To find the length of the side of a rhombus, we use the formula for its area involving diagonals and the property that diagonals of a rhombus bisect each other at right angles. This forms right-angled triangles, allowing us to use the Pythagorean theorem.
Recall the area formula for a rhombus:
The area of a rhombus is given by the formula:
where \(d_1\) and \(d_2\) are the lengths of the diagonals.
Substitute the given values:
We are given that the Area is \(96 \text{ cm}^2\) and one diagonal, say \(d_1\), is \(16 \text{ cm}\). Let the other diagonal be \(d_2\).
Calculate the length of the second diagonal (\(d_2\)):
\[ 96 = 8 \times d_2 \]\[ d_2 = \frac{96}{8} \]\[ d_2 = 12 \text{ cm} \]Use the properties of a rhombus and the Pythagorean theorem:
The diagonals of a rhombus bisect each other at right angles. This means they divide the rhombus into four congruent right-angled triangles. If 's' is the side length of the rhombus, and the diagonals are \(d_1\) and \(d_2\), then the legs of each right-angled triangle are \(\frac{d_1}{2}\) and \(\frac{d_2}{2}\), and the hypotenuse is 's'.
Calculate half the lengths of the diagonals:
\[ \frac{d_1}{2} = \frac{16}{2} = 8 \text{ cm} \]\[ \frac{d_2}{2} = \frac{12}{2} = 6 \text{ cm} \]Apply the Pythagorean theorem:
For one of the right-angled triangles:
Correct Option: D) 10 cm
The question asks to identify the predominant soil type found in the Deccan Plateau region of India. This requires knowledge of the geographical distribution of major soil types in India.
Correct Option: C) Black soil — The Deccan Plateau, particularly the Deccan Trap region, is extensively covered by Black soil (Regur soil). This soil is formed from the weathering of basaltic rocks and is known for its dark color, high clay content, and fertility, especially for cotton cultivation.
To find the cost of 56 books, we first need to determine the cost of a single book. We can do this by dividing the total cost of 10 books by the number of books. Once we have the cost of one book, we can multiply it by 56 to find the total cost of 56 books.
Given that the cost of 10 books is ₹560.
Cost of 1 book \[= \frac{\text{Total cost of 10 books}}{\text{Number of books}}\]
\[ \text{Cost of 1 book} = \frac{560}{10} = \text{₹}56 \]
Now that we know the cost of one book is ₹56, we can find the cost of 56 books by multiplying the cost per book by 56.
Cost of 56 books \( = \text{Cost of 1 book} \times 56\)
\[ \text{Cost of 56 books} = 56 \times 56 \]
\[ 56 \times 56 = 3136 \]
So, the cost of 56 such books is ₹3,136.
Correct Option: D) ₹3,136
To find the missing number in the series, we need to identify the pattern or rule governing the sequence of numbers. This usually involves looking at the differences between consecutive terms.
Thus, the missing number in the series is 39.
Correct Option: B) 39
The question asks for the location where the World Sustainable Development Summit (WSDS) 2019 was organized. This is a factual question requiring knowledge of major international conferences and their venues.
B) New Delhi, India: The World Sustainable Development Summit (WSDS) 2019 was organized in New Delhi, India, from February 11-13, 2019. The theme for the 2019 summit was "Attaining the 2030 Agenda: Delivering on Our Promise".
What was the rank of India in the list of world's most competitive economies in the IMD World Competitiveness Rankings 2019?
The question asks for India's rank in the IMD World Competitiveness Rankings for the year 2019. This requires recalling specific factual information related to international economic reports and rankings.
Correct Option: C) 43 rd
India was ranked 43rd in the IMD World Competitiveness Rankings 2019, improving its position from the previous year.
Two walls of equal height are on either side of a road, which is 100 m wide. At a point on the road two stairs are along the two walls in such a way that they indicate the two angles of elevation from that point at 60° and 30°. The length of the longer stair is:
75 m
50\(\sqrt{3}\)m
\(\frac{50\sqrt{3}}{3}\) m
50 m
This problem involves trigonometry, specifically the tangent function, to relate the height of the walls, the angles of elevation, and the distances from the point on the road to the base of each wall. We will set up two right-angled triangles and solve for the height of the walls and the lengths of the stairs.
Let the height of the two walls be \(h\). Let the point on the road be P. Let the total width of the road be \(W = 100 \text{ m}\).
Let the distance from point P to the base of the first wall be \(x\). Then the distance from point P to the base of the second wall will be \((W - x)\), which is \((100 - x)\).
