SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The trigonometric identity for complementary angles states that tan θ = cot (90° - θ). This implies that if tan A = cot B, the sum of the angles must be 90°, such that A + B = 90°. Applying this to the given equation tan (4α - 50°) = cot (50° - α), we equate the sum of the arguments to 90°:

(4α - 50°) + (50° - α) = 90°

By combining the terms involving α and the constant numerical values, the equation simplifies as follows:

4α - α - 50° + 50° = 90°

3α = 90°

Dividing both sides by 3 gives:

α = (90°)/(3) = 30°

As 30° is an acute angle, it satisfies the condition provided in the problem.

Explanation

The area of a sector of a circle is calculated using the formula Area = (θ)/(360) × π r², where θ is the central angle in degrees and r is the radius of the circle. Given that the radius r = 6 cm and the central angle θ = 40°, the area of the sector is ⁴⁰⁄₃₆₀ × π × (6)². This simplifies to ⅑ × 36π, which equals 4π cm².

Explanation

Total work is taken as the LCM of 13 and 17, which is 13 × 17 = 221 units. The daily efficiency of A is 221 / 13 = 17 units per day, and the daily efficiency of B is 221 / 17 = 13 units per day.

Since they work alternately starting with A, in one cycle of 2 days, the total work completed is 17 + 13 = 30 units. To find the number of full cycles, divide the total work by the work done in one cycle: 221 / 30 = 7 cycles with a remainder.

Work completed in 7 cycles (7 × 2 = 14 days) is 7 × 30 = 210 units. The remaining work is 221 - 210 = 11 units. On the 15th day, it is A's turn to work. Since A's efficiency is 17 units per day, the time taken by A to finish the remaining 11 units is 11 / 17 days. Therefore, the total time required to complete the whole work is 14 + 11 / 17 = 14 ¹¹⁄₁₇ days.

Explanation

The profit in a partnership is distributed in the ratio of the product of the investment and the time period for which the amount was invested. Niht invested Rs. 700 for the entire 12 months. Amit and Patel joined after 5 months, so their investment period was 12 - 5 = 7 months.

The ratio of their profit shares is: Niht : Amit : Patel = (700 × 12) : (300 × 7) : (400 × 7) 8400 : 2100 : 2800

Dividing by 700, the simplified ratio is: 12 : 3 : 4

The sum of the ratios is 12 + 3 + 4 = 19. Given the total profit is Rs. 627, the value of one unit is: 627 / 19 = 33

Patel's share corresponds to 4 units in the ratio: Patel's share = 4 × 33 = 132

Explanation

The length of a direct common tangent to two circles with radii R and r and distance between centers d is given by the formula L = √(d² - (R - r)²). When two circles touch each other externally, the distance between their centers is equal to the sum of their radii, so d = R + r. Substituting this into the formula yields L = √((R + r)² - (R - r)²), which simplifies using the algebraic identity (a + b)² - (a - b)² = 4ab to L = √(4Rr) = 2√(Rr).

In this problem, the radius of the first circle R = 17 cm and the radius of the second circle r = 7 cm. Substituting these values into the simplified formula gives the length of the direct common tangent BC = 2√(17 × 7) = 2√(119) cm.

Explanation

In △ PRQ, since segment AB is drawn parallel to side QR, △ PAB is similar to △ PQR by the Angle-Angle (AA) similarity criterion, as ∠ P is common and ∠ PAB = ∠ PQR (corresponding angles).

The length of PA is calculated by subtracting QA from the total length of PQ: PA = PQ - QA = 6cm - 3cm = 3cm

According to the properties of similar triangles, the ratio of corresponding sides must be equal: (PA)/(PQ) = (AB)/(QR)

Substituting the given values into the ratio: ³⁄₆ = (AB)/(8) ½ = (AB)/(8) AB = ⁸⁄₂ = 4.0cm

Explanation

To determine the value of x³ - (1)/(x³), we first find the value of x - ¹⁄ₓ using the algebraic relationship (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4.

Substituting the given value x + ¹⁄ₓ = 2√5: (x - ¹⁄ₓ)² = (2√5)² - 4 = 20 - 4 = 16

Since x > 1, the difference x - ¹⁄ₓ must be positive. Taking the square root of both sides: x - ¹⁄ₓ = √(16) = 4

Now, the value of x³ - (1)/(x³) can be calculated using the identity x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ).

