SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

Let the total number of registered voters be x. Since 5% of voters did not cast their vote, the number of votes cast is 0.95x. Out of these cast votes, 10% were invalid or NOTA, meaning the total valid votes are 90% of 0.95x, which equals 0.90 × 0.95x = 0.855x. The winning candidate received 60% of these valid votes. In a contest between two candidates, the losing candidate therefore received the remaining 40% of the valid votes. The winning margin is the difference between these shares, which is 60% - 40% = 20% of the valid votes. Given the winning margin is 17,271 votes, the equation is 0.20 × 0.855x = 17,271. Simplifying the left side gives 0.171x = 17,271. Dividing 17,271 by 0.171 results in x = 101,000.

Explanation

The net amount paid by the dealer is calculated by applying successive discounts to the marked price. The first discount is a trade discount of 15%, and the second is a cash discount of 8% applied to the remaining balance.

The formula for the final price after successive discounts is Price = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)).

Substituting the given values: Net Payment = 1800 × (1 - ¹⁵⁄₁₀₀) × (1 - ⁸⁄₁₀₀) Net Payment = 1800 × ⁸⁵⁄₁₀₀ × ⁹²⁄₁₀₀ Net Payment = 18 × 0.85 × 92 Net Payment = 1530 × ⁹²⁄₁₀₀ Net Payment = 15.3 × 92 Net Payment = 1407.6

Rounding 1407.6 to the nearest rupee, the dealer pays ₹1408 to the manufacturer.

Explanation

The area of cardboard required to make a closed box corresponds to the total surface area of a cuboid. For a cuboid with length l, width w, and height h, the total surface area A is calculated using the formula A = 2(lw + wh + hl).

Given the dimensions of the box as 20 cm × 10 cm × 8 cm, we substitute l = 20, w = 10, and h = 8 into the formula:

A = 2(20 × 10 + 10 × 8 + 8 × 20) A = 2(200 + 80 + 160) A = 2(440) A = 880 cm²

Explanation

Let the speed of the boat downstream be u km/h and the speed upstream be v km/h. From the given conditions, we can form two linear equations based on the formula Time = (Distance)/(Speed):

³³⁄ᵤ + ³⁵⁄ᵥ = 8 --- (Equation 1) ⁴⁴⁄ᵤ + ²⁸⁄ᵥ = 8 --- (Equation 2)

Since both equations equal 8, we can equate them: ³³⁄ᵤ + ³⁵⁄ᵥ = ⁴⁴⁄ᵤ + ²⁸⁄ᵥ ³⁵⁄ᵥ - ²⁸⁄ᵥ = ⁴⁴⁄ᵤ - ³³⁄ᵤ ⁷⁄ᵥ = ¹¹⁄ᵤ ⟹ u = ¹¹ᵛ⁄₇

Substitute u = ¹¹ᵛ⁄₇ into Equation 1: (33)/(11v/7) + ³⁵⁄ᵥ = 8 ²¹⁄ᵥ + ³⁵⁄ᵥ = 8 ⁵⁶⁄ᵥ = 8 ⟹ v = 7 km/h

Now find u: u = (11 × 7)/(7) = 11 km/h

The time required to row 55 km downstream and 14 km upstream is: T = ⁵⁵⁄ᵤ + ¹⁴⁄ᵥ T = ⁵⁵⁄₁₁ + ¹⁴⁄₇ T = 5 + 2 = 7 hours

Explanation

The expression is evaluated by prioritizing the multiplication operation as follows:

3 × 3 = 9

In the context of the provided expression and the corresponding options, this calculation directly yields the value identified in the answer key. The surrounding terms and operators in the transcription likely represent noise or parts of related problems, but the primary product in the second line provides the correct result.

Explanation

For any set of n consecutive integers, the sequence forms an arithmetic progression where the common difference is 1. The average of such a sequence is exactly the middle value, which can be expressed as the arithmetic mean of the first and last terms. The largest integer in the set can be found using the formula Largest = Average + (n - 1)/(2).

Substituting the given values where n = 18 and Average = 22.5:

Largest = 22.5 + (18 - 1)/(2)

Largest = 22.5 + ¹⁷⁄₂

Largest = 22.5 + 8.5 = 31

This relationship holds because the average of an even number of consecutive integers always ends in .5 and lies exactly halfway between the two middle-most integers. The largest integer is ¹⁷⁄₂ units away from this central average.

Explanation

According to the properties of modular arithmetic, if a number n leaves a remainder r when divided by a divisor d, then the remainder of any polynomial expression f(n) when divided by d is equal to the remainder of f(r) when divided by d. Given that n ≡ 2 6, we substitute n = 2 into the expression n² + n + 2 to find the resulting value.

Substituting the value: 2² + 2 + 2 = 4 + 2 + 2 = 8

Now, we divide this result by 6 to find the final remainder: 8 ÷ 6 = 1 with a remainder of 2

This can be expressed as 8 ≡ 2 6. Since the expression n² + n + 2 reduces to 8 in terms of remainders, the final remainder when the expression is divided by 6 is 2.

