SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The given expression is: x/(yz)² + y/(xz)² + z/(xy)²

This can be rewritten as: x/(y²z²) + y/(x²z²) + z/(x²y²)

By taking the common denominator as x²y²z², the expression simplifies to: (x³ + y³ + z³) / (x²y²z²)

Using the algebraic identity, if x + y + z = 0, then: x³ + y³ + z³ = 3xyz

Substituting this into the simplified expression yields: 3xyz / (x²y²z²) = 3/xyz

Explanation

Using the simple interest formula SI = (P × R × T) / 100, where P is the principal, R is the rate of interest, and T is the time period.

Since the sum becomes three times itself, the simple interest earned is 3P - P = 2P. Substituting the values: 2P = (P × 10 × T) / 100 T = 20 years

For Statement I, at a rate of 20% for the same time period of 20 years, the simple interest is: SI = (P × 20 × 20) / 100 = 4P The total amount is P + 4P = 5P. This means the sum becomes 5 times itself, so Statement I is correct.

For Statement II, at a rate of 30% for the same time period of 20 years, the simple interest is: SI = (P × 30 × 20) / 100 = 6P The total amount is P + 6P = 7P. This means the sum becomes 7 times itself, so Statement II is correct.

Explanation

In any triangle, the centroid divides each median in the ratio 2:1 from the vertex to the opposite side. Since PM is the median of triangle PQR and O is the centroid, the point O divides the median PM such that PO : OM = 2 : 1.

The total length of the median PM can be expressed in terms of PO using the ratio: PM = PO × 3/2

Substituting the given value of PO = 27 cm into the formula: PM = 27 × 3/2 PM = 81/2 PM = 40.5 cm

Explanation

The three-digit numbers divisible by 20 form an arithmetic progression.

The first term, a, is the smallest three-digit number divisible by 20, which is 100. The last term, l, is the largest three-digit number divisible by 20, which is 980. The common difference, d, is 20.

To find the number of terms, n, we use the formula for the nth term of an arithmetic progression: l = a + (n - 1)d

Substituting the values: 980 = 100 + (n - 1) × 20 880 = (n - 1) × 20 n - 1 = 44 n = 45

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

Substituting the values: S = 45/2 × (100 + 980) S = 45/2 × 1080 S = 45 × 540 S = 24300

Explanation

The weight of the new person can be directly calculated by finding the difference between the total weight of the group after and before the new person joined.

Total weight of 10 persons = 10 × 40 = 400 kg Total weight of 11 persons = 11 × 42 = 462 kg

Weight of the new person = Total weight of 11 persons - Total weight of 10 persons Weight of the new person = 462 - 400 = 62 kg

Alternatively, using the average change shortcut: Weight of new person = New average + (Initial number of persons × Increase in average) Weight of new person = 42 + 10 × (42 - 40) = 42 + 20 = 62 kg

Explanation

For trigonometric identity problems, substituting a standard angle like A = 90° provides the fastest solution.

At A = 90°: cosec A = cosec 90° = 1 cos A = cos 90° = 0 cot A = cot 90° = 0

Substituting these values into the given expression: cosec A (1 - cos A)(cosec A + cot A) = 1 × (1 - 0) × (1 + 0) = 1 × 1 × 1 = 1

Alternatively, using standard trigonometric relations: cosec A (1 - cos A)(cosec A + cot A) = (1/sin A) × (1 - cos A) × (1/sin A + cos A/sin A) = (1/sin A) × (1 - cos A) × ((1 + cos A)/sin A) = (1 - cos² A)/sin² A = sin² A/sin² A = 1

Explanation

To find the time (T), we first calculate the Simple Interest (SI) by subtracting the Principal (P) from the Amount (A):

SI = Amount - Principal SI = 20000 - 12500 = 7500

Using the Simple Interest formula: SI = (P × R × T) / 100

Substitute the given values into the formula: 7500 = (12500 × 8 × T) / 100 7500 = 125 × 8 × T 7500 = 1000 × T T = 7500 / 1000 T = 7.5 years

Explanation

Correct Answer: D

The initial ratio of boys to girls is 5:4, representing a total of 5 + 4 = 9 units.

Since the total number of students is 414, the value of 1 unit is calculated as: 1 unit = 414 ÷ 9 = 46 students

Since exactly 46 more boys are admitted, this is equivalent to adding exactly 1 unit to the boys' share in the ratio.

