If \( a^{3}+b^{3}+c^{3}-3abc=0 \), then find the value of \(\left(\frac{a^2}{bc} + \frac{b^2}{ac} - 3\right)\).
\(-\frac{c^2}{ab}\)
\(-\frac{c^2}{bc}\)
\(-\frac{c^3}{ba}\)
\(-\frac{c}{a}\)
The problem provides an algebraic identity and asks for the value of an expression. We will use the given identity \(a^3+b^3+c^3-3abc=0\) to deduce the relationship between \(a, b,\) and \(c\), and then substitute this relationship into the expression \[\left(\frac{a^2}{bc} + \frac{b^2}{ac} - 3\right)\] to simplify it.
Recall the algebraic identity:
\[ a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca) \]Given that \(a^3+b^3+c^3-3abc=0\), we can write:
\[ (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = 0 \]This implies that either \(a+b+c=0\) or \(a^2+b^2+c^2-ab-bc-ca=0\).
Consider the second case: \(a^2+b^2+c^2-ab-bc-ca=0\). This can be rewritten as:
\[ \frac{1}{2}[(a-b)^2+(b-c)^2+(c-a)^2] = 0 \]For this equation to hold, each term must be zero, which means \(a-b=0\), \(b-c=0\), and \(c-a=0\). This implies \(a=b=c\).
If \(a=b=c\), then \(a+b+c = a+a+a = 3a\). If \(a=b=c=0\), then the original expression \(a^3+b^3+c^3-3abc=0\) holds, but the expression \[\left(\frac{a^2}{bc} + \frac{b^2}{ac} - 3\right)\] would involve division by zero. Therefore, we assume \(a, b, c\) are not all zero.
If \(a=b=c\) (and not all zero), then \(a+b+c \neq 0\). In this case, the condition \(a^3+b^3+c^3-3abc=0\) implies \(a=b=c\). Let's evaluate the given expression under this condition:
\[ \frac{a^2}{bc} + \frac{b^2}{ac} - 3 \]Substitute \(b=a\) and \(c=a\):
\[ \frac{a^2}{a \cdot a} + \frac{a^2}{a \cdot a} - 3 = \frac{a^2}{a^2} + \frac{a^2}{a^2} - 3 = 1 + 1 - 3 = -1 \]Now consider the first case: \(a+b+c=0\). This implies \(c = -(a+b)\).
Substitute \(c = -(a+b)\) into the expression \[\left(\frac{a^2}{bc} + \frac{b^2}{ac} - 3\right)\]:
\[ \frac{a^2}{b(-(a+b))} + \frac{b^2}{a(-(a+b))} - 3 \]\[ = \frac{a^2}{-ab-b^2} + \frac{b^2}{-a^2-ab} - 3 \]\[ = \frac{a^2}{-b(a+b)} + \frac{b^2}{-a(a+b)} - 3 \]Find a common denominator, which is \(-ab(a+b)\):
\[ = \frac{a^2 \cdot a}{-ab(a+b)} + \frac{b^2 \cdot b}{-ab(a+b)} - 3 \]\[ = \frac{a^3 + b^3}{-ab(a+b)} - 3 \]Recall the sum of cubes formula: \(a^3+b^3 = (a+b)(a^2-ab+b^2)\). Substitute this into the expression:
\[ = \frac{(a+b)(a^2-ab+b^2)}{-ab(a+b)} - 3 \]Assuming \(a+b \neq 0\), we can cancel \((a+b)\):
\[ = \frac{a^2-ab+b^2}{-ab} - 3 \]\[ = -\left(\frac{a^2}{ab} - \frac{ab}{ab} + \frac{b^2}{ab}\right) - 3 \]\[ = -\left(\frac{a}{b} - 1 + \frac{b}{a}\right) - 3 \]\[ = -\frac{a}{b} + 1 - \frac{b}{a} - 3 \]\[ = -\frac{a}{b} - \frac{b}{a} - 2 \]We know \(c = -(a+b)\), so \(c^2 = (a+b)^2 = a^2+2ab+b^2\). From this, \(a^2+b^2 = c^2-2ab\).