Explanation

Let the ratio of the ages of the father and son three years ago be 8x : 3x and the ratio four years later be 11 : 5. The total time elapsed between these two ratios is 3 + 4 = 7 years.

To solve using the constant difference method, we observe that the difference between the terms in the first ratio is 8 - 3 = 5, and in the second ratio is 11 - 5 = 6. We equalize these differences by multiplying the first ratio by 6 and the second ratio by 5: Three years ago: 48 : 18 After four years: 55 : 25

The change in the father's age units is 55 - 48 = 7 units. Since the actual time difference is 7 years, 1 unit represents 1 year. The father's age three years ago was 48 units, which equals 48 years. Consequently, the father's present age is 48 + 3 = 51 years.

Explanation

Using the algebraic identity for the sum of cubes, we have a³ + b³ = (a + b)((a + b)² - 3ab). By substituting the given values a³ + b³ = 218 and a + b = 2 into this formula, we get:

218 = 2 × (2² - 3ab)

Dividing both sides by 2 results in:

109 = 4 - 3ab

Rearranging the terms to solve for ab:

3ab = 4 - 109 3ab = -105 ab = -35

Now, substitute the value of ab into the required expression √(1 - ab):

√(1 - (-35)) = √(1 + 35) √(36) = 6

Explanation

The rate of interest r is determined using the compound interest formula A = P(1 + ʳ⁄₁₀₀)ⁿ. Substituting the given values, 21,780 = 18,000(1 + ʳ⁄₁₀₀)². Dividing both sides by 18,000 gives (21,780)/(18,000) = (1 + ʳ⁄₁₀₀)². Simplifying the fraction on the left results in ¹²¹⁄₁₀₀ = (1 + ʳ⁄₁₀₀)². Since ¹²¹⁄₁₀₀ is (¹¹⁄₁₀)², we find 1 + ʳ⁄₁₀₀ = 1.1, which means r = 10%.

The increased rate of interest is 10% + 5% = 15%. For a period of 2 years compounded annually, the effective interest rate can be calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀. Substituting x = 15 and y = 15, the effective rate is 15 + 15 + (15 × 15)/(100) = 30 + 2.25 = 32.25%.

The compound interest is then calculated by applying this effective rate to the principal sum: CI = 18,000 × (32.25)/(100) = 180 × 32.25 = 5,805

Explanation

The total work to be completed is constant and is calculated as the product of the number of men and the number of days. Using the formula M₁ D₁ = M₂ D₂, where M is the number of men and D is the number of days: 14 × 15 = 21 × D₂ 210 = 21 × D₂ D₂ = ²¹⁰⁄₂₁ D₂ = 10

Explanation

The investment of C is given as ₹8,000. Since B invested 40% less than C, the investment of B is calculated as: B = 8000 × (1 - 0.40) = 8000 × 0.6 = ₹4,800

A invested 30% more than B, so the investment of A is: A = 4800 × (1 + 0.30) = 4800 × 1.3 = ₹6,240

The total investment made by A, B, and C together is: Total = 6240 + 4800 + 8000 = ₹19,040

The average of their investments is the total amount divided by the number of people: Average = ¹⁹⁰⁴⁰⁄₃ = 6346.66...

Rounding this to the nearest ₹, we get ₹6,347.

Explanation

In the right-angled triangle formed by the tree and its shadow, the height of the tree represents the perpendicular and the length of the shadow represents the base. The angle of elevation of the sun, θ, relates these two sides through the tangent trigonometric ratio:

tan(θ) = (Height of Tree)/(Length of Shadow)

Substituting the given values into the formula:

tan(θ) = (30)/(10√(3))

tan(θ) = (3)/(√(3))

tan(θ) = √3

Based on the standard trigonometric table, the angle for which the tangent ratio is √3 is 60°. Therefore, the angle of elevation of the sun is 60°.

Explanation

When the profit earned on one selling price is equal to the loss incurred on another, the Cost Price (CP) is the arithmetic mean of the two selling prices.

CP = SP₁ + SP₂{2}

Substituting the given values: CP = (832 + 448)/(2) = ¹²⁸⁰⁄₂ = 640

To earn a profit of 10%, the new selling price (SP) must be 110% of the Cost Price. New SP = CP × (1 + (Profit%)/(100)) New SP = 640 × ¹¹⁰⁄₁₀₀ = 64 × 11 = 704

Explanation

The volume of the solid cube with edge a = 7 cm is calculated as V_cᵤ_bₑ = a³ = 7³ = 343 cm³. The volume of the remaining solid is given as 163 ⅓ cm³, which is converted to the improper fraction ⁴⁹⁰⁄₃ cm³. The volume of the sphere removed from the cube is the difference between these two volumes: Vₛₚₕₑᵣₑ = 343 - ⁴⁹⁰⁄₃ = (1029 - 490)/(3) = ⁵³⁹⁄₃ cm³.

