The question asks for the total number of elements known as of 2022. This is a factual question related to the field of chemistry and general science, specifically concerning the periodic table of elements.
D) 118. As of 2022, the periodic table officially recognizes 118 elements. This includes elements from hydrogen (atomic number 1) to Oganesson (atomic number 118).
If 4 sin²θ = 1 and θ is an acute angle, then the value of cos²θ + tan²θ = ________.
\(\frac{13}{12}\)
1
\(\frac{5}{13}\)
\(\frac{1}{4}\)
The problem provides an equation involving \(\sin^2\theta\) and states that \(\theta\) is an acute angle. We need to find the value of an expression involving \(\cos^2\theta\) and \(\tan^2\theta\). The strategy is to first solve for \(\sin\theta\), then use trigonometric identities to find \(\cos\theta\) and \(\tan\theta\), and finally substitute these values into the required expression.
Given the equation:
\[ 4 \sin^2\theta = 1 \]Solve for \(\sin^2\theta\):
\[ \sin^2\theta = \frac{1}{4} \]Since \(\theta\) is an acute angle, \(\sin\theta\) must be positive. Take the square root of both sides:
\[ \sin\theta = \sqrt{\frac{1}{4}} = \frac{1}{2} \]We know that for an acute angle, \(\sin\theta = \frac{1}{2}\) implies \(\theta = 30^\circ\). However, we don't need to find \(\theta\) explicitly. We can use the identity \(\sin^2\theta + \cos^2\theta = 1\) to find \(\cos^2\theta\):
\[ \cos^2\theta = 1 - \sin^2\theta \]\[ \cos^2\theta = 1 - \frac{1}{4} \]\[ \cos^2\theta = \frac{4-1}{4} = \frac{3}{4} \]Next, find \(\tan^2\theta\). We know that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), so \(\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}\):
\[ \tan^2\theta = \frac{\frac{1}{4}}{\frac{3}{4}} \]\[ \tan^2\theta = \frac{1}{4} \times \frac{4}{3} = \frac{1}{3} \]Finally, substitute the values of \(\cos^2\theta\) and \(\tan^2\theta\) into the expression \(\cos^2\theta + \tan^2\theta\):
\[ \cos^2\theta + \tan^2\theta = \frac{3}{4} + \frac{1}{3} \]To add these fractions, find a common denominator, which is 12:
\[ \cos^2\theta + \tan^2\theta = \frac{3 \times 3}{4 \times 3} + \frac{1 \times 4}{3 \times 4} \]\[ \cos^2\theta + \tan^2\theta = \frac{9}{12} + \frac{4}{12} \]\[ \cos^2\theta + \tan^2\theta = \frac{9+4}{12} = \frac{13}{12} \]A) The calculated value of \(\cos^2\theta + \tan^2\theta\) is \(\frac{13}{12}\).
The question asks to identify the Jain principle that means non-acquisition. This requires knowledge of the five main vows (Mahavratas) in Jainism and their respective meanings.
Correct Option: A) Aparigraha is the Jain principle that means non-acquisition or non-possession. It advocates for limiting one's possessions and desires, recognizing that excessive attachment to material things leads to suffering and hinders spiritual progress.
The question asks for the chemical formula of marble. Marble is a metamorphic rock composed primarily of a specific mineral. Identifying this mineral and its chemical composition will lead to the correct formula.
Correct Option: B) \(\text{CaCO}_3\) is the chemical formula for calcium carbonate, which is the main component of marble.
The question asks for the definition of the focal length of a spherical mirror. We need to recall the standard terminology and definitions used in optics for spherical mirrors.
Correct Option: A) F and pole. The focal length is precisely defined as the distance between the principal focus (F) and the pole (P) of the mirror.
To simplify the given expression, we will follow the order of operations, often remembered by the acronym BODMAS/PEMDAS: Brackets/Parentheses, Orders/Exponents, Division/Multiplication (from left to right), and Addition/Subtraction (from left to right). The 'of' operation is treated as multiplication and is performed after brackets but before other multiplications and divisions.
A) 431 is the correct result after applying the BODMAS rule systematically.
The question asks to identify one of the contrasting pairs of characters studied by Gregor Mendel in his experiments with pea plants. To answer this, recall Mendel's seven pairs of contrasting traits.
D) purple and white flowers
This is one of the seven contrasting pairs of characters (flower color) studied by Mendel, where purple was dominant over white.
A man buys a second-hand car for Rs 3,30,000 and spends Rs 40,000 on its repair. What should be its selling price if he wants to earn a 15% profit?
To find the selling price, we first need to calculate the total cost price of the car, which includes the purchase price and the repair cost. Then, we will calculate the desired profit amount (15% of the total cost price) and add it to the total cost price to get the selling price.
The total cost price is the sum of the buying price and the repair cost.
\[ \text{Total CP} = \text{Buying Price} + \text{Repair Cost} \]\[ \text{Total CP} = \text{Rs } 3,30,000 + \text{Rs } 40,000 \]\[ \text{Total CP} = \text{Rs } 3,70,000 \]The man wants to earn a 15% profit on the total cost price.
\[ \text{Profit Amount} = 15\% \text{ of Total CP} \]\[ \text{Profit Amount} = \frac{15}{100} \times 3,70,000 \]\[ \text{Profit Amount} = 15 \times 3,700 \]\[ \text{Profit Amount} = \text{Rs } 55,500 \]The selling price is the total cost price plus the profit amount.
\[ \text{Selling Price (SP)} = \text{Total CP} + \text{Profit Amount} \]\[ \text{SP} = 3,70,000 + 55,500 \]\[ \text{SP} = \text{Rs } 4,25,500 \]Thus, the selling price should be Rs 4,25,500 to earn a 15% profit.
B) 4,25,500
To find the total surface area of a closed cube, we need to calculate the area of each of its six faces and sum them up. Since all faces of a cube are identical squares, we can find the area of one face and multiply it by six.
Correct Option: D) The total surface area of the closed cube is \(384 \, \text{cm}^2\).
A shopkeeper has five deliveries of different varieties of dry fruits—cashew nuts, almonds, pistachios, walnuts, and pine nuts—each delivery planned on a different date, viz. 1st, 6th, 12th, 20th and 30th of July 2022. The delivery of pine nuts is planned exactly after the delivery of almonds, which is planned exactly after the delivery of pistachios. The delivery of cashew nuts is planned exactly between the deliveries of pine nuts and walnuts. If the delivery of walnuts is the last delivery planned, on which date is the delivery of almonds planned?
The problem requires us to determine the delivery date of almonds based on a series of sequential delivery conditions and a fixed last delivery. We will list the dry fruits and their delivery dates, then use the given clues to establish the order of deliveries and match them to the dates.
Correct Option: D) 6 th July