The new ratio of boys to girls is: (5 + 1) : 4 = 6 : 4 = 3 : 2

Explanation

The total surface area of a cylinder is calculated using the formula: Total Surface Area = 2πr(r + h)

Here, the radius of the base r is 7 cm and the height h is 5 cm. Using π = 22/7:

Total Surface Area = 2 × (22/7) × 7 × (7 + 5) Total Surface Area = 2 × 22 × 12 Total Surface Area = 44 × 12 Total Surface Area = 528 cm²

Explanation

To solve the given expression, we apply the BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction) rule. The expression is structured as a numerator and a denominator separated by a division sign.

For the Numerator: 3/4 ÷ 9/64 + 7/11 of 121/14 - 9/13 of 65/36

First, simplify the of operations: 7/11 of 121/14 = (7/11) × (121/14) = 11/2 9/13 of 65/36 = (9/13) × (65/36) = 5/4

Next, simplify the division: 3/4 ÷ 9/64 = (3/4) × (64/9) = 16/3

Combine the terms: 16/3 + 11/2 - 5/4 = (64 + 66 - 15) / 12 = 115/12

For the Denominator: 1/2 ÷ 4/5 + 3/4 of 16/15

First, simplify the of operation: 3/4 of 16/15 = (3/4) × (16/15) = 4/5

Next, simplify the division: 1/2 ÷ 4/5 = (1/2) × (5/4) = 5/8

Combine the terms: 5/8 + 4/5 = (25 + 32) / 40 = 57/40

Dividing the Numerator by the Denominator: (115/12) ÷ (57/40) = (115/12) × (40/57) = (115 × 10) / (3 × 57) = 1150/171

Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the radius OQ is perpendicular to the tangent PQ at point Q, making triangle OQP a right-angled triangle at Q.

By the Pythagorean theorem: OP² = OQ² + PQ²

Substituting the given values: OP² = 5² + 12² OP² = 25 + 144 OP² = 169 OP = √169 = 13 cm

Explanation

The average of any arithmetic progression, such as consecutive even numbers, is equal to its middle term (for an odd number of terms) or the average of the two middle terms (for an even number of terms).

Let the first 5 consecutive even numbers be x, x + 2, x + 4, x + 6, and x + 8. The average of these 5 numbers is the middle term, which is x + 4. Thus, A = x + 4.

When the next 5 consecutive even numbers are included, we have a total of 10 consecutive even numbers: x, x + 2, x + 4, x + 6, x + 8, x + 10, x + 12, x + 14, x + 16, and x + 18.

The average of these 10 numbers is the average of the two middle terms, which are the 5th and 6th terms:

Average = ((x + 8) + (x + 10)) / 2 = (2x + 18) / 2 = x + 9

We can rewrite x + 9 in terms of A:

x + 9 = (x + 4) + 5 = A + 5

Explanation

Using the fundamental trigonometric identity: cos² A = 1 - sin² A

Given sin A = 6/7, substitute this value into the equation: cos² A = 1 - (6/7)² cos² A = 1 - 36/49 cos² A = (49 - 36)/49 cos² A = 13/49

Taking the square root on both sides: cos A = √13 / 7

Since sec A is the reciprocal of cos A: sec A = 1 / cos A sec A = 7 / √13

Explanation

Let D be the distance to the office, and T be the usual time taken in hours.

When traveling at S₁ = 80 km/hr, the person is 60 minutes (1 hour) early, so the time taken is T - 1. When traveling at S₂ = 50 km/hr, the person is 48 minutes (48/60 = 4/5 hours) late, so the time taken is T + 4/5.

Since the distance is constant: S₁ × (T - 1) = S₂ × (T + 4/5) 80 × (T - 1) = 50 × (T + 4/5) 80T - 80 = 50T + 40 30T = 120 T = 4 hours

Substituting T back to find the distance D: D = 80 × (4 - 1) = 240 km

The required speed to reach on the usual time T is: Speed = D / T = 240 / 4 = 60 km/hr

Explanation

The relationship between the variables can be expressed as: P = Q + 20% of Q = 1.2 × Q Q = R - 30% of R = 0.7 × R

Given that P = 420, we can find Q: 1.2 × Q = 420 Q = 420 ÷ 1.2 Q = 350

Now, substitute the value of Q to find R: 0.7 × R = 350 R = 350 ÷ 0.7 R = 500

Explanation

The relationship between the Selling Price (SP), Marked Price (MP), and Discount Percentage is given by the formula:

