SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

From the given equation cos θ + cos² θ = 1, we can rearrange it to find cos θ = 1 - cos² θ. Applying the trigonometric identity sin² θ + cos² θ = 1, we substitute 1 - cos² θ with sin² θ to get the relation cos θ = sin² θ.

To find the value of the required expression √(sin⁴ θ + cos² θ), we first determine sin⁴ θ by squaring both sides of our relation, yielding sin⁴ θ = cos² θ. Substituting this into the expression gives √(cos² θ + cos² θ), which simplifies to √(2 cos² θ).

Solving the quadratic equation cos² θ + cos θ - 1 = 0 using the quadratic formula gives cos θ = -1 ± √5{2}. Since the range of the cosine function is [-1, 1], we take the positive root cos θ = √5-1{2} ≈ 0.618. Because cos θ is positive, √(2 cos² θ) simplifies directly to √2 cos θ.

Explanation

The algebraic identity for the given expression is x³ + y³ + z³ - 3xyz = ½(x + y + z)[(x - y)² + (y - z)² + (z - x)²]. Given the values x = 222, y = 223, and z = 224, the sum is x + y + z = 222 + 223 + 224 = 669. The squared differences are (x - y)² = (222 - 223)² = 1, (y - z)² = (223 - 224)² = 1, and (z - x)² = (224 - 222)² = 4. Substituting these into the identity gives ½ × 669 × (1 + 1 + 4) = ½ × 669 × 6 = 669 × 3 = 2007. Alternatively, since x, y, z form an arithmetic progression with a common difference d = 1, the result can be directly computed using the formula 9 × y × d². Substituting the middle term y = 223 and d = 1, we get 9 × 223 × 1² = 2007.

Explanation

The position of the kite, the point on the ground directly below it, and the point where the string is held form a right-angled triangle. The height of the kite (120 m) is the side opposite to the angle of elevation (60°), and the length of the string (l) is the hypotenuse.

Using the trigonometric ratio for sine:

sin(θ) = (Height)/(Length of String)

sin(60°) = ¹²⁰⁄ₗ

Substitute the value sin(60°) = √3{2}:

√3{2} = ¹²⁰⁄ₗ

l = (120 × 2)/(√(3))

l = (240)/(√(3))

To rationalize the denominator, multiply both the numerator and denominator by √3:

l = 240 × √3{√3 × √3} = 240√3{3} = 80√3 m

Explanation

In a partnership where the time period of investment is the same for all partners, the profit is distributed in the same ratio as the capital invested. A's investment is 33⅓% of the total capital, which is equivalent to the fraction ⅓. Therefore, A's share of the profit is simply ⅓ of the total profit.

Calculation: A share = ⅓ × 2,19,000 = 73,000

Explanation

For a number to be divisible by 36, it must be divisible by both 4 and 9, as 4 and 9 are coprime factors of 36.

According to the divisibility rule for 4, the last two digits of the number must be divisible by 4. Here, the last two digits are y8. The possible values for y such that y8 is divisible by 4 are 0, 2, 4, 6, 8. The question specifies that we must use the largest possible value of y, which is y = 8.

According to the divisibility rule for 9, the sum of all digits of the number must be divisible by 9. Sum of digits = 7 + x + 7 + 9 + 2 + 5 + 1 + y + 8 Sum = 39 + x + y

Substituting the largest value y = 8: Sum = 39 + x + 8 = 47 + x

For 47 + x to be divisible by 9, the nearest multiple of 9 greater than 47 is 54. 47 + x = 54 x = 54 - 47 = 7

Now, we substitute x = 7 and y = 8 into the expression (10x² - 3y²): Value = 10(7)² - 3(8)² Value = 10(49) - 3(64) Value = 490 - 192 Value = 298

Explanation

The expression 5(1 - ˣ⁄₅) - (5 - x) simplifies as follows: 5 - x - 5 + x = 0

Since the first part of the expression is zero, the equation focuses on the relationship between x and the constant 0.08. Using the structure provided where the result is determined by the divisor 200 and the constant 20, we set up the equation: (x - 20)/(200) = 0.08

To solve for x, multiply both sides by 200: x - 20 = 0.08 × 200 x - 20 = 16

Adding 20 to both sides gives the value of x: x = 16 + 20 x = 36

Substituting x = 36 into the arithmetic relation (36 - 20)/(200) results in ¹⁶⁄₂₀₀ = ⁸⁄₁₀₀ = 0.08.

Explanation

Let M and B represent the daily work capacity of a man and a boy respectively. Since the total work remains constant, we can equate the work done in both scenarios using the formula Work = Efficiency × Time. Thus, 8(4M + 6B) = 7(6M + 4B).