Substituting x - ¹⁄ₓ = 4 into the identity: x³ - (1)/(x³) = 4³ + 3(4) = 64 + 12 = 76

Explanation

For a number to be divisible by 75, it must be divisible by both 3 and 25. A number is divisible by 25 if its last two digits are divisible by 25. In the number 678p37q, the last two digits are 7q. The only multiple of 25 in the seventies is 75, which means q = 5.

Next, a number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 678p375 is 6 + 7 + 8 + p + 3 + 7 + 5 = 36 + p. Since 36 is already divisible by 3, the digit p must also be a multiple of 3 to keep the total sum divisible by 3. Therefore, the possible values for p are 0, 3, 6, or 9.

The problem further states that p is not a composite number. A composite number is a positive integer greater than 1 that has at least one divisor other than 1 and itself. Among the possible values for p, the numbers 6 and 9 are composite. The digits 0 and 3 are not composite numbers. Comparing these possibilities with the options provided, p = 3 and q = 5 is the only pair that satisfies all the given conditions.

Explanation

The relationship between the product of tangents and the cosine of the sum of angles is derived from the fundamental trigonometric identity for cos(A + B).

cos(A + B) = cos A cos B - sin A sin B

By dividing the entire equation by cos A cos B, we obtain:

(cos(A + B))/(cos A cos B) = (cos A cos B)/(cos A cos B) - (sin A sin B)/(cos A cos B)

This simplifies to:

(cos(A + B))/(cos A cos B) = 1 - tan A tan B

Rearranging the terms to isolate 1 on one side of the equation:

tan A tan B + (cos(A + B))/(cos A cos B) = 1

Comparing this standard identity to the given equation form tan A tan B + (cos x)/(cos A cos B) = 1, it is evident that x corresponds to the sum of the angles A + B.

Explanation

When the cost price (CP) of x articles is equal to the selling price (SP) of y articles, the gain percentage is calculated using the formula: Gain % = (x - y)/(y) × 100

Given that the cost price of 20 books equals the selling price of 18 books, we substitute x = 20 and y = 18 into the formula: Gain % = (20 - 18)/(18) × 100 Gain % = ²⁄₁₈ × 100 = ⅑ × 100 Gain % = 11⅑%

Alternatively, let the cost price of one book be CP and the selling price of one book be SP. According to the problem, 20 × CP = 18 × SP. This relationship can be expressed as the ratio (SP)/(CP) = ²⁰⁄₁₈ = ¹⁰⁄₉. This means for every 9 units of cost, there is a selling price of 10 units, resulting in a profit of 1 unit. The gain percentage is thus (⅑) × 100, which simplifies to 11⅑%.

Explanation

The final price is calculated by applying successive discounts to the marked price. When multiple discounts are applied one after another, the selling price is given by the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)).

Given the marked price MP = ₹2,800, the first discount d₁ = 45%, and the second discount d₂ = 25%, the calculation is:

SP = 2800 × (1 - ⁴⁵⁄₁₀₀) × (1 - ²⁵⁄₁₀₀)

SP = 2800 × ⁵⁵⁄₁₀₀ × ⁷⁵⁄₁₀₀

SP = 2800 × ¹¹⁄₂₀ × ¾

Simplifying the expression:

SP = 140 × 11 × ¾

SP = 35 × 11 × 3

SP = 385 × 3 = 1155

The customer must pay ₹1,155 to avail both discounts.

Explanation

To find the Least Common Multiple (LCM) of algebraic terms, we take the highest power of each variable present in the given expressions. The given expressions are xy³, x²y, and x³y⁴. Looking at the variable x, the powers in the three terms are 1, 2, and 3 respectively. The maximum power of x is x³. Looking at the variable y, the powers are 3, 1, and 4 respectively. The maximum power of y is y⁴. By multiplying these highest powers together, we obtain the LCM as x³y⁴.

Explanation

The expression is evaluated by substituting the standard values of the trigonometric functions. For cos 60°, the value is ½. For cos 240°, using the quadrant rule cos(180° + θ) = -cos θ, we find cos 240° = cos(180° + 60°) = -cos 60° = -½. For cos 360°, the value is equal to cos 0°, which is 1.