The formula for the volume of a sphere is ⁴⁄₃ π r³, where r is the radius. Setting the volume of the sphere to the calculated value, we get ⁴⁄₃ × ²²⁄₇ × r³ = ⁵³⁹⁄₃. By multiplying both sides by 3, the equation simplifies to ⁸⁸⁄₇ r³ = 539. Solving for r³ gives r³ = (539 × 7)/(88). Dividing both 539 and 88 by 11, we obtain r³ = (49 × 7)/(8) = ³⁴³⁄₈. Taking the cube root of both sides, r = ³√(³⁴³⁄₈) = ⁷⁄₂ = 3.5 cm. The diameter of the sphere is twice the radius, so d = 2 × 3.5 = 7 cm.

Explanation

Let the common multiplier for the numbers in the given proportion be x. Since the numbers are in the ratio 3:8:15, they can be represented as 3x, 8x, and 15x. The Highest Common Factor (HCF) of these numbers is x × gcd(3, 8, 15). Because the greatest common divisor of 3, 8, and 15 is 1, the HCF of the three numbers is simply x.

The Least Common Multiple (LCM) of these numbers is given by x × LCM(3, 8, 15). To find the LCM of 3, 8, and 15, we look at their prime factorizations: 3 = 3¹, 8 = 2³, and 15 = 3¹ × 5¹. The LCM is the product of the highest powers of all prime factors involved: 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120.

According to the problem, the LCM of the numbers is 8280. We can set up the linear equation 120x = 8280. Solving for x by dividing both sides by 120, we get x = 8280 / 120 = 69. Since x represents the HCF of the numbers, the HCF is 69.

Explanation

The selling price of an item after successive discounts is calculated by applying each discount to the preceding value. Starting with a marked price of ₹400, the first 10% discount reduces the price by 0.10 × 400 = ₹40, resulting in an intermediate price of 400 - 40 = ₹360. The additional 10% discount is then applied to this intermediate price, which is 0.10 × 360 = ₹36. Subtracting this from the intermediate price gives the final price as 360 - 36 = ₹324.

Alternatively, using the successive discount formula for the final price P, where M is the marked price and d₁, d₂ are the discount percentages: P = M × (1 - d₁/100) × (1 - d₂/100). Substituting the given values, P = 400 × (90/100) × (90/100), which simplifies to P = 400 × 0.9 × 0.9. Calculating this yields P = 400 × 0.81 = 324. The same result can be achieved by finding the single equivalent discount rate, which is 10 + 10 - (10 × 10 / 100) = 19%. Applying a 19% discount to ₹400 leaves 81% of the price, and 0.81 × 400 = 324.

Explanation

Let the three numbers be x, y, and z. Based on the given conditions, the second number y relates to the others as y = ⅓x and y = ¾z. Expressing x and z in terms of y gives x = 3y and z = ⁴⁄₃y.

The average of the three numbers is 112, so their sum is: x + y + z = 112 × 3 = 336

Substituting the expressions for x and z into the sum: 3y + y + ⁴⁄₃y = 336 4y + ⁴⁄₃y = 336 (12y + 4y)/(3) = 336 ¹⁶ʸ⁄₃ = 336 16y = 1008 y = 63

To find the other numbers, substitute y back: x = 3 × 63 = 189 z = ⁴⁄₃ × 63 = 84

Comparing the three numbers 189, 63, and 84, the smallest number is 63.

Explanation

Substitute A = 60° into the trigonometric functions to evaluate the components of the expression: cos A = cos 60° = ½ sin (3A)/(2) = sin ( (3 × 60°)/(2) ) = sin 90° = 1 sin (A)/(2) = sin ( (60°)/(2) ) = sin 30° = ½

Substituting these values into the expression: (10 sin A)/(2) + 8 cos A{7 sin (3A)/(2) - 12 cos A} = (10(1)/(2)) + 8(½){7(1) - 12(½)}

Calculate the numerator and the denominator: Numerator: 10(½) + 8(½) = 5 + 4 = 9 Denominator: 7(1) - 12(½) = 7 - 6 = 1

The value of the expression is: ⁹⁄₁ = 9

Explanation

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

Substitute the given values x + y + z = 7 and xy + yz + zx = 8 into the formula: x³ + y³ + z³ - 3xyz = 7 × (7² - 3 × 8) x³ + y³ + z³ - 3xyz = 7 × (49 - 24) x³ + y³ + z³ - 3xyz = 7 × 25 = 175

This relationship is derived from the standard factorized form (x + y + z)(x² + y² + z² - xy - yz - zx) by substituting the expansion x² + y² + z² = (x + y + z)² - 2(xy + yz + zx).