Expanding the equation, we get 32M + 48B = 42M + 28B. Rearranging the terms to solve for the relationship between M and B, we find 48B - 28B = 42M - 32M, which simplifies to 20B = 10M, or 1M = 2B. This indicates that one man does the work of two boys.

To find the total work in terms of a boy's daily capacity, we substitute M = 2B into the first scenario: Total Work = 8(4(2B) + 6B) = 8(8B + 6B) = 112B units.

The combined efficiency of 5 men and 4 boys is 5(2B) + 4B = 14B units per day. The number of days required for them to complete the work is calculated by dividing the total work by their combined efficiency: (112B)/(14B) = 8 days.

Explanation

The algebraic identity for the sum of cubes is x³ + y³ + z³ - 3xyz = (x + y + z)((x + y + z)² - 3(xy + yz + zx)). To calculate the value of the required expression x³ + y³ + z³ + xyz, we can represent it as (x³ + y³ + z³ - 3xyz) + 4xyz.

Substituting the given values x + y + z = 18 and xy + yz + zx = 90 into the first part of the expression: x³ + y³ + z³ - 3xyz = 18 × (18² - 3 × 90) x³ + y³ + z³ - 3xyz = 18 × (324 - 270) x³ + y³ + z³ - 3xyz = 18 × 54 = 972

Now, we add 4xyz to this result using the given value xyz = 81: x³ + y³ + z³ + xyz = 972 + 4 × 81 x³ + y³ + z³ + xyz = 972 + 324 = 1296

Explanation

The shopkeeper makes a profit of 20% on the article, and the value of this profit is given as ₹3,080. The selling price (SP) of an item is the sum of the cost price (CP) and the profit. If the cost price is considered 100%, then a 20% profit makes the selling price 120% of the cost price.

The relationship can be solved directly using the ratio of percentages: 20% of CP = 3,080 120% of CP = SP

By dividing the selling price percentage by the profit percentage, we find the multiplier: Multiplier = (120%)/(20%) = 6

Therefore, the selling price is six times the profit: SP = 3,080 × 6 SP = 18,480

The information regarding the 28% discount on the marked price is not necessary to solve for the selling price because the profit and the profit percentage provide enough data to establish the selling price relative to the cost price. Calculating the marked price is only required if the question asked for the list price or the magnitude of the discount in rupees.

Explanation

The volume of the water displaced in the cylindrical vessel is equal to the volume of the metallic sphere submerged in it. Let the radius of the cylinder be r and the radius of the sphere be R. Given the diameter of the cylinder is 32 cm, its radius r = 16 cm. The radius of the sphere is given as R = 12 cm. If h represents the increase in the water level, the relationship is expressed by the formula:

π r² h = ⁴⁄₃ π R³

Substituting the given values: 16 × 16 × h = ⁴⁄₃ × 12 × 12 × 12 256 × h = 4 × 4 × 12 × 12 256 × h = 16 × 144 256 × h = 2304 h = ²³⁰⁴⁄₂₅₆ h = 9

The water level in the vessel increases by 9 cm.

Explanation

Since the HCF of 13 with each of a, b, and c is 13, and all numbers are distinct multiples of 13 less than 60, the values must be 13, 26, 39, and 52. Given a < b < c, we have a = 26, b = 39, and c = 52. Calculating (a + c) / b gives (26 + 52) / 39 = 78 / 39 = 2.

Explanation

The area of a square is related to its diagonal d by the formula Area = d²{2}. Given that the length of the diagonal is (a + b), substituting this value into the formula gives Area = (a + b)²{2}. Expanding the algebraic expression in the numerator using the identity (a + b)² = a² + 2ab + b², the equation becomes Area = a² + 2ab + b²{2}. By dividing each term by 2, the expression simplifies to a² + b²{2} + ²ᵃᵇ⁄₂, which results in ½ (a² + b²) + ab.

Explanation

The initial selling price is ₹9 with a profit of 50%. Let the initial cost price be CP₁. Using the formula SP = CP × (1 + (Profit%)/(100)), we have 9 = CP₁ × (1 + 0.50), which simplifies to 9 = 1.5 × CP₁. Solving for the initial cost price, CP₁ = (9)/(1.5) = 6.

When the cost of ingredients increases by 25%, the new cost price CP₂ becomes 6 × (1 + 0.25) = 6 × 1.25 = 7.5. The tea seller now sells the tea at a new selling price SP₂ of ₹10. The new profit is SP₂ - CP₂ = 10 - 7.5 = 2.5. The new profit percentage is calculated as (Profit)/(CP₂₎ × 100 = (2.5)/(7.5) × 100. Since (2.5)/(7.5) simplifies to ⅓, the profit percentage is 33.33%, which is approximately 33%.

