Section - Mathematics
Simplify: 4((\( \frac{6}{4} \))x² − 25x + 12) − 6(x² + 5x − 13).
To simplify the given algebraic expression, we need to distribute the constants into the parentheses and then combine like terms. This involves basic arithmetic operations and careful handling of signs.
Correct Option: A) \(-130x + 126\)
To determine which number divides 386937355, we will use divisibility rules for each of the given options. Divisibility rules provide a quick way to check if a number is divisible by another number without performing long division.
C) 5 — The number 386937355 ends with the digit 5. According to the divisibility rule for 5, any number ending in 0 or 5 is divisible by 5. Thus, 386937355 is divisible by 5.
This problem involves profit sharing in a partnership, where profit is distributed in proportion to the product of the capital invested and the time for which it was invested. We will set up equations based on the given information about investments, time periods, and profit shares, and then solve for the unknown capital amounts.
Let N's capital be \(\text{₹}x\). Then M's capital is \(\text{₹}(x + 79000)\).
M invests for 7 months, and N invests for 6 months.
The ratio of their profits will be equal to the ratio of (Capital \(\times\) Time).
\[ \frac{\text{M's Profit}}{\text{N's Profit}} = \frac{(x + 79000) \times 7}{x \times 6} \]The total profit is \(\text{₹}6705\). M's share is \(\text{₹}745\) more than N's share.
Let M's profit be \(P_M\) and N's profit be \(P_N\).
\[ P_M + P_N = 6705 \]\[ P_M = P_N + 745 \]Substitute \(P_M\) in the first equation:
\[ (P_N + 745) + P_N = 6705 \]\[ 2P_N + 745 = 6705 \]\[ 2P_N = 6705 - 745 \]\[ 2P_N = 5960 \]\[ P_N = \frac{5960}{2} \]\[ P_N = 2980 \]Now find M's profit:
\[ P_M = P_N + 745 \]\[ P_M = 2980 + 745 \]\[ P_M = 3725 \]Now use the profit ratio equation:
\[ \frac{3725}{2980} = \frac{(x + 79000) \times 7}{x \times 6} \]Simplify the fraction \(\frac{3725}{2980}\). Both are divisible by 5:
\[ \frac{745}{596} = \frac{(x + 79000) \times 7}{x \times 6} \]Cross-multiply and solve for \(x\):
\[ 745 \times x \times 6 = 596 \times (x + 79000) \times 7 \]\[ 4470x = 4172(x + 79000) \]\[ 4470x = 4172x + 4172 \times 79000 \]\[ 4470x - 4172x = 329588000 \]\[ 298x = 329588000 \]\[ x = \frac{329588000}{298} \]\[ x = 1106000 \]This is N's capital. M's capital is \(x + 79000\).
\[ \text{M's Capital} = 1106000 + 79000 \]\[ \text{M's Capital} = 1185000 \]D) ₹11,85,000
To solve the given expression, we need to follow the order of operations, often remembered by the acronym BODMAS/PEMDAS. This means we prioritize Brackets/Parentheses, Orders/Exponents, Division/Multiplication (from left to right), and Addition/Subtraction (from left to right).
First, evaluate the innermost bracket: \((5 - 4)\).
\[ 5 - 4 = 1 \]Substitute this value back into the expression:
\[ 682 \div \{20 + 2 \times (1)\} \]Next, evaluate the multiplication inside the curly braces: \(2 \times 1\).
\[ 2 \times 1 = 2 \]Substitute this value back into the expression:
\[ 682 \div \{20 + 2\} \]Now, evaluate the addition inside the curly braces: \(20 + 2\).
\[ 20 + 2 = 22 \]Finally, perform the division:
\[ 682 \div 22 \]To perform the division:
\[ \frac{682}{22} = \frac{341}{11} \]Dividing 341 by 11:
\[ 341 \div 11 = 31 \]Thus, the value of the expression is 31.
Correct Option: A) 31
We need to find the percentage by which the marked price (MP) is higher than the cost price (CP). We are given information about a discount, a free item offer, and the overall profit percentage. We will use the concept of cost price, selling price, marked price, discount, and profit percentage to solve this problem. We'll consider the transaction for a pack of 15 books.
Let the Cost Price (CP) of one book be \(C\). Then the Cost Price of 15 books is \(15C\).
The seller gives one book free for buying every pack of 15 books. This means the customer pays for 15 books but receives 16 books. So, the effective number of books sold for the price of 15 is 16. The cost to the seller for these 16 books is \(16C\).