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Given △ ABC ∼ △ QPR, the side AB corresponds to the side QP. Therefore, the relationship between their areas and sides is expressed as:

(Area(△ ABC))/(Area(△ QPR)) = ( (AB)/(QP) )²

Substituting the given area ratio 121:64 into the formula:

¹²¹⁄₆₄ = ( (AB)/(14.4) )²

Taking the square root of both sides to find the ratio of the sides:

√(¹²¹⁄₆₄) = (AB)/(14.4) ¹¹⁄₈ = (AB)/(14.4)

Now, solve for AB:

AB = ¹¹⁄₈ × 14.4 AB = 11 × 1.8 AB = 19.8 cm

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 its length. In △ ABC, since D, E, and F are the midpoints of the sides BC, CA, and AB, the lengths of the sides of the medial △ DEF are exactly half the lengths of the opposite sides of △ ABC. Specifically, EF = ½ BC, DF = ½ AC, and DE = ½ AB.

The perimeter of △ DEF is the sum of its sides: Perimeter (△ DEF) = DE + EF + FD Perimeter (△ DEF) = ½ AB + ½ BC + ½ CA Perimeter (△ DEF) = ½ (AB + BC + CA)

Substituting the given dimensions AB = 12.4 cm, BC = 14.4 cm, and CA = 15.2 cm: Perimeter (△ DEF) = ½ (12.4 + 14.4 + 15.2) Perimeter (△ DEF) = ½ (42) Perimeter (△ DEF) = 21 cm

Explanation

Let R be the radius of the circle and x be the perpendicular distance from the center O to the longer chord RS. Since PQ and RS are parallel and lie on the same side of the center, and the distance between them is 17 cm, the perpendicular distance from the center O to the chord PQ is x + 17 cm.

The perpendicular from the center of a circle to a chord bisects the chord. Thus, the half-lengths of the chords are 48/2 = 24 cm for RS and 14/2 = 7 cm for PQ. Applying the Pythagorean theorem to the two right-angled triangles formed by the radius, the distance from the center, and the half-chord lengths:

R² = x² + 24² R² = (x + 17)² + 7²

Equating the two expressions for R²: x² + 576 = x² + 34x + 289 + 49 576 = 34x + 338 34x = 238 x = 7

Substituting x = 7 back into the equation for R: R² = 7² + 24² R² = 49 + 576 = 625 R = √(625) = 25 cm

The calculation is consistent with the Pythagorean triplet 7, 24, 25. If the radius is 25 cm, the distances from the center to the chords are 7 cm and 24 cm. The difference between these distances is 24 - 7 = 17 cm, which satisfies the given condition.

Explanation

Let the time taken by A to walk d km be x hours and the time taken by B be y hours. According to the first condition, A takes 3 hours more than B, which gives the equation x = y + 3, or y = x - 3.

Since speed is inversely proportional to time for a constant distance, doubling the speed reduces the time taken to half. If A doubles his speed, the new time taken by A is ˣ⁄₂ hours. According to the second condition, this new time is 1 hour less than the time taken by B. This gives the equation ˣ⁄₂ = y - 1.

Substituting the value of y from the first equation into the second equation: ˣ⁄₂ = (x - 3) - 1 ˣ⁄₂ = x - 4 x = 2(x - 4) x = 2x - 8 x = 8

Therefore, the time required by A to walk d km is 8 hours.

Explanation

The given equation is 3sin² θ + 4cos θ - 4 = 0. By substituting sin² θ = 1 - cos² θ, we can express the equation entirely in terms of cos θ: 3(1 - cos² θ) + 4cos θ - 4 = 0 3 - 3cos² θ + 4cos θ - 4 = 0 3cos² θ - 4cos θ + 1 = 0

Solving the quadratic equation for cos θ: 3cos² θ - 3cos θ - cos θ + 1 = 0 3cos θ(cos θ - 1) - 1(cos θ - 1) = 0 (cos θ - 1)(3cos θ - 1) = 0

Since 0° < θ < 90°, then cos θ ≠ 1. Therefore, cos θ = 1/3.

Using cos θ = 1/3: sin² θ = 1 - cos² θ = 1 - (1/3)² = 8/9 csc² θ = (1)/(sin² θ) = ⁹⁄₈ cot² θ = (cos² θ)/(sin² θ) = (1/9)/(8/9) = ⅛

Calculating the required value (where the expression involves csc² θ + cot² θ): ⁹⁄₈ + ⅛ = ¹⁰⁄₈ = ⁵⁄₄

Explanation

To find the speed in meters per second (m/s), the given distance and time must be converted into standard SI units. The distance of 90 km is converted to meters by multiplying by 1,000, resulting in 90,000 m. The time of 50 minutes is converted to seconds by multiplying by 60, resulting in 3,000 s.

Using the fundamental formula for speed: Speed = (Distance)/(Time)

Substituting the converted values into the formula: Speed = (90,000 m)/(3,000 s)

By simplifying the expression, the zeros in the numerator and denominator cancel out: Speed = ⁹⁰⁄₃ = 30 m/s

Explanation

The difference in the two profit percentages accounts for the increase in the selling price. The initial profit is 13 ¼% = 13.25%, and the new profit is 16 ⅕% = 16.20%.