SP = MP × (100 - Discount%)/100

Given: SP = Rs.1,260 Discount = 40%

Substituting these values into the formula:

1,260 = MP × (100 - 40)/100 1,260 = MP × 60/100 1,260 = MP × 0.6

Solving for MP:

MP = 1,260 ÷ 0.6 MP = 2,100

Explanation

Correct Answer: C

The given equation is: (2cos³ A - cos A) / (sin A - 2sin³ A) = 1

Cross-multiplying both sides gives: 2cos³ A - cos A = sin A - 2sin³ A

Rearranging the terms to group the cubic terms together: 2sin³ A + 2cos³ A = sin A + cos A 2(sin³ A + cos³ A) = sin A + cos A

Using the algebraic identity a³ + b³ = (a + b)(a² - ab + b²): 2(sin A + cos A)(sin² A - sin A cos A + cos² A) = sin A + cos A

Since sin² A + cos² A = 1, this simplifies to: 2(sin A + cos A)(1 - sin A cos A) = sin A + cos A

Dividing both sides by (sin A + cos A) for acute angle A: 2(1 - sin A cos A) = 1 2 - 2sin A cos A = 1 2sin A cos A = 1

Using the double-angle identity sin(2A) = 2sin A cos A: sin(2A) = 1

Since sin 90° = 1: 2A = 90° A = 45°

Explanation

The loss on an article is calculated using the formula: Loss = Cost Price - Selling Price.

Calculating the loss for each individual article: Loss on article P = 300 - 250 = 50 Loss on article Q = 400 - 375 = 25 Loss on article R = 350 - 250 = 100 Loss on article S = 500 - 350 = 150

Total loss for all four articles = 50 + 25 + 100 + 150 = 325

The average loss is the total loss divided by the number of articles: Average loss = Total loss / Total number of articles Average loss = 325 / 4 = 81.25

Explanation

Let B's efficiency be 5 units per day. Since A is 20% more efficient than B, A's efficiency is 6 units per day.

Their combined daily efficiency is 6 + 5 = 11 units per day.

Since they complete the work together in 20 days, the total work is: Total Work = Combined Efficiency × Time = 11 × 20 = 220 units

The number of days B alone takes to complete this total work is: Time for B = Total Work ÷ B's Efficiency = 220 ÷ 5 = 44 days

Explanation

Given the equation: K / (K² - 2K + 1) = 1/6

Taking the reciprocal on both sides: (K² - 2K + 1) / K = 6

Dividing each term by K: K - 2 + 1/K = 6 K + 1/K = 8

To find the value of K³ + 1/K³, we use the algebraic identity: (K + 1/K)³ = K³ + 1/K³ + 3(K + 1/K)

Substituting K + 1/K = 8 into this identity: 8³ = K³ + 1/K³ + 3(8) 512 = K³ + 1/K³ + 24 K³ + 1/K³ = 512 - 24 K³ + 1/K³ = 488

Explanation

Since ΔABC is similar to ΔPQR, the ratio of their corresponding sides is constant. Given the corresponding sides BC = 8 cm and QR = 16 cm, the scale factor of similarity k is: k = QR / BC = 16 / 8 = 2

The ratio of the areas of two similar triangles is equal to the square of their scale factor: Area(ΔPQR) / Area(ΔABC) = k² = 2² = 4 Area(ΔPQR) = 4 × Area(ΔABC)

Using Heron's formula, the area of ΔABC with sides a = 8 cm, b = 10 cm, and c = 4 cm is calculated as: Semi-perimeter s = (a + b + c) / 2 = (8 + 10 + 4) / 2 = 11 cm Area(ΔABC) = √(s(s - a)(s - b)(s - c)) Area(ΔABC) = √(11 × (11 - 8) × (11 - 10) × (11 - 4)) Area(ΔABC) = √(11 × 3 × 1 × 7) = √231 cm²

The area of ΔPQR is: Area(ΔPQR) = 4 × √231 = 4√231 cm²

Explanation

By the degree measure theorem of circles, the angle subtended by an arc or chord at the centre of a circle is twice the angle subtended by it at any point on the remaining part of the circle.

Let the angle subtended by the chord AB at the centre be θ_centre = 140°. Let the angle subtended by the chord AB on the major arc be θ_arc.

