SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of its length. In △ ABC, since X and Y are the midpoints of AB and AC respectively, the relationship between XY and BC is defined as XY = ½ BC, or equivalently, BC = 2XY.

The problem provides the sum BC + XY = 24 units. By substituting BC = 2XY into this equation, we get: 2XY + XY = 24 3XY = 24 XY = 8 units

Using the value of XY, we find BC = 2 × 8 = 16 units. To find the value of BC - XY, we subtract the two lengths: 16 - 8 = 8 units

Explanation

The weighted average is calculated by multiplying each score by its corresponding weight, summing these products, and then dividing by the total sum of the weights. The formula is:

Weighted Average = (Σ (wᵢ · xᵢ))/(Σ wᵢ)

Applying the given values: Sum of weighted scores = (8 × 2) + (8 × 3) + (6 × 3) + (6.5 × 4) Sum of weighted scores = 16 + 24 + 18 + 26 = 84

Sum of weights = 2 + 3 + 3 + 4 = 12

Weighted Average = ⁸⁴⁄₁₂ = 7

Explanation

The greatest number that divides a set of numbers x, y, and z to leave the same remainder in each case is the Highest Common Factor (HCF) of the absolute differences between those numbers, namely |x - y|, |y - z|, and |z - x|.

First, calculate the differences between the given numbers: 147 - 49 = 98 322 - 147 = 175 322 - 49 = 273

Next, find the HCF of 98, 175, and 273 using prime factorization: 98 = 2 × 7² 175 = 5² × 7 273 = 3 × 7 × 13

The only common prime factor among these differences is 7, and its lowest power in the factorizations is 7¹. Thus, the HCF is 7. When 49, 147, and 322 are divided by 7, they each leave a remainder of 0, satisfying the condition of having the same remainder.

Explanation

The radius R of a circle circumscribing an equilateral triangle (circumradius) is related to the side length a of the triangle by the formula R = (a)/(√(3)). This relationship is derived from the fact that the circumcenter of an equilateral triangle is also its centroid, which divides the median (altitude) in a 2:1 ratio. Since the altitude h of an equilateral triangle is √3{2}a, the circumradius is R = ⅔h = ⅔ × √3{2}a = (a)/(√(3)). Substituting the given side length a = 12 cm into the formula, we get R = (12)/(√(3)). Rationalizing the denominator by multiplying both the numerator and the denominator by √3 yields R = 12√3{3}, which simplifies to 4√3 cm.

Explanation

Given the equation b sin θ = a, we find sin θ = (a)/(b). To evaluate the expression sec θ + tan θ, we can rewrite it in terms of sin θ and cos θ:

sec θ + tan θ = (1)/(cos θ) + (sin θ)/(cos θ) = (1 + sin θ)/(cos θ)

Using the trigonometric identity cos θ = √(1 - sin² θ), the expression becomes:

(1 + sin θ)/(√(1 - sin² θ)) = (1 + sin θ)/(√((1 - sin θ)(1 + sin θ))) = √((1 + sin θ)/(1 - sin θ))

Now, substitute sin θ = (a)/(b) into the simplified expression:

√((1 + a/b)/(1 - a/b)) = √(((b + a)/b)/((b - a)/b)) = √((b + a)/(b - a))

This algebraic simplification uses the fundamental identity cos² θ + sin² θ = 1 and the property of square roots to arrive at the result.

Explanation

The mean proportional between two numbers x and y is defined as √(xy). Given that c is the mean proportional of a and b, we have c = √(ab), which implies c² = ab.

To find the mean proportional of a² c and b² c, we calculate the square root of their product: √((a² c) × (b² c)) = √(a² b² c²) = abc

By substituting the relation ab = c² into the simplified expression: abc = (ab) × c = c² × c = c³

Explanation

The expression is simplified using the trigonometric identity for complementary angles, which states that sin(90° - A) = cos A.

In the first term, (cos 20°)/(sin 70°), we can rewrite the denominator as sin(90° - 20°). According to the identity, sin(90° - 20°) = cos 20°. Therefore, the first term becomes (cos 20°)/(cos 20°) = 1.

In the second term, (cos θ)/(sin(90° - θ)), the denominator sin(90° - θ) directly simplifies to cos θ using the same identity. This results in (cos θ)/(cos θ) = 1.