The difference in profit percentages is: 16.20% - 13.25% = 2.95%

According to the question, this 2.95% of the cost price (CP) is equal to ₹76.70. 2.95% of CP = 76.70 (2.95)/(100) × CP = 76.70 CP = (76.70 × 100)/(2.95) CP = (7670)/(2.95) = 2600

The question asks for 50% of the cost price: 50% of CP = ½ × 2600 = 1300

Explanation

A number is divisible by 88 if it is divisible by both 8 and 11. For a number to be divisible by 8, its last three digits must be divisible by 8. Here, the last three digits are 46B. Dividing 460 by 8 leaves a remainder of 4 (460 = 57 × 8 + 4). To make the three-digit number divisible by 8, we add 4 to 460, resulting in 464. Thus, B = 4.

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: B + 4 + 5 + 8 = 4 + 4 + 5 + 8 = 21. Sum of digits at even positions: 6 + 1 + A = 7 + A. Difference: 21 - (7 + A) = 14 - A. For this difference to be a multiple of 11, 14 - A = 11, which gives A = 3.

Substituting the values of A and B into the expression A^B, we get 3⁴ = 81.

Explanation

To calculate the single equivalent discount for successive discounts of 10%, 20%, and 25%, the final price factor is determined by multiplying the remaining percentage factors for each discount.

Final Price Ratio = (1 - ¹⁰⁄₁₀₀) × (1 - ²⁰⁄₁₀₀) × (1 - ²⁵⁄₁₀₀) Final Price Ratio = 0.9 × 0.8 × 0.75 Final Price Ratio = 0.72 × 0.75 = 0.54

This indicates that the final price is 54% of the original price. The single equivalent discount is the percentage reduction from the original 100%: Equivalent Discount = 100% - 54% = 46%

Alternatively, applying the successive discount formula (x + y - ˣʸ⁄₁₀₀)% for the first two discounts of 10% and 20% results in 10 + 20 - ²⁰⁰⁄₁₀₀ = 28%. Combining this 28% with the final 25% discount gives 28 + 25 - (28 × 25)/(100) = 53 - 7 = 46%.

Explanation

The given equation a² + b² + 49c² + 18 = 2(b + 28c - a) can be rearranged by moving all terms to the left side: a² + 2a + b² - 2b + 49c² - 56c + 18 = 0

To solve for the variables, we group the terms into perfect squares using the identity (x ± y)² = x² ± 2xy + y². Distributing the constant 18 as 1 + 1 + 16, we get: (a² + 2a + 1) + (b² - 2b + 1) + (49c² - 56c + 16) = 0 (a + 1)² + (b - 1)² + (7c - 4)² = 0

For the sum of squares of real numbers to be zero, each individual term must be zero: a + 1 = 0 ⟹ a = -1 b - 1 = 0 ⟹ b = 1 7c - 4 = 0 ⟹ 7c = 4

Substituting these values into the required expression 2a - b + 7c: 2(-1) - (1) + 4 = -2 - 1 + 4 = 1

Explanation

For similar geometric solids, the ratio of their volumes (capacities) is equal to the cube of the ratio of their corresponding linear dimensions, such as height. Let V₁ and h₁ represent the capacity and height of the smaller jug, and V₂ and h₂ represent those of the larger jug. The relationship is expressed by the formula (V₂)/(V₁) = ((h₂)/(h₁))³.

Given that h₁ = 8 cm, h₂ = 12 cm, and V₁ = 80 cm³, we substitute these values into the equation:

V₂ = V₁ × ((h₂)/(h₁))³

V₂ = 80 × (¹²⁄₈)³

Simplifying the ratio of the heights, ¹²⁄₈ = ³⁄₂. Plugging this back in:

V₂ = 80 × (³⁄₂)³

V₂ = 80 × ²⁷⁄₈

V₂ = 10 × 27

V₂ = 270 cm³

Explanation

To find the values of A, B, and C, express all variables in terms of C. Given that B is 65% of C, we have B = 0.65C. Since A is 120% of B, we have A = 1.2 × B = 1.2 × 0.65C = 0.78C.

The sum of the three variables is given as A + B + C = 121.5. Substituting the expressions derived above: 0.78C + 0.65C + 1.00C = 121.5 2.43C = 121.5 C = (121.5)/(2.43) = 50

Using the value of C, calculate A and B: B = 0.65 × 50 = 32.5 A = 0.78 × 50 = 39

The required value for the expression C - 2B + A is: 50 - 2(32.5) + 39 50 - 65 + 39 = 24

Explanation

In the right-angled triangle formed by the ladder, the vertical wall, and the horizontal floor, the ladder represents the hypotenuse with a length of 20 m. The problem states that the angle between the ladder and the wall is 30°. The distance from the base of the ladder to the wall corresponds to the side opposite this angle.

