The problem involves finding an angle in a triangle using given angles and the property that the sum of angles in a triangle is 180 degrees. We will use the perpendicular condition to find one angle, then use the angle sum property in different triangles to deduce the required angle.
Analyze the given information:
Find \(\angle C\) in \(\triangle BDC\):
Since \(BD \perp AC\), \(\triangle BDC\) is a right-angled triangle with \(\angle BDC = 90^\circ\).
The sum of angles in \(\triangle BDC\) is \(180^\circ\).
\[ \angle C + \angle DBC + \angle BDC = 180^\circ \]\[ \angle C + 39^\circ + 90^\circ = 180^\circ \]\[ \angle C + 129^\circ = 180^\circ \]\[ \angle C = 180^\circ - 129^\circ \]\[ \angle C = 51^\circ \]Find \(\angle AEB\) using \(\triangle AEC\):
Consider \(\triangle AEC\). We know \(\angle CAE = 16^\circ\) and \(\angle ACE = \angle C = 51^\circ\).
The angle \(\angle AEB\) is an exterior angle to \(\triangle AEC\) at vertex \(E\).
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
\[ \angle AEB = \angle CAE + \angle ACE \]\[ \angle AEB = 16^\circ + 51^\circ \]\[ \angle AEB = 67^\circ \]C) \(67^\circ\) is the correct measure for \(\angle AEB\), as calculated by using the exterior angle property of a triangle.
To find the average speed for the entire journey, we need to calculate the total distance traveled and the total time taken. The average speed is then the total distance divided by the total time. Since the journey is divided into three equal parts in terms of distance, we can assume a total distance and calculate the time for each part.
Let the total distance of the journey be \(D\) km. Since the journey is divided into three equal parts, each part will have a distance of \(\frac{D}{3}\) km.
Calculate the time taken for each part of the journey:
For the first one-third of the journey:
\[ \text{Distance}_1 = \frac{D}{3} \]\[ \text{Speed}_1 = 2 \text{ km/hr} \]\[ \text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{D/3}{2} = \frac{D}{6} \text{ hours} \]For the next one-third of the journey:
\[ \text{Distance}_2 = \frac{D}{3} \]\[ \text{Speed}_2 = 87 \text{ km/hr} \]\[ \text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{D/3}{87} = \frac{D}{261} \text{ hours} \]For the remaining one-third of the journey:
\[ \text{Distance}_3 = \frac{D}{3} \]\[ \text{Speed}_3 = 53 \text{ km/hr} \]\[ \text{Time}_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{D/3}{53} = \frac{D}{159} \text{ hours} \]Calculate the total time taken for the entire journey:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3 \]\[ \text{Total Time} = \frac{D}{6} + \frac{D}{261} + \frac{D}{159} \]\[ \text{Total Time} = D \left( \frac{1}{6} + \frac{1}{261} + \frac{1}{159} \right) \]Find a common denominator or calculate the sum of fractions:
To simplify, we can use the formula for average speed when distance is divided into equal parts:
\[ \text{Average Speed} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \dots + \frac{1}{v_n}} \]where \(n\) is the number of parts and \(v_i\) are the speeds for each part.In this case, \(n=3\), \(v_1=2\), \(v_2=87\), \(v_3=53\).
\[ \text{Average Speed} = \frac{3}{\frac{1}{2} + \frac{1}{87} + \frac{1}{53}} \]Calculate the sum of the reciprocals:
\[ \frac{1}{2} = 0.5 \]\[ \frac{1}{87} \approx 0.011494 \]\[ \frac{1}{53} \approx 0.018868 \]\[ \text{Sum} = 0.5 + 0.011494 + 0.018868 = 0.530362 \]Calculate the average speed:
\[ \text{Average Speed} = \frac{3}{0.530362} \approx 5.6565 \text{ km/hr} \]Rounding off to 1 decimal place, the average speed is \(5.7\) km/hr.
Correct Option: B) 5.7
Find the value of 5-23 × 16-23 × 21-23.
\(\frac{1}{1725^{23}}\)
\(\frac{1}{1680^{23}}\)
\(\frac{1}{1726^{23}}\)
\(\frac{1}{1634^{23}}\)
The problem requires us to simplify an expression involving multiplication of numbers raised to the same negative power. We will use the exponent rule \((a \times b \times c)^m = a^m \times b^m \times c^m\) and the rule for negative exponents \(a^{-n} = \frac{1}{a^n}\).
The given expression is \(5^{-23} \times 16^{-23} \times 21^{-23}\).
