To simplify the given algebraic expression, we need to first distribute the multiplication by \(-3\) into the second set of parentheses and then combine like terms. This involves careful handling of signs.
\(x^2 + 5x - 12 - 3(x^2 - 2x + 9)\)
Multiply \(-3\) by \(x^2\), \(-2x\), and \(+9\).
\[ -3(x^2 - 2x + 9) = (-3)(x^2) + (-3)(-2x) + (-3)(9) \]\[ = -3x^2 + 6x - 27 \]The expression now becomes:
\[ x^2 + 5x - 12 - 3x^2 + 6x - 27 \]Group terms with \(x^2\), terms with \(x\), and constant terms.
\[ (x^2 - 3x^2) + (5x + 6x) + (-12 - 27) \]Combining these results, we get:
\[ -2x^2 + 11x - 39 \]Correct Option: C) -2x² + 11x - 39
The question asks to identify the specific Article of the Indian Constitution that declares Hindi in Devanagari script as the official language of the Union. This requires knowledge of the constitutional provisions related to official languages.
Correct Option: C) Article 343(1) is the correct answer because it directly specifies that "The official language of the Union shall be Hindi in Devanagari script."
The problem presents a number series, and we need to identify the pattern to find the missing number. We will analyze the relationship between consecutive terms to determine the rule governing the series.
The differences are 10, 20, 40, 80. It is clear that each subsequent difference is double the previous difference.
Following this pattern, the next difference should be \(80 \times 2 = 160\).
To find the next term in the series, add this predicted difference to the last given term:
\[ 158 + 160 = 318 \]D) 318 is the correct answer. The pattern involves adding a difference that doubles with each step.
The question asks about the main aim of an agreement made by UNEP in 1987. This refers to the Montreal Protocol, which was a significant international treaty aimed at addressing ozone depletion.
D) To freeze the production of CFCs at 1986 level. The Montreal Protocol, signed in 1987, specifically targeted the reduction and eventual elimination of ozone-depleting substances, with the immediate goal being to freeze the production of CFCs at their 1986 levels.
The question asks us to evaluate two statements regarding Sodium hydroxide (NaOH). We need to determine if Statement-I about its preparation and Statement-II about its use are true or false.
Correct Option: D) Both the statements are true.
The question asks to identify the in-house messaging application developed by the Indian Army for the military sector. This requires knowledge of recent initiatives and applications launched by the Indian armed forces.
Correct Option: D) ASIGMA
The question asks us to evaluate two statements related to homologous series in organic chemistry. We need to determine if each statement is individually correct and then choose the option that accurately reflects the correctness of both statements.
B) Both the statements are correct. As explained above, Statement I correctly defines a homologous series, and Statement II accurately describes the gradation in physical properties observed as molecular mass increases within such a series.
If x + \( \frac{1}{x} \) = 4, then x2 + \( \frac{1}{x^2} \) =?
The problem asks us to find the value of \(x^2 + \frac{1}{x^2}\) given the equation \(x + \frac{1}{x} = 4\). We can solve this by squaring the given equation and then rearranging the terms to isolate the desired expression.
We are given the equation:
\[ x + \frac{1}{x} = 4 \]To find \(x^2 + \frac{1}{x^2}\), we can square both sides of the given equation:
\[ \left(x + \frac{1}{x}\right)^2 = (4)^2 \]Expand the left side using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a=x\) and \(b=\frac{1}{x}\):
\[ x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 16 \]Simplify the middle term \(2 \cdot x \cdot \frac{1}{x}\):
\[ x^2 + 2 + \frac{1}{x^2} = 16 \]Now, isolate \(x^2 + \frac{1}{x^2}\) by subtracting 2 from both sides of the equation:
\[ x^2 + \frac{1}{x^2} = 16 - 2 \]Perform the subtraction:
\[ x^2 + \frac{1}{x^2} = 14 \]Thus, the value of \(x^2 + \frac{1}{x^2}\) is 14.
A) 14 is the correct answer, as derived from the step-by-step calculation.
If 23k - 5 = 4096, then the value of k is:
To find the value of \(k\), we need to solve the given exponential equation. The key is to express both sides of the equation with the same base. Once the bases are the same, we can equate the exponents and solve for \(k\).
Identify the given equation:
\(2^{3k - 5} = 4096\)
Express the right-hand side (4096) as a power of 2:
We need to find an integer \(x\) such that \(2^x = 4096\). We can do this by repeatedly dividing 4096 by 2:
Counting the number of times we divided by 2, we find that \(4096 = 2^{12}\).
Rewrite the equation with the same base:
Now the equation becomes:
\(2^{3k - 5} = 2^{12}\)
Equate the exponents:
Since the bases are the same, the exponents must be equal:
\(3k - 5 = 12\)
Solve for \(k\):
Add 5 to both sides:
\(3k = 12 + 5\)
\(3k = 17\)
Divide by 3:
\(k = \frac{17}{3}\)
B) \( \frac{17}{3} \) is the correct value of \(k\).
This problem involves applying a set of given symbol substitutions to a mathematical expression and then evaluating the modified expression using the order of operations (BODMAS/PEMDAS).
B) \(93\) is the correct answer, as derived by applying the given symbol changes and then following the order of operations.