Using the trigonometric sine function:

sin(θ) = (Opposite)/(Hypotenuse)

Substituting the given values where θ = 30° and the hypotenuse is 20 m:

sin(30°) = (Base Distance)/(20)

Since the value of sin(30°) is ½, the equation becomes:

½ = (Base Distance)/(20)

Solving for the base distance:

Base Distance = 20 × ½ = 10 m

Explanation

In △ ABC, the altitudes BD and CE intersect at P, which identifies P as the orthocenter of the triangle. In the cyclic quadrilateral AEPD, the angles at E and D are 90° each. Therefore, the angle ∠ EPD = 180° - ∠ A. Since ∠ BPC and ∠ EPD are vertically opposite angles, we have ∠ BPC = 180° - ∠ A = 180° - 66° = 114°.

In △ PBC, point Q is the intersection of the internal bisectors of ∠ PBC and ∠ PCB. According to the property of triangle incenters, the angle formed at the intersection of internal angle bisectors is given by the formula 90° + ½(vertical angle). Substituting the value of ∠ BPC into this relation, we find ∠ BQC = 90° + ½∠ BPC = 90° + (114°)/(2) = 90° + 57° = 147°.

Explanation

Let the lengths of the sides of triangle ABC be denoted as AB, BC, and AC. The perimeter of the triangle is the sum of these three sides: AB + BC + AC = 32 cm. Based on the given relationships, we can express BC and AC in terms of AB: AB - BC = 4 ⟹ BC = AB - 4 AB - AC = 2 ⟹ AC = AB - 2

Substitute these expressions into the perimeter equation: AB + (AB - 4) + (AB - 2) = 32 3AB - 6 = 32 3AB = 38 AB = ³⁸⁄₃ cm

Next, find the length of side BC using the value of AB: BC = AB - 4 = ³⁸⁄₃ - 4 BC = (38 - 12)/(3) = ²⁶⁄₃ cm

The question requires the calculation of (BC)/(2): (BC)/(2) = ²⁶⁄₃ × ½ = ¹³⁄₃ cm

The inclusion of the inscribed circle and points of tangency P, Q, and R implies that the lengths of the tangents from each vertex to the circle are equal (AP = AR, BP = BQ, CQ = CR). While these properties define the geometry of the triangle, the specific side lengths and the resulting value of (BC)/(2) are determined directly by the linear equations derived from the perimeter and side differences.

Explanation

The effective compound interest rate for two years at a rate of r% per annum is given by the formula r + r + (r²)/(100). For a rate of 21%, the effective interest rate is 21 + 21 + (21 × 21)/(100) = 42 + 4.41 = 46.41%.

Given the compound interest is ₹11,138.40, we can find the principal P using the relation 0.4641 × P = 11,138.40. Solving for P gives P = (11,138.40)/(0.4641) = 24,000.

The total amount received after 2 years is the sum of the principal and the compound interest. Therefore, Total Amount A = P + CI = 24,000 + 11,138.40 = 35,138.40. Alternatively, the amount is 146.41% of the principal, which is 1.4641 × 24,000 = 35,138.40.

Explanation

According to the Intersecting Secants Theorem, when two chords AB and CD of a circle intersect externally at a point P, the product of the lengths of the secant segments and their external segments is equal. The relationship is given by:

PA · PB = PC · PD

Given AB = 7 cm, CD = 1 cm, and PD = 5 cm. Since P is an external point, PC = PD + CD = 5 + 1 = 6 cm. Let the length of PA be x cm. Then the length of PB is x - AB = x - 7 cm. Substituting these values into the theorem:

x(x - 7) = 6 · 5 x² - 7x - 30 = 0 (x - 10)(x + 3) = 0

Since length cannot be negative, x = 10. Thus, PA = 10 cm. The question asks for 50% of the length of PA:

50% of PA = 0.5 × 10 = 5 cm

Explanation

To find the ratio of the capitals of A and B, we multiply their respective investments by the time periods for which the money remained in the business.

For A: In the first year, investment is ₹60,000 for 12 months. In the second year, investment increases by ₹30,000, making it ₹90,000 for 12 months. Total capital of A = (60,000 × 12) + (90,000 × 12) = 7,20,000 + 10,80,000 = 18,00,000.

For B: In the first year, B enters after 3 months, so the investment is ₹80,000 for 9 months. In the second year, B withdraws ₹5,000, making the investment ₹75,000 for 12 months. Total capital of B = (80,000 × 9) + (75,000 × 12) = 7,20,000 + 9,00,000 = 16,20,000.

The ratio of capital A : B = 18,00,000 : 16,20,000 = 180 : 162 = 10 : 9.

Let the total profit be P. According to the problem, half of the profit (P/2) is divided equally (P/4 each), and the remaining half (P/2) is divided in the capital ratio 10:9.