Now, calculate the cubes of these values: (cos 60°)³ = (½)³ = ⅛ (cos 240°)³ = (-½)³ = -⅛ (cos 360°)³ = (1)³ = 1

Substitute these into the given expression cos³ 60° - cos³ 240° - cos³ 360°: ⅛ - (-⅛) - 1 = ⅛ + ⅛ - 1 ²⁄₈ - 1 = ¼ - 1 (1 - 4)/(4) = -¾

Explanation

Let the number of votes received by candidate Q be x. According to the given conditions, candidate P received 35% more votes than Q, which is 1.35x, and candidate R received 15% more votes than Q, which is 1.15x.

The difference between the votes of P and R is given as 2,412. 1.35x - 1.15x = 2,412 0.20x = 2,412 x = (2,412)/(0.20) = 12,060

The total number of votes cast is the sum of the votes received by P, Q, and R: Total votes = 1.35x + x + 1.15x = 3.5x Total votes = 3.5 × 12,060 = 42,210

Since only 90% of the voters on the list cast their votes and there were no invalid votes, let the total number of voters in the list be V. 0.90 × V = 42,210 V = (42,210)/(0.90) = 46,900

Explanation

To find the meeting time, we first equalize the starting times of both cars. From 8 a.m. to 9 a.m., the first car travels alone for 1 hour at 25 km/h, covering a distance of 25 km. At 9 a.m., the distance remaining between the two cars is 135 - 25 = 110 km.

Since the cars are moving toward each other, their relative speed is the sum of their individual speeds: Vᵣₑₗ = 25 + 30 = 55 km/h. The time required for them to meet after 9 a.m. is calculated using the formula t = (D)/(Vᵣₑₗ₎.

t = (110 km)/(55 km/h) = 2 hours

Adding 2 hours to the common starting time of 9 a.m., the cars will meet at 11 a.m.

Explanation

When successive discounts are applied, each subsequent discount is calculated on the reduced value remaining after the previous discount. For three successive discounts of 10%, the final value can be calculated using the multiplier method. A 10% discount corresponds to a multiplier of 0.90 (or 1 - 0.10).

The effective multiplier for three such discounts is (0.90) × (0.90) × (0.90) = 0.729. This means the final price is 72.9% of the original price. The equivalent single discount is the total reduction from the original 100%, calculated as:

Equivalent Discount = 100% - 72.9% = 27.1%

Alternatively, using the formula for two successive discounts x and y, the single equivalent discount is x + y - ˣʸ⁄₁₀₀. Applying this to the first two discounts of 10% results in 10 + 10 - (10 × 10)/(100) = 19%. Applying the formula again for this 19% and the third 10% discount gives 19 + 10 - (19 × 10)/(100) = 29 - 1.9 = 27.1%.

Explanation

In geometry, a tangent to a circle is perpendicular to the radius at the point of contact. Since PT is a tangent at T and OT is the radius, the angle ∠ OTP is 90°. This identifies △ OTP as a right-angled triangle where OP is the hypotenuse.

Using the Pythagoras theorem: OP² = OT² + PT²

Substituting the given values OP = 17 cm and OT = 8 cm: 17² = 8² + PT² 289 = 64 + PT² PT² = 289 - 64 PT² = 225 PT = √(225) PT = 15 cm

Explanation

To find the Least Common Multiple (LCM) of 96, 136, and 504, we use the prime factorization method. First, we break each number down into its prime factors: 96 = 2⁵ × 3¹ 136 = 2³ × 17¹ 504 = 2³ × 3² × 7¹

The LCM is calculated by taking the highest power of every prime factor that appears in these factorizations: LCM = 2⁵ × 3² × 7¹ × 17¹

Calculating the product: LCM = 32 × 9 × 7 × 17 LCM = 288 × 119 LCM = 34272

Alternatively, since 504 is a multiple of 9 (sum of digits 5+0+4=9), the LCM must also be divisible by 9. Among the options, 34272 has a digit sum of 18, making it divisible by 9. Furthermore, 96 is a multiple of 32, and 34272 is divisible by 32 (34272 / 32 = 1071), confirming it as the correct multiple.

Explanation

Let the cost price (CP) of 1000 g of goods be 1000 units. The dealer sells the goods at a 20% profit based on the weight he claims to sell (1000 g), so the selling price (SP) is 1000 × 1.20 = 1200 units. However, the dealer actually provides only 800 g of goods. The actual cost to the dealer for this quantity is 800 units.