Since all terms have the same exponent \(-23\), we can combine the bases:
\[ 5^{-23} \times 16^{-23} \times 21^{-23} = (5 \times 16 \times 21)^{-23} \]First, multiply \(5 \times 16\):
\[ 5 \times 16 = 80 \]Next, multiply the result by \(21\):
\[ 80 \times 21 = 1680 \]Now the expression becomes:
\[ (1680)^{-23} \]Using this rule, we can rewrite the expression with a positive exponent:
\[ (1680)^{-23} = \frac{1}{1680^{23}} \]B) The simplified value of the expression is \(\frac{1}{1680^{23}}\), which matches option B.
To determine which number divides 814242541, we will apply divisibility rules for each option provided. We will check for divisibility by 8, 12, 7, and 15.
Correct Option: C) The number 814242541 is divisible by 7, as shown by the divisibility test for 7, which resulted in 70, a multiple of 7.
We are given the total cost price of two speakers and the percentage loss/gain on each. We also know that there was no overall loss or gain. We need to find the cost price of the first speaker. We can set up equations based on the given information and solve for the unknown cost prices.
Let the cost price of the first speaker be \(CP_1\) and the cost price of the second speaker be \(CP_2\).
The total cost price of the two speakers is ₹1,200.
\[ CP_1 + CP_2 = 1200 \quad \text{(Equation 1)} \]The first speaker was sold at a loss of 8%. The loss amount on the first speaker is \(8\%\) of \(CP_1\).
\[ \text{Loss}_1 = 0.08 \times CP_1 \]The second speaker was sold at a gain of 22%. The gain amount on the second speaker is \(22\%\) of \(CP_2\).
\[ \text{Gain}_2 = 0.22 \times CP_2 \]On the whole, there was neither a loss nor a gain. This means the total loss equals the total gain.
\[ \text{Loss}_1 = \text{Gain}_2 \]\[ 0.08 \times CP_1 = 0.22 \times CP_2 \]From the above equation, we can express \(CP_2\) in terms of \(CP_1\):
\[ CP_2 = \frac{0.08}{0.22} \times CP_1 \]\[ CP_2 = \frac{8}{22} \times CP_1 \]\[ CP_2 = \frac{4}{11} \times CP_1 \quad \text{(Equation 2)} \]Substitute Equation 2 into Equation 1:
\[ CP_1 + \frac{4}{11} CP_1 = 1200 \]Combine the terms with \(CP_1\):
\[ \left(1 + \frac{4}{11}\right) CP_1 = 1200 \]\[ \left(\frac{11+4}{11}\right) CP_1 = 1200 \]\[ \frac{15}{11} CP_1 = 1200 \]Solve for \(CP_1\):
\[ CP_1 = 1200 \times \frac{11}{15} \]\[ CP_1 = 80 \times 11 \]\[ CP_1 = 880 \]The cost price of the first speaker is ₹880.
Correct Option: C) 880
Pipe A can fill a tank in 17 hours, pipe B can fill the same tank in 21 hours and pipe C can fill the same tank in 19 hours. The time taken by them to fill the same tank if they operate together is:
13 \( \frac{309}{1079} \) hours
5 \( \frac{309}{1079} \) hours
6 \( \frac{309}{1079} \) hours
11 \( \frac{309}{1079} \) hours
To find the combined time taken by multiple pipes to fill a tank, we first calculate the individual filling rates of each pipe. The rate is the reciprocal of the time taken to fill the tank. Then, we sum these individual rates to get the combined filling rate. Finally, the total time taken to fill the tank when all pipes operate together is the reciprocal of the combined rate.
Calculate the filling rate of Pipe A:
Pipe A fills the tank in 17 hours. So, in one hour, Pipe A fills \(\frac{1}{17}\) of the tank.
\[ \text{Rate of A} = \frac{1}{17} \text{ tank/hour} \]Calculate the filling rate of Pipe B:
Pipe B fills the tank in 21 hours. So, in one hour, Pipe B fills \(\frac{1}{21}\) of the tank.
\[ \text{Rate of B} = \frac{1}{21} \text{ tank/hour} \]Calculate the filling rate of Pipe C:
Pipe C fills the tank in 19 hours. So, in one hour, Pipe C fills \(\frac{1}{19}\) of the tank.
\[ \text{Rate of C} = \frac{1}{19} \text{ tank/hour} \]Calculate the combined filling rate of A, B, and C:
When all three pipes operate together, their rates add up.
