SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

By the inscribed angle theorem, the angle subtended by a chord at the centre of a circle is twice the angle subtended by the same chord at any point on the circumference.

Angle at circumference = ½ × Angle at centre

Given that the angle subtended by the chord at the centre is 180°, the angle subtended on the circumference is:

½ × 180° = 90°

Alternatively, a chord that subtends an angle of 180° at the centre of a circle is a diameter. According to Thales's theorem, the angle subtended by a diameter at any point on the circumference of a circle is always a right angle, which is equal to 90°.

Explanation

Using the algebraic identity (a - b)(a + b) = a² - b², the terms can be multiplied sequentially:

First, multiply the first two factors: (k + 1/k)(k - 1/k) = k² - 1/k²

Next, multiply this result by the third factor: (k² - 1/k²)(k² + 1/k²) = (k²)² - (1/k²)² = k⁴ - 1/k⁴

Finally, multiply this result by the fourth factor: (k⁴ - 1/k⁴)(k⁴ + 1/k⁴) = (k⁴)² - (1/k⁴)² = k⁸ - 1/k⁸

Explanation

The time duration of the journey from 10:00 a.m. to 2:30 p.m. is 4.5 hours.

The speed of the train is given in meters per second (m/s) and is converted to kilometers per hour (km/h) using the conversion factor 18/5: Speed = 40 × 18/5 = 144 km/h

Using the standard formula for distance: Distance = Speed × Time Distance = 144 × 4.5 = 648 km

Explanation

The relation between expenditure, price, and consumption is given by the formula: Expenditure = Price × Consumption

Let the initial price be P, initial consumption be C, and initial expenditure be E, so that E = P × C.

The price increases by 25% and the expenditure decreases by 40%. New Price (P') = 1.25P New Expenditure (E') = 0.60E

Let the new consumption be C'. Using the formula E' = P' × C': 0.60E = 1.25P × C' 0.60(P × C) = 1.25P × C' C' = (0.60 / 1.25) × C C' = 0.48C

The percentage decrease in consumption is: ((C - C') / C) × 100 = ((C - 0.48C) / C) × 100 = 0.52 × 100 = 52%

Explanation

To find the ratio of juice and water in the new mixture, choose a convenient equal volume for both vessels. The sum of the ratio terms for the first vessel is 3 + 5 = 8, and for the second vessel is 3 + 1 = 4. The least common multiple of 8 and 4 is 8.

Let the capacity of each vessel be 8 units.

In the first vessel: Juice = 3 units Water = 5 units

In the second vessel: Juice = 3/4 × 8 = 6 units Water = 1/4 × 8 = 2 units

When the contents of both vessels are mixed together: Total Juice = 3 + 6 = 9 units Total Water = 5 + 2 = 7 units

The ratio of juice and water in the new mixture is 9:7.

Explanation

The value of a³ + b³ can be determined using the standard algebraic identity:

a³ + b³ = (a + b)³ - 3ab(a + b)

Substituting the given values, a + b = 6 and ab = 4, into the identity:

a³ + b³ = 6³ - 3 × 4 × 6

Calculating the terms:

a³ + b³ = 216 - 72

a³ + b³ = 144

Explanation

According to the circle theorem, the angle subtended by a chord at the centre is twice the angle subtended by it on the major arc of the circle.

If the angle subtended at the centre is θ, then the angle subtended on the major arc is θ/2.