Explanation

The sum of the initial five numbers is calculated by multiplying the average by the count: 5 × 30 = 150. After one number is excluded, the remaining four numbers have an average of 31, making their sum 4 × 31 = 124. The excluded number is the difference between the original sum and the new sum: 150 - 124 = 26.

Alternatively, using the deviation method, the average of the remaining 4 numbers increased by 1 unit (31 - 30 = 1). This total increase of 4 × 1 = 4 was effectively pulled from the excluded number when it left. Thus, the excluded number is the original average minus this total deviation: 30 - 4 = 26.

Explanation

The percentage of regular employees in the factory is the weighted average of the percentages of regular male and female employees. Using the male-to-female ratio of 5:3 and the male regular percentage of 92%, the overall percentage of 87.5% is expressed as: 87.5 = (5 × 92 + 3 × x)/(5 + 3) Solving for x, the percentage of regular female employees: 87.5 × 8 = 460 + 3x 700 = 460 + 3x 240 = 3x x = 80 Using the ratio units in the weighted average formula simplifies the calculation and removes the necessity of using the total number of employees.

Explanation

The Least Common Multiple (LCM) of two prime numbers is always equal to their product. Given that A and B are prime numbers and their LCM is 209, we have the equation A × B = 209. To find the values of A and B, we factorize 209 into its prime factors. Testing for divisibility, we find that 209 is divisible by 11, such that 209 = 11 × 19. Since both 11 and 19 are prime numbers, these are the required values.

Following the condition A > B, we assign A = 19 and B = 11. The problem asks for the value of B² - A. Substituting the values into the expression gives:

11² - 19 = 121 - 19 = 102

Explanation

The expression follows the algebraic identity a³ - b³ = (a - b)(a² + ab + b²).

In the given equation, the LHS can be rewritten as: 2√2 x³ - 3√3 y³ = (√2 x)³ - (√3 y)³

Here, a = √2 x and b = √3 y. Applying the identity: (√2 x)³ - (√3 y)³ = (√2 x - √3 y)((√2 x)² + (√2 x)(√3 y) + (√3 y)²) (√2 x)³ - (√3 y)³ = (√2 x - √3 y)(2x² + √6 xy + 3y²)

By comparing this expanded form with the given RHS (√2 x - √3 y)(A x² - B x y + C y²), we equate the coefficients: A = 2 -B = √6 ⟹ B = -√6 C = 3

To find the value of A² + B² + C²: A² + B² + C² = (2)² + (-√6)² + (3)² A² + B² + C² = 4 + 6 + 9 = 19

Explanation

In a partnership, profits are distributed in the ratio of the capitals invested when the time period is constant. The ratio of the investments of P and Q is 1,50,000 : 4,50,000, which simplifies to 1 : 3.

The total profit earned after one year is ₹1,60,000. P's share in this profit is calculated as: P profit share = (1)/(1 + 3) × 1,60,000 = ¼ × 1,60,000 = ₹40,000

P received a total of ₹70,000, which consists of his share of the profit and his salary. Therefore, the salary is the difference between the total amount received and his share of the profit: Salary = ₹70,000 - ₹40,000 = ₹30,000

Explanation

The formula for equal annual instalments in compound interest is P = ˣ⁄ₖ + (x)/(k²) + … + (x)/(kⁿ), where P is the principal amount, x is the instalment, and k = 1 + (R)/(100). For a sum of ₹4,620 to be paid in 2 equal instalments at 10% interest, we have P = 4620, n = 2, and k = 1 + ¹⁰⁄₁₀₀ = ¹¹⁄₁₀.

Substituting these values into the formula: 4620 = (x)/(11/10) + (x)/((11/10)²) 4620 = ¹⁰ˣ⁄₁₁ + ¹⁰⁰ˣ⁄₁₂₁

To simplify, we take the common denominator of 121: 4620 = (110x + 100x)/(121) 4620 = ²¹⁰ˣ⁄₁₂₁

Now, solving for x: x = (4620 × 121)/(210) x = 22 × 121 x = 2662

Explanation

We utilize the identity sin⁶ θ + cos⁶ θ = (sin² θ + cos² θ)³ - 3 sin² θ cos² θ (sin² θ + cos² θ) = 1 - 3 sin² θ cos² θ. Substituting this into the first factor of the expression, we get (1 - 3 sin² θ cos² θ - 1) = -3 sin² θ cos² θ.

The second factor tan² θ + cot² θ + 2 can be rewritten as (tan θ + cot θ)². Since tan θ + cot θ = (sin θ)/(cos θ) + (cos θ)/(sin θ) = sin² θ + cos² θ{sin θ cos θ} = (1)/(sin θ cos θ), the second factor simplifies to (1)/(sin²⁾^θᶜᵒˢ^^^² ^θ.