The seller gains 35% profit. The profit is calculated on the actual cost incurred by the seller for the items given away. So, the Selling Price (SP) for 16 books (which the customer gets for the price of 15) is:
\[ \text{SP} = \text{CP} \times \left(1 + \frac{\text{Profit \%}}{100}\right) \]
\[ \text{SP} = 16C \times \left(1 + \frac{35}{100}\right) = 16C \times \frac{135}{100} = 16C \times \frac{27}{20} = \frac{108C}{5} \]
The seller gives a 4% discount on the marked price of the books for which the customer pays. The customer pays for 15 books. Let the Marked Price (MP) of one book be \(M\). The Marked Price of 15 books is \(15M\).
The Selling Price (SP) after a 4% discount on the marked price of 15 books is:
\[ \text{SP} = 15M \times \left(1 - \frac{\text{Discount \%}}{100}\right) \]
\[ \text{SP} = 15M \times \left(1 - \frac{4}{100}\right) = 15M \times \frac{96}{100} = 15M \times \frac{24}{25} = \frac{360M}{25} = \frac{72M}{5} \]
Now, we equate the two expressions for SP:
\[ \frac{108C}{5} = \frac{72M}{5} \]
Multiply both sides by 5:
\[ 108C = 72M \]
Divide both sides by 36:
\[ 3C = 2M \]
Express \(M\) in terms of \(C\):
\[ M = \frac{3}{2}C \]
\[ M = 1.5C \]
To find by what percentage the marked price is higher than the cost price, we use the formula:
\[ \text{Percentage higher} = \frac{\text{MP} - \text{CP}}{\text{CP}} \times 100 \]
\[ \text{Percentage higher} = \frac{1.5C - C}{C} \times 100 \]
\[ \text{Percentage higher} = \frac{0.5C}{C} \times 100 \]
\[ \text{Percentage higher} = 0.5 \times 100 = 50\% \]
D) 50%
To find the profit, we need to calculate the total cost price and the total selling price. The cost price is given directly. For the selling price, we first determine the number of mirrors remaining after some broke and then multiply by the selling price per mirror. Finally, profit is calculated as Selling Price - Cost Price.
Calculate the Cost Price (CP):
The problem states that 83 mirrors were purchased for ₹83.
Calculate the number of mirrors remaining:
Initially, there were 83 mirrors. 22 mirrors broke in transit.
Calculate the Selling Price (SP):
The trader sold the remaining 61 mirrors at ₹3.60 each.
Calculate the Profit:
Profit is the difference between the Selling Price and the Cost Price.
B) ₹136.60 is the calculated profit.
To find the percentage of literate females, we first need to calculate the total number of females, the total number of literate people, and the number of literate males. From these, we can deduce the number of literate females and then express it as a percentage of the total female population.
Calculate the number of females:
Given total population = \(362000\)
Number of males = \(181000\)
\[ \text{Number of females} = \text{Total population} - \text{Number of males} \] \[ \text{Number of females} = 362000 - 181000 = 181000 \]Calculate the total number of literate people:
47% of the total population is literate.
\[ \text{Total literate people} = 0.47 \times 362000 \] \[ \text{Total literate people} = 170140 \]Calculate the number of literate males:
35% of males are literate.
\[ \text{Number of literate males} = 0.35 \times \text{Number of males} \] \[ \text{Number of literate males} = 0.35 \times 181000 \] \[ \text{Number of literate males} = 63350 \]Calculate the number of literate females:
\[ \text{Number of literate females} = \text{Total literate people} - \text{Number of literate males} \] \[ \text{Number of literate females} = 170140 - 63350 \] \[ \text{Number of literate females} = 106790 \]Calculate the percentage of literate females:
\[ \text{Percentage of literate females} = \left( \frac{\text{Number of literate females}}{\text{Number of females}} \right) \times 100 \] \[ \text{Percentage of literate females} = \left( \frac{106790}{181000} \right) \times 100 \] \[ \text{Percentage of literate females} = 0.590 \times 100 \] \[ \text{Percentage of literate females} = 59\% \]Correct Option: D) 59%
To find the overall average runs scored, we need to calculate the total runs scored in all matches and divide it by the total number of matches played. We will calculate the total runs for the first set of matches and the second set of matches separately, then sum them up.
Calculate total runs in the first 23 matches:
Given that the average runs in 23 matches is 40.
Calculate total runs in the next 10 matches:
Given that the average runs in the next 10 matches is 16.
Calculate the total runs in all 33 matches:
\[ \text{Total runs (all 33 matches)} = \text{Runs in first 23 matches} + \text{Runs in next 10 matches} \] \[ = 920 + 160 = 1080 \]Calculate the total number of matches:
\[ \text{Total matches} = 23 + 10 = 33 \]Calculate the overall average runs scored:
\[ \text{Overall average runs} = \frac{\text{Total runs}}{\text{Total matches}} \] \[ = \frac{1080}{33} \] \[ \approx 32.7272... \]Round off to two decimal places:
\[ \text{Overall average runs} \approx 32.73 \]Correct Option: D) 32.73
The problem involves finding the price of the cheapest item given the average price of three items and their price ratio. We will first calculate the total price of the three items using the average. Then, we will use the given ratio to find the individual prices and identify the cheapest one.