Using the given condition that their sum is 180°: θ + θ/2 = 180° 3θ/2 = 180° θ = 180° × 2/3 θ = 120°

Explanation

The centroid Z of triangle JKL divides the median KY of length 12 cm in the ratio 2:1. KZ = (2/3) × 12 = 8 cm

Since JX is a median to the side KL, X is the midpoint of KL. KX = KL / 2 = 22 / 2 = 11 cm

The medians JX and KY intersect at Z at right angles, making triangle KZX a right-angled triangle at Z. Using Pythagoras' theorem in triangle KZX: KX² = KZ² + ZX² 11² = 8² + ZX² 121 = 64 + ZX² ZX² = 57 ZX = √57 cm

The centroid Z divides the median JX in the ratio 2:1. JX = 3 × ZX = 3√57 cm

Explanation

The selling price of an item after a discount is calculated using the formula:

Selling Price = Marked Price × (1 - Discount Rate)

Given: Marked Price = ₹480 Discount Rate = 5% = 0.05

Substituting these values:

Selling Price = 480 × (1 - 0.05) = 480 × 0.95 = ₹456

Alternatively, the discount amount is:

5% of 480 = (5/100) × 480 = ₹24

Subtracting this discount from the marked price yields the final selling price:

480 - 24 = ₹456

Explanation

Using the algebraic identity for the difference of squares: A² - B² = (A + B)(A - B)

Substitute the given values of A and B into the identity: A + B = (tan θ + sin θ) + (tan θ - sin θ) = 2tan θ A - B = (tan θ + sin θ) - (tan θ - sin θ) = 2sin θ

Multiplying these two results gives: A² - B² = (2tan θ) × (2sin θ) = 4tan θ sin θ

Explanation

By Pythagoras' theorem in right-angled triangle ABC with ∠C = 90°: AB² = AC² + BC²

Substituting the given values, AB = 25 cm and BC = 20 cm: 25² = AC² + 20² 625 = AC² + 400 AC² = 225 AC = 15 cm

For angle A, the adjacent side is AC and the hypotenuse is AB. The secant of angle A is defined as the ratio of the hypotenuse to the adjacent side: sec A = AB / AC sec A = 25 / 15 sec A = 5/3

Explanation

Using the average formula, Total Marks = Average × Number of Students.

The total marks obtained by all 80 students is 80 × 50 = 4000.

The number of students who failed is 30, which means the number of students who passed is 80 - 30 = 50.

The total marks obtained by the passed students is 50 × 55 = 2750.

The total marks obtained by the failed students is the difference between the overall total marks and the total marks of the passed students, which is 4000 - 2750 = 1250.

The average marks of the failed students is 1250 / 30 = 41.66.

Explanation

Let Y take t days to complete the wall alone. Since X takes 50% more time than Y, the time taken by X alone is 1.5t days.

The daily work rates of Y and X are 1/t and 1/(1.5t) which is 2/(3t) respectively.

Together, they complete the wall in 15 days, which means their combined daily work rate is 1/15.

1/t + 2/(3t) = 1/15

5/(3t) = 1/15

3t = 75

t = 25 days

The time taken by X alone to complete the wall is 1.5t = 1.5 × 25 = 37.5 days.

Explanation

The two-digit numbers divisible by 9 form an arithmetic progression starting from 18 and ending at 99, with a common difference of 9.

The number of terms, n, in this progression is determined by the formula for the n-th term: l = a + (n - 1)d

Substituting the first term a = 18, the last term l = 99, and the common difference d = 9: 99 = 18 + (n - 1) × 9 81 = (n - 1) × 9 n - 1 = 9 n = 10

The sum of an arithmetic progression is given by the formula: Sₙ = n/2 × (a + l)

Substituting the values: S₁₀ = 10/2 × (18 + 99) S₁₀ = 5 × 117 S₁₀ = 585

Explanation

To find the value of A2 / A1, first calculate the values of A1 and A2 from the given table.

Total production of bikes by all five companies = 700 + 450 + 500 + 250 + 300 = 2200 Total production of trucks by all five companies = 500 + 300 + 200 + 600 + 650 = 2250

A1 is the difference between the total production of bikes and trucks: A1 = 2250 - 2200 = 50

A2 is the average production of bikes by companies D, F, and H: A2 = (700 + 500 + 300) / 3 = 1500 / 3 = 500

Now, calculate the ratio of A2 to A1: A2 / A1 = 500 / 50 = 10

Explanation

The inradius r of an equilateral triangle with side length a is given by the formula:

r = a / (2√3)

Given that the side of the equilateral triangle is a = 12 cm, we substitute this value into the formula:

r = 12 / (2√3)

Simplifying the expression:

r = 6 / √3

Rationalizing the denominator:

r = (6 × √3) / 3

r = 2√3 cm

Explanation

The initial sum of the ages of the 6 girls is 6x.