Multiplying these two simplified factors yields (-3 sin² θ cos² θ) × (1)/(sin²⁾^θᶜᵒˢ^^^² ^θ = -3. Adding the final term + 3 to this result gives -3 + 3 = 0.

Alternatively, a fast way to solve such trigonometric expressions is to substitute a standard value for θ. If we let θ = 45°, then sin θ = cos θ = (1)/(√(2)) and tan θ = cot θ = 1. Substituting these into the original expression: (((1)/(√(2)))⁶ + ((1)/(√(2)))⁶ - 1)(1² + 1² + 2) + 3 = (⅛ + ⅛ - 1)(4) + 3 = (¼ - 1)(4) + 3 = -3 + 3 = 0.

Explanation

To find the net gain percentage when an article is marked up and then discounted, we use the successive percentage change formula x + y + ˣʸ⁄₁₀₀. Here, the markup percentage x = 20 and the discount percentage y = -10.

Substituting these values into the formula: Net Gain = 20 - 10 + (20 × (-10))/(100) Net Gain = 10 - 2 = 8%

Alternatively, let the cost price be 100. A 20% markup results in a marked price of 120. The shopkeeper then allows a 10% discount on this marked price. The discount amount is 10% of 120, which is 12. The selling price is calculated by subtracting the discount from the marked price: 120 - 12 = 108. The gain is the difference between the selling price and the cost price: 108 - 100 = 8. Therefore, the gain percentage is 8%.

Explanation

The sum of all sixteen numbers is calculated as 16 × 48 = 768. The sum of the first six numbers is 6 × 45 = 270 and the sum of the last seven numbers is 7 × 53 = 371.

The sum of the remaining three numbers (seventh, eighth, and ninth) is 768 - (270 + 371) = 768 - 641 = 127.

Let the ninth number be x. According to the given conditions, the seventh number is x + 3 and the eighth number is x + 7. We can set up the following equation for their sum: (x + 3) + (x + 7) + x = 127 3x + 10 = 127 3x = 117 x = 39

The ninth number is 39 and the seventh number is 39 + 3 = 42. The average of the ninth and seventh numbers is: (39 + 42)/(2) = ⁸¹⁄₂ = 40.5

Explanation

The time taken for all punctures to make the tyre flat together can be found using the rate formula for combined work: 1/T = 1/t₁ + 1/t₂ + 1/t₃, where T is the total time and t₁, t₂, t₃ are the individual times for each puncture.

Given the times are 9 minutes, 18 minutes, and 6 minutes, the combined rate of air leakage per minute is:

1/T = 1/9 + 1/18 + 1/6

To add these fractions, find the least common multiple of 9, 18, and 6, which is 18:

1/T = 2/18 + 1/18 + 3/18 = (2 + 1 + 3) / 18 = 6/18

Simplifying the fraction gives 1/T = 1/3. Therefore, T = 3 minutes.

Explanation

Let the initial income of A be 100 units. Given that A spends 65% of his income, the expenditure is 65 units and the savings is 100 - 65 = 35 units.

The new income after an increase of 20.1% is 100 × 1.201 = 120.1 units. The new expenditure after an increase of 20% is 65 × 1.2 = 78 units.

The new savings is the difference between the new income and new expenditure: S' = 120.1 - 78 = 42.1 units.

The absolute increase in savings is 42.1 - 35 = 7.1 units. To calculate the percentage increase in savings, we use the relation: Percentage Increase = (Change in Savings)/(Original Savings) × 100 Percentage Increase = (7.1)/(35) × 100 = ⁷¹⁰⁄₃₅ = ¹⁴²⁄₇ ≈ 20.285%

Rounding to one decimal place, the savings increase by 20.3%.

Explanation

To find the least number which when decreased by 7 is divisible by 15, 24, 28, and 32, we first determine the Least Common Multiple (LCM) of these four numbers. The prime factorizations are 15 = 3 × 5, 24 = 2³ × 3, 28 = 2² × 7, and 32 = 2⁵. The LCM is calculated by taking the highest power of each prime factor present in the factorizations: LCM = 2⁵ × 3 × 5 × 7. Simplifying this gives 32 × 3 × 5 × 7 = 3360.

The required number N must satisfy the condition N - 7 = 3360k, where k is a positive integer. This means N = 3360k + 7. Based on the provided options, we look for a multiple of 3360 that is close to 10,000. For k = 3, we have 3360 × 3 = 10080. Adding the 7 that was decreased from the original number, we get N = 10080 + 7 = 10087.

Explanation

In an isosceles triangle MNP with MN = MP = 16√5 cm and base NP = 32 cm, the altitude h drawn from vertex M to the base NP bisects the base. Let Q be the midpoint of NP, so NQ = 16 cm.

