Simplify (7x − 1)³ + (7x + 1)³
To simplify the given expression \((7x - 1)^3 + (7x + 1)^3\), we can use the algebraic identity for the sum of cubes, \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\), or the binomial expansion formulas for \((a-b)^3\) and \((a+b)^3\). Using the binomial expansion and then combining like terms is often more straightforward for expressions of this form.
For \((7x - 1)^3\), let \(a = 7x\) and \(b = 1\):
\[ (7x - 1)^3 = (7x)^3 - 3(7x)^2(1) + 3(7x)(1)^2 - (1)^3 \] \[ = 343x^3 - 3(49x^2) + 21x - 1 \] \[ = 343x^3 - 147x^2 + 21x - 1 \quad \text{(Equation 1)} \]For \((7x + 1)^3\), let \(a = 7x\) and \(b = 1\):
\[ (7x + 1)^3 = (7x)^3 + 3(7x)^2(1) + 3(7x)(1)^2 + (1)^3 \] \[ = 343x^3 + 3(49x^2) + 21x + 1 \] \[ = 343x^3 + 147x^2 + 21x + 1 \quad \text{(Equation 2)} \]Correct Option: D) \(686x^3 + 42x\)
The problem requires us to modify a given number based on rules for its odd and even digits and then count how many digits in the new number appear more than once.
C) Four. The digits 2, 4, 6, and 8 appear more than once in the new number 4624820486.
This problem involves tracking movement in different directions and calculating the final displacement from the starting point. We can visualize Surya's path by breaking down each segment of his journey and then determine his final position relative to the start.
D) 3 km. Surya's final position is 3 km North of the starting point. To return to the starting point, he must run 3 km South.
Each of six teachers, J, K, L, M, N and O, has their class on six different days of a week starting from Monday and ending on Saturday of the same week, but not necessarily in the same order. Only two teachers have classes before M’s class. J has the class on Tuesday, and O has the class on Saturday. Only one teacher has the class after the class of K. N’s class is not between the classes of K and M. Who among the following has the class on Monday?
This is a logical reasoning problem involving scheduling. We need to deduce the class schedule for six teachers (J, K, L, M, N, O) over six days (Monday to Saturday) based on the given clues. We will use a table to organize the information and fill in the schedule step-by-step.
| Day | Teacher |
|---|---|
| Monday | |
| Tuesday | |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday |
| Day | Teacher |
|---|---|
| Monday | |
| Tuesday | J |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday |
| Day | Teacher |
|---|---|
| Monday | |
| Tuesday | J |
| Wednesday | |
| Thursday | |
| Friday | |
| Saturday | O |
| Day | Teacher |
|---|---|
| Monday | |
| Tuesday | J |
| Wednesday | M |
| Thursday | |
| Friday | |
| Saturday | O |
| Day | Teacher |
|---|---|
| Monday | |
| Tuesday | J |
| Wednesday | M |
| Thursday | |
| Friday | K |
| Saturday | O |
| Day | Teacher |
|---|---|
| Monday | N |
| Tuesday | J |
| Wednesday | M |
| Thursday | L |
| Friday | K |
| Saturday | O |
C) N
The question asks to identify the metal that forms a green-colored coating due to corrosion. This phenomenon is commonly known as patination or tarnishing, where a metal reacts with elements in the environment to form a stable layer on its surface.
C) Copper — Copper forms a distinctive green layer called patina, which is a mixture of basic copper carbonate and sometimes basic copper sulfate, when it corrodes in the presence of moist air and carbon dioxide.
Select the option that is related to the fifth number in the same way as the second number is related to the first number and the fourth number is related to the third number.
Series:
2 : 18 :: 4 : 74 :: 9 :?
The problem asks us to find the missing number in a series based on a specific relationship between pairs of numbers. We need to identify the pattern that relates the first number to the second, and the third number to the fourth, then apply the same pattern to the fifth number to find the sixth.
Let the first number be \(x\) and the second number be \(y\). Here, \(x = 2\) and \(y = 18\).
We can try to find a relationship involving powers and arithmetic operations. Let's consider \(x^3\) and \(x^2\):
\[ 2^3 = 8 \] \[ 2^2 = 4 \]If we add these, \(8 + 4 = 12\), which is not 18.
Let's try \(x^4\) and \(x^2\):
\[ 2^4 = 16 \] \[ 2^2 = 4 \]If we add these, \(16 + 4 = 20\), which is close to 18.
Let's try another combination: \(x^3 + (x \times \text{some number})\) or \(x^3 + x^2 + x\).
Consider the pattern \(x^3 + x^2 + x\):
\[ 2^3 + 2^2 + 2 = 8 + 4 + 2 = 14 \]This is not 18.
Consider the pattern \(x^4 + x^2 - x\):
\[ 2^4 + 2^2 - 2 = 16 + 4 - 2 = 18 \]This matches the first pair. So, the pattern seems to be \(x^4 + x^2 - x\).
Here, \(x = 4\). Applying the derived pattern:
\[ x^4 + x^2 - x = 4^4 + 4^2 - 4 \] \[ 4^4 = 256 \] \[ 4^2 = 16 \] \[ 256 + 16 - 4 = 272 - 4 = 268 \]This does not match 74. So, the pattern \(x^4 + x^2 - x\) is incorrect.
Let's re-evaluate the pattern for 2 : 18.
Another common pattern involves \(x^3\) or \(x^2\) multiplied by some factor or added to some factor.
Consider \(x^3 + (\text{some number})\):
\[ 2^3 = 8 \]To get 18, we need to add 10. So, \(x^3 + 10\). Let's check this for the second pair.