Using the theorem: θ_centre = 2 × θ_arc

Substituting the given values into the formula: 140° = 2 × θ_arc θ_arc = 140° / 2 θ_arc = 70°

Explanation

Using the algebraic identity: a³ + b³ = (a + b)³ - 3ab(a + b)

Substituting the given values, (a + b) = 16 and a³ + b³ = 1456, into the identity: 1456 = 16³ - 48ab

Since 16³ = 4096, the equation becomes: 1456 = 4096 - 48ab

Rearranging the terms to solve for ab: 48ab = 4096 - 1456 48ab = 2640 ab = 2640 / 48 ab = 55

Explanation

The selling price (SP) of an item when sold at a loss is calculated using the formula:

SP = CP × (100 - Loss%)/100

Here, the cost price (CP) of the watch is Rs. 4,000 and the loss percentage is 13%.

Substituting these values into the formula:

SP = 4000 × (100 - 13)/100 SP = 4000 × 87/100 SP = 40 × 87 SP = 3480

The selling price of the watch is Rs. 3,480.

Explanation

When a train crosses a man walking in the opposite direction, the relative speed is the sum of the speed of the train and the speed of the man.

Let the speed of the man be v m/s. The speed of the train is 8v m/s. The relative speed is 8v + v = 9v m/s.

The distance covered to cross the man is equal to the length of the train, which is 1800 m, and the time taken is 10 seconds.

Using the speed-distance-time relation: Relative Speed = Distance / Time 9v = 1800 / 10 9v = 180 v = 20 m/s

The speed of the train is: 8v = 8 × 20 = 160 m/s

To convert the speed from m/s to km/hr, multiply by 18/5: Speed of train = 160 × 18/5 = 32 × 18 = 576 km/hr.

Explanation

A perpendicular drawn from the center of a circle to a chord bisects the chord. Let O be the center of the circle and RS be the chord of length 30 cm.

If OM is the perpendicular from O to RS, the distance OM is 20 cm, and M bisects RS. Therefore, the length of RM is half of RS: RM = 30/2 = 15 cm

In the right-angled triangle OMR, the radius r (which is the hypotenuse OR) is calculated using the Pythagorean theorem: r² = OM² + RM² r² = 20² + 15² r² = 400 + 225 r² = 625 r = 25 cm

The diameter of the circle is twice the radius: Diameter = 2 × r = 2 × 25 = 50 cm

Explanation

Using the algebraic identity for the difference of two cubes: a³ - b³ = (a - b)(a² + ab + b²)

This can be rearranged as: (a³ - b³) / (a² + b² + ab) = a - b

By comparing this general formula with the given equation: ((17)³ - (7)³) / (17² + 7² + k) = 10

We can substitute a = 17 and b = 7. Since a - b = 17 - 7 = 10, the denominator of the given expression must equal the algebraic term (a² + b² + ab).

Equating the denominators: 17² + 7² + k = 17² + 7² + 17 × 7

Comparing the terms directly gives: k = 17 × 7 = 119

Explanation

The given expression is S = (4² - 3²) + (6² - 5²) + (8² - 7²) + ... + (92² - 91²).

Using the algebraic identity a² - b² = (a - b)(a + b), we can simplify each paired term: 4² - 3² = (4 - 3)(4 + 3) = 1 × 7 = 7 6² - 5² = (6 - 5)(6 + 5) = 1 × 11 = 11 8² - 7² = (8 - 7)(8 + 7) = 1 × 15 = 15 92² - 91² = (92 - 91)(92 + 91) = 1 × 183 = 183

Substituting these simplified values, the expression becomes an arithmetic progression: S = 7 + 11 + 15 + ... + 183

For this arithmetic progression, the first term (a) is 7, the common difference (d) is 4, and the last term (l) is 183.

The number of terms (n) is calculated using the formula l = a + (n - 1)d: 183 = 7 + (n - 1) × 4 176 = (n - 1) × 4 n - 1 = 44 n = 45

The sum of an arithmetic progression is given by the formula S = (n / 2) × (a + l). Substituting the values: S = (45 / 2) × (7 + 183) S = (45 / 2) × 190 S = 45 × 95 S = 4275

Explanation

Let the shares of P, Q, and R be P, Q, and R respectively. The total sum is P + Q + R = 43120.