When the 4 new girls join, the sum of their ages is: (x - 2) + (x + 4) + (x + 8) + (x + 10) = 4x + 20

The new total sum of the ages for all 10 girls is: 6x + 4x + 20 = 10x + 20

The new average age of the class is the total sum of ages divided by the total number of girls: New Average = (10x + 20) / 10 = x + 2

Explanation

In triangle DEF, since MN is parallel to EF, triangle DMN is similar to triangle DEF.

By the property of similar triangles, the ratio of their corresponding sides is equal: DM / DE = MN / EF

Substituting the given values MN : EF = 2 : 5 and DE = 60 cm: DM / 60 = 2 / 5 DM = 60 × 2/5 DM = 24 cm

The length of ME is the difference between DE and DM: ME = DE - DM ME = 60 - 24 ME = 36 cm

Explanation

Let the common tangent at the point of contact Z intersect the direct common tangent XY at point M.

The lengths of tangents drawn from an external point to a circle are equal. For the first circle, the tangents from M are MX and MZ, so MX = MZ. For the second circle, the tangents from M are MY and MZ, so MY = MZ.

Combining these two relations gives MX = MY = MZ. Since MX = MY, M is the midpoint of XY, which means MZ = XY/2.

In triangle XZY, the median ZM from vertex Z to the side XY is equal to half of the side XY. In any triangle, if the median to a side is equal to half of that side, the angle opposite to that side is a right angle. Therefore, the angle subtended at Z, which is ∠XZY, is 90°.

Explanation

Using the standard algebraic identity: (x + y + z)(x² + y² + z² - xy - yz - zx) = x³ + y³ + z³ - 3xyz

By substituting x = a, y = b, and z = 2c into this identity, the expression becomes: (a + b + 2c)(a² + b² + (2c)² - ab - b(2c) - (2c)a)

Expanding and simplifying this using the right side of the identity yields: a³ + b³ + (2c)³ - 3(a)(b)(2c) = a³ + b³ + 8c³ - 6abc

Explanation

The time taken by X to complete the work is 18 days. Therefore, the work done by X in one day is 1/18.

The time taken by Y to complete the work is half of the time taken by X, which is 18 ÷ 2 = 9 days. Thus, the work done by Y in one day is 1/9.

When working together, the part of the work they can complete in one day is the sum of their individual one-day works: 1/18 + 1/9 = 1/18 + 2/18 = 3/18 = 1/6.

Explanation

To find the value of the expression, substitute a convenient angle, such as θ = 45°.

For θ = 45°: tan 45° = 1 sec 45° = √2 cot 45° = 1 cosec 45° = √2

Substituting these values into the given expression: (1 + tan θ - sec θ)(1 + cot θ + cosec θ) = (1 + 1 - √2)(1 + 1 + √2) = (2 - √2)(2 + √2)

Using the algebraic identity (a - b)(a + b) = a² - b²: = 2² - (√2)² = 4 - 2 = 2

Explanation

The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. Rearranging this formula gives the radius r = C / 2π.

The difference between the radii of the two circles is: r₂ - r₁ = (C₂ - C₁) / 2π

Given the circumferences C₁ = 77 cm and C₂ = 154 cm, and using the value of π as 22/7: r₂ - r₁ = (154 - 77) / (2 × 22/7) r₂ - r₁ = 77 / (44/7) r₂ - r₁ = (77 × 7) / 44 r₂ - r₁ = (7 × 7) / 4 r₂ - r₁ = 49 / 4 r₂ - r₁ = 12.25 cm

Explanation

Let the Marked Price of the watch be MP and the Cost Price be CP.