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \] \[ \text{Combined Rate} = \frac{1}{17} + \frac{1}{21} + \frac{1}{19} \]To add these fractions, find a common denominator, which is the product of 17, 21, and 19.
\[ 17 \times 21 \times 19 = 357 \times 19 = 6783 \] \[ \text{Combined Rate} = \frac{21 \times 19}{17 \times 21 \times 19} + \frac{17 \times 19}{17 \times 21 \times 19} + \frac{17 \times 21}{17 \times 21 \times 19} \] \[ \text{Combined Rate} = \frac{399}{6783} + \frac{323}{6783} + \frac{357}{6783} \] \[ \text{Combined Rate} = \frac{399 + 323 + 357}{6783} \] \[ \text{Combined Rate} = \frac{1079}{6783} \text{ tank/hour} \]Calculate the total time taken to fill the tank:
The time taken is the reciprocal of the combined rate.
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{6783}{1079} \text{ hours} \]Convert the improper fraction to a mixed fraction:
Divide 6783 by 1079:
\[ 6783 \div 1079 \approx 6.286 \]Perform the division:
\[ 6783 = 6 \times 1079 + 309 \]So, the mixed fraction is \(6 \frac{309}{1079}\) hours.
Correct Option: C) 6 \( \frac{309}{1079} \) hours
By selling an article at \( \frac{4}{14} \) of its actual selling price, Akash incurs a loss of 30%. If he sells it at 54% of its actual selling price, then the profit percentage is:
We need to find the profit percentage when an article is sold at 54% of its actual selling price. We are given the loss percentage when it is sold at a fraction of its actual selling price. We will use the concept of Cost Price (CP), Selling Price (SP), and Actual Selling Price (ASP) to solve this problem.
Let the Actual Selling Price (ASP) of the article be \( \text{Rs. } x \).
According to the problem, Akash sells the article at \( \frac{4}{14} \) of its actual selling price. So, the first Selling Price (\( \text{SP}_1 \)) is:
\[ \text{SP}_1 = \frac{4}{14} \times x = \frac{2}{7}x \]He incurs a loss of 30% at this selling price. This means \( \text{SP}_1 \) is 70% of the Cost Price (CP).
\[ \text{SP}_1 = \text{CP} - 0.30 \times \text{CP} = 0.70 \times \text{CP} \]Substitute the value of \( \text{SP}_1 \):
\[ \frac{2}{7}x = 0.70 \times \text{CP} \]\[ \frac{2}{7}x = \frac{7}{10} \times \text{CP} \]Now, express CP in terms of x:
\[ \text{CP} = \frac{2}{7}x \times \frac{10}{7} = \frac{20}{49}x \]Next, we need to find the profit percentage if he sells it at 54% of its actual selling price. Let this new selling price be \( \text{SP}_2 \).
\[ \text{SP}_2 = 0.54 \times x \]Now, calculate the profit:
\[ \text{Profit} = \text{SP}_2 - \text{CP} \]\[ \text{Profit} = 0.54x - \frac{20}{49}x \]\[ \text{Profit} = \frac{54}{100}x - \frac{20}{49}x \]\[ \text{Profit} = \frac{27}{50}x - \frac{20}{49}x \]Find a common denominator (50 * 49 = 2450):
\[ \text{Profit} = \frac{27 \times 49 - 20 \times 50}{2450}x \]\[ \text{Profit} = \frac{1323 - 1000}{2450}x \]\[ \text{Profit} = \frac{323}{2450}x \]Calculate the profit percentage:
\[ \text{Profit Percentage} = \frac{\text{Profit}}{\text{CP}} \times 100 \]\[ \text{Profit Percentage} = \frac{\frac{323}{2450}x}{\frac{20}{49}x} \times 100 \]\[ \text{Profit Percentage} = \frac{323}{2450} \times \frac{49}{20} \times 100 \]\[ \text{Profit Percentage} = \frac{323 \times 49 \times 100}{2450 \times 20} \]\[ \text{Profit Percentage} = \frac{323 \times 49 \times 100}{49000} \]\[ \text{Profit Percentage} = \frac{323 \times 49}{490} \]\[ \text{Profit Percentage} = \frac{323}{10} = 32.3\% \]Correct Option: D) 32.3%
To find the least number of square tiles required to cover a rectangular floor, we need to find the largest possible size for each square tile. This is achieved by finding the Highest Common Factor (HCF) of the dimensions of the floor, after converting them to the same unit (centimeters for easier calculation with integers).
Convert dimensions to centimeters:
The dimensions of the floor are 12.15 m and 11.34 m.