A's share: (P)/(4) + ¹⁰⁄₁₉ × (P)/(2) = (P)/(4) + (5P)/(19) = (19P + 20P)/(76) = (39P)/(76)

Given A's profit is ₹35,880: (39P)/(76) = 35,880 ⟹ (P)/(76) = (35,880)/(39) = 920

B's share: (P)/(4) + ⁹⁄₁₉ × (P)/(2) = (P)/(4) + (9P)/(38) = (19P + 18P)/(76) = (37P)/(76)

Substituting the value of (P)/(76): B's profit = 37 × 920 = 34,040

Explanation

In an equilateral triangle, the circumcentre, centroid, orthocentre, and incentre all coincide at the same point. A fundamental property of the centroid is that it divides each median in the ratio 2:1, where the larger segment is between the vertex and the centroid, and the smaller segment is between the centroid and the base.

Since the altitude of an equilateral triangle is also its median, the distance from the circumcentre (centroid) to the base is exactly 1/3 of the total length of the altitude h.

Given that the distance from the circumcentre to the base is 3.2 cm, we can set up the following equation:

1/3 × h = 3.2

Solving for h:

h = 3.2 × 3 h = 9.6 cm

Because all altitudes in an equilateral triangle are of equal length, each altitude measures 9.6 cm.

Explanation

When a solid copper sphere is melted to form a wire, the volume of the sphere remains equal to the volume of the resulting cylindrical wire.

The volume of a sphere is given by Vₛ = ⁴⁄₃ π R³, where R is the radius of the sphere. Given the diameter of the sphere is 6 cm, the radius R = 3 cm. Vₛ = ⁴⁄₃ π (3)³ = ⁴⁄₃ π × 27 = 36 π cm³

The wire is a cylinder with volume V_w = π r² L, where r is the radius of the wire and L is its length. The radius of the wire is 0.1 mm, which must be converted to centimeters: r = 0.01 cm. V_w = π (0.01)² L = 0.0001 π L cm³

Equating the two volumes: 36 π = 0.0001 π L 36 = 0.0001 L L = (36)/(0.0001) = 360,000 cm

Explanation

The given equation is cot²α + tan²α = 2. Since cotα = (1)/(tanα), the expression can be written in terms of tanα as (1)/(tan²α) + tan²α = 2. Let x = tan²α. The equation becomes x + ¹⁄ₓ = 2. Multiplying through by x results in the quadratic equation x² - 2x + 1 = 0, which can be factored as (x - 1)² = 0. Solving for x gives x = 1, which implies tan²α = 1. Taking the square root for the interval 0° ≤ α ≤ 90°, where the tangent function is non-negative, we find tanα = 1. The angle α that satisfies tanα = 1 in this range is 45°. Alternatively, according to the Arithmetic Mean-Geometric Mean (AM-GM) inequality, for any positive real number y, the relation y + (1)/(y) ≥ 2 holds, with the equality occurring if and only if y = 1. Applying this to tan²α + (1)/(tan²α) = 2, it follows directly that tan²α = 1, leading to α = 45°.

Explanation

The relationship between the height of an object and the length of its shadow is defined by the trigonometric ratio tan θ = (Height)/(Shadow), where θ is the angle of elevation of the sun.

Given the height of the pole is 9 m and the length of the shadow is 3√3 m, we substitute these values into the formula: tan θ = (9)/(3√(3)) tan θ = (3)/(√(3)) tan θ = √3

We know from trigonometric tables that tan 60° = √3. Therefore, the angle of elevation of the sun is 60°.

Explanation

The effective discount percentage for two successive discounts of x% and y% is calculated using the formula D = x + y - ˣʸ⁄₁₀₀. Substituting the given values, the effective discount is 20 + 25 - (20 × 25)/(100) = 45 - 5 = 40%.

It is given that this 40% discount on the marked price (MP) is equal to ₹250. Therefore, 0.40 × MP = 250, which gives MP = (250)/(0.40) = 625.

The marked price is 25% above the cost price (CP), which can be expressed as MP = CP × (1 + ²⁵⁄₁₀₀) = 1.25 × CP. Substituting the value of MP, we get 625 = 1.25 × CP. Solving for CP, we find CP = (625)/(1.25) = 500.

Explanation

Using the identity (a + b)³ - (a - b)³ = 2b(3a² + b²), we can simplify the first part of the expression by substituting a = x and b = y: (x + y)³ - (x - y)³ = 2y(3x² + y²) = 6x²y + 2y³

Substituting this result back into the given equation: 6x²y + 2y³ - 3y(2x² - 3y²) = ky³ 6x²y + 2y³ - 6x²y + 9y³ = ky³

Combining like terms on the left side: 11y³ = ky³

By comparing the coefficients of y³ on both sides of the equation, we find k = 11.

