We need to evaluate each given statement about geometric figures (parallelogram, rectangle, kite) to determine if it is always true based on their definitions and properties.
A parallelogram is a quadrilateral with two pairs of parallel sides. A fundamental property of parallelograms is that their diagonals intersect at their midpoints, meaning each diagonal divides the other into two equal parts. This statement is always true.
A rectangle is a parallelogram with all four angles equal to 90°. By definition, all angles in a rectangle are right angles. Since opposite angles in any parallelogram are equal, and all angles in a rectangle are 90°, it follows that opposite angles are indeed equal and each measures 90°. This statement is always true.
A kite is a quadrilateral where two pairs of equal-length sides are adjacent to each other. A key property of a kite is that its diagonals are perpendicular to each other. One diagonal is the perpendicular bisector of the other. This statement is always true.
Since all three statements (a), (b), and (c) are always true, the option that includes all three is the correct answer.
Correct Option: B) All three of (a), (b) and (c)
This question involves a double displacement reaction between two ionic compounds. We need to identify the products formed and determine which one is insoluble in water, thus forming a precipitate.
D) Barium sulfate is formed as an insoluble product in the reaction between potassium sulfate and barium chloride, hence it forms a precipitate.
The value of 2 - \(\frac{\cos^{2}\theta}{1+\sin\theta}+\frac{1+\sin\theta}{\cos\theta}-\frac{\cos\theta}{1-\sin\theta}\) is:
The problem requires simplifying a trigonometric expression. We will use fundamental trigonometric identities such as \(\sin^2\theta + \cos^2\theta = 1\) and algebraic manipulation to simplify the given expression step-by-step.
The given expression is:
\[ 2 - \frac{\cos^{2}\theta}{1+\sin\theta} + \frac{1+\sin\theta}{\cos\theta} - \frac{\cos\theta}{1-\sin\theta} \]First, simplify the term \( -\frac{\cos^{2}\theta}{1+\sin\theta} \). We know that \(\cos^2\theta = 1 - \sin^2\theta = (1-\sin\theta)(1+\sin\theta)\).
\[ -\frac{\cos^{2}\theta}{1+\sin\theta} = -\frac{(1-\sin\theta)(1+\sin\theta)}{1+\sin\theta} = -(1-\sin\theta) = -1+\sin\theta \]Now, consider the last two terms: \[+\frac{1+\sin\theta}{\cos\theta} - \frac{\cos\theta}{1-\sin\theta}\]. Combine them by finding a common denominator, which is \(\cos\theta(1-\sin\theta)\).
\[ \frac{1+\sin\theta}{\cos\theta} - \frac{\cos\theta}{1-\sin\theta} = \frac{(1+\sin\theta)(1-\sin\theta) - \cos\theta \cdot \cos\theta}{\cos\theta(1-\sin\theta)} \]Simplify the numerator:
\[ (1+\sin\theta)(1-\sin\theta) - \cos^2\theta = (1-\sin^2\theta) - \cos^2\theta \]\[ = \cos^2\theta - \cos^2\theta = 0 \]So, the combined term becomes:
\[ \frac{0}{\cos\theta(1-\sin\theta)} = 0 \]Substitute the simplified terms back into the original expression:
\[ 2 + (-1+\sin\theta) + 0 \]\[ = 2 - 1 + \sin\theta \]\[ = 1 + \sin\theta \]Thus, the value of the given expression is \(1 + \sin\theta\).
D) 1 + sin \(\theta\) is the correct answer, as derived from the simplification steps.
To solve this problem, we need to recall the condition for a quadratic equation to have equal roots. For a standard quadratic equation of the form \(Ax^2 + Bx + C = 0\), the roots are equal if and only if its discriminant, \(\Delta = B^2 - 4AC\), is equal to zero.
Thus, the value of \(a\) for which the quadratic equation has equal roots is 4.
D) 4 — As calculated, setting the discriminant to zero yields \(a = 4\).
The question asks for the name of India's first pan-India toll-free helpline (14567) launched in September 2021 by the Ministry of Social Justice and Empowerment, which provides information, guidance, emotional support, and immediate assistance in cases of abuse, specifically targeting a vulnerable demographic. Identifying the target group based on the Ministry and the nature of support helps determine the correct name.
Correct Option: A) Elder line is the correct name for the pan-India toll-free helpline 14567, launched by the Ministry of Social Justice and Empowerment in September 2021, aimed at providing support and assistance to senior citizens.
We are given a quadratic equation and one of its roots. We need to find the other root. We can use the properties of roots of a quadratic equation, specifically the product of roots formula, or substitute the given root into the equation to find the value of 'a' and then solve for the roots.