According to the question, the watch is sold at 4/5 of the marked price: Selling Price (SP) = 4/5 × MP

There is a loss of 10% on selling the watch, which means the Selling Price is 90% of the Cost Price: SP = 90% of CP = 9/10 × CP

Equating both expressions for the Selling Price: 4/5 × MP = 9/10 × CP

Rearranging the terms to find the ratio of the Marked Price to the Cost Price: MP / CP = (9/10) × (5/4) MP / CP = 45/40 MP / CP = 9/8

The ratio of the marked price to the cost price of the watch is 9:8.

Explanation

The relationship between distance, speeds, and time difference is given by the formula: Distance = (S₁ × S₂ / |S₁ - S₂|) × Δt

Where S₁ is the usual speed, S₂ is the reduced speed, and Δt is the difference in travel time.

Given: Distance = 280 km Speed difference |S₁ - S₂| = 14 km/hr Time difference Δt = 60 minutes = 1 hour

Let the usual speed be S km/hr. The reduced speed is S - 14 km/hr.

Substituting these values into the formula: 280 = (S × (S - 14) / 14) × 1 280 × 14 = S × (S - 14) 3920 = S(S - 14)

Substituting S = 70: 70 × (70 - 14) = 70 × 56 = 3920

This satisfies the equation, confirming the usual speed is 70 km/hr.

Explanation

Using the trigonometric co-function identity, sin(90° - θ) = cos θ.

Since 69° can be written as 90° - 21°, we have: sin 69° = sin(90° - 21°) = cos 21°

Substituting this value into the given expression: sin² 21° + sin² 69° = sin² 21° + cos² 21°

According to the fundamental Pythagorean identity, sin² θ + cos² θ = 1. Therefore: sin² 21° + cos² 21° = 1

Explanation

Let the total sum of money be P.

The first portion of the sum is P/2, lent at a rate of 20% per annum. The remaining sum is P - P/2 = P/2. The second portion is 2/5 of this remaining sum, which is 2/5 × P/2 = P/5, lent at a rate of 25% per annum. The third portion is the remainder of the total sum, which is P - P/2 - P/5 = 3P/10, lent at a rate of 10% per annum.

The total simple interest earned over 2 years is the sum of the interest earned from each portion. Using the formula Interest = Principal × Rate × Time / 100:

Interest from the first portion = (P/2 × 20 × 2)/100 = 0.20P Interest from the second portion = (P/5 × 25 × 2)/100 = 0.10P Interest from the third portion = (3P/10 × 10 × 2)/100 = 0.06P

Summing these individual interests gives the total simple interest received: 0.20P + 0.10P + 0.06P = 0.36P

Given that the total interest is Rs. 360: 0.36P = 360 P = 360 / 0.36 = 1000

Explanation

The average weight of the group can be calculated using the total sum method.

Sum of weights of the initial 15 persons = 15 × 30 = 450 kg

When a new person weighing 78 kg joins, the total weight of the 16 persons becomes: 450 + 78 = 528 kg

The new average weight of the 16 persons is the total weight divided by the number of persons: 528 / 16 = 33 kg

Alternatively, using the deviation method, the new person brings an excess weight of 78 - 30 = 48 kg over the initial average. Distributing this excess weight equally among all 16 persons increases the average by 48 / 16 = 3 kg. Thus, the new average is 30 + 3 = 33 kg.

Explanation

Since Ram neither earns a profit nor suffers a loss on the entire transaction, the profit earned on the first watch must be equal to the loss incurred on the second watch.

Let CP₁ be the cost price of the watch sold at 25% profit, and CP₂ be the cost price of the watch sold at 10% loss.