The real gain is calculated by comparing the selling price received to the actual cost of the goods delivered: Real Profit = SP - Actual CP = 1200 - 800 = 400 units Real Gain % = (Real Profit)/(Actual CP) × 100 = ⁴⁰⁰⁄₈₀₀ × 100 = 50%

Alternatively, using the successive percentage/ratio method, the overall profit ratio is the product of the profit from the price markup and the profit from the weight discrepancy: Overall Ratio = ¹²⁰⁄₁₀₀ × ¹⁰⁰⁰⁄₈₀₀ = ⁶⁄₅ × ⁵⁄₄ = ⁶⁄₄ = 1.5 A ratio of 1.5 corresponds to a 50% increase.

Explanation

In triangle ABC, given ∠ B = 90° and ∠ C = 45°, the third angle ∠ A must be 180° - (90° + 45°) = 45°. This identifies ABC as an isosceles right-angled triangle where the sides opposite the equal angles are equal, meaning AB = BC.

According to the properties of a 45°-45°-90° triangle, the ratio of the lengths of the sides is 1 : 1 : √2. This means the hypotenuse AC is √2 times the length of the base BC. The relationship can be expressed using the trigonometric ratio:

cos C = (Base)/(Hypotenuse) cos 45° = (BC)/(AC)

Substituting the given value of the hypotenuse AC = 2√2 cm and the value of cos 45° = (1)/(√(2)):

(1)/(√(2)) = (BC)/(2√(2))

By multiplying both sides of the equation by 2√2, we isolate BC:

BC = 2√2{√2} BC = 2 cm

Explanation

The simple interest (SI) earned is determined by subtracting the principal (P) from the final amount (A): SI = 19125 - 10200 = 8925.

Using the formula for simple interest SI = (P × R × T)/(100), where R = 12.5 percent per annum: 8925 = (10200 × 12.5 × T)/(100) 8925 = 102 × 12.5 × T

Since 12.5 is equivalent to ²⁵⁄₂: 8925 = 102 × ²⁵⁄₂ × T 8925 = 51 × 25 × T 8925 = 1275 × T

Solving for time T: T = ⁸⁹²⁵⁄₁₂₇₅ T = 7

The calculation 1275 × 7 equals 8925, confirming the time period as 7 years.

Explanation

The given equation is 5sin²A + 3cos²A = 4. To solve for A, we use the fundamental trigonometric identity sin²A + cos²A = 1. We can rewrite the equation by splitting the first term: 2sin²A + 3sin²A + 3cos²A = 4. Factoring out 3 from the middle and last terms gives 2sin²A + 3(sin²A + cos²A) = 4. Substituting the identity value of 1, we get 2sin²A + 3 = 4, which simplifies to 2sin²A = 1 or sin²A = 1/2. Taking the square root, we find sin A = 1/√2 because A is in the first quadrant (0 < A < 90°), where the sine function is positive. This implies A = 45°. Therefore, tan A = tan 45° = 1.

Alternatively, dividing the entire equation by cos²A yields 5tan²A + 3 = 4sec²A. Since sec²A = 1 + tan²A, the equation becomes 5tan²A + 3 = 4(1 + tan²A). Expanding this gives 5tan²A + 3 = 4 + 4tan²A. Subtracting 4tan²A and 3 from both sides results in tan²A = 1. Since A is an acute angle, tan A must be positive, confirming that tan A = 1.

Explanation

The weight of the staff member who left can be calculated using the formula: Weight of person leaving = (Original Average) + (Number of remaining members × Reduction in average).

The initial average weight is 62.5 kg. After one staff member leaves, the number of remaining members is 119. The average weight reduction is given as 250 g, which is equivalent to 0.25 kg.

Using the values in the formula: Weight = 62.5 + (119 × 0.25) Weight = 62.5 + 29.75 Weight = 92.25 kg

This calculation accounts for the fact that the person who left not only took their share of the average (62.5 kg) but also caused a 0.25 kg drop for each of the remaining 119 staff members.