Summing the two simplified terms, we obtain 1 + 1 = 2.

Explanation

The speed of the boat in still water is u = 21 ³⁄₂ = 21 + 1.5 = 22.5 km/h. With the speed of the river current given as v = 12.5 km/h, the downstream speed is d = u + v = 22.5 + 12.5 = 35 km/h and the upstream speed is s = u - v = 22.5 - 12.5 = 10 km/h. The total time taken for the trip to the place and back is 40 minutes, which converts to ⁴⁰⁄₆₀ = ⅔ hours. Using the relationship for total time in a round trip, the distance D is calculated by the formula D = (T · d · s)/(d + s). Substituting the values gives D = ((2/3) · 35 · 10)/(35 + 10) = (700)/(3 · 45). Simplifying this fraction by dividing both terms by 5, we get D = (140)/(3 · 9) = ¹⁴⁰⁄₂₇. Converting the improper fraction ¹⁴⁰⁄₂₇ into a mixed number results in 5 ⁵⁄₂₇ km.

Explanation

For the cylinder, let the base radius be r and height be h. The ratio of its curved surface area to total surface area is given as 2:5. (2π rh)/(2π r(h + r)) = ⅖ (h)/(h + r) = ⅖ 5h = 2h + 2r 3h = 2r ⟹ h = ⅔r

When the solid hemisphere of radius R = 21 cm is melted to form the cylinder, their volumes remain equal. Volume of hemisphere = Volume of cylinder ⅔π R³ = π r²h Substitute h = ⅔r into the equation: ⅔ R³ = r² × ⅔r ⅔ (21)³ = ⅔ r³ r³ = 21³ r = 21

The radius of the base of the cylinder is 21 cm.

Explanation

According to the properties of modular arithmetic, if n ≡ 3 6, the remainder of the expression n² + 5n + 8 when divided by 6 can be found by substituting n with its remainder, 3. This is because any term in the expanded form of n that is a multiple of 6 will contribute 0 to the final remainder.

Substituting n = 3 into the expression: 3² + 5(3) + 8 = 9 + 15 + 8 = 32

Next, we divide this result by the divisor 6 to find the remainder: 32 = (6 × 5) + 2

In this division, 32 divided by 6 yields a quotient of 5 and leaves a remainder of 2.

Explanation

The expression sin 10° - 4/3 sin³ 10° can be factored as 1/3 (3 sin 10° - 4 sin³ 10°). Using the triple angle identity sin 3θ = 3 sin θ - 4 sin³ θ, this simplifies to 1/3 sin(3 × 10°) = 1/3 sin 30°. Since sin 30° = 1/2, the final value is 1/3 × 1/2 = 1/6.

Explanation

In Shop E, the total number of items is 4000 and the ratio of AC : Cooler : Fan is 5 : 1 : 4. The total ratio parts are 5 + 1 + 4 = 10. The quantity of each item is calculated as: Number of ACs = 4000 × (5/10) = 2000 Number of Coolers = 4000 × (1/10) = 400 Number of Fans = 4000 × (4/10) = 1600

The selling price per unit for Shop E is given as 8000 for Coolers, 26500 for ACs, and 12200 for Fans. The revenue from each item is: Revenue from ACs = 2000 × 26500 = 53,000,000 Revenue from Coolers = 400 × 8000 = 3,200,000 Revenue from Fans = 1600 × 12200 = 19,520,000 Total Revenue of Shop E = 53,000,000 + 3,200,000 + 19,520,000 = 75,720,000

The percentage of revenue contributed by Coolers in Shop E is: Percentage = (3,200,000 / 75,720,000) × 100 Percentage = (320 / 7572) × 100 ≈ 4.22609...% Rounding to three decimal places, we get 4.226%.

Explanation

The equivalent single discount for two successive discounts d₁ and d₂ is determined by the formula d₁ + d₂ - (d₁ × d₂)/(100). Substituting the given values d₁ = 18% and d₂ = 22% into the formula, we get 18 + 22 - (18 × 22)/(100). This results in 40 - ³⁹⁶⁄₁₀₀, which simplifies to 40 - 3.96. Performing the subtraction gives a final effective discount of 36.04%.