Profit on first watch = Loss on second watch 25% of CP₁ = 10% of CP₂ 25 × CP₁ = 10 × CP₂ CP₁ / CP₂ = 10 / 25 = 2 / 5

The ratio of the cost prices of the two watches is 2:5.

The total cost price of both watches is Rs. 3500. Sum of the ratio terms = 2 + 5 = 7 units. 7 units = Rs. 3500 1 unit = Rs. 500

Cost price of the first watch = 2 × 500 = Rs. 1000 Cost price of the second watch = 5 × 500 = Rs. 2500

Explanation

According to the divisibility rule of 11, a number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is 0 or a multiple of 11.

For the number 5A72B: Sum of digits at odd positions from right to left = B + 7 + 5 = 12 + B Sum of digits at even positions from right to left = 2 + A

The difference between these sums is: (12 + B) - (2 + A) = 10 + B - A

For the number to be divisible by 11, this difference must be a multiple of 11, represented as 11k where k is an integer: 10 + B - A = 11k

Since A and B are single-digit integers from 0 to 9, the difference B - A must lie between -9 and 9. Setting k = 1: 10 + B - A = 11 B - A = 1

Explanation

To find the combined ratio P:Q:R:S, the common terms between the ratios are made equal.

First, combine P:Q = 2:3 and Q:R = 4:5. The common term is Q. To make the Q term equal, multiply P:Q by 4 and Q:R by 3: P:Q = 8:12 Q:R = 12:15 This gives the combined ratio P:Q:R = 8:12:15.

Next, combine P:Q:R = 8:12:15 with R:S = 2:1. The common term is R. To make the R term equal, multiply P:Q:R by 2 and R:S by 15: P:Q:R = 16:24:30 R:S = 30:15 This gives the combined ratio P:Q:R:S = 16:24:30:15.

Extracting the parts for P, R, and S from this combined ratio yields P:R:S = 16:30:15.

Explanation

The given equation is K + 1/K + 2 = 0.

Multiplying the entire equation by K gives: K² + 2K + 1 = 0 (K + 1)² = 0

Solving for K yields K = -1, which satisfies the given condition K < 0.

Substitute K = -1 into the expression K¹⁷ + 1/K¹¹: (-1)¹⁷ + 1/(-1)¹¹

Since both 17 and 11 are odd exponents, any negative number raised to these powers remains negative: (-1)¹⁷ = -1 (-1)¹¹ = -1

Substituting these values into the expression: -1 + 1/(-1) = -1 - 1 = -2

Explanation

According to the angle in a semicircle theorem, a chord subtends a 90° angle on the circumference of a circle if and only if the chord is the diameter of that circle.

Therefore, chord AB is the diameter of the circle.

Diameter d = 16 cm

The radius r of the circle is half of the diameter: r = d / 2 r = 16 / 2 r = 8 cm

Explanation

The sum of the given expression 1/2 + 3/4 + 1/4 + 3/2 + 1/4 + 1/4 is equal to 14/4, which simplifies to 7/2 or 3.5. Since none of the provided options (126/5, 116/5, 49/8, 89/6) match this value, no correct option is available.

Explanation

Let the smaller number be S and the larger number be L.

According to the given condition, 20% of the smaller number is equal to 14% of the larger number: 0.20 × S = 0.14 × L 20 × S = 14 × L S = (14 / 20) × L S = 0.7 × L

The sum of both numbers is given as 8942: S + L = 8942

Substituting the value of S in terms of L into the sum equation: 0.7 × L + L = 8942 1.7 × L = 8942 L = 8942 / 1.7 L = 5260

Explanation

The total surface area of a cube with side length a is given by the formula: Total Surface Area = 6a²

Given that the total surface area is 384 cm², we can find the side length of the cube: 6a² = 384 a² = 64 a = 8 cm

The volume of a cube is given by the formula: Volume = a³

Substituting the side length into the volume formula: Volume = 8³ = 512 cm³

Explanation

Wages are distributed in proportion to the efficiency of the workers when they work together for the same duration. Since efficiency is inversely proportional to the time taken to complete the work individually:

Ratio of efficiency of Mohit to Rohit = 1/10 : 1/15 = 15 : 10 = 3 : 2

The total wage of Rs. 4400 is divided in the ratio 3 : 2.