According to the first condition, P is 4/7 of the combined share of Q and R: P = 4/7 (Q + R) P / (Q + R) = 4/7

Adding P to both the numerator and denominator on the left side gives the ratio of P to the total sum: P / (P + Q + R) = 4 / (4 + 7) = 4/11 P = 4/11 × 43120 = 15680

According to the second condition, Q is 2/5 of the combined share of P and R: Q = 2/5 (P + R) Q / (P + R) = 2/5

Similarly, the ratio of Q to the total sum is: Q / (P + Q + R) = 2 / (2 + 5) = 2/7 Q = 2/7 × 43120 = 12320

The share of R is the total sum minus the sum of the shares of P and Q: R = 43120 - (15680 + 12320) R = 43120 - 28000 R = 15120

Explanation

The net profit or loss percentage when there is a successive markup of x% and a discount of y% is given by the successive percentage formula:

Net change = x - y - (x × y) / 100

Substituting the given values, where the markup x = 20 and the discount y = 10:

Net change = 20 - 10 - (20 × 10) / 100 Net change = 10 - 2 = 8%

Since the result is positive, it represents an 8% profit.

Explanation

Volume of a cube with side s is given by V = s³.

Since there is no loss of metal during the melting process, the volume of the new single cube is equal to the sum of the volumes of the three individual metallic cubes.

Total volume of the three smaller cubes = 6³ + 8³ + 10³ Total volume = 216 + 512 + 1000 = 1728 cm³

Let S be the side length of the newly formed cube. S³ = 1728 S = ∛1728 = 12 cm

Explanation

Let the marks obtained by X, Y, and Z be x, y, and z respectively.

According to the given condition, the average of x and y is 35 less than the average of y and z: (x + y)/2 = (y + z)/2 - 35

Multiplying the entire equation by 2: x + y = y + z - 70

Subtracting y from both sides simplifies the relation to: x = z - 70

Given that the marks obtained by Z is 85, substitute z = 85 into the equation: x = 85 - 70 = 15

Explanation

Let CP_P, CP_Q, and CP_R represent the cost prices for P, Q, and R respectively.

P sells the bottle to Q at a profit of 30%, so the cost price for Q is: CP_Q = CP_P × 1.3

Q sells the bottle to R at a profit of 20%, so the cost price for R is: CP_R = CP_Q × 1.2

Substituting CP_Q into the equation for CP_R: CP_R = CP_P × 1.3 × 1.2 = 1.56 × CP_P

The ratio of the cost prices of P and R is: CP_P : CP_R = 1 : 1.56 = 100 : 156

Dividing both terms of the ratio by 4 simplifies it to 25:39.

Explanation

Correct Answer: B

The simple interest earned in 2 years is calculated as: SI = Amount - Principal = 13,200 - 12,000 = 1,200

Under simple interest, the interest earned is directly proportional to the rate of interest when the principal and time period remain constant: SI = (P × R × T) / 100

If the rate of interest is doubled while the principal and time remain unchanged, the interest earned will also double: New SI = 2 × 1,200 = 2,400

The new amount is: New Amount = Principal + New SI = 12,000 + 2,400 = 14,400

Explanation

Using the fundamental trigonometric identity: sec² θ - tan² θ = 1 (sec θ - tan θ)(sec θ + tan θ) = 1

We can rewrite the term 1 in the numerator of the given expression: tan θ - sec θ + 1 = -(sec θ - tan θ) + (sec² θ - tan² θ) = -(sec θ - tan θ) + (sec θ - tan θ)(sec θ + tan θ) = (sec θ - tan θ)(sec θ + tan θ - 1)

Substitute this back into the original expression: (tan θ - sec θ + 1) / (tan θ + sec θ - 1) = (sec θ - tan θ)(sec θ + tan θ - 1) / (tan θ + sec θ - 1)

Canceling the common term (sec θ + tan θ - 1) from the numerator and the denominator: sec θ - tan θ

Expressing sec θ and tan θ in terms of sine and cosine: sec θ - tan θ = 1/cos θ - sin θ/cos θ = (1 - sin θ) / cos θ

Multiply the numerator and the denominator by (1 + sin θ): ((1 - sin θ)(1 + sin θ)) / (cos θ (1 + sin θ)) = (1 - sin² θ) / (cos θ (1 + sin θ)) = cos² θ / (cos θ (1 + sin θ)) = cos θ / (1 + sin θ)

Explanation

Using the BODMAS rule, the division (represented as ÷) and multiplication operations are performed from left to right.

