Section - Mathematics
The value of 2² + \(\sqrt{8^2}\) − 2\(\sqrt{4}\) − 11 is
To find the value of the given expression, we need to simplify each term using the order of operations (PEMDAS/BODMAS) and properties of exponents and square roots. We will evaluate powers, square roots, and then perform addition and subtraction from left to right.
The final value of the expression is \(-3\).
Correct Option: D) -3
We are given that the product of two consecutive positive natural numbers is 272. We need to find the greater of these two numbers. We can solve this by forming a quadratic equation or by checking the options.
Alternatively, we can check the options. If the greater number is 17, then the smaller number is 16. Their product is \(16 \times 17 = 272\), which matches the given condition.
Correct Option: C) 17
To solve this problem, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we first perform operations inside Brackets/Parentheses, then Orders/Exponents, then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right).
Thus, the value of the expression is 145.
Correct Option: A) 145
To find the smallest number that should be added to 81661 so that the sum is completely divisible by 73, we need to perform division. We will divide 81661 by 73 and find the remainder. The difference between the divisor (73) and the remainder will be the smallest number that needs to be added to make the sum perfectly divisible.
We perform long division:
\[\begin{array}{r} 1118 \ 73 \overline{\smash{)} 81661} \ -73\downarrow \ \hline 86 \ -73\downarrow \ \hline 136 \ -73\downarrow \ \hline 631 \ -584 \ \hline 47 \end{array}\]From the division, the quotient is 1118 and the remainder is 47.
For the sum to be completely divisible by 73, the remainder should be 0. The current remainder is 47. To make the remainder 0, we need to add a number such that when it's added to 47, the sum is 73 (or a multiple of 73). The smallest such number would be the difference between the divisor and the remainder.
\[ \text{Number to be added} = \text{Divisor} - \text{Remainder} \]\[ \text{Number to be added} = 73 - 47 \]\[ \text{Number to be added} = 26 \]If we add 26 to 81661, we get \(81661 + 26 = 81687\). Now, let's divide 81687 by 73:
\[ 81687 \div 73 = 1119 \] (with a remainder of 0).This confirms that 26 is the smallest number to be added.
Correct Option: C) 26
The problem involves finding an angle in a trapezium. We will use properties of parallel lines, angles in a triangle, and isosceles triangles to solve this. The key is to find the angles within triangle ACD using the given information and then determine \(\angle ADC\).
Given that ABCD is a trapezium with \(BC \parallel AD\). Also, \(\angle ABC = 113°\) and \(\angle BAC = 26°\).
Since \(BC \parallel AD\), the sum of consecutive interior angles is \(180°\). Therefore, \(\angle ABC + \angle BAD = 180°\).
Substitute the given value of \(\angle ABC\):
\[ 113° + \angle BAD = 180° \]
\[ \angle BAD = 180° - 113° = 67° \]
We know that \(\angle BAD = \angle BAC + \angle CAD\). We are given \(\angle BAC = 26°\).
Substitute the values to find \(\angle CAD\):
\[ 67° = 26° + \angle CAD \]
\[ \angle CAD = 67° - 26° = 41° \]
Now consider \(\triangle ACD\). We are given that \(AC = CD\). This means \(\triangle ACD\) is an isosceles triangle.
In an isosceles triangle, the angles opposite the equal sides are equal. So, \(\angle CAD = \angle CDA\) (or \(\angle ADC\)).
Therefore, \(\angle ADC = 41°\).
C) 41° is the correct answer, as calculated by finding \(\angle CAD\) and using the property of the isosceles triangle \(ACD\).
A) 168° — This value is much larger than any angle in \(\triangle ACD\) could be, and doesn't follow from the properties of the trapezium or triangle.
B) 165° — Similar to option A, this value is too large and does not result from correct application of geometric principles.
D) 166° — This value is also excessively large and incorrect. It seems to be a distractor, possibly from incorrect calculations or assumptions about the angles.
To find the discount amount, we need to calculate the specified percentage of the original price. The original price of the printer is given, and the trade discount percentage is also provided. We will use the formula for calculating a percentage of a number.