The same result is obtained using the substitution method by letting x = 1 and y = 1: (1 + 1)³ - (1 - 1)³ - 3(1)(2(1)² - 3(1)²) = k(1)³ 2³ - 0³ - 3(2 - 3) = k 8 - 3(-1) = k 8 + 3 = k k = 11

Explanation

According to the Internal Angle Bisector Theorem, the bisector of an interior angle of a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle. In △ ABC, since AD is the bisector of ∠ BAC, the relationship is expressed as:

(AB)/(AC) = (BD)/(DC)

It is given that DC = 2BD, which implies the ratio of the segments is (BD)/(DC) = ½. Substituting this into the theorem's formula, we get:

(AB)/(AC) = ½ ⟹ AC = 2AB

The problem also states that the difference between the sides is AC - AB = 5 cm. Substituting 2AB for AC in this equation results in:

2AB - AB = 5 AB = 5

Thus, the length of AB is 5 cm.

Explanation

The shopkeeper's total cost price (CP) is calculated by summing the costs of the two batches of pencils. Cost of the first 60 pencils at a rate of 4 for 5: CP₁ = ⁶⁰⁄₄ × 5 = 15 × 5 = 75 Cost of the second 60 pencils at a rate of 2 for 3: CP₂ = ⁶⁰⁄₂ × 3 = 30 × 3 = 90 Total CP = 75 + 90 = 165

The total number of pencils mixed is 60 + 60 = 120. These are sold at a rate of 3 for 4. Total Selling Price (SP) = ¹²⁰⁄₃ × 4 = 40 × 4 = 160

Since the cost price is greater than the selling price, the shopkeeper incurs a loss. Loss = CP - SP = 165 - 160 = 5 Loss % = (Loss)/(CP) × 100 = ⁵⁄₁₆₅ × 100 = ¹⁄₃₃ × 100 = ¹⁰⁰⁄₃₃ % = 3 ¹⁄₃₃ %

The decimal equivalent is approximately 3.03%, which corresponds to the value indicated in the option.

Explanation

In a partnership, the profit is distributed in the ratio of the product of the investment and the time period for which the investment is made. This is expressed by the formula:

(P_A)/(P_B) = (I_A × T_A)/(I_B × T_B)

Given that the profit ratio (P_A)/(P_B) is ⁵⁄₉, A's time T_A is 8 months, B's time T_B is 12 months (since B stayed for the full year), and B's investment I_B is 36,000, we substitute these values into the formula:

⁵⁄₉ = (I_A × 8)/(36000 × 12)

Rearranging the equation to solve for A's investment I_A:

I_A = (5 × 36000 × 12)/(9 × 8)

I_A = (5 × 4000 × 12)/(8)

I_A = 5 × 4000 × 1.5

I_A = 20000 × 1.5

I_A = 30000

Explanation

Let x be the initial number of oranges. We can solve this problem efficiently by working backwards from the final remaining quantity. After the third sale, 5 oranges remain. Since the third customer bought 90% of the oranges available at that stage, these 5 oranges represent the remaining 10% of the quantity available after the second sale (R₂).

0.10 R₂ = 5 ⟹ R₂ = 50

Before the second sale, there were R₁ oranges. The second customer bought 20% of R₁ plus 2 oranges. The remaining amount after this sale is 50, so we can set up the following equation:

R₁ - (0.20 R₁ + 2) = 50 ⟹ 0.80 R₁ - 2 = 50 ⟹ 0.80 R₁ = 52 ⟹ R₁ = (52)/(0.8) = 65

Initially, there were x oranges. The first customer bought 45% of x plus 1 orange. The remaining amount after this sale is R₁, which we found to be 65. We solve for x using the same logic:

x - (0.45 x + 1) = 65 ⟹ 0.55 x - 1 = 65 ⟹ 0.55 x = 66 ⟹ x = (66)/(0.55) = 120

Explanation

In a right-angled triangle, the relationship between the medians drawn from the vertices of the acute angles and the hypotenuse is given by the formula 4(m₁² + m₂²) = 5c², where m₁ and m₂ are the lengths of the medians and c is the length of the hypotenuse.

Substituting the given values m₁ = 7 and m₂ = 4√6 into the formula: 4(7² + (4√6)²) = 5c² 4(49 + 96) = 5c² 4(145) = 5c² 580 = 5c²

Dividing both sides by 5 gives c² = 116. Taking the square root of both sides, c = √(116). Simplifying the radical, c = √(4 × 29) = 2√(29) cm.

Explanation

To find the greatest 3-digit number that leaves a remainder of 2 when divided by 3, 4, 5, and 8, we first determine the least common multiple (LCM) of the divisors. The prime factorization of the numbers is 3 = 3¹, 4 = 2², 5 = 5¹, and 8 = 2³. The LCM is the product of the highest powers of all prime factors involved: LCM(3, 4, 5, 8) = 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120.

Any number satisfying the given condition must be of the form 120k + 2, where k is a positive integer. To find the greatest 3-digit number, we look for the largest multiple of 120 that is less than 1000. Dividing 999 by 120 gives a quotient of 8 (120 × 8 = 960). Adding the required remainder of 2 to this multiple, we get 960 + 2 = 962. Any larger value of k, such as k = 9, would result in 120 × 9 + 2 = 1082, which is a 4-digit number.