For 4 : 74, if \(x=4\), then \(4^3 + 10 = 64 + 10 = 74\). This matches!
So, the pattern is \(x^3 + 10\).
Here, \(x = 9\). Applying the pattern \(x^3 + 10\):
\[ 9^3 + 10 \] \[ 9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729 \] \[ 729 + 10 = 739 \]Therefore, the missing number is 739.
Correct Option: C) 739
A sphere and a cube have the same surface area. The ratio of the volume of the sphere to that of the cube is:
\(\sqrt{\pi}\) : \(\sqrt{6}\)
\(\sqrt{6}\) : \(\sqrt{\pi}\)
\(\sqrt{\pi}\) : 4\(\sqrt{3}\)
4\(\sqrt{3}\) : \(\sqrt{\pi}\)
To find the ratio of the volumes of a sphere and a cube with the same surface area, we need to first express their surface areas and volumes in terms of their respective radii and side lengths. Then, we equate their surface areas to establish a relationship between the radius of the sphere and the side length of the cube. Finally, we use this relationship to find the ratio of their volumes.
Let \(r\) be the radius of the sphere and \(a\) be the side length of the cube.
Surface Area of Sphere (\(SA_S\)):
\[ SA_S = 4\pi r^2 \]Surface Area of Cube (\(SA_C\)):
\[ SA_C = 6a^2 \]Given that the surface areas are equal:
\[ 4\pi r^2 = 6a^2 \]From this equation, we can find a relationship between \(r\) and \(a\):
\[ r^2 = \frac{6a^2}{4\pi} = \frac{3a^2}{2\pi} \] \[ r = a \sqrt{\frac{3}{2\pi}} \]Volume of Sphere (\(V_S\)):
\[ V_S = \frac{4}{3}\pi r^3 \]Volume of Cube (\(V_C\)):
\[ V_C = a^3 \]Now, substitute the expression for \(r\) into the volume of the sphere:
\[ V_S = \frac{4}{3}\pi \left( a \sqrt{\frac{3}{2\pi}} \right)^3 \] \[ V_S = \frac{4}{3}\pi a^3 \left( \frac{3}{2\pi} \right)^{3/2} \] \[ V_S = \frac{4}{3}\pi a^3 \frac{3\sqrt{3}}{(2\pi)\sqrt{2\pi}} \] \[ V_S = \frac{4}{3}\pi a^3 \frac{3\sqrt{3}}{2\pi\sqrt{2\pi}} \] \[ V_S = \frac{2 a^3 \sqrt{3}}{\sqrt{2\pi}} \] \[ V_S = \frac{2 a^3 \sqrt{3}}{\sqrt{2}\sqrt{\pi}} = \frac{\sqrt{2} a^3 \sqrt{3}}{\sqrt{\pi}} = \frac{a^3 \sqrt{6}}{\sqrt{\pi}} \]The ratio of the volume of the sphere to that of the cube is \(V_S : V_C\):
\[ \frac{V_S}{V_C} = \frac{\frac{a^3 \sqrt{6}}{\sqrt{\pi}}}{a^3} \] \[ \frac{V_S}{V_C} = \frac{\sqrt{6}}{\sqrt{\pi}} \]So, the ratio is \(\sqrt{6} : \sqrt{\pi}\).
B) \(\sqrt{6}\) : \(\sqrt{\pi}\) is the correct ratio derived from the calculations.
The question asks about the subject matter of the Twelfth Schedule of the Indian Constitution. This requires knowledge of the various schedules and their respective functions as outlined in the Constitution.
C) urban local bodies — The Twelfth Schedule, added by the 74th Constitutional Amendment Act of 1992, deals with the powers, authority, and responsibilities of Municipalities (urban local bodies). It contains 18 functional items over which these bodies have jurisdiction.
Determine the positive value of k for which the equations x2 + kx + 64 = 0 and x2 - 8x + k = 0 will both have real roots.
For a quadratic equation \(ax^2 + bx + c = 0\) to have real roots, its discriminant (\(D\)) must be greater than or equal to zero, i.e., \(D = b^2 - 4ac \ge 0\). We will apply this condition to both given quadratic equations and find the range of \(k\) for each. The value of \(k\) that satisfies both conditions will be the answer.
Analyze the first equation: \(x^2 + kx + 64 = 0\)
Here, \(a=1\), \(b=k\), and \(c=64\). For real roots, the discriminant must be non-negative:
\[ D_1 = k^2 - 4(1)(64) \ge 0 \] \[ k^2 - 256 \ge 0 \] \[ k^2 \ge 256 \] Taking the square root of both sides, we get: \[ |k| \ge 16 \] This implies \(k \ge 16\) or \(k \le -16\).Analyze the second equation: \(x^2 - 8x + k = 0\)
Here, \(a=1\), \(b=-8\), and \(c=k\). For real roots, the discriminant must be non-negative:
\[ D_2 = (-8)^2 - 4(1)(k) \ge 0 \] \[ 64 - 4k \ge 0 \] \[ 64 \ge 4k \] \[ k \le \frac{64}{4} \] \[ k \le 16 \]Find the common value of k:
We need a value of \(k\) that satisfies both conditions:
We are looking for the positive value of \(k\). Combining \(k \ge 16\) and \(k \le 16\), the only value that satisfies both is \(k=16\).
Correct Option: C) 16
This problem can be solved by setting up a system of two linear equations based on the given information. We will assign variables to the father's and son's ages, form equations, and then solve them simultaneously to find the son's age.
A) 30 years is the calculated present age of the son.