The discount amount is calculated as a percentage of the original price.
\[ \text{Discount Amount} = \text{Trade Discount Percentage} \times \text{Original Price} \] \[ \text{Discount Amount} = 9\% \times \text{₹}32,000 \]To convert the percentage to a decimal, divide by 100:
\[ 9\% = \frac{9}{100} = 0.09 \]Now, substitute this value into the formula:
\[ \text{Discount Amount} = 0.09 \times \text{₹}32,000 \] \[ \text{Discount Amount} = \text{₹}2,880 \]Thus, the discount amount is ₹2,880.
B) ₹2,880 is the correct discount amount calculated from 9% of ₹32,000.
Pipe A can fill a tank in 14 hours, pipe B can fill the same tank in 24 hours and pipe C can fill the same tank in 17 hours. The time taken by them to fill the same tank if they operate together is:
5 \( \frac{401}{491} \) hours
11 \( \frac{401}{491} \) hours
10 \( \frac{401}{491} \) hours
2 \( \frac{401}{491} \) hours
To find the time taken by multiple pipes working together to fill a tank, we first calculate the individual filling rates of each pipe. Then, we sum these rates to find the combined filling rate. Finally, the total time taken is the reciprocal of the combined rate.
Calculate the filling rate of each pipe:
The rate at which a pipe fills a tank is the reciprocal of the time it takes to fill the tank alone.
Calculate the combined filling rate of all three pipes:
When pipes work together, their individual rates add up.
\[ \text{Combined Rate} = \text{Rate}_A + \text{Rate}_B + \text{Rate}_C \]\[ \text{Combined Rate} = \frac{1}{14} + \frac{1}{24} + \frac{1}{17} \]Find a common denominator for the fractions:
The least common multiple (LCM) of 14, 24, and 17. Since 17 is a prime number, the LCM will be \(14 \times 24 \times 17\).
LCM \(= 2^3 \times 3 \times 7 \times 17 = 8 \times 3 \times 7 \times 17 = 24 \times 7 \times 17 = 168 \times 17 = 2856\)
Add the fractions:
\[ \text{Combined Rate} = \frac{2856/14}{2856} + \frac{2856/24}{2856} + \frac{2856/17}{2856} \]\[ \text{Combined Rate} = \frac{204}{2856} + \frac{119}{2856} + \frac{168}{2856} \]\[ \text{Combined Rate} = \frac{204 + 119 + 168}{2856} \]\[ \text{Combined Rate} = \frac{491}{2856} \text{ tank per hour} \]Calculate the total time taken:
The total time taken to fill the tank is the reciprocal of the combined rate.
\[ \text{Time} = \frac{1}{\text{Combined Rate}} \]\[ \text{Time} = \frac{2856}{491} \text{ hours} \]Convert the improper fraction to a mixed number:
Divide 2856 by 491:
\[ 2856 \div 491 \]\[ 2856 = 5 \times 491 + 401 \]\[ 2856 = 2455 + 401 \]\[ \text{Time} = 5 \frac{401}{491} \text{ hours} \]A) \(5 \frac{401}{491}\) hours is the correct time taken for all three pipes to fill the tank together.
The problem involves calculating the cost price of two items given their total cost, individual profit/loss percentages, and the condition of no overall profit or loss. We can set up equations based on the cost prices and the profit/loss amounts to solve for the unknown cost price.
Let the cost price of the first frock be \(C_1\) and the cost price of the second frock be \(C_2\).