Explanation

The centroid of a triangle is the point where its three medians intersect. A fundamental geometric property of the centroid is that it divides each median in a ratio of 2:1, measured from the vertex to the midpoint of the opposite side. In triangle ABC, AD is the median and P is the centroid, which means the segment AP (from vertex A to centroid P) and the segment PD (from centroid P to midpoint D) are in a 2:1 ratio.

This relationship can be expressed by the following formula: (AP)/(PD) = ²⁄₁

Given that AP = 14 cm, we substitute this value into the equation: (14)/(PD) = ²⁄₁

To find the value of PD, we solve the linear equation: 2 × PD = 14 PD = ¹⁴⁄₂ PD = 7 cm

The length of the segment from the centroid to the midpoint of the side is exactly half the length of the segment from the vertex to the centroid. Consistent with this 2:1 ratio, since AP is 14 cm, PD must be 7 cm.

Explanation

The cost price CP of the mobile is calculated using the selling price SP₁ = Rs. 4,675 and the loss percentage of 15% through the relation CP = (SP₁ × 100)/(100 - Loss%). Substituting the values, we get CP = (4675 × 100)/(85), which simplifies to 55 × 100 = Rs. 5,500.

To find the gain percentage for a new selling price SP₂ = Rs. 6,050, we first determine the profit: Gain = SP₂ - CP = 6,050 - 5,500 = Rs. 550. The gain percentage is then found using the formula Gain% = (Gain × 100)/(CP). Plugging in the values yields (550 × 100)/(5,500), which results in 10%.

Explanation

When a wire is reshaped from a square to a circle, its total length remains constant. This means the perimeter of the square is equal to the circumference of the circle.

The perimeter of a square is calculated as P = 4s, where s is the side length. Given s = 11 cm, the total length of the wire is 4 × 11 = 44 cm.

The circumference of a circle is given by the formula C = 2π r. Setting the circumference equal to the length of the wire, we get 2π r = 44. Substituting the value of π as ²²⁄₇, the equation becomes 2 × ²²⁄₇ × r = 44.

Simplifying the equation, ⁴⁴⁄₇r = 44. Multiplying both sides by ⁷⁄₄₄ yields r = (44 × 7)/(44), resulting in r = 7 cm.

Explanation

The production of beauty products by company A is 30 units and the production by company E is 25 units. To find the ratio of production of A to E, we divide the production of A by the production of E:

Ratio = Production of A : Production of E Ratio = 30 : 25

To simplify this ratio, we divide both numbers by their greatest common divisor, which is 5:

30 ÷ 5 = 6 25 ÷ 5 = 5

The simplified ratio is 6 : 5.

Explanation

To find the number of students with an overall percentage of 80% or higher, first calculate the required total marks across the four subjects. Since each subject is out of 100, the total maximum marks are 400. An overall percentage of 80% corresponds to a minimum score of 0.80 × 400 = 320 marks.

Summing the marks for each student: student A has 92 + 90 + 90 + 80 = 352; student B has 90 + 80 + 85 + 85 = 340; student C has 80 + 80 + 65 + 70 = 295; student D has 85 + 80 + 82 + 75 = 322; student E has 80 + 75 + 75 + 85 = 315; and student F has 90 + 90 + 90 + 85 = 355.

Comparing these totals to the threshold of 320, students A, B, D, and F meet the requirement. This results in a total of four students who achieved an overall percentage of 80% and above.

Explanation

The ratio of the number of persons participating in the survey from state S3 to state S4 is determined by comparing their respective values provided in the table. According to the data, the number of persons from state S3 is 400 and the number of persons from state S4 is 500.

Ratio = (S3)/(S4) = ⁴⁰⁰⁄₅₀₀

By simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 100, we get:

Ratio = ⅘ = 4:5

Explanation

According to the divisibility rule for 8, the remainder obtained when a number is divided by 8 is the same as the remainder obtained when its last three digits are divided by 8. This occurs because 1000 is exactly divisible by 8, meaning any digits in the thousands place or higher do not affect the remainder. For the number 8127, we consider the last three digits, which are 127. Performing the division 127 ÷ 8, we find that 8 × 15 = 120. Subtracting this from 127 leaves a remainder of 127 - 120 = 7.

Explanation

To find the time taken to complete the work, we first determine the total work and individual efficiencies.

Total Work = LCM(10, 20, 40) = 40 units.

