0 \]\[ x^2 + y^2 - 14x - 12y + 76 = 0 \]This does not match any of the options.B) The equation \( x^{2} + y^{2} - 8x - 6y + 16 = 0 \) is obtained when the center is \((4, 3)\) and the radius is \(3\). This center satisfies the condition \(k = h - 1\) (i.e., \(3 = 4 - 1\)) and the circle passes through \((7, 3)\) as shown in the step-by-step derivation.
To find the fourth proportional to three given numbers \(a\), \(b\), and \(c\), we set up a proportion where the ratio of the first two numbers is equal to the ratio of the third number to the fourth proportional. If the fourth proportional is \(x\), then the relationship is \(a : b :: c : x\), which can be written as a fraction equation \[\frac{a}{b} = \frac{c}{x}\].
A) 25 is the correct answer because when we set up the proportion \(3/5 = 15/x\) and solve for \(x\), we get \(x = 25\).
If A's salary is 50% more than that of B, then by what percentage is B's salary less than that of A?
23 \( \frac{1}{3} \) %
13 \( \frac{1}{3} \) %
33 \( \frac{1}{3} \) %
30 \( \frac{1}{3} \) %
This problem involves percentage calculations. We are given that A's salary is 50% more than B's salary. We need to find out by what percentage B's salary is less than A's salary. This requires setting up a relationship between the salaries and then calculating the percentage difference relative to A's salary.
Assume B's salary: Let's assume B's salary is \(x\).
Calculate A's salary: A's salary is 50% more than B's salary. So, A's salary will be:
\[ \text{A's Salary} = x + 50\% \text{ of } x \]\[ \text{A's Salary} = x + \frac{50}{100} \times x \]\[ \text{A's Salary} = x + 0.5x \]\[ \text{A's Salary} = 1.5x \]Find the difference between A's and B's salaries:
\[ \text{Difference} = \text{A's Salary} - \text{B's Salary} \]\[ \text{Difference} = 1.5x - x \]\[ \text{Difference} = 0.5x \]Calculate the percentage by which B's salary is less than A's salary: To find this percentage, we compare the difference to A's salary (since we are asking "less than that of A").
\[ \text{Percentage less} = \frac{\text{Difference}}{\text{A's Salary}} \times 100\% \]\[ \text{Percentage less} = \frac{0.5x}{1.5x} \times 100\% \]\[ \text{Percentage less} = \frac{0.5}{1.5} \times 100\% \]\[ \text{Percentage less} = \frac{1}{3} \times 100\% \]\[ \text{Percentage less} = 33 \frac{1}{3}\% \]C) The calculation shows that B's salary is \(33 \frac{1}{3}\%\) less than that of A. This matches option C.
The question asks which atmospheric gas can be 'fixed' by lightning. This refers to the process of converting an atmospheric gas into a more reactive form, often a compound, that can be utilized by living organisms or deposited in the soil/water. We need to identify which of the given gases undergoes such a transformation due to the high energy provided by lightning.
C) Nitrogen — Lightning provides the necessary energy to break the strong triple bond in atmospheric nitrogen (\(\text{N}_2\)), allowing it to react with oxygen to form nitrogen oxides. These compounds are then converted into nitrates, which are essential nutrients for plants. This process is a significant part of the natural nitrogen cycle.