Given that the total cost price of the two frocks is ₹1,200, we have:
\[ C_1 + C_2 = 1200 \quad \text{(Equation 1)} \]The first frock was sold at a loss of 10%. The loss amount on the first frock is:
\[ \text{Loss}_1 = 10\% \text{ of } C_1 = \frac{10}{100} C_1 = 0.1 C_1 \]The second frock was sold at a gain of 15%. The gain amount on the second frock is:
\[ \text{Gain}_2 = 15\% \text{ of } C_2 = \frac{15}{100} C_2 = 0.15 C_2 \]On the whole, there was neither a loss nor a gain. This means the total loss equals the total gain. In this case, the loss from the first frock is exactly offset by the gain from the second frock.
\[ \text{Loss}_1 = \text{Gain}_2 \]\[ 0.1 C_1 = 0.15 C_2 \quad \text{(Equation 2)} \]From Equation 2, we can express \(C_2\) in terms of \(C_1\):
\[ C_2 = \frac{0.1}{0.15} C_1 = \frac{10}{15} C_1 = \frac{2}{3} C_1 \]Substitute this expression for \(C_2\) into Equation 1:
\[ C_1 + \frac{2}{3} C_1 = 1200 \]Combine the terms involving \(C_1\):
\[ \left(1 + \frac{2}{3}\right) C_1 = 1200 \]\[ \left(\frac{3}{3} + \frac{2}{3}\right) C_1 = 1200 \]\[ \frac{5}{3} C_1 = 1200 \]Solve for \(C_1\):
\[ C_1 = 1200 \times \frac{3}{5} \]\[ C_1 = 240 \times 3 \]\[ C_1 = 720 \]The cost price of the first frock is ₹720.
Correct Option: D) 720
This problem involves calculating the relative speed of two trains moving towards each other and determining their meeting time. We need to first calculate the duration of travel for each train to find their respective speeds. Since the trains travel the same distance, we can use the concept of relative speed to find the time they meet.
Calculate the travel time for Train A:
Train A leaves at 8:05 a.m. and reaches at 3:05 p.m. on the same day.
Calculate the travel time for Train B:
Train B leaves at 10:05 a.m. and reaches at 3:05 p.m. on the same day.
Assume the distance between stations M and N:
Let the distance between station M and station N be \(D\) km.
Calculate the speed of Train A:
\[ \text{Speed of A } (S_A) = \frac{\text{Distance}}{\text{Time}} = \frac{D}{7} \text{ km/hr} \]Calculate the speed of Train B:
\[ \text{Speed of B } (S_B) = \frac{\text{Distance}}{\text{Time}} = \frac{D}{5} \text{ km/hr} \]Determine the head start of Train A:
Train A starts at 8:05 a.m. and Train B starts at 10:05 a.m. So, Train A travels alone for \(10:05 \text{ a.m.} - 8:05 \text{ a.m.} = 2 \text{ hours}\).
Calculate the distance covered by Train A during its head start:
\[ \text{Distance covered by A in 2 hours} = S_A \times 2 = \frac{D}{7} \times 2 = \frac{2D}{7} \text{ km} \]Calculate the remaining distance when both trains are moving:
At 10:05 a.m., Train A has covered \(\frac{2D}{7}\) km. The remaining distance between the trains is:
Calculate the relative speed of the trains:
Since the trains are moving towards each other, their relative speed is the sum of their individual speeds.
Calculate the time taken for the trains to meet after 10:05 a.m.:
\[ \text{Time to meet} = \frac{\text{Remaining Distance}}{\text{Relative Speed}} = \frac{\frac{5D}{7}}{\frac{12D}{35}} \]\[ = \frac{5D}{7} \times \frac{35}{12D} = \frac{5 \times 5}{12} = \frac{25}{12} \text{ hours} \]Convert the meeting time into hours and minutes:
\[ \frac{25}{12} \text{ hours} = 2 \frac{1}{12} \text{ hours} \]\[ = 2 \text{ hours } + \left(\frac{1}{12} \times 60\right) \text{ minutes} \]\[ = 2 \text{ hours } 5 \text{ minutes} \]Determine the actual meeting time:
The trains start moving towards each other at 10:05 a.m. They meet 2 hours and 5 minutes later.
A) The calculated meeting time is 12:10 p.m.
To find the surface area of a cuboid, we need to sum the areas of all its six faces. A cuboid has three pairs of identical rectangular faces. The formula for the total surface area of a cuboid with length \(l\), breadth \(b\), and height \(h\) is \(2(lb + bh + hl)\).
Correct Option: C) 158 cm²