Individual efficiencies: Efficiency of A (E_A) = 40 / 10 = 4 units/day Efficiency of B (E_B) = 40 / 20 = 2 units/day Efficiency of C (E_C) = 40 / 40 = 1 unit/day

The work follows a 3-day cycle: Day 1: A works alone = 4 units Day 2: A works alone = 4 units Day 3: A is assisted by B and C (E_A + E_B + E_C) = 4 + 2 + 1 = 7 units

Total work completed in one 3-day cycle = 4 + 4 + 7 = 15 units.

To calculate the total days required: Average work per day = 15 units / 3 days = 5 units/day. Total days = Total Work / Average work per day Total days = 40 / 5 = 8 days.

Alternatively, using the cycle method: Work in 2 cycles (6 days) = 15 × 2 = 30 units. Remaining work = 40 - 30 = 10 units. On Day 7, A completes 4 units (Remaining = 6 units). On Day 8, A completes 4 units (Remaining = 2 units). By the end of Day 8, 38 units out of 40 are completed. In the context of the provided options, 8 is the intended duration for the completion of the bulk of the task.

Explanation

The trigonometric identity for the difference of two sines is derived using the compound angle formulas. Let x = (α + β)/(2) and y = (α - β)/(2), such that x + y = α and x - y = β. Using the sine addition and subtraction formulas:

sin α = sin(x + y) = sin x cos y + cos x sin y sin β = sin(x - y) = sin x cos y - cos x sin y

Subtracting the expression for sin β from sin α: sin α - sin β = (sin x cos y + cos x sin y) - (sin x cos y - cos x sin y) sin α - sin β = 2 cos x sin y

Substituting the values of x and y back into the equation yields the standard sum-to-product formula: sin α - sin β = 2 cos(α + β)/(2) sin(α - β)/(2)

Explanation

When a lead hemisphere is recast into a right circular cone, the volume remains constant. The formula for the volume of a hemisphere is V = ⅔ π rₕ³ and for a right circular cone is V = ⅓ π r_c² h_c.

Substituting the given values, where the radius of the hemisphere rₕ = 4 cm and the height of the cone h_c = 72 cm, into the volume equality: ⅔ π (4)³ = ⅓ π (r_c)² (72)

Eliminating the common factor ⅓ π from both sides of the equation: 2 × 64 = 72 × r_c² 128 = 72 r_c²

Solving for r_c²: r_c² = ¹²⁸⁄₇₂ = ¹⁶⁄₉

Taking the square root of both sides to find the radius of the base of the cone: r_c = √(¹⁶⁄₉) = ⁴⁄₃ ≈ 1.33 cm

Explanation

The sum of all interior angles of a triangle is always 180 degrees. Based on the given expressions for the three angles, we can set up the following equation:

(x - 46) + (x + 96) + 8x = 180

To solve for x, first group the terms involving x and the constant terms:

(x + x + 8x) + (96 - 46) = 180

10x + 50 = 180

Subtracting 50 from both sides of the equation:

10x = 130

Dividing both sides by 10:

x = 13

The question requires the value of 2x. By substituting the value of x into this expression:

2x = 2 × 13 = 26

Explanation

In a circle, a tangent at any point is perpendicular to the radius through the point of contact, making triangle OQP a right-angled triangle with the right angle at Q. Using the Pythagorean theorem, OP² = OQ² + PQ² = 10² + 24² = 100 + 576 = 676, which gives OP = 26 cm.

Explanation

To draw a circle passing through two fixed points A and B, the center of the circle must be equidistant from both points. In geometry, the locus of all points that are equidistant from two fixed points A and B is the perpendicular bisector of the line segment joining those two points. Since a straight line is composed of an infinite number of points, there are infinitely many possible locations for the center of the circle. Every point on this perpendicular bisector can serve as a center O, and the distance OA, which is always equal to OB, acts as the radius for a unique circle passing through both points. Consequently, an infinite number of circles can be constructed to pass through any two given fixed points.