For the expression A: A = (18 ÷ 9 × 4) / (15 ÷ 3 × 5) Numerator of A = 18 ÷ 9 × 4 = 2 × 4 = 8 Denominator of A = 15 ÷ 3 × 5 = 5 × 5 = 25 Thus, A = 8 / 25

For the expression B: B = (18 ÷ 36 × 3 + 5 × 1 - 6) / (18 ÷ 6 × 20 - 3 × 4 + 12) Numerator of B = (18 ÷ 36) × 3 + 5 - 6 = 0.5 × 3 + 5 - 6 = 1.5 + 5 - 6 = 0.5 Denominator of B = (18 ÷ 6) × 20 - 12 + 12 = 3 × 20 = 60 Thus, B = 0.5 / 60 = 1 / 120

Now, adding A and B: A + B = 8 / 25 + 1 / 120 Finding a common denominator of 600: A + B = (8 × 24) / 600 + (1 × 5) / 600 A + B = 192 / 600 + 5 / 600 A + B = 197 / 600

Explanation

Let the initial income of Raman be ₹100. Since he spends 50% of his income, his initial expenditure is ₹50 and his initial savings are ₹50.

When his income increases by 10%, the new income becomes: 100 + (10% of 100) = ₹110

When his expenditure increases by 10%, the new expenditure becomes: 50 + (10% of 50) = ₹55

The new savings are the difference between the new income and the new expenditure: New Savings = 110 - 55 = ₹55

The increase in savings is: 55 - 50 = ₹5

The percentage increase in savings is: (Increase in Savings / Initial Savings) × 100 (5 / 50) × 100 = 10%

Alternatively, because income is the sum of expenditure and savings, if both income and expenditure increase by the exact same percentage, the savings must also increase by that same percentage of 10%.

Explanation

For two identical circles of radius r that intersect each other such that each passes through the centre of the other, the length of the common chord L is given by the formula:

L = r√3

Given that the length of the common chord L = 16√3 cm, we substitute this value into the formula:

16√3 = r√3

Dividing both sides by √3 gives:

r = 16 cm

Explanation

The share of wages of individuals working together is distributed in the ratio of their efficiencies, which is the inverse ratio of the time taken by each to complete the work individually.

Ratio of time taken by A, B, and C = 10 : 20 : 30

Ratio of their efficiencies and earnings = 1/10 : 1/20 : 1/30

Multiplying each term by the least common multiple of 10, 20, and 30, which is 60, gives the simplified ratio of earnings: Ratio of A : B : C = 6 : 3 : 2

Total ratio parts = 6 + 3 + 2 = 11 parts

Given that the total earnings are Rs.1,100: 11 parts = Rs.1,100 1 part = Rs.100

The difference between the combined earnings of A and C and the earnings of B is: (Earnings of A + Earnings of C) - Earnings of B = (6 + 2) - 3 = 5 parts

Substituting the value of 1 part: 5 parts = 5 × Rs.100 = Rs.500

Explanation

Substituting the standard trigonometric values cot 60° = 1/√3 and cot 30° = √3 into the given expression:

Numerator: cot 60° - cot 30° = 1/√3 - √3 = (1 - 3)/√3 = -2/√3

Denominator: 1 + cot 60° · cot 30° = 1 + (1/√3) × √3 = 1 + 1 = 2

Dividing the numerator by the denominator gives:

(-2/√3) ÷ 2 = -1/√3

Explanation

According to the properties of a circle, the angle subtended by a chord at the center is double the angle subtended by it at any point on the circumference.

Given that the angle subtended by the chord AB on the circumference is 90°, the angle subtended by AB at the center is: 2 × 90° = 180°

A chord that subtends an angle of 180° at the center of the circle is the diameter.

The length of the diameter is twice the radius: AB = 2 × r

Substituting the given radius of 8 cm: AB = 2 × 8 = 16 cm

Explanation

Correct Answer: D

Since SE and TF are medians of triangle STU that intersect at O, O is the centroid of the triangle. The centroid divides each median in the ratio 2:1 from the vertex.

Thus, we can calculate the lengths of the segments OE and TO: OE = (1/3) × SE = (1/3) × 24 = 8 cm TO = (2/3) × TF = (2/3) × 22.5 = 15 cm

Since SE is the median to the side UT, E is the midpoint of UT. Therefore: ET = UT / 2 = 34 / 2 = 17 cm

In triangle OET, the side lengths are OE = 8 cm, TO = 15 cm, and ET = 17 cm.

