This problem involves trigonometry, specifically angles of elevation and depression. We can draw a diagram to represent the situation and use trigonometric ratios (tangent) to find the unknown height of the tower.
Let AB be the cliff of height 25 m. Let CD be the tower. Let P be the point of observation at the top of the cliff (A). Draw a horizontal line AE from A parallel to the ground BC. Let the height of the tower be \(h\). So, \(CD = h\).
The angle of elevation of the top of the tower (D) from A is \(\theta\). So, \(\angle DAE = \theta\).
The angle of depression of the foot of the tower (C) from A is also \(\theta\). So, \(\angle EAC = \theta\).
Since AE is parallel to BC, \(\angle ACB = \angle EAC = \theta\) (alternate interior angles).
In right-angled triangle \(\triangle ABC\):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \] \[ \tan \theta = \frac{25}{x} \quad \text{(Equation 1)} \]In right-angled triangle \(\triangle AED\):
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{DE}{AE} \] \[ \tan \theta = \frac{y}{x} \quad \text{(Equation 2)} \]From Equation 1 and Equation 2:
\[ \frac{25}{x} = \frac{y}{x} \]Since \(x \neq 0\), we can cancel \(x\) from both sides:
\[ y = 25 \text{ m} \]The height of the tower \(CD = CE + ED = AB + y\).
\[ CD = 25 \text{ m} + 25 \text{ m} \] \[ CD = 50 \text{ m} \]Correct Option: D) 50 m
The question asks for the specific name of the skill development programme launched by the Government of India. This requires knowledge of various government schemes related to skill development.
Correct Option: D) Pradhan Mantri Kaushal Vikas Yojana is the flagship skill development scheme of the Ministry of Skill Development and Entrepreneurship, Government of India. It aims to enable a large number of Indian youth to take up industry-relevant skill training that will help them in securing a better livelihood.
The question asks to identify the painter of "In Memoriam," a significant artwork related to the 1857 Mutiny in India. This requires knowledge of prominent artists and their works associated with this historical event.
Correct Option: D) Joseph Noel Paton is the correct answer. His painting "In Memoriam" (1858) is a poignant depiction of British women and children huddled together, awaiting their fate during the Indian Mutiny of 1857. It reflects the Victorian sentiment and anxieties surrounding the events.
To find the ratio A : D from given consecutive ratios A : B, B : C, and C : D, we can multiply these ratios. This method allows for the cancellation of intermediate terms (B and C) to directly obtain the ratio of A to D.
Write down the given ratios in fractional form:
\[ \frac{A}{B} = \frac{15}{16} \]\[ \frac{B}{C} = \frac{8}{25} \]\[ \frac{C}{D} = \frac{20}{9} \]To find A : D, multiply these fractions:
\[ \frac{A}{D} = \frac{A}{B} \times \frac{B}{C} \times \frac{C}{D} \]Substitute the given values into the equation:
\[ \frac{A}{D} = \frac{15}{16} \times \frac{8}{25} \times \frac{20}{9} \]Perform the multiplication and simplify the expression by canceling common factors:
Cancel 8 from 16: \(16 \div 8 = 2\)
Cancel 5 from 15 and 25: \(15 \div 5 = 3\), \(25 \div 5 = 5\)
Cancel 5 from 20: \(20 \div 5 = 4\)
Cancel 2 from 4: \(4 \div 2 = 2\)
Cancel 3 from 9: \(9 \div 3 = 3\)
The expression becomes:
\[ \frac{A}{D} = \frac{\cancel{15}^{\text{3}}}{\cancel{16}^{\text{2}}} \times \frac{\cancel{8}^{\text{1}}}{\cancel{25}^{\text{5}}} \times \frac{\cancel{20}^{\text{4}}}{\cancel{9}^{\text{3}}} \]\[ \frac{A}{D} = \frac{3}{2} \times \frac{1}{5} \times \frac{4}{3} \]\[ \frac{A}{D} = \frac{\cancel{3}^{\text{1}}}{2} \times \frac{1}{\cancel{5}^{\text{1}}} \times \frac{\cancel{4}^{\text{2}}}{\cancel{3}^{\text{1}}} \]\[ \frac{A}{D} = \frac{1}{1} \times \frac{1}{1} \times \frac{2}{1} \]\[ \frac{A}{D} = \frac{2}{3} \]Therefore, A : D = 2 : 3.
Correct Option: D) 2 : 3
We are given two equations involving \(x\) and \(y\): \(x^3 + y^3 = 35\) and \(x + y = 5\). We need to find the value of \(x^4 + y^4\). We can use algebraic identities to find intermediate values like \(xy\) and then use these to calculate \(x^4 + y^4\).