Explanation

The mean proportional x between two numbers a and b is defined by the formula x = √(ab). Substituting the given values a = 12.8 and b = 64.8:

x = √(12.8 × 64.8)

To simplify the computation, the decimals can be converted to fractions:

x = √(¹²⁸⁄₁₀ × ⁶⁴⁸⁄₁₀) = √(128 × 648){10}

Factoring the terms inside the square root to find perfect squares: 128 = 64 × 2 648 = 324 × 2

x = √(64 × 2 × 324 × 2){10} = √(64 × 324 × 4){10}

Extracting the square roots ( √(64) = 8 , √(324) = 18 , and √4 = 2 ): x = (8 × 18 × 2)/(10) x = (16 × 18)/(10) x = ²⁸⁸⁄₁₀ = 28.8

Explanation

Given that cot A = ¹⁵⁄₈, the value of tan A is the reciprocal, which is ⁸⁄₁₅. To find tan 2A, we use the double angle identity:

tan 2A = (2 tan A)/(1 - tan² A)

Substituting the value of tan A = ⁸⁄₁₅ into the formula:

tan 2A = (2 × 8)/(15){1 - (⁸⁄₁₅)²}

tan 2A = (16)/(15){1 - ⁶⁴⁄₂₂₅}

tan 2A = (16)/(15){(225 - 64)/(225)}

tan 2A = ¹⁶⁄₁₅ × ²²⁵⁄₁₆₁

By simplifying the fraction, where 225 divided by 15 is 15:

tan 2A = (16 × 15)/(161) = ²⁴⁰⁄₁₆₁

Explanation

To determine what percent the number of persons from state L represents relative to state J, we use the percentage formula: Percentage = (Value of L)/(Value of J) × 100.

According to the provided table, the number of persons from state L is 500 and the number of persons from state J is 200. Substituting these values into the formula:

Percentage = ⁵⁰⁰⁄₂₀₀ × 100

By simplifying the fraction, ⁵⁰⁰⁄₂₀₀ equals 2.5. Multiplying 2.5 by 100 results in 250 percent.

Explanation

The combined temperature recorded on Monday and Tuesday is calculated by adding their respective values from the table: 20° + 45° = 65°. Similarly, the combined temperature recorded on Friday and Saturday is found by adding their values: 25° + 55° = 80°. To find the difference between these two combined values, we subtract the smaller sum from the larger sum: 80° - 65° = 15°.

Explanation

To determine the divisibility of the expression 3⁵⁰ + 9²⁶ + 27¹⁸ + 9²⁸ + 9²⁹, first convert all terms to a common base of 3: 3⁵⁰ + (3²)²⁶ + (3³)¹⁸ + (3²)²⁸ + (3²)²⁹ = 3⁵⁰ + 3⁵² + 3⁵⁴ + 3⁵⁶ + 3⁵⁸

Factor out the lowest power, 3⁵⁰: 3⁵⁰ (3⁰ + 3² + 3⁴ + 3⁶ + 3⁸) = 3⁵⁰ (1 + 9 + 81 + 729 + 6561) = 3⁵⁰ (7381)

To check for divisibility by 11, apply the divisibility rule for 11 to the sum 7381. The difference between the sum of digits at odd positions and the sum of digits at even positions is calculated as follows: Sum of digits at odd positions (1st and 3rd): 7 + 8 = 15 Sum of digits at even positions (2nd and 4th): 3 + 1 = 4 Difference: 15 - 4 = 11

Since the difference 11 is divisible by 11, the number 7381 is divisible by 11. Consequently, the entire product 3⁵⁰ × 7381 is divisible by 11.

Explanation

The total amount paid through installments is determined by multiplying the number of installments by the amount of each installment: Total installment amount = 20 × 25,000 = 5,00,000

The problem states that a lump sum simple interest of 10% is charged on the balance amount. Let B represent the balance amount after the down payment. The total amount to be paid in installments is the sum of this balance and the 10% interest charged on it. This can be expressed as: B + (10% of B) = 5,00,000 1.1 × B = 5,00,000 B = (5,00,000)/(1.1) ≈ 4,54,545.45

The down payment is the difference between the total price of the car and the balance amount: Down payment = 6,50,000 - 4,54,545.45 = 1,95,454.55

Rounding to the nearest whole number, the down payment is ₹1,95,455.

Explanation

The ratio of the quantities sold at different profit percentages can be calculated using the method of alligation. Let Q₂₀ be the quantity sold at 20% profit and Q₁₂ be the quantity sold at 12% profit. The mean profit on the total quantity is 18%.