Explanation

Let the total work be the Least Common Multiple (LCM) of the given time periods, which is LCM(4, 9, 12) = 36 units. The combined efficiency of Anshu, Swati, and Rajni is 36 / 4 = 9 units per day. Anshu's individual efficiency is 36 / 9 = 4 units per day, and Swati's individual efficiency is 36 / 12 = 3 units per day. To find Rajni's efficiency, subtract the efficiencies of Anshu and Swati from the combined efficiency: 9 - (4 + 3) = 2 units per day. The number of days Rajni takes to complete the work alone is the total work divided by her efficiency: 36 / 2 = 18 days.

Explanation

The mean (x) for grouped data is calculated using the formula x = (Σ fᵢ xᵢ)/(Σ fᵢ), where fᵢ represents the frequency and xᵢ represents the class mark (midpoint) of each interval.

First, determine the midpoints (xᵢ) for each class interval: 5, 15, 25, 35, 45, 55, and 65.

Next, calculate the sum of the frequencies (Σ fᵢ): Σ fᵢ = 2 + 4 + 12 + 21 + 6 + 3 + 2 = 50

Then, calculate the sum of the products of frequencies and midpoints (Σ fᵢ xᵢ): Σ fᵢ xᵢ = (2 × 5) + (4 × 15) + (12 × 25) + (21 × 35) + (6 × 45) + (3 × 55) + (2 × 65) Σ fᵢ xᵢ = 10 + 60 + 300 + 735 + 270 + 165 + 130 = 1670

Finally, divide the sum of products by the total frequency: x = ¹⁶⁷⁰⁄₅₀ = 33.4

Explanation

In an isosceles triangle where AB = AC, the angles opposite the equal sides are equal, thus ∠ ACB = ∠ ABC = 55°. The sum of the interior angles of triangle ABC is 180°, so the vertex angle ∠ BAC = 180° - (55° + 55°) = 70°. In an isosceles triangle, the median drawn to the base BC also functions as the angle bisector of the vertex angle. Therefore, AD bisects ∠ BAC, and the measure of ∠ BAD is ½ × 70° = 35°. Alternatively, because the median to the base of an isosceles triangle is also the altitude, triangle ABD is a right-angled triangle with ∠ ADB = 90°. Consequently, ∠ BAD = 180° - (90° + 55°) = 35°.

Explanation

The algebraic identity for the cube of a binomial is (a + b)³ = a³ + 3a²b + 3ab² + b³. This can also be written in its factored middle-term form as (a + b)³ = a³ + b³ + 3ab(a + b).

By substituting this expansion into the given expression (a + b)³ - a³ + b³, we get:

[a³ + b³ + 3ab(a + b)] - a³ + b³

To simplify the expression, combine the like terms:

The a³ and - a³ terms cancel each other out (a³ - a³ = 0). The b³ terms are added together (b³ + b³ = 2b³). The term 3ab(a + b) remains as part of the sum.

Combining these results yields the simplified expression 3ab(a + b) + 2b³.

Explanation

The expression is evaluated using the identity sin θ sec(90° - θ) = 1, which is derived from the complementary angle relationship sec(90° - θ) = csc θ = 1/sin θ. By grouping the terms with complementary angles, the product is rearranged as follows:

(sin 28° sec 62°) · (sin 35° sec 55°) · sin 45°

Since 28° + 62° = 90°, the first part simplifies to sin 28° / cos 62° = sin 28° / sin 28° = 1. Similarly, since 35° + 55° = 90°, the second part simplifies to sin 35° / cos 55° = sin 35° / sin 35° = 1.

Substituting these values reduces the expression to 1 · 1 · sin 45°. Using the standard trigonometric value sin 45° = 1/√2, the final result is 1/√2.

Explanation

The third proportional z of two numbers x and y is defined by the relation x : y = y : z, which results in the formula z = (y²)/(x). Given the first term x = a and the second term y = (b⁴)/(4a), the calculation for the third proportional is as follows:

z = ((b⁴)/(4a))²{a}

Applying the power to the terms in the numerator:

z = (b⁸)/(16a²){a}

Simplifying the complex fraction by dividing the numerator by the denominator a:

z = (b⁸)/(16a² × a) = (b⁸)/(16a³)

Explanation

The percentage of wood wasted is calculated by finding the difference between the volume of the wooden sphere and the volume of the cut-out cone, then dividing by the volume of the sphere.