Total parts = 3 + 2 = 5 parts

Difference in their shares = 3 - 2 = 1 part

Difference in the amount received = (1/5) × 4400 = Rs. 880

Explanation

Let the original cost price of the article be x. Since it is sold at a 10 percent loss, the original selling price is 0.9x. When the cost price is decreased by Rs.5 and the selling price is increased by Rs.8, the new cost price is x - 5 and the new selling price is 0.9x + 8. According to the problem, the new selling price yields a 10 percent profit, so 0.9x + 8 = 1.1(x - 5), which solves to x = 67.5. Thus, the original selling price is 0.9 * 67.5 = Rs.60.75.

Explanation

The given equation is sin B - cos A = 0, which can be rewritten as sin B = cos A.

Using the complementary angle identity of trigonometry, cos A = sin(90° - A).

Substituting this into the equation gives sin B = sin(90° - A).

Comparing the angles yields B = 90° - A, which simplifies to A + B = 90°.

Explanation

To find the value of A + B, we simplify the expressions for A and B using the BODMAS rule:

For expression A: Numerator of A = 36 ÷ 18 × 10 = 2 × 10 = 20 Denominator of A = 36 ÷ (3 + 1) = 36 ÷ 4 = 9 A = 20/9

For expression B: Numerator of B = 10 + 15 × 5 + 6 × 2 - 5 = 10 + 75 + 12 - 5 = 92 Denominator of B = 15 ÷ 5 × 25 - 2 × 3 + 6 = 3 × 25 - 6 + 6 = 75 B = 92/75

Now, we calculate the sum (A + B): A + B = 20/9 + 92/75

The least common multiple of 9 and 75 is 225: A + B = (20 × 25)/225 + (92 × 3)/225 A + B = 500/225 + 276/225 A + B = (500 + 276)/225 = 776/225

Explanation

For any arithmetic progression with an odd number of terms, the average is equal to the middle term. For the given 9 consecutive odd numbers, the average is the 5th term, which is 27.

When we include the previous and next odd numbers, we add one term at the beginning and one term at the end of the sequence. This symmetric addition of two terms preserves the balance around the center. The new sequence consists of 11 consecutive odd numbers, and its middle term remains 27.

Using the formula for the average of an arithmetic progression, average = (first term + last term) / 2. If the original sequence has first term a and last term l, then (a + l) / 2 = 27. The new sequence has first term a - 2 and last term l + 2. The new average is ((a - 2) + (l + 2)) / 2 = (a + l) / 2 = 27.

Explanation

Let O be the center of the concentric circles. Draw a perpendicular line from O to the line PQRS, meeting it at point M. Since a line drawn from the center perpendicular to a chord bisects the chord, OM is perpendicular to both the chord PS of the larger circle and the chord QR of the smaller circle.

For the larger circle, M is the midpoint of PS. Thus, PM = PS / 2 = 48 / 2 = 24 cm.

In the right-angled triangle OMP, the hypotenuse OP is the radius of the larger circle, which is 26 cm. Applying Pythagoras' theorem: OM² = OP² - PM² OM² = 26² - 24² OM² = 676 - 576 OM² = 100 OM = 10 cm

For the smaller circle, M is the midpoint of QR. Thus, QM = QR / 2 = 40 / 2 = 20 cm.