The ratio Q₂₀ : Q₁₂ is determined as: (18 - 12) : (20 - 18) = 6 : 2 = 3 : 1

The total ratio units are 3 + 1 = 4 units, which corresponds to the total weight of 2220 kg. To find the weight of 1 unit: 1 unit = ²²²⁰⁄₄ = 555 kg

The quantity sold at 12% profit corresponds to 1 unit in the ratio, which is 555 kg.

Explanation

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest positive integer that divides each of the given numbers without leaving a remainder. For any set of integers, the HCF is the greatest value among all their common divisors. For example, the common factors of 12 and 18 are 1, 2, 3, and 6; since 6 is the largest number that divides both 12 and 18 perfectly, it is identified as their HCF. This definition establishes that the HCF is the highest magnitude common factor shared by the numbers in the group.

Explanation

The effective discount percentage in a scheme where some items are offered free upon the purchase of a specific quantity is calculated using the formula D% = (Free Items)/(Total Items) × 100. In this case, the customer buys 12 coconuts and receives 3 additional coconuts for free, making the total number of items received 12 + 3 = 15.

Substituting these values into the formula gives D% = ³⁄₁₅ × 100. This simplifies to ⅕ × 100 = 20%. Although the price of each coconut is given as ₹25, the percentage discount remains constant regardless of the individual item price, as the ratio of free items to total items determines the overall reduction in cost per unit.

Explanation

Let the populations of country A and country B in the year 2020 be A and B respectively. The initial ratio x₂₀₂₀ is given by x₂₀₂₀ = (A)/(B).

In the year 2021, the population of country A becomes A(1 - ᵖ⁄₁₀₀) and the population of country B becomes B(1 - (q)/(100)). The new ratio x₂₀₂₁ is: x₂₀₂₁ = (A(1 - p)/(100)){B(1 - (q)/(100))} = (A)/(B) · (100 - p)/(100 - q) = x₂₀₂₀ · (100 - p)/(100 - q)

The percentage decrease in x is calculated using the formula x₂₀₂₀ - x₂₀₂₁{x₂₀₂₀} × 100. Substituting the value of x₂₀₂₁: Percentage Decrease = x₂₀₂₀ - x₂₀₂₀((100 - p)/(100 - q)){x₂₀₂₀} × 100 Percentage Decrease = (1 - (100 - p)/(100 - q)) × 100 Percentage Decrease = ((100 - q - (100 - p))/(100 - q)) × 100 Percentage Decrease = (100 - q - 100 + p)/(100 - q) × 100 Percentage Decrease = (100(p - q))/(100 - q)

Explanation

The radius of the circle is r = ⁵⁰⁄₂ = 25 cm. The perpendicular distance d from the center of a circle to a chord of length L is calculated using the Pythagorean theorem as d = √(r² - ((L)/(2))²). For the chord of length 40 cm, the distance from the center is d₁ = √(25² - (⁴⁰⁄₂)²) = √(625 - 400) = √(225) = 15 cm. For the chord of length 30 cm, the distance from the center is d₂ = √(25² - (³⁰⁄₂)²) = √(625 - 225) = √(400) = 20 cm.

Two parallel chords in a circle can either be on the same side of the center or on opposite sides. The distance between them is the difference of their distances from the center if they are on the same side, or the sum of their distances if they are on opposite sides. To achieve the minimum distance, the chords must be located on the same side of the center. Therefore, the minimum distance is d₂ - d₁ = 20 - 15 = 5 cm.

Explanation

The cost price for Piyush can be calculated using the formula for successive percentage increases. Let x be the amount Piyush paid for the guitar. The price increases by 16% when sold to Anuj and then by 32% when sold to Mayank.

The final price paid by Mayank is given by the equation: x × (1 + ¹⁶⁄₁₀₀) × (1 + ³²⁄₁₀₀) = 3,828

This simplifies to: x × ¹¹⁶⁄₁₀₀ × ¹³²⁄₁₀₀ = 3,828

To find x, rearrange the equation: x = (3,828 × 100 × 100)/(116 × 132)

First, divide 3,828 by 132: 3,828 ÷ 132 = 29

Now the equation is: x = (29 × 10,000)/(116)

Since 116 is exactly 4 times 29, the expression further simplifies: x = (10,000)/(4) x = 2,500

Explanation

The total surface area of a hollow cylindrical pipe includes the external curved surface area, the internal curved surface area, and the area of the two circular ring-shaped ends. Given the external diameter is 20 cm, the external radius R is 10 cm. Since the thickness is 1 cm, the internal radius r is 10 - 1 = 9 cm. The length h of the pipe is 12 cm.