Since \(1 \text{ m} = 100 \text{ cm}\), we convert the dimensions:
\[ \text{Length} = 12.15 \text{ m} = 12.15 \times 100 \text{ cm} = 1215 \text{ cm} \] \[ \text{Width} = 11.34 \text{ m} = 11.34 \times 100 \text{ cm} = 1134 \text{ cm} \]Find the HCF of the dimensions:
The side length of the largest possible square tile will be the HCF of 1215 cm and 1134 cm. We can use the prime factorization method or Euclidean algorithm.
Using prime factorization:
\[ 1215 = 3 \times 405 = 3 \times 3 \times 135 = 3 \times 3 \times 3 \times 45 = 3 \times 3 \times 3 \times 3 \times 15 = 3^5 \times 5 \] \[ 1134 = 2 \times 567 = 2 \times 3 \times 189 = 2 \times 3 \times 3 \times 63 = 2 \times 3 \times 3 \times 3 \times 21 = 2 \times 3 \times 3 \times 3 \times 3 \times 7 = 2 \times 3^4 \times 7 \]The common prime factors are \(3^4\).
\[ \text{HCF}(1215, 1134) = 3^4 = 81 \]So, the side length of each square tile is 81 cm.
Calculate the number of tiles along each dimension:
Number of tiles along the length:
\[ N_{\text{length}} = \frac{\text{Floor Length}}{\text{Tile Side}} = \frac{1215 \text{ cm}}{81 \text{ cm}} = 15 \]Number of tiles along the width:
\[ N_{\text{width}} = \frac{\text{Floor Width}}{\text{Tile Side}} = \frac{1134 \text{ cm}}{81 \text{ cm}} = 14 \]Calculate the total number of tiles:
\[ \text{Total Tiles} = N_{\text{length}} \times N_{\text{width}} = 15 \times 14 = 210 \]Correct Option: C) 210
The problem involves simple interest calculations for two different investments. We are given the total principal amount, the two different interest rates, the time period, and the condition that the interests earned from both investments are equal. We need to find the amount invested at the higher interest rate.
Let the total investment be \(P = \text{₹}22,400\).
Let the amount invested at 13% per annum be \(P_1\).
Then, the amount invested at 3% per annum will be \(P_2 = P - P_1 = \text{₹}(22,400 - P_1)\).
The time period for both investments is \(T = 7\) years.
The simple interest formula is \(I = \frac{P \times R \times T}{100}\).
Interest from the first investment (at 13%):
\[ I_1 = \frac{P_1 \times 13 \times 7}{100} \]Interest from the second investment (at 3%):
\[ I_2 = \frac{(22,400 - P_1) \times 3 \times 7}{100} \]According to the problem, the interests earned are equal: \(I_1 = I_2\).
Set up the equation:
\[ \frac{P_1 \times 13 \times 7}{100} = \frac{(22,400 - P_1) \times 3 \times 7}{100} \]We can cancel out the common terms (\(7/100\)) from both sides:
\[ P_1 \times 13 = (22,400 - P_1) \times 3 \]Simplify and solve for \(P_1\):
\[ 13 P_1 = 22,400 \times 3 - 3 P_1 \]\[ 13 P_1 = 67,200 - 3 P_1 \]\[ 13 P_1 + 3 P_1 = 67,200 \]\[ 16 P_1 = 67,200 \]\[ P_1 = \frac{67,200}{16} \]\[ P_1 = 4,200 \]So, the sum invested at 13% per annum is ₹4,200.
B) The calculation shows that the amount invested at 13% per annum is ₹4,200.
To find the median of a given set of data, the first step is to arrange the data in ascending or descending order. Once ordered, the median is the middle value if the number of data points is odd. If the number of data points is even, the median is the average of the two middle values.
The given data set is: 57, 96, 60, 91, 69, 50, 72, 16, 78, 73, 55.
Counting the numbers, we find there are 11 data points. So, \(n = 11\).
16, 50, 55, 57, 60, 69, 72, 73, 78, 91, 96.
Since \(n\) is an odd number (11), the median is the \( \left( \frac{n+1}{2} \right)^{\text{th}} \) term.
\[ \text{Median position} = \left( \frac{11+1}{2} \right)^{\text{th}} = \left( \frac{12}{2} \right)^{\text{th}} = 6^{\text{th}} \text{ term} \]Looking at the ordered data, the 6th term is 69.
Ordered data: 16, 50, 55, 57, 60, 69, 72, 73, 78, 91, 96.
Thus, the median of the given data is 69.
C) 69 is the correct median value as it is the middle term in the sorted data set.