For the first wall, the angle of elevation is \(60^\circ\). Using the tangent function:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x} \] \[ \sqrt{3} = \frac{h}{x} \implies h = x\sqrt{3} \quad \text{(Equation 1)} \]For the second wall, the angle of elevation is \(30^\circ\). Using the tangent function:
\[ \tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{100 - x} \] \[ \frac{1}{\sqrt{3}} = \frac{h}{100 - x} \implies h = \frac{100 - x}{\sqrt{3}} \quad \text{(Equation 2)} \]Equate Equation 1 and Equation 2 to find \(x\):
\[ x\sqrt{3} = \frac{100 - x}{\sqrt{3}} \] \[ x\sqrt{3} \cdot \sqrt{3} = 100 - x \] \[ 3x = 100 - x \] \[ 4x = 100 \] \[ x = 25 \text{ m} \]Now, find the height \(h\) using \(x = 25 \text{ m}\) in Equation 1:
\[ h = 25\sqrt{3} \text{ m} \]The length of the stairs can be found using the sine function or by the Pythagorean theorem. Let \(L_1\) be the length of the stair with the \(60^\circ\) angle and \(L_2\) be the length of the stair with the \(30^\circ\) angle.
For the first stair (angle \(60^\circ\), adjacent side \(x=25\)):
\[ \cos(60^\circ) = \frac{x}{L_1} \implies L_1 = \frac{x}{\cos(60^\circ)} = \frac{25}{1/2} = 50 \text{ m} \]Alternatively, using sine:
\[ \sin(60^\circ) = \frac{h}{L_1} \implies L_1 = \frac{h}{\sin(60^\circ)} = \frac{25\sqrt{3}}{\sqrt{3}/2} = 50 \text{ m} \]For the second stair (angle \(30^\circ\), adjacent side \(100-x = 75\)):
\[ \cos(30^\circ) = \frac{100 - x}{L_2} \implies L_2 = \frac{75}{\cos(30^\circ)} = \frac{75}{\sqrt{3}/2} = \frac{150}{\sqrt{3}} = \frac{150\sqrt{3}}{3} = 50\sqrt{3} \text{ m} \]Alternatively, using sine:
\[ \sin(30^\circ) = \frac{h}{L_2} \implies L_2 = \frac{h}{\sin(30^\circ)} = \frac{25\sqrt{3}}{1/2} = 50\sqrt{3} \text{ m} \]Comparing the lengths: \(L_1 = 50 \text{ m}\) and \(L_2 = 50\sqrt{3} \text{ m}\). Since \(\sqrt{3} \approx 1.732\), \(L_2 \approx 50 \times 1.732 = 86.6 \text{ m}\).
The longer stair is \(L_2 = 50\sqrt{3} \text{ m}\).
Correct Option: B) \(50\sqrt{3}\)m
The question asks about the general effect of temperature on the rate constant of a chemical reaction. This is a common observation in chemical kinetics, often explained by the Arrhenius equation. We need to recall the general rule regarding temperature and reaction rates.
B) 10 degrees — A \(10^\circ\text{C}\) rise in temperature is a common approximation for the doubling of the rate constant of many chemical reactions. This observation is consistent with the Arrhenius equation, which shows an exponential dependence of the rate constant on temperature.
The question asks to identify the kingdom that Lord Dalhousie referred to as "a cherry". This phrase is famously associated with his policy of annexation and specifically refers to a kingdom he annexed on the pretext of misgovernance.
C) Kingdom of Awadh — Lord Dalhousie famously described the Kingdom of Awadh as "a cherry that will drop into our mouth one day" before its annexation in 1856 on the pretext of misgovernance.
To identify a right-angled triangle, we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. If the sides of a triangle are \(a\), \(b\), and \(c\), and \(c\) is the longest side, then for it to be a right-angled triangle, the condition \(a^2 + b^2 = c^2\) must be satisfied.
Given sides: \(a = 7 \text{ cm}\), \(b = 24 \text{ cm}\), \(c = 25 \text{ cm}\).
Here, the longest side is \(c = 25 \text{ cm}\).
Calculate the sum of the squares of the two shorter sides:
\[ a^2 + b^2 = (7)^2 + (24)^2 \]\[ a^2 + b^2 = 49 + 576 \]\[ a^2 + b^2 = 625 \]Calculate the square of the longest side:
\[ c^2 = (25)^2 \]\[ c^2 = 625 \]Since \(a^2 + b^2 = c^2\) (\(625 = 625\)), the triangle with sides 7 cm, 24 cm, and 25 cm is a right-angled triangle.