Let's re-evaluate the expression \(\frac{a^2-ab+b^2}{-ab} - 3\):
\[ = \frac{(a^2+b^2)-ab}{-ab} - 3 \]Substitute \(a^2+b^2 = c^2-2ab\):
\[ = \frac{(c^2-2ab)-ab}{-ab} - 3 \]\[ = \frac{c^2-3ab}{-ab} - 3 \]\[ = \frac{c^2}{-ab} - \frac{3ab}{-ab} - 3 \]\[ = -\frac{c^2}{ab} + 3 - 3 \]\[ = -\frac{c^2}{ab} \]This result is consistent for both cases (when \(a=b=c\) and when \(a+b+c=0\)). If \(a=b=c\), then \(c^2/ab = a^2/a^2 = 1\), so \(-c^2/ab = -1\), which matches our earlier calculation. The expression \(-c^2/ab\) is the general solution.
This is a blood relations problem. We need to trace the relationships described in the statement step-by-step, starting from the last part of the statement and working backward to determine the relationship between the woman and the speaker.
A) Father-in-law's sister. As derived, the woman is the sister of the speaker's father-in-law. This makes her the father-in-law's sister.
The question asks to identify the Indian state that does not share a border with Myanmar. This requires knowledge of the geographical location of Indian states in the Northeast and their international borders.
Assam does not share a border with Myanmar. It is located to the west of Nagaland and Manipur, and its easternmost border is with Nagaland and Manipur, not directly with Myanmar. Assam shares international borders with Bhutan and Bangladesh.
The question asks about the subject of a Geographical Indication (GI) tag. A GI tag is a sign used on items that have a specific geographical origin and possess qualities or a reputation due to that origin. Understanding the definition and purpose of GI tags is key to answering this question.
D) Products — A Geographical Indication (GI) is primarily associated with products (agricultural, natural, manufactured, or handicrafts) that derive their unique characteristics and reputation from their specific geographical origin and traditional methods of production developed over centuries.
The question asks about the number of states where assembly elections were declared by the Election Commission of India in March 2016, to be held in April-May 2016. This requires knowledge of major electoral events in India during that period.
C) 5. The Election Commission of India declared assembly elections in five states/union territories (Assam, West Bengal, Tamil Nadu, Kerala, and Puducherry) in March 2016, scheduled for April-May 2016.
This problem involves the concept of work and time, specifically how the number of workers, their efficiency, and the time taken to complete a job are related. We can use the formula that relates these quantities: \(\text{Work} = \text{Number of Persons} \times \text{Efficiency} \times \text{Time}\). Since the job is the same in both scenarios, the total work done is constant.
Define variables for the first scenario:
Define variables for the second scenario:
Equate the total work done in both scenarios:
Since the job is the same, the total work done is equal in both cases:
\[ 10 \times E \times X = 20 \times (2E) \times 5 \]Solve for X:
First, simplify the right side of the equation:
\[ 10 \times E \times X = 20 \times 2 \times E \times 5 \] \[ 10 \times E \times X = 200 \times E \]Now, divide both sides by \(10 \times E\). (We can cancel \(E\) as efficiency cannot be zero).
\[ X = \frac{200 \times E}{10 \times E} \] \[ X = \frac{200}{10} \] \[ X = 20 \]C) 20 is the correct answer. The value of X is 20 days.
P can do a piece of work in 60 days & Q can do the same work in 50 days. Find the ratio of working efficiencies of P to Q.
\( \frac{1}{3} \)
\( \frac{6}{5} \)
\( \frac{5}{6} \)
\( \frac{4}{5} \)
To find the ratio of working efficiencies, we first need to determine the individual efficiency of P and Q. Efficiency is inversely proportional to the time taken to complete a work. If a person completes a work in 'n' days, their one day's work (efficiency) is \( \frac{1}{n} \) of the total work.
Thus, the ratio of working efficiencies of P to Q is \( \frac{5}{6} \).
Correct Option: C) \( \frac{5}{6} \)
This problem can be solved using basic trigonometry. We are given the height of the kite (opposite side) and the length of the string (hypotenuse). We need to find the angle of inclination, which is the angle between the string and the horizontal ground. The sine function relates the opposite side and the hypotenuse.
In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \]We need to find the angle whose sine is \(1/2\). From standard trigonometric values, we know that:
\[ \sin(30^\circ) = \frac{1}{2} \]Therefore, the inclination of the string to the horizontal ground is \(30^\circ\).
Correct Option: D) 30
The question asks to identify the vitamin to which Retinol is mainly related. This requires knowledge of the common names and chemical forms of various vitamins.
Correct Option: A) Retinol is the primary active form of Vitamin A in the human body. It plays a vital role in vision, immune function, and cell differentiation.
This is a blood relation question. To solve such problems, we break down the statement into smaller parts and identify the relationships step-by-step, usually starting from the end of the statement or a clear reference point.
Correct Option: B) Paternal aunt