In the right-angled triangle OMQ, the hypotenuse OQ is the radius of the smaller circle, which is x. Applying Pythagoras' theorem: OQ² = OM² + QM² x² = 10² + 20² x² = 100 + 400 x² = 500 x = 10√5 cm

Explanation

Total sales of company A over all years is: Total A = 350 + 550 + 750 + 650 + 450 = 2750

Total sales of company B over all years is: Total B = 1150 + 850 + 950 + 350 + 250 = 3550

L1 is the difference between the total sales of A and B: L1 = 3550 - 2750 = 800

L2 is the average sales of company A: L2 = 2750 / 5 = 550

The ratio of L1 to L2 is: L1 : L2 = 800 : 550 = 16 : 11

Explanation

The weighted average weight of all students is calculated using the formula:

Average weight = (N_g × W_g + N_b × W_b) / (N_g + N_b)

where N_g and N_b represent the number of girls and boys, and W_g and W_b represent their respective average weights.

Given that the average weight of girls is 30 kg, the average weight of boys is 36 kg, and the number of girls is twice the number of boys (N_g = 2 × N_b):

Average weight = (2 × N_b × 30 + N_b × 36) / (2 × N_b + N_b)

Average weight = (60 × N_b + 36 × N_b) / (3 × N_b)

Average weight = 96 × N_b / (3 × N_b)

Average weight = 96 / 3 = 32 kg

Explanation

By definition, tan F is the ratio of the perpendicular to the base of a right-angled triangle, which is given as 15/8.

Using the Pythagorean theorem, the hypotenuse h is calculated as: h = √(15² + 8²) h = √(225 + 64) h = √289 h = 17

Using the definitions of sine and cosine: sin F = perpendicular/hypotenuse = 15/17 cos F = base/hypotenuse = 8/17

Adding these two values: sin F + cos F = 15/17 + 8/17 = 23/17

Explanation

Correct Answer: C

The effective compound interest rate for 2 years at an annual interest rate of R% is given by the formula:

Effective Rate = R + R + (R × R)/100

Substituting R = 20 into the formula:

Effective Rate = 20 + 20 + (20 × 20)/100 = 44%

The compound interest on the sum is Rs. 4180, which represents 44% of the principal sum.

Sum = (Compound Interest × 100) / Effective Rate

Sum = (4180 × 100) / 44 = 9500

Explanation

The length L of a chord that subtends an angle θ at the center of a circle with radius r is given by the formula:

L = 2r sin(θ/2)

Substituting the given values, r = 14 cm and θ = 180°:

L = 2 × 14 × sin(180°/2) L = 28 × sin(90°) L = 28 × 1 = 28 cm

Alternatively, a chord that subtends an angle of 180° at the center of a circle passes through the center, making it the diameter of the circle. The length of the diameter is twice the radius:

L = 2r = 2 × 14 = 28 cm

Explanation

In right-angled triangle XYZ, the angle Y is 90 degrees, XY = 30 cm, and the hypotenuse XZ = 34 cm.

Using the Pythagorean theorem to find the third side YZ: YZ = √(XZ² - XY²) YZ = √(34² - 30²) YZ = √(1156 - 900) YZ = √256 = 16 cm

The length of the altitude YN perpendicular to the hypotenuse XZ in a right-angled triangle is given by the relation: YN = (XY × YZ) / XZ

Substituting the values into the formula: YN = (30 × 16) / 34 YN = 480 / 34 YN = 240/17 cm

Explanation

According to the divisibility rule of 9, a number is completely divisible by 9 if the sum of its individual digits is a multiple of 9.

For the number 572X41, the sum of its digits is: 5 + 7 + 2 + X + 4 + 1 = 19 + X

The smallest multiple of 9 that is greater than 19 is 27.

Setting the sum of the digits equal to 27: 19 + X = 27 X = 8

Explanation

Let the three numbers be represented as 2x, 3x, and 5x, where x is a common multiplier.

The smallest number is 2x and the largest number is 5x.

The sum of the largest and the smallest number is: 5x + 2x = 7x = 9800

Solving for x: x = 9800 ÷ 7 = 1400

The difference between the largest and the smallest number is: 5x - 2x = 3x

Substituting the value of x: Difference = 3 × 1400 = 4200