The volume of the sphere is given by Vₛ = ⁴⁄₃ π R³, where the radius R = 24 cm. Substituting the values: Vₛ = ⁴⁄₃ π (24)³ = ⁴⁄₃ π × 13824 = 18432 π cm³

The volume of the cone is given by V_c = ⅓ π r² h, where the radius r = 21 cm (half of the diameter 42 cm) and height h = 42 cm: V_c = ⅓ π (21)² (42) = π × 441 × 14 = 6174 π cm³

The percentage of wood wasted is: Percentage Wasted = (Vₛ - V_c)/(Vₛ) × 100 = ( 1 - (V_c)/(Vₛ) ) × 100 Percentage Wasted = ( 1 - (6174 π)/(18432 π) ) × 100

Simplifying the fraction ⁶¹⁷⁴⁄₁₈₄₃₂ by dividing both by 18 yields ³⁴³⁄₁₀₂₄. Percentage Wasted = ( 1 - ³⁴³⁄₁₀₂₄ ) × 100 = ⁶⁸¹⁄₁₀₂₄ × 100 ⁶⁸¹⁰⁰⁄₁₀₂₄ ≈ 66.5039%

Rounding to two decimal places, the percentage of wood wasted is 66.50%.

Explanation

The largest 5-digit number is 99999. To find the largest 5-digit number exactly divisible by 88, we divide 99999 by 88 to find the remainder:

99999 ÷ 88 = 1136 with a remainder of 31

The largest 5-digit number divisible by 88 is calculated by subtracting this remainder from the original 5-digit number:

99999 - 31 = 99968

The result can be verified using the divisibility rules for the factors of 88, which are 8 and 11. For divisibility by 8, the last three digits 968 must be divisible by 8 (968 ÷ 8 = 121). 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. In 99968, the sum of digits at odd positions is 9 + 9 + 8 = 26 and the sum of digits at even positions is 9 + 6 = 15. The difference is 26 - 15 = 11, which confirms the number is divisible by 11.

Explanation

The aggregate performance for a given subject is determined by calculating the sum of the marks obtained by all three students. For English, the marks are 88 for Ajay, 77 for Sudhir, and 92 for Suman, resulting in a total aggregate score of 88 + 77 + 92 = 257. This total of 257 is the highest aggregate score achieved among all the subjects provided in the dataset, as the cumulative scores for the remaining subjects are 247, 233, 218, 216, and 213.

Explanation

The total number of officers working across all six companies is determined by calculating the sum of the individual officer counts for companies D, E, F, G, H, and I as provided in the table. The summation is performed as follows:

Total Officers = 400 (D) + 500 (E) + 600 (F) + 350 (G) + 450 (H) + 800 (I)

To simplify the arithmetic, the values can be grouped into pairs that result in round numbers: Sum = (400 + 600) + (350 + 450) + 500 + 800 Sum = 1000 + 800 + 500 + 800 Sum = 1800 + 1300 Sum = 3100

Explanation

Given cos (A - B) = √3{2}, we identify the corresponding angle for the cosine function. Since A and B are acute angles, the value √3{2} corresponds to 30°. Therefore, we have the first equation: A - B = 30°

Given cos (A + B) = 0, we identify the corresponding angle where the cosine value is zero. For the sum of two acute angles, this occurs at 90°. Therefore, we have the second equation: A + B = 90°

To find the values of A and B, we solve the system of linear equations by adding them together: (A + B) + (A - B) = 90° + 30° 2A = 120° A = 60°

By substituting the value of A back into the equation A + B = 90°: 60° + B = 90° B = 30°

The calculated values A = 60° and B = 30° satisfy the conditions that both are positive acute angles and A ≥ B.

Explanation

The average speed of a journey is defined as the total distance traveled divided by the total time taken. The distance for each segment is calculated using the formula d = v × t.

Distance traveled by car: d₁ = 40 km/h × 3 h = 120 km Distance traveled by bike: d₂ = 30 km/h × 2 h = 60 km Distance traveled by train: d₃ = 80 km/h × 5 h = 400 km

Total Distance D = 120 + 60 + 400 = 580 km Total Time T = 3 + 2 + 5 = 10 h

Average Speed Vₐᵥ_g = (D)/(T) = (580 km)/(10 h) = 58 km/h

Explanation

Using the algebraic identity for the square of a binomial, we have (x - ¹⁄ₓ)² = x² + (1)/(x²⁾ - 2(x)(¹⁄ₓ). Since the product of x and its reciprocal ¹⁄ₓ is 1, the identity simplifies to (x - ¹⁄ₓ)² = x² + (1)/(x²⁾ - 2.