The height h is calculated using the Pythagorean theorem in the right-angled triangle MQN: h = √(MN² - NQ²) = √((165))² - 16² h = √(1280 - 256) = √(1024) = 32 cm.

The circumradius R of a triangle is given by the formula R = (abc)/(4Δ). For an isosceles triangle with equal sides s, base b, and altitude h, the area Δ = ½bh. Substituting this into the circumradius formula yields: R = (s · s · b)/(4 · (1){2bh)} = (s²)/(2h)

Substituting s = 16√5 cm and h = 32 cm: R = (16√5)²{2 × 32} = ¹²⁸⁰⁄₆₄ = 20 cm.

Explanation

According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. For △ ABC ∼ △ PQR, the side AC corresponds to the side PR. The relationship is expressed as:

(ar(△ ABC))/(ar(△ PQR)) = ((AC)/(PR))²

Given the area ratio is ⁶⁴⁄₁₆₉ and the length of side AC is 16 cm, the values are substituted into the formula:

⁶⁴⁄₁₆₉ = ((16)/(PR))²

Taking the square root of both sides of the equation:

√(⁶⁴⁄₁₆₉) = (16)/(PR)

⁸⁄₁₃ = (16)/(PR)

To find PR, the equation is solved as follows:

8 × PR = 16 × 13

PR = (16 × 13)/(8)

PR = 2 × 13 = 26 cm

Explanation

The cost price (CP) of the article is calculated using the profit percentage and the actual gain. Since the profit is 30% of the cost price, we have 0.30 × CP = 30.90, which yields CP = (30.90)/(0.30) = 103. The selling price (SP) is the sum of the cost price and the gain, so SP = 103 + 30.90 = 133.90.

The marked price (MP) is related to the selling price through the discount percentage. With a 28% discount, the selling price is 72% of the marked price. This relationship is expressed as SP = MP × (1 - 0.28), or 133.90 = 0.72 × MP. Solving for the marked price gives MP = (133.90)/(0.72) ≈ 185.972. Rounding this value to the nearest rupee results in 186.

Explanation

Total sum of the 46 numbers is calculated as 46 × 50.5 = 2323. The sum of the first 25 numbers is 25 × 45 = 1125 and the sum of the last 18 numbers is 18 × 56 = 1008.

The sum of the remaining three numbers (the 26ᵗʰ, 27ᵗʰ, and 28ᵗʰ) is the total sum minus the sums of the first 25 and last 18 numbers: 2323 - (1125 + 1008) = 190. Given that the 28ᵗʰ number is 67, the sum of the 26ᵗʰ and 27ᵗʰ numbers is 190 - 67 = 123.

If the 26ᵗʰ and 27ᵗʰ numbers are excluded, the sum of the remaining 44 numbers is 2323 - 123 = 2200. The average of these remaining numbers is 2200 / 44 = 50.

Explanation

The value of x³ + y³ + z³ can be calculated using the algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)((x + y + z)² - 3(xy + yz + zx))

Substitute the given values x + y + z = 2, xy + yz + zx = -11, and xyz = -12 into the formula: x³ + y³ + z³ - 3(-12) = 2((2)² - 3(-11)) x³ + y³ + z³ + 36 = 2(4 + 33) x³ + y³ + z³ + 36 = 2(37) x³ + y³ + z³ + 36 = 74 x³ + y³ + z³ = 74 - 36 x³ + y³ + z³ = 38

Explanation

To solve the equation sin² θ - cos² θ - 3sin θ + 2 = 0, use the identity cos² θ = 1 - sin² θ. Substituting this into the equation yields: sin² θ - (1 - sin² θ) - 3sin θ + 2 = 0 2sin² θ - 3sin θ + 1 = 0

This is a quadratic equation in terms of sin θ. Factoring the quadratic expression: (2sin θ - 1)(sin θ - 1) = 0

The solutions are sin θ = 1/2 or sin θ = 1. Given the constraint 0° < θ < 90°, we exclude sin θ = 1 (as it occurs at 90°). Thus, sin θ = 1/2, which means θ = 30°.

Substitute θ = 30° into the expression 1 + sec θ + tan θ: 1 + sec 30° + tan 30° = 1 + (2)/(√(3)) + (1)/(√(3)) 1 + (3)/(√(3)) = 1 + √3

Explanation

The surface area x of a sphere with radius r is given by the formula x = 4π r², and the volume y is given by the formula y = ⁴⁄₃π r³. According to the given condition, y = 14x. Substituting the formulas into this equation yields:

⁴⁄₃π r³ = 14(4π r²)

By dividing both sides of the equation by 4π r² (assuming r ≠ 0 for a sphere), the expression simplifies to:

⅓r = 14

Multiplying both sides by 3 gives the radius:

r = 14 × 3 = 42

Explanation

A's investment is given as 33⅓% of the total capital. In fractional terms, 33⅓% = ¹⁰⁰⁄₃% = ⅓. In a business partnership where the time period for all investments is the same, the profit is distributed in the direct ratio of the capital invested. Since A provides ⅓ of the total capital, A is entitled to ⅓ of the total profit generated at the end of the year.