A quadratic equation is generally represented as \(Ax^2 + Bx + C = 0\).
The given equation is \(x^2 - (a - 1)x + 10 = 0\).
Here, \(A = 1\), \(B = -(a - 1)\), and \(C = 10\).
For a quadratic equation \(Ax^2 + Bx + C = 0\), if \(\alpha\) and \(\beta\) are the roots, then the product of the roots is given by \(\alpha \cdot \beta = \frac{C}{A}\).
Let the given root be \(\alpha = 5\), and let the other root be \(\beta\).
We have \(C = 10\) and \(A = 1\).
So, \(5 \cdot \beta = \frac{10}{1}\)
\(5 \cdot \beta = 10\)
\(\beta = \frac{10}{5}\)
\(\beta = 2\)
A) 2 is the other root of the equation.
The question asks to identify the hormone responsible for helping the body adjust to stress, particularly during strong emotional states like anger, embarrassment, or worry. This points to the body's "fight or flight" response, which is mediated by specific hormones.
Correct Option: A) Adrenalin (also known as Epinephrine) is a hormone produced by the adrenal glands. It plays a central role in the "fight or flight" response. When a person experiences stress, anger, embarrassment, or worry, adrenalin is rapidly released, preparing the body to react to perceived threats. It increases heart rate, constricts blood vessels, dilates air passages, and redirects blood flow to muscles, all of which help the body cope with acute stress.
The question asks about the individual to whom Annie Besant handed over the Central Hindu College. This requires knowledge of the historical development of educational institutions in India and the roles of prominent social and educational reformers.
C) Madan Mohan Malviya — Annie Besant handed over the Central Hindu College to Madan Mohan Malviya, who then used it as the foundation for the establishment of the Banaras Hindu University in 1916.
To find the power of the electric equipment, we first need to calculate the current flowing through it. Current is defined as the rate of flow of charge. Once the current is known, we can use the formula for power, which is the product of voltage and current.
Current is the charge flowing per unit time. The formula is:
\[ I = \frac{Q}{t} \]Substitute the given values:
\[ I = \frac{15 \text{ C}}{5 \text{ s}} = 3 \text{ A} \]Power in an electrical circuit is given by the product of voltage and current. The formula is:
\[ P = V \times I \]Substitute the calculated current and given voltage:
\[ P = 20 \text{ V} \times 3 \text{ A} = 60 \text{ W} \]The unit of power is Watts (W).
Correct Option: C) 60 watt
To find the ratio of copper and tin in mixture C, we need to calculate the total amount of copper and the total amount of tin when equal quantities of mixtures A and B are combined. We will determine the fractional composition of copper and tin in each mixture first, then sum them up for the final mixture.
Mixture A: Copper and Tin in the ratio 1 : 2.
Total parts in Mixture A = \(1 + 2 = 3\).
Fraction of Copper in A = \(\frac{1}{3}\).
Fraction of Tin in A = \(\frac{2}{3}\).
Mixture B: Copper and Tin in the ratio 1 : 3.
Total parts in Mixture B = \(1 + 3 = 4\).
Fraction of Copper in B = \(\frac{1}{4}\).
Fraction of Tin in B = \(\frac{3}{4}\).
Combining equal quantities: Let's assume we take 1 unit of Mixture A and 1 unit of Mixture B. (The actual quantity doesn't matter as long as they are equal, as we are looking for a ratio).
Amount of Copper from Mixture A = \[1 \times \frac{1}{3} = \frac{1}{3}\].
Amount of Tin from Mixture A = \[1 \times \frac{2}{3} = \frac{2}{3}\].
Amount of Copper from Mixture B = \[1 \times \frac{1}{4} = \frac{1}{4}\].
Amount of Tin from Mixture B = \[1 \times \frac{3}{4} = \frac{3}{4}\].
Total Copper and Total Tin in Mixture C:
Total Copper in C = (Copper from A) + (Copper from B)
\[ = \frac{1}{3} + \frac{1}{4} = \frac{4 + 3}{12} = \frac{7}{12} \]Total Tin in C = (Tin from A) + (Tin from B)
\[ = \frac{2}{3} + \frac{3}{4} = \frac{8 + 9}{12} = \frac{17}{12} \]Ratio of Copper to Tin in Mixture C:
\[ \text{Copper} : \text{Tin} = \frac{7}{12} : \frac{17}{12} \]Multiplying both sides by 12 to remove the denominators:
\[ \text{Copper} : \text{Tin} = 7 : 17 \]Correct Option: A) 7 : 17 is the correct ratio of copper and tin in mixture C.