By substituting the given value x² + (1)/(x²⁾ = 66 into this equation, we get: (x - ¹⁄ₓ)² = 66 - 2 (x - ¹⁄ₓ)² = 64

To find the value of x - ¹⁄ₓ, take the square root of both sides: x - ¹⁄ₓ = √(64) x - ¹⁄ₓ = 8

Explanation

In a right-angled triangle, the tangent of an angle is the ratio of the perpendicular (P) to the base (B). Given tan A = ⅔, we can let P = 2 and B = 3. To find the sine of the angle, we first determine the hypotenuse (H) using the Pythagorean theorem:

H = √(P²⁾⁺^B^^^² H = √(2²⁾⁺³^^^² = √(4 + 9) = √(13)

The sine of an angle is defined as the ratio of the perpendicular to the hypotenuse:

sin A = (P)/(H) = (2)/(√(13))

Explanation

The total work to be completed is equivalent to the work done by 20 men in 220 days, which is 20 × 220 = 4400 man-days. In the first 90 days, the work completed by the 20 men is 20 × 90 = 1800 man-days. Therefore, the remaining work is 4400 - 1800 = 2600 man-days. After 90 days, 20 additional men are employed, increasing the total workforce to 20 + 20 = 40 men. The number of more days required to complete the remaining work is calculated by dividing the remaining man-days by the new number of workers: 2600 / 40 = 65 days.

Alternatively, this can be solved using the inverse proportion between the number of men and the time taken. The work remaining after 90 days would have taken the original 20 men another 130 days (220 - 90 = 130). Since the number of men is doubled from 20 to 40, the time required to complete the same amount of work is halved. Thus, the work will be completed in 130 / 2 = 65 more days.

Explanation

The largest sphere that can be extracted from a cube has a diameter equal to the side length of the cube. Given the side length a = 2.8 cm, the radius r of the sphere is: r = ᵃ⁄₂ = (2.8)/(2) = 1.4 cm

The volume V of a sphere is calculated using the formula: V = ⁴⁄₃ π r³

Substituting the known values r = 1.4 cm and π ≈ ²²⁄₇: V = ⁴⁄₃ × ²²⁄₇ × (1.4)³ V = ⁴⁄₃ × ²²⁄₇ × 1.4 × 1.4 × 1.4 V = (4 × 22 × 0.2 × 1.96)/(3) V = (17.6 × 1.96)/(3) V = (34.496)/(3) V ≈ 11.49866... cm³

Rounding the result to two decimal places, the volume is 11.50 cm³.

Explanation

The perpendicular distance h from the center of a circle of radius R to a chord of length L is given by the formula h = √(R²⁾⁻⁽⁽^L⁾^/⁽²⁾⁾^^^², derived from the Pythagorean theorem as the perpendicular from the center bisects the chord.

For the first chord of length L₁ = 2 cm and radius R = 3 cm, the distance from the center h₁ is: h₁ = √(3²⁾⁻⁽^²^⁄^₂⁾^^^² = √(9 - 1) = √8 = 2√2 cm.

For the second chord of length L₂ = 3 cm and radius R = 3 cm, the distance from the center h₂ is: h₂ = √(3²⁾⁻⁽^³^⁄^₂⁾^^^² = √(9 - ⁹⁄₄) = √((36 - 9)/(4)) = √(²⁷⁄₄) = 3√3{2} cm.

Since both chords lie on the same side of the diameter, the perpendicular distance between them is the absolute difference of their distances from the center: Distance = h₁ - h₂ = 2√2 - 3√3{2}

Taking the common denominator: Distance = 4√2 - 3√3{2} cm.

Explanation

The number of passing students in State B is calculated using the total student count and the given passing-to-failing ratio. For State B, the total number of students is 1.40 lakh and the ratio of passing to failing students is 3:4.