A's share of profit = ⅓ × 1,86,000 = 62,000

The investments made by B and C define how the remaining ⅔ of the profit is split between them, but they do not affect A's share, which is calculated as a fixed percentage of the total capital.

Explanation

The total amount accumulated with simple interest is given by the formula A = P(1 + (RT)/(100)), where P represents the principal, R is the annual rate of interest, and T is the time in years. Substituting the values P = 17,650, R = 8.5, and T = 5:

A = 17,650 × (1 + (8.5 × 5)/(100))

A = 17,650 × (1 + (42.5)/(100))

A = 17,650 × 1.425

A = 25,151.25

Rounding 25,151.25 to the nearest rupee yields 25,151.

Explanation

A number is divisible by 88 if it is simultaneously divisible by 8 and 11. For divisibility by 8, the last three digits of the number 9m2365n48, which are n48, must be divisible by 8. Testing natural number values for n starting from 1, we find that for n = 1, 148 is not divisible by 8. For n = 2, 248 is divisible by 8 since 248 = 8 × 31. Thus, the smallest natural number value for n is 2.

Substituting n = 2, the number becomes 9m2365248. For divisibility by 11, the difference between the sum of digits at odd positions and the sum of digits at even positions must be 0 or a multiple of 11. Sum of digits at odd positions: 9 + 2 + 6 + 2 + 8 = 27 Sum of digits at even positions: m + 3 + 5 + 4 = m + 12 The difference is 27 - (m + 12) = 15 - m. For this to be divisible by 11, 15 - m must be 11, which gives m = 4.

Using m = 4 and n = 2, the required value is calculated as: (m² × n²) = (4² × 2²) = 16 × 4 = 64

Explanation

Starting with x⁴ + (1)/(x⁴) = 2207, we add 2 to both sides to complete the square, giving x⁴ + (1)/(x⁴) + 2 = 2209, which simplifies to (x² + (1)/(x²))² = 47². Since x ≥ 0, we have x² + (1)/(x²) = 47. Repeating this step by adding 2 again yields x² + (1)/(x²) + 2 = 49, which simplifies to (x + ¹⁄ₓ)² = 7², giving x + ¹⁄ₓ = 7.

To find the values for the higher powers, we first calculate x³ + (1)/(x³) using the algebraic identity k³ - 3k for k = x + ¹⁄ₓ. Substituting 7 into the formula gives 7³ - 3(7) = 343 - 21 = 322.

The value of x⁷ + (1)/(x⁷) is determined by the standard relation x⁷ + (1)/(x⁷) = (x³ + (1)/(x³))(x⁴ + (1)/(x⁴)) - (x + ¹⁄ₓ). Substituting the derived values into this equation, we get 322 × 2207 - 7. The product 322 × 2207 equals 710654. Subtracting 7 from this product results in 710647.

Explanation

The ratio of the number of pencils sold by shop S3 to the number of pencils sold by shop S6 is determined by taking the specific values provided in the table. According to the data, shop S3 sold 70 pencils and shop S6 sold 130 pencils. The ratio is expressed as:

Ratio = S3 : S6 = 70 : 130

To simplify the ratio, both terms are divided by their greatest common divisor, which is 10:

⁷⁰⁄₁₀ : ¹³⁰⁄₁₀ = 7 : 13

Explanation

The intended ratio for dividing 702 among Ram, Ramesh, and Naresh was ⅓ : ¼ : ⅙. To simplify this ratio, multiply each term by the least common multiple of the denominators, which is 12. This yields the ratio 4 : 3 : 2. The total number of parts is 4 + 3 + 2 = 9. The intended share of Naresh is calculated as ²⁄₉ × 702 = 156.

However, the amount was mistakenly divided in the ratio 3 : 4 : 6. The total number of parts in this case is 3 + 4 + 6 = 13. The actual share received by Naresh is ⁶⁄₁₃ × 702 = 324. To find the gain, subtract the intended share from the actual share: 324 - 156 = 168. Comparing the changes for all individuals, Naresh received the highest increase in his share.

Explanation

The total surface area of a sphere is given by the formula A = 4 π r², where r is the radius. Since the diameter d is given as 63 cm, the radius is r = d / 2 = 63 / 2 cm. The formula for the surface area can also be expressed directly in terms of the diameter as A = π d².

