We are given the sides of a right-angled triangle in terms of a variable 'x'. We need to find the area of the triangle. The key to solving this problem is to use the Pythagorean theorem to find the value of 'x', and then calculate the lengths of the perpendicular sides (base and height) to determine the area.
Identify the sides of the right triangle:
The given sides are \((x - 13)\), \((x - 26)\), and \(x\). In a right-angled triangle, the hypotenuse is always the longest side. Comparing the expressions, \(x\) must be the hypotenuse because \((x - 13) < x\) and \((x - 26) < x\).
So, let the sides be:
Apply the Pythagorean theorem:
For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, \(a^2 + b^2 = c^2\).
Substituting the given expressions:
\[ (x - 26)^2 + (x - 13)^2 = x^2 \]Expand and simplify the equation:
Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
Rearrange into a quadratic equation:
Subtract \(x^2\) from both sides:
Solve the quadratic equation for x:
We can solve this by factoring. We need two numbers that multiply to 845 and add up to -78. Let's find the factors of 845:
So, \(845 = 5 \times 13 \times 13 = 65 \times 13\).
Notice that \(65 + 13 = 78\). So, the numbers are -65 and -13.
Check the validity of x values:
If \(x = 13\), then one side would be \((x - 13) = 13 - 13 = 0\), which is not possible for a triangle. Also, \((x - 26) = 13 - 26 = -13\), which is also not possible.
Therefore, \(x = 65\) is the only valid solution.
Calculate the lengths of the sides:
Substitute \(x = 65\) into the expressions for the sides:
These sides form a Pythagorean triplet \((39, 52, 65)\), which is a multiple of the basic triplet \((3, 4, 5)\) (i.e., \(13 \times 3, 13 \times 4, 13 \times 5\)).
Calculate the area of the triangle:
The area of a right-angled triangle is given by \(\frac{1}{2} \times \text{base} \times \text{height}\). The base and height are the two perpendicular sides.
Correct Option: B) 1014
If tanθ = \( \frac{7}{8} \), then evaluate \(\frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)(\cot\theta)}\).
\( \frac{8}{7} \)
\( \frac{7}{8} \)
\( \frac{49}{64} \)
\( \frac{64}{49} \)
The problem asks us to evaluate a trigonometric expression given the value of \(\tan\theta\). We can simplify the given expression using trigonometric identities and then substitute the value of \(\tan\theta\).
The given expression is:
\[ \frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)(\cot\theta)} \]Use the algebraic identity \((a+b)(a-b) = a^2 - b^2\):
\[ (1+\sin\theta)(1-\sin\theta) = 1^2 - \sin^2\theta = 1 - \sin^2\theta \] \[ (1+\cos\theta)(1-\cos\theta) = 1^2 - \cos^2\theta = 1 - \cos^2\theta \]Substitute these into the expression:
\[ \frac{1 - \sin^2\theta}{ (1 - \cos^2\theta)(\cot\theta) } \]Use the fundamental trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\), which implies \(1 - \sin^2\theta = \cos^2\theta\) and \(1 - \cos^2\theta = \sin^2\theta\):
\[ \frac{\cos^2\theta}{ (\sin^2\theta)(\cot\theta) } \]Recall that \(\cot\theta = \frac{\cos\theta}{\sin\theta}\):
\[ \frac{\cos^2\theta}{ (\sin^2\theta)\left(\frac{\cos\theta}{\sin\theta}\right) } \]Simplify the denominator:
\[ (\sin^2\theta)\left(\frac{\cos\theta}{\sin\theta}\right) = \sin\theta \cdot \cos\theta \]Substitute this back into the expression:
\[ \frac{\cos^2\theta}{\sin\theta \cdot \cos\theta} \]Cancel out one \(\cos\theta\) term from the numerator and denominator:
\[ \frac{\cos\theta}{\sin\theta} \]This simplifies to \(\cot\theta\):
\[ \cot\theta \]We are given \(\tan\theta = \frac{7}{8}\). Since \(\cot\theta = \frac{1}{\tan\theta}\), we can find the value of \(\cot\theta\):
\[ \cot\theta = \frac{1}{\frac{7}{8}} = \frac{8}{7} \]The problem can be solved by setting up a linear equation based on the given information. We need to represent the unknown number with a variable and then translate the verbal description into a mathematical equation. Once the equation is formed, we can solve for the variable.
Thus, the unknown number is 94.
Correct Option: A) 94
The sides of a rectangular field are 169 m and 154 m long. Its area is equal to the area of a circular field. What is the circumference (in m) of the circular field?
Take π = \( \frac{22}{7} \)
The problem requires us to find the circumference of a circular field given that its area is equal to the area of a rectangular field. We will first calculate the area of the rectangular field, then use this area to find the radius of the circular field, and finally calculate its circumference.
