The problem asks us to find the missing number in a given series. We need to identify the pattern or rule governing the sequence of numbers.
Correct Option: B) 6
The question asks for the technique used to determine the depth of the sea and locate underwater objects. This involves understanding how different wave phenomena are used in various applications.
C) SONAR — SONAR (Sound Navigation And Ranging) is a technique that uses sound propagation (usually underwater, as in submarine navigation) to navigate, communicate with or detect objects on or under the surface of the water, such as other vessels, icebergs, and the seabed. It works by emitting ultrasonic waves and measuring the time it takes for the echoes to return, allowing for the calculation of distance and mapping of underwater features.
This question tests fundamental knowledge of chemical reactions, specifically the reaction between metals and acids. Understanding the general reaction pattern for metals with acids is key to identifying the gas produced.
Correct Option: C) Hydrogen is the gas typically produced when reactive metals react with acids. This is a standard chemical reaction taught in basic chemistry.
If cot θ = \( \frac{a}{b} \), then find \(\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}\).
\(\frac{b}{a}\)
\(\frac{a}{b}\)
\(\frac{a^2}{b^2}\)
\(\frac{a-b}{a+b}\)
The problem provides the value of \(\cot \theta\) and asks us to find the value of an expression involving \(\sin \theta\) and \(\cos \theta\). The most straightforward approach is to convert the given expression into terms of \(\cot \theta\) or \(\tan \theta\), as we already know the value of \(\cot \theta\).
We are given that \(\cot \theta = \frac{a}{b}\).
We need to find the value of the expression \[\frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}\].
To convert this expression into terms of \(\cot \theta\), we can divide both the numerator and the denominator by \(\sin \theta\). This is a common technique when dealing with expressions involving both \(\sin \theta\) and \(\cos \theta\).
\[ \frac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta} = \frac{\frac{\cos\theta}{\sin\theta} - \frac{\sin\theta}{\sin\theta}}{\frac{\cos\theta}{\sin\theta} + \frac{\sin\theta}{\sin\theta}} \]We know that \(\frac{\cos\theta}{\sin\theta} = \cot \theta\) and \(\frac{\sin\theta}{\sin\theta} = 1\).
Substitute these identities into the expression:
\[ \frac{\cot\theta - 1}{\cot\theta + 1} \]Now, substitute the given value of \(\cot \theta = \frac{a}{b}\) into this simplified expression:
\[ \frac{\frac{a}{b} - 1}{\frac{a}{b} + 1} \]To simplify the complex fraction, find a common denominator for the terms in the numerator and the denominator:
Numerator: \[\frac{a}{b} - 1 = \frac{a}{b} - \frac{b}{b} = \frac{a-b}{b}\]
Denominator: \[\frac{a}{b} + 1 = \frac{a}{b} + \frac{b}{b} = \frac{a+b}{b}\]
Substitute these back into the expression:
\[ \frac{\frac{a-b}{b}}{\frac{a+b}{b}} \]Cancel out the common denominator \(b\):
\[ \frac{a-b}{a+b} \]Correct Option: D) \(\frac{a-b}{a+b}\)
To find the volume of the cube, we first need to determine its side length. We are given that the numerical value of the surface area of the cube is equal to the numerical value of its volume. We will set up an equation using the formulas for surface area and volume of a cube and solve for the side length. Once the side length is found, we can calculate the volume.
Correct Option: C) 216 cubic units
This is a syllogism problem. We can solve it by drawing Venn diagrams for the given statements and then checking which conclusions logically follow from these diagrams.
Now let's evaluate each conclusion based on these relationships:
Therefore, conclusions I and IV logically follow.
Correct Option: C) Only conclusions I and IV follow
P, Q and R enter into a partnership investing Rs.35,000, Rs.45,000 and Rs.55,000. Find their respective share in the annual profit of Rs.40,500.
Rs.10,500, Rs.13,500, Rs.18,500
Rs.10,500, Rs.13,500, Rs.19,500
Rs.10,500, Rs.13,500, Rs.17,500
Rs.10,500, Rs.13,500, Rs.16,500
In a partnership, when partners invest capital for the same duration (implied as annual profit here), their share of the profit is directly proportional to the ratio of their investments. We will first find the ratio of their investments, then distribute the total profit according to this ratio.
The ratio of investments for P, Q, and R is:
\[ \text{P} : \text{Q} : \text{R} = 35,000 : 45,000 : 55,000 \]Divide each term by 1,000:
\[ 35 : 45 : 55 \]Now, divide each term by the greatest common divisor, which is 5:
\[ \frac{35}{5} : \frac{45}{5} : \frac{55}{5} \]\[ 7 : 9 : 11 \]So, the ratio of their investments is \(7:9:11\).
Sum of ratios = \(7 + 9 + 11 = 27\)
The total annual profit is Rs. 40,500.
\(10,500 + 13,500 + 16,500 = 40,500\). This matches the given total profit.
D) Rs.10,500, Rs.13,500, Rs.16,500
The question asks to identify the Mughal emperor who was also known by the title "Alamgir." This requires knowledge of the regnal names or titles adopted by prominent Mughal rulers.
Correct Option: C) Aurangzeb was known by the title "Alamgir." He assumed this title upon his accession to the throne in 1658 and ruled for nearly 50 years, expanding the Mughal Empire to its greatest extent.
If \((x-k)\) is a factor of a polynomial \(P(x)\), then according to the Factor Theorem, \(P(k) = 0\). We will use this theorem for both given factors to form a system of two linear equations in terms of 'a' and 'b', and then solve for 'a' and 'b'.
Identify the polynomial and its factors:
The given polynomial is \(P(x) = x^2 + ax + b\).
The factors are \((x - 1)\) and \((x + 3)\).
Apply the Factor Theorem for the first factor \((x - 1)\):
If \((x - 1)\) is a factor, then \(P(1) = 0\).
\[ (1)^2 + a(1) + b = 0 \]\[ 1 + a + b = 0 \]\[ a + b = -1 \quad \text{(Equation 1)} \]Apply the Factor Theorem for the second factor \((x + 3)\):
If \((x + 3)\) is a factor, then \(P(-3) = 0\).
\[ (-3)^2 + a(-3) + b = 0 \]\[ 9 - 3a + b = 0 \]\[ -3a + b = -9 \quad \text{(Equation 2)} \]Solve the system of linear equations:
We have two equations:
1) \(a + b = -1\)
2) \(-3a + b = -9\)
Subtract Equation 2 from Equation 1:
\[ (a + b) - (-3a + b) = -1 - (-9) \]\[ a + b + 3a - b = -1 + 9 \]\[ 4a = 8 \]\[ a = \frac{8}{4} \]\[ a = 2 \]Substitute the value of 'a' back into Equation 1 to find 'b':
Using \(a + b = -1\):
\[ 2 + b = -1 \]\[ b = -1 - 2 \]\[ b = -3 \]State the values of 'a' and 'b':
So, \(a = 2\) and \(b = -3\).
Correct Option: B) 2, -3
The values of 'a' and 'b' are 2 and -3 respectively, which matches option B.