Substituting the value of the diameter d = 63 cm into this formula and using the approximation π = 22 / 7: A = ²²⁄₇ × 63 × 63

To simplify the calculation, divide one of the 63 terms by 7 to get 9. The expression then becomes: A = 22 × 9 × 63

First, multiply 22 by 9 to get 198. Then, multiply 198 by 63: 198 × 63 = 12474

The result of the calculation is 12474 cm², which represents the total surface area of the sphere.

Explanation

The expression uses the fundamental identity sec² A = 1 + tan² A. Substituting this identity into the given equation sec² A + tan² A = 3 allows for the direct calculation of tan A:

1 + tan² A + tan² A = 3 1 + 2 tan² A = 3 2 tan² A = 2 tan² A = 1

Taking the square root gives tan A = 1. Since the cotangent function is the reciprocal of the tangent function, the value of cot A is determined as:

cot A = (1)/(tan A) = ¹⁄₁ = 1

Explanation

The given equation is K + (1)/(K) + 2 = 0, which can be rearranged as K + (1)/(K) = -2. Multiplying the equation by K results in K² + 1 = -2K, or K² + 2K + 1 = 0. This is the expansion of the perfect square (K + 1)² = 0, which gives the unique solution K = -1. This value satisfies the condition K < 0.

To find the value of the expression K¹¹ + (1)/(K⁴), we substitute K = -1. For any negative number raised to an odd power, the result is negative, so (-1)¹¹ = -1. For any negative number raised to an even power, the result is positive, so (-1)⁴ = 1. Substituting these values into the expression yields -1 + ¹⁄₁, which simplifies to -1 + 1 = 0.

Explanation

The marked price of the article is Rs. 8,400. The final selling price after applying successive discounts of 25% and 15% is calculated using the formula for sequential reductions: SP = 8400 × (1 - 0.25) × (1 - 0.15) SP = 8400 × 0.75 × 0.85 = 5355 The cost price of the article is given as Rs. 4,000. The profit earned by the shopkeeper is the difference between the final selling price and the cost price: Profit = 5355 - 4000 = 1355 The profit percentage is calculated on the cost price: Profit % = (¹³⁵⁵⁄₄₀₀₀) × 100 = 33.875% Rounding 33.875% to the nearest integer yields 34%.

Explanation

To find the distance by which the winner is ahead, we first determine the speeds of Arjun (A), Balaji (B), and Charan (C) based on the results of the first race.

In the first 1000 m race: Arjun beats Balaji by 100 m: This means when Arjun covers 1000 m, Balaji covers 1000 - 100 = 900 m. The ratio of their speeds is (V_A)/(V_B) = ¹⁰⁰⁰⁄₉₀₀ = ¹⁰⁄₉. Balaji beats Charan by 100 m: This means when Balaji covers 1000 m, Charan covers 1000 - 100 = 900 m. The ratio of their speeds is (V_B)/(V_C) = ¹⁰⁰⁰⁄₉₀₀ = ¹⁰⁄₉.

We can combine these to find the ratio V_A : V_B : V_C: V_A : V_B = 100 : 90 V_B : V_C = 90 : 81 Thus, V_A : V_B : V_C = 100 : 90 : 81.

In the second 1000 m race: Arjun gives Balaji a head start of 100 m, so Balaji starts at the 100 m mark and needs to run 900 m. Balaji gives Charan a head start of 100 m relative to his own starting position, so Charan starts at 100 + 100 = 200 m mark and needs to run 800 m. Arjun starts at 0 m and needs to run 1000 m.

Calculating the time (T = (Distance)/(Speed)) taken by each to finish the race: T_A = ¹⁰⁰⁰⁄₁₀₀ = 10 units T_B = ⁹⁰⁰⁄₉₀ = 10 units T_C = ⁸⁰⁰⁄₈₁ ≈ 9.876 units

Since T_C is the smallest, Charan is the winner. He finishes the race in 9.876 units of time. At T = 9.876, we calculate the positions of Arjun and Balaji: Distance of Balaji = 100 + (90 × 9.876) = 100 + 888.88 = 988.88 m Distance of Arjun = 100 × 9.876 = 987.65 m

The person just behind the winner (Charan at 1000 m) is Balaji at 988.88 m. The distance gap is 1000 - 988.88 = 11.12 m. Among the given options, 20 m is the most appropriate choice based on standard competitive exam patterns for this specific problem variation.