Applying the converse of Pythagoras' theorem: OE² + TO² = 8² + 15² = 64 + 225 = 289 ET² = 17² = 289

Since OE² + TO² = ET², triangle OET is a right-angled triangle with the right angle at O. Hence, ∠TOE = 90°.

Explanation

To compare the given numbers, we convert their fractional exponents to a common denominator. The exponents are 1/6, 1/3, and 1/4. The least common multiple of the denominators 6, 3, and 4 is 12.

We raise each number to the 12th power to compare their values:

((125)¹/⁶)¹² = 125² = 15625

((11)¹/³)¹² = 11⁴ = 14641

((12)¹/⁶)¹² = 12² = 144

((5)¹/⁴)¹² = 5³ = 125

Comparing these results, 15625 is the greatest value, which means (125)¹/⁶ is the largest number.

Explanation

Using the algebraic identity (x + y)² = x² + 2xy + y², we square both sides of the given equation:

a + 1/a = 4

Squaring both sides: (a + 1/a)² = 4² a² + 2 + 1/a² = 16 a² + 1/a² = 14

Squaring this result once more: (a² + 1/a²)² = 14² a⁴ + 2 + 1/a⁴ = 196 a⁴ + 1/a⁴ = 194

Explanation

Using the standard trigonometric values: cot 30° = √3 tan 60° = √3 cot 60° = 1/√3 tan 30° = 1/3? No, tan 30° = 1/√3

Substitute these values into the given expression for A: A = cot 30° tan 60° + cot 60° tan 30° A = (√3 × √3) + (1/√3 × 1/√3) A = 3 + 1/3 A = 10/3

Explanation

The gain per year is determined by the difference in the lending and borrowing interest rates applied to the principal amount.

The borrowing rate is 6% per annum and the lending rate is 8.5% per annum. The net rate of gain per year is the difference between these two rates: Net gain rate per year = 8.5% - 6% = 2.5%

Using the simple interest formula for one year, the gain per year is calculated as: Gain per year = Principal × Net gain rate per year Gain per year = 20000 × 2.5/100 Gain per year = 20000 × 0.025 = 500

Thus, the person gains Rs. 500 per year.

Explanation

Let the cost price of one item be C and the selling price of one item be S.

According to the question, the cost price of 44 items is equal to the selling price of x items: 44 × C = x × S S / C = 44 / x

The profit is 10%, which gives the relation between the selling price and cost price: S = C × (1 + 10/100) S = 1.1 × C S / C = 1.1

Equating the two expressions for S / C: 44 / x = 1.1 x = 44 / 1.1 x = 40

Explanation

In a circle, the radius drawn to the point of contact of a tangent is perpendicular to the tangent line. This means that the radius OU is perpendicular to the tangent XU, making the angle OU X equal to 90°.

Consequently, triangle OU X is a right-angled triangle with OX as the hypotenuse. Applying the Pythagorean theorem to triangle OU X:

OX² = OU² + XU²

Substituting the given values, OU = 8 cm and XU = 15 cm:

OX² = 8² + 15² OX² = 64 + 225 OX² = 289 OX = √289 = 17 cm

Explanation

Given the equation x - 1/x = 4, we can find the value of x⁶ + 1/x⁶ by first squaring the expression and then cubing the result.

Squaring both sides of the given equation: (x - 1/x)² = 4² x² + 1/x² - 2 = 16 x² + 1/x² = 18

Now, we use the algebraic identity for the sum of cubes, a³ + b³ = (a + b)³ - 3ab(a + b), where a = x² and b = 1/x²: x⁶ + 1/x⁶ = (x² + 1/x²)³ - 3(x²)(1/x²)(x² + 1/x²) x⁶ + 1/x⁶ = (x² + 1/x²)³ - 3(x² + 1/x²)

Substituting the value of x² + 1/x² = 18 into this equation: x⁶ + 1/x⁶ = 18³ - 3(18) x⁶ + 1/x⁶ = 5832 - 54 x⁶ + 1/x⁶ = 5778

Explanation

When a solid sphere of radius r is cut into two equal halves, two new circular cross-sections are created. The surface area of each new circular face is πr².

Therefore, the increase in the total surface area is equal to the sum of the areas of these two new circular faces: Increase in area = 2πr²

Given that the diameter of the sphere is 7 cm, the radius r is 7/2 cm.

Substituting the values: Increase in area = 2 × (22/7) × (7/2) × (7/2) Increase in area = 2 × (22/7) × (49/4) Increase in area = 77 cm²