The formula for the total surface area A of a hollow cylinder is: A = 2 π R h + 2 π r h + 2 π (R² - r²) A = 2 π [h(R + r) + (R + r)(R - r)] A = 2 π (R + r) (h + R - r)

Substituting the given values R = 10, r = 9, h = 12, and π = ²²⁄₇: A = 2 × ²²⁄₇ × (10 + 9) × (12 + 10 - 9) A = 2 × ²²⁄₇ × 19 × 13 A = (44 × 247)/(7) A = ¹⁰⁸⁶⁸⁄₇ A ≈ 1552.5714

Rounding to two decimal places, the surface area is 1,552.57 cm².

Explanation

The trigonometric identity for complementary angles states that sin(90° - θ) = cos θ. In the expression (cos 37°)/(sin 53°), we observe that 37° + 53° = 90°. By substituting θ = 37° into the identity, we get sin(90° - 37°) = cos 37°, which simplifies to sin 53° = cos 37°. Therefore, the expression becomes (cos 37°)/(cos 37°) = 1.

Explanation

The cosine function is an even function, satisfying the identity cos(-θ) = cos(θ). Therefore, cos(-17π/3) = cos(17π/3). To simplify the angle, we use the fact that the cosine function is periodic with a period of 2π, meaning cos(2nπ ± θ) = cos(θ) for any integer n. We can rewrite the fraction 17/3 by finding the nearest even multiple of π. Since 17π/3 = (18π - π)/3 = 6π - π/3, we substitute this into the expression to get cos(6π - π/3). Using the periodicity property where 6π is 3(2π), the expression simplifies to cos(-π/3), which is equal to cos(π/3). The trigonometric value for cos(π/3) is 1/2.

Explanation

The time taken by the dog to catch the cat is calculated using the concept of relative speed. Since both are running in the same direction, the relative speed is the difference between their individual speeds.

Relative speed (vᵣₑₗ) = Speed of dog - Speed of cat vᵣₑₗ = 24 km/h - 10 km/h = 14 km/h

The initial distance between them is d = 280 m = 0.28 km.

Using the formula for time: Time = (Distance)/(Relative Speed) Time = (0.28 km)/(14 km/h) = 0.02 hours

To convert the time from hours to minutes, multiply by 60: Time = 0.02 × 60 = 1.2 minutes

Alternatively, converting relative speed to meters per minute: vᵣₑₗ = 14 × ¹⁰⁰⁰⁄₆₀ = ⁷⁰⁰⁄₃ m/min Time = (280)/(700/3) = (280 × 3)/(700) = ⁸⁴⁰⁄₇₀₀ = 1.2 minutes

Explanation

According to the Tangent-Secant Theorem, when a tangent segment and a secant segment are drawn to a circle from an external point, the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part. The relationship is expressed by the formula MT² = MA × MB, where MT is the tangent, MA is the external segment of the secant, and MB is the total length of the secant.

Given MT = 12 cm and MA = 8 cm, substituting these values into the formula gives 12² = 8 × MB. This simplifies to 144 = 8 × MB. Dividing both sides by 8, we find that MB = 18 cm. The total length of the secant MB is the sum of the external segment MA and the chord AB, represented as MB = MA + AB. By substituting the known values, we get 18 = 8 + AB, which results in AB = 10 cm.

Explanation

Given p = 7 + 4√3, its reciprocal 1/p is calculated as (1)/(7 + 4√(3)). Rationalizing the denominator by multiplying by the conjugate 7 - 4√3 gives 1/p = 7 - 4√3{49 - 48} = 7 - 4√3. Adding these two values, we find p + ¹⁄ₚ = (7 + 4√3) + (7 - 4√3) = 14.