The average price of three items is ₹5,325.
Total price = Average price × Number of items
\[ \text{Total price} = 5325 \times 3 = \text{₹}15975 \]The prices are in the ratio 3 : 5 : 7. Let the prices of the three items be \(3x\), \(5x\), and \(7x\) respectively.
The sum of the individual prices must equal the total price calculated in step 1.
\[ 3x + 5x + 7x = 15975 \]\[ 15x = 15975 \]The cheapest item corresponds to the smallest ratio part, which is \(3x\).
Price of the cheapest item = \(3x = 3 \times 1065\)
\[ \text{Price of the cheapest item} = 3195 \]Correct Option: C) 3,195
The price of the cheapest item is ₹3,195.
If secθ + tanθ = s, then find the value of \(\frac{s^2-1}{s^2+1}\).
The problem asks us to find the value of an expression involving \(s\), where \(s\) is defined in terms of \(\text{sec}\theta\) and \(\text{tan}\theta\). We will use the fundamental trigonometric identity \(\text{sec}^2\theta - \text{tan}^2\theta = 1\) to simplify the expression.
We are given the relation:
\[ \text{sec}\theta + \text{tan}\theta = s \]Recall the trigonometric identity:
\[ \text{sec}^2\theta - \text{tan}^2\theta = 1 \]Factor the left side of the identity using the difference of squares formula \((a^2 - b^2) = (a-b)(a+b)\):
\[ (\text{sec}\theta - \text{tan}\theta)(\text{sec}\theta + \text{tan}\theta) = 1 \]Substitute the given value \((\text{sec}\theta + \text{tan}\theta) = s\) into the factored identity:
\[ (\text{sec}\theta - \text{tan}\theta)s = 1 \]Solve for \((\text{sec}\theta - \text{tan}\theta)\):
\[ \text{sec}\theta - \text{tan}\theta = \frac{1}{s} \]Now we have two equations:
Equation (1): \( \text{sec}\theta + \text{tan}\theta = s \)
Equation (2): \( \text{sec}\theta - \text{tan}\theta = \frac{1}{s} \)
Subtract Equation (2) from Equation (1):
\[ (\text{sec}\theta + \text{tan}\theta) - (\text{sec}\theta - \text{tan}\theta) = s - \frac{1}{s} \]\[ \text{sec}\theta + \text{tan}\theta - \text{sec}\theta + \text{tan}\theta = \frac{s^2 - 1}{s} \]\[ 2\text{tan}\theta = \frac{s^2 - 1}{s} \]Solve for \(\text{tan}\theta\):
\[ \text{tan}\theta = \frac{s^2 - 1}{2s} \]Add Equation (1) and Equation (2):
\[ (\text{sec}\theta + \text{tan}\theta) + (\text{sec}\theta - \text{tan}\theta) = s + \frac{1}{s} \]\[ \text{sec}\theta + \text{tan}\theta + \text{sec}\theta - \text{tan}\theta = \frac{s^2 + 1}{s} \]\[ 2\text{sec}\theta = \frac{s^2 + 1}{s} \]Solve for \(\text{sec}\theta\):
\[ \text{sec}\theta = \frac{s^2 + 1}{2s} \]We need to find the value of \(\frac{s^2-1}{s^2+1}\). We can express this in terms of \(\text{tan}\theta\) and \(\text{sec}\theta\):
\[ \frac{s^2-1}{s^2+1} = \frac{2s \cdot \text{tan}\theta}{2s \cdot \text{sec}\theta} \]\[ \frac{s^2-1}{s^2+1} = \frac{\text{tan}\theta}{\text{sec}\theta} \]Substitute the definitions of \(\text{tan}\theta\) and \(\text{sec}\theta\) in terms of \(\text{sin}\theta\) and \(\text{cos}\theta\):
\[ \frac{\text{tan}\theta}{\text{sec}\theta} = \frac{\frac{\text{sin}\theta}{\text{cos}\theta}}{\frac{1}{\text{cos}\theta}} \]\[ \frac{\text{tan}\theta}{\text{sec}\theta} = \frac{\text{sin}\theta}{\text{cos}\theta} \cdot \text{cos}\theta \]\[ \frac{\text{tan}\theta}{\text{sec}\theta} = \text{sin}\theta \]Therefore,
\[ \frac{s^2-1}{s^2+1} = \text{sin}\theta \]Correct Option: C) sinθ