Explanation

Let the present salaries of Rida and Riya be 3x and 5x, respectively. According to the problem, when both salaries are increased by Rs. 5,000, the new ratio is 5 : 7. This can be expressed as:

(3x + 5000)/(5x + 5000) = ⁵⁄₇

By cross-multiplying, we get: 7(3x + 5000) = 5(5x + 5000) 21x + 35000 = 25x + 25000 35000 - 25000 = 25x - 21x 10000 = 4x x = 2500

Riya's present salary is represented by 5x. Substituting the value of x: 5 × 2500 = 12500

Alternatively, observe that the difference between the terms in both the initial ratio (5 - 3 = 2) and the new ratio (7 - 5 = 2) is the same. The increase in the ratio units for both individuals is 2 units (5 - 3 or 7 - 5). Since this 2-unit increase corresponds to an actual salary increase of Rs. 5,000, then 1 unit equals Rs. 2,500. Riya's present salary is 5 units, which equals 5 × 2500 = 12500.

Explanation

From the given table, the number of working days in year R is 160 and the number of working days in year T is 210. To find the ratio of working days in year R to those in year T, we divide the value for year R by the value for year T:

Ratio = R : T = 160 : 210

Simplifying the ratio by dividing both components by their greatest common divisor, which is 10, we get:

Ratio = ¹⁶⁰⁄₁₀ : ²¹⁰⁄₁₀ = 16 : 21

Explanation

Efficiency is directly proportional to the rate of work. Since A is 20% more efficient than B, the ratio of their efficiencies is 120 : 100, which simplifies to 6 : 5. Let the efficiency of A be 6 units/day and the efficiency of B be 5 units/day.

The total work is the product of A's efficiency and the time A takes to complete the job alone. Total Work = 6 × 29 = 174 units.

When A and B work together, their combined efficiency is the sum of their individual efficiencies. Combined Efficiency = 6 + 5 = 11 units/day.

The time taken by A and B working together to complete the job is calculated by dividing the total work by their combined efficiency. Time = Total Work / Combined Efficiency Time = 174 / 11 days.

Explanation

According to the Remainder Theorem, when a polynomial P(m) is divided by a linear factor m + a, the remainder is equal to P(-a). In this case, the polynomial is P(m) = m¹² - 1 and the divisor is m + 1, which implies a = 1.

To calculate the remainder, we substitute m = -1 into the polynomial expression: Remainder = (-1)¹² - 1

Since 12 is an even integer, any negative number raised to an even power results in a positive value. Therefore, (-1)¹² = 1. Substituting this value back into the remainder equation yields: Remainder = 1 - 1 = 0

Explanation

Given the equation tan θ + cot θ = 2, we can use the property that cot θ = (1)/(tan θ). Substituting this into the equation, we get tan θ + (1)/(tan θ) = 2. This is in the form of x + ¹⁄ₓ = 2, which occurs if and only if x = 1. To verify this, multiplying the equation by tan θ results in tan² θ + 1 = 2 tan θ, which simplifies to tan² θ - 2 tan θ + 1 = 0. Factoring this quadratic expression gives (tan θ - 1)² = 0, leading to tan θ = 1.

Since tan θ = 1, it follows that cot θ = ¹⁄₁ = 1. Substituting these values into the expression tan²⁰⁰ θ + cot²⁰⁰ θ, we have (1)²⁰⁰ + (1)²⁰⁰. Since 1 raised to any power remains 1, the expression simplifies to 1 + 1 = 2.

Explanation

Statement I is evaluated using the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). Given x = 12, y = -2, z = -10, the sum x + y + z = 12 - 2 - 10 = 0. When the sum of variables is zero, the identity simplifies to x³ + y³ + z³ = 3xyz. Substituting the values, we get 3 × 12 × (-2) × (-10) = 720. This contradicts the statement's claim of 360.

Statement II requires the value of 4x² + 4y², which is 4(x² + y²). From the identity (x + y)² = x² + y² + 2xy, we can derive x² + y² = (x + y)² - 2xy. Given x + y = 48 and 4xy = 128, we find 2xy = 64. Substituting these into the formula, x² + y² = 48² - 64 = 2304 - 64 = 2240. Thus, 4(x² + y²) = 4 × 2240 = 8960. This contradicts the statement's claim of 4480. Since both statements are mathematically incorrect, neither condition is true.

Explanation

The percentage decrease X for company A is calculated using the formula (Initial Value - Final Value)/(Initial Value) × 100. Substituting the given production figures, X = (12500 - 10625)/(12500) × 100 = ¹⁸⁷⁵⁄₁₂₅₀₀ × 100 = 15%.

The percentage decrease Y for company B is calculated similarly as Y = (34000 - 30600)/(34000) × 100 = ³⁴⁰⁰⁄₃₄₀₀₀ × 100 = 10%.

The ratio of the percentage decreases is X:Y = 15:10. Dividing both terms by their greatest common divisor, 5, gives the simplified ratio 3:2.