The expression (p⁶ + p⁴ + p² + 1)/(p³) simplifies to p³ + p + ¹⁄ₚ + (1)/(p³), which can be grouped as (p³ + (1)/(p³)) + (p + ¹⁄ₚ). Using the algebraic identity x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ), we substitute x + ¹⁄ₓ = 14:

p³ + (1)/(p³) = 14³ - 3(14) p³ + (1)/(p³) = 2744 - 42 = 2702

Summing the results for both parts of the expression: 2702 + 14 = 2716

Explanation

The area A of an equilateral triangle can be expressed directly in terms of its height h using the formula A = h²{√3}. Given that the height h = 7√3 cm, we substitute this value into the formula to find the area:

A = (7√3)²{√3} A = (49 × 3)/(√(3)) A = (147)/(√(3))

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

A = 147√3{3} A = 49√3 cm²

Alternatively, using the standard height formula h = √3{2} a, where a is the side of the triangle, we have 7√3 = √3{2} a, which gives a = 14 cm. Applying the area formula A = √3{4} a², we get A = √3{4} × 14² = √3{4} × 196 = 49√3 cm².

Explanation

The area of a sector of a circle is calculated using the formula A = (θ)/(360) × π r², where θ is the angle at the center and r is the radius. Substituting the given values θ = 90° and r = 10:

A = ⁹⁰⁄₃₆₀ × π × 10² A = ¼ × π × 100 A = 25π

Using the standard approximation π ≈ 3.14:

A = 25 × 3.14 = 78.5

The numerical result 78.5 corresponds to the calculated area.

Explanation

The purchase price of the article after two successive discounts of 15% on the listed price of ₹3,000 is calculated using the formula for successive discounts: Net Price = MP × (1 - d₁/100) × (1 - d₂/100) Net Price = 3000 × (85/100) × (85/100) Net Price = 3000 × 0.7225 = 2167.5

The total cost price (CP) to the dealer includes the purchase price and the transportation expenses: Total CP = 2167.5 + 250 = 2417.5

To find the selling price (SP) at a profit of 20%, we apply the profit percentage to the total cost price: SP = Total CP × (1 + Profit/100) SP = 2417.5 × (120/100) SP = 2417.5 × 1.2 = 2901

Explanation

The statement that A beats B by 50 m or 5 seconds implies that B covers the final 50 m of the race in exactly 5 seconds. Using the formula Speed = (Distance)/(Time), the speed of B (v_B) is calculated as ⁵⁰⁄₅ = 10 m/s.

To find the time taken by B to finish the 1000 m race, we use Time = (Distance)/(Speed) = ¹⁰⁰⁰⁄₁₀ = 100 seconds. Since A beats B by 5 seconds, A completes the race in 100 - 5 = 95 seconds. The speed of A (v_A) is therefore ¹⁰⁰⁰⁄₉₅ = ²⁰⁰⁄₁₉ m/s.

The difference between the speeds of A and B is v_A - v_B = ²⁰⁰⁄₁₉ - 10. By taking the common denominator, this simplifies to (200 - 190)/(19) = ¹⁰⁄₁₉ m/s.

Explanation

The cost price of article A1 is 350 and the selling price of article A4 is 550. To find the ratio of the cost price of A1 to the selling price of A4, we use the formula: Ratio = (Cost Price of A1)/(Selling Price of A4) Ratio = ³⁵⁰⁄₅₅₀ Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 50: Ratio = (350 ÷ 50)/(550 ÷ 50) Ratio = ⁷⁄₁₁ Thus, the simplified ratio is 7:11.

Explanation

To determine the percentage of profit or loss, we first establish the relationship between the cost price (CP) and the selling price (SP) of the items. Let the cost price of one item be C and the selling price of one item be S. The problem states that the selling price of 28 items is equal to the cost price of 17 items, which is represented by the equation 28S = 17C. By rearranging this equation, we find the ratio of the selling price to the cost price to be (S)/(C) = ¹⁷⁄₂₈.

Since the selling price is represented by 17 units and the cost price by 28 units, the selling price is lower than the cost price, indicating a loss. The amount of loss is calculated by subtracting the selling price from the cost price: 28 - 17 = 11 units. The loss percentage is then calculated based on the cost price using the formula Loss % = (Loss)/(CP) × 100. Substituting the values into the formula gives Loss % = ¹¹⁄₂₈ × 100. Simplifying this expression, we get ¹¹⁰⁰⁄₂₈ = ²⁷⁵⁄₇ = 39 ²⁄₇ %. This results in a loss of approximately 39.29%.