The sides of the rectangular field are given as 169 m and 154 m.
\[ \text{Area of rectangle} = \text{length} \times \text{width} \]\[ \text{Area of rectangle} = 169 \text{ m} \times 154 \text{ m} \]\[ \text{Area of rectangle} = 26026 \text{ m}^2 \]It is given that the area of the circular field is equal to the area of the rectangular field.
\[ \text{Area of circle} = \text{Area of rectangle} \]\[ \pi r^2 = 26026 \text{ m}^2 \]We are given \( \pi = \frac{22}{7} \).
\[ \frac{22}{7} \times r^2 = 26026 \]\[ r^2 = \frac{26026 \times 7}{22} \]\[ r^2 = 1183 \times 7 \]\[ r^2 = 8281 \]\[ r = \sqrt{8281} \]\[ r = 91 \text{ m} \]The formula for the circumference of a circle is \( 2\pi r \).
\[ \text{Circumference} = 2 \times \frac{22}{7} \times 91 \]\[ \text{Circumference} = 2 \times 22 \times \frac{91}{7} \]\[ \text{Circumference} = 2 \times 22 \times 13 \]\[ \text{Circumference} = 44 \times 13 \]\[ \text{Circumference} = 572 \text{ m} \]C) The calculated circumference of the circular field is 572 m, which matches option C.
The question asks us to identify the actor who received the 2025 International Goya Award for his contributions to global cinema. We need to recall the recipient of the award and understand its significance in recognising international film personalities.
Understand the International Goya Award:
The International Goya Award is presented by the Academy of Cinematographic Arts and Sciences of Spain.
It recognises international film personalities who have made outstanding contributions to cinema and helped connect audiences across cultures.
Identify the 2025 award recipient:
American actor and producer Richard Gere received the International Goya Award at the 39th Goya Awards ceremony, held on 8 February 2025 in Granada, Spain.
The award recognised his contributions to international cinema and his longstanding commitment to social causes.
Understand Richard Gere's contributions to cinema:
Richard Gere is internationally recognised for his performances in several notable films, including:
His performances across different genres have contributed to his international recognition and influence in cinema.
Understand the significance of the honour:
Richard Gere became the fourth recipient of the International Goya Award.
The award was presented to him by Spanish actor Antonio Banderas during the ceremony.
It acknowledged Gere's cinematic achievements and his contributions to humanitarian and social causes.
Determine the correct answer:
The actor honoured with the 2025 International Goya Award was Richard Gere.
Therefore, Option B is the correct answer.
Correct Option: B) Richard Gere — American actor and producer Richard Gere received the 2025 International Goya Award at the 39th Goya Awards in Granada, Spain, in recognition of his outstanding contributions to global cinema and his commitment to social causes. Hence, Option B is the correct answer.
A) Sigourney Weaver: This is incorrect because Sigourney Weaver received the International Goya Award in 2024, not 2025.
C) Cate Blanchett: This is incorrect because Cate Blanchett received the inaugural International Goya Award in 2022, not 2025.
D) Juliette Binoche: This is incorrect because Juliette Binoche received the International Goya Award in 2023, not 2025.
The problem asks us to identify the letter-cluster that does not fit the pattern followed by the other three, based on the English alphabetical order. We need to analyze the positional values of the letters in each cluster and find a common rule.
C) PRV — The pattern for PRV is P (+2) R (+4) V. This is different from the pattern observed in the other three options.
Simplify: \( \sqrt[6]{729} + \sqrt[4]{65536} + \sqrt[3]{5832} + \sqrt{1764} \)
To simplify the given expression, we need to calculate the value of each root term separately and then sum them up. This involves finding the 6th root, 4th root, cube root, and square root of the given numbers. We can do this by prime factorization or by recognizing common powers of small integers.
The simplified value of the expression is 79.
Correct Option: D) 79
This is a logical reasoning problem involving the arrangement of boxes. We need to deduce the correct order of seven boxes based on the given clues and then determine the number of boxes below E.
This means A is immediately above D, and no other box is above A. So, the top two positions are fixed:
Position 7: A
Position 6: D
Since D is at position 6, there must be four boxes below D, and then F. This means F is at position 1 (the bottommost position).
Position 7: A
Position 6: D
Position 5: _
Position 4: _
Position 3: _
Position 2: _
Position 1: F
This means E and B are together as (E, B) in that order. Possible slots are (5,4), (4,3), (3,2).
Possible pairs for (C, _, _, G) or (G, _, _, C) are (5,1) or (4,1) if we consider the remaining slots. Let's list the remaining empty slots: 5, 4, 3, 2.
This is a crucial negative clue.
We have slots 5, 4, 3, 2 remaining. The pair (E, B) must occupy two consecutive slots. The pair (C, G) with two boxes in between means they must occupy slots like (5, 2) or (4, 1) or (3, 0) which is not possible. So, C and G must be in slots 5 and 2 (or vice versa).
If C is at 5 and G is at 2, then slots 4 and 3 are left for (E, B). So, E is at 4 and B is at 3. This arrangement is:
Position 7: A
Position 6: D
Position 5: C
Position 4: E
Position 3: B
Position 2: G
Position 1: F
Let's check all conditions:
This arrangement satisfies all conditions.
From the final arrangement, E is at position 4. The boxes below E are B, G, and F. There are 3 boxes below E.
Correct Option: D) 3
The problem involves identifying a pattern in letter sequences and numbers. We need to analyze the relationship between the letters and the numbers in the given pairs and then apply the same logic to find the missing term.
Correct Option: D) PM −1
The question asks about a specific milestone achieved by Virat Kohli in the 2025 IPL season as of April 15, 2025. This requires knowledge of recent cricketing records and statistics, specifically related to the Indian Premier League (IPL).
Correct Option: C) Registering a combined total of 1000 fours and sixes in IPL history