Use the identity for the sum of cubes:
The identity for the sum of cubes is \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Applying this to our given equation:
\[ x^3 + y^3 = (x+y)(x^2 - xy + y^2) \]Substitute the given values:
We know \(x^3 + y^3 = 35\) and \(x + y = 5\). Substitute these into the identity:
\[ 35 = 5(x^2 - xy + y^2) \]Solve for \(x^2 - xy + y^2\):
Divide both sides by 5:
\[ \frac{35}{5} = x^2 - xy + y^2 \]\[ 7 = x^2 - xy + y^2 \quad \text{(Equation 1)} \]Relate \(x^2 + y^2\) to \((x+y)^2\):
We know the identity \((x+y)^2 = x^2 + 2xy + y^2\). From this, we can write \(x^2 + y^2 = (x+y)^2 - 2xy\).
Substitute this into Equation 1:
Replace \(x^2 + y^2\) in Equation 1:
\[ 7 = ((x+y)^2 - 2xy) - xy \]\[ 7 = (x+y)^2 - 3xy \]Substitute \(x+y=5\) and solve for \(xy\):
Substitute \(x+y=5\) into the equation:
\[ 7 = (5)^2 - 3xy \]\[ 7 = 25 - 3xy \]\[ 3xy = 25 - 7 \]\[ 3xy = 18 \]\[ xy = \frac{18}{3} \]\[ xy = 6 \]Calculate \(x^2 + y^2\):
Using \(x^2 + y^2 = (x+y)^2 - 2xy\):
\[ x^2 + y^2 = (5)^2 - 2(6) \]\[ x^2 + y^2 = 25 - 12 \]\[ x^2 + y^2 = 13 \]Calculate \(x^4 + y^4\):
We can find \(x^4 + y^4\) by squaring \(x^2 + y^2\):
\[ (x^2 + y^2)^2 = (x^2)^2 + 2x^2y^2 + (y^2)^2 \]\[ (x^2 + y^2)^2 = x^4 + y^4 + 2(xy)^2 \]Rearrange to solve for \(x^4 + y^4\):
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2 \]Substitute the values of \(x^2 + y^2\) and \(xy\):
We found \(x^2 + y^2 = 13\) and \(xy = 6\). Substitute these values:
\[ x^4 + y^4 = (13)^2 - 2(6)^2 \]\[ x^4 + y^4 = 169 - 2(36) \]\[ x^4 + y^4 = 169 - 72 \]\[ x^4 + y^4 = 97 \]B) 97
The task is to identify the word-pair that has a different relationship compared to the other three. We need to analyze the relationship between the words in each pair (synonyms, antonyms, cause-effect, etc.) and find the odd one out.
D) Negligent : Attentive — "Negligent" means careless, while "Attentive" means careful or observant. These words are antonyms, representing opposite qualities.
The question asks to identify a tiger reserve from the given options. This requires knowledge of the designated tiger reserves in India.
Correct Option: A) Jim Corbett National Park is indeed a prominent tiger reserve in India, being the first one declared under Project Tiger.
The question asks to identify which of the given options is NOT a web browser. To answer this, we need to know the definition of a web browser and recognize which options fit this definition and which do not.
Correct Option: A) Google is primarily a search engine and a technology company, not a web browser. While Google develops the Chrome browser, "Google" itself refers to the company or its search engine service.
The question asks about the visual defect caused by an excessive curvature of the eye lens. We need to identify which refractive error corresponds to this condition.
Correct Option: D) myopia
Myopia, also known as nearsightedness, occurs when the eye lens has an excessive curvature (or the eyeball is too long). This causes light rays from distant objects to converge and form an image in front of the retina, leading to blurred vision for distant objects.
To find the correct mean, we first need to calculate the incorrect sum of all items using the given incorrect mean and the number of items. Then, we adjust this sum by subtracting the wrongly taken values and adding the correct values. Finally, we divide the corrected sum by the total number of items to get the correct mean.
The mean is calculated as the sum of all items divided by the number of items. So, the sum of items is the mean multiplied by the number of items.
\[ \text{Incorrect Sum} = \text{Incorrect Mean} \times \text{Number of Items} \]Given:
Two items were wrongly taken:
The total value that was wrongly included in the sum is the sum of the wrongly taken values.
\[ \text{Sum of Wrong Values} = 83 + 76 = 159 \]The total value that should have been included is the sum of the correct values.
\[ \text{Sum of Correct Values} = 38 + 46 = 84 \]The net change required in the sum is:
\[ \text{Correction} = \text{Sum of Correct Values} - \text{Sum of Wrong Values} \] \[ \text{Correction} = 84 - 159 = -75 \]Adjust the incorrect sum by adding the correction.
\[ \text{Correct Sum} = \text{Incorrect Sum} + \text{Correction} \] \[ \text{Correct Sum} = 6750 + (-75) = 6750 - 75 = 6675 \]Divide the correct sum by the number of items.
\[ \text{Correct Mean} = \frac{\text{Correct Sum}}{\text{Number of Items}} \] \[ \text{Correct Mean} = \frac{6675}{150} \] \[ \text{Correct Mean} = 44.5 \]A) 44.5 is the correct mean after adjusting for the errors in the calculation.