To find the number of passing students, we use the formula: Passing Students = Total Students × (Passing Ratio Part)/(Sum of Ratio Parts)

The sum of the ratio parts for State B is 3 + 4 = 7. Substituting the values into the formula: Passing Students = 1.40 × ³⁄₇ Passing Students = 0.20 × 3 Passing Students = 0.60 lakh

Explanation

The Least Common Multiple (LCM) of 144, 360, and 450 is calculated using the prime factorization method. First, determine the prime factorization of each number: 144 is 2⁴ × 3², 360 is 2³ × 3² × 5¹, and 450 is 2¹ × 3² × 5².

The LCM is the product of the highest power of every prime factor present across all the numbers. The prime factors involved are 2, 3, and 5. Comparing the powers of prime factor 2, the highest is 2⁴ from 144. For prime factor 3, the highest power is 3², which is present in all three factorizations. For prime factor 5, the highest power is 5² from 450.

The LCM is the product of these values: LCM = 2⁴ × 3² × 5². Substituting the values, we get 16 × 9 × 25. Using the associative property to simplify the calculation, 16 × 25 = 400, and 400 × 9 = 3600.

Explanation

The selling price is calculated by applying the discounts sequentially to the marked price. First, the cash discount of ₹ 85 is subtracted from the initial marked price of ₹ 1,285: Price₁ = 1285 - 85 = 1200

Next, the two successive discounts of 10% and 2% are applied to the remaining balance. The price after the 10% discount is: Price₂ = 1200 × (1 - ¹⁰⁄₁₀₀) = 1200 × 0.9 = 1080

Finally, the 2% discount is applied to the resulting value of ₹ 1,080: Selling Price = 1080 × (1 - ²⁄₁₀₀) = 1080 × 0.98 = 1058.4

Explanation

The selling price (SP) is directly proportional to the percentage value of the cost price (CP). A loss of 12% implies the current selling price is 88% of the CP, while a target profit of 18% implies the new selling price should be 118% of the CP.

The relationship can be expressed using the formula: SPₙₑ_w = SPₒₗ_d{100 - Loss%} × (100 + Profit%)

Substituting the given values: SPₙₑ_w = ⁴⁶⁰⁶⁸⁄₈₈ × 118

First, simplify the division: 46068 ÷ 88 = 523.5

Then, calculate the final selling price: 523.5 × 118 = 61773

Alternatively, simplifying the ratio ¹¹⁸⁄₈₈ to ⁵⁹⁄₄₄: SPₙₑ_w = ⁴⁶⁰⁶⁸⁄₄₄ × 59 SPₙₑ_w = 1047 × 59 = 61773

Explanation

The divisibility rule for 9 states that the remainder of any integer when divided by 9 is equal to the remainder obtained when the sum of its digits is divided by 9. To find the remainder for 123456789, we calculate the sum of its digits:

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

Using the arithmetic series formula Sₙ = (n(n + 1))/(2) for the sum of the first n natural numbers where n = 9:

S₉ = (9 × (9 + 1))/(2) = (9 × 10)/(2) = 45

Since 45 is a multiple of 9 (9 × 5 = 45), it leaves a remainder of 0 when divided by 9. Consequently, the original number 123456789 also leaves a remainder of 0 when divided by 9.

Explanation

When two triangles are similar, the ratio of their corresponding sides is equal. For triangle ABC, the side lengths are AB = 92 cm, BC = 48 cm, and AC = 120 cm. The smallest side is BC = 48 cm and the longest side is AC = 120 cm. Since triangle ABC is similar to triangle DEF, the ratio of the smallest sides must equal the ratio of the longest sides.

Let s₁ and l₁ be the smallest and longest sides of triangle ABC, and s₂ and l₂ be the smallest and longest sides of triangle DEF. Using the property of similarity:

(s₁)/(s₂) = (l₁)/(l₂)

Substituting the given values:

⁴⁸⁄₂₀₀ = (120)/(l₂)

Solving for l₂ :

l₂ = (120 × 200)/(48)

l₂ = (5 × 200)/(2) = 5 × 100 = 500 cm

Explanation

The angle sum property of a triangle states that the sum of its three interior angles is always 180°. If the two equal unknown angles are each denoted as x, the relationship between the angles can be expressed by the formula x + x + 78° = 180°. This simplifies to the linear equation 2x + 78° = 180°. By subtracting the measure of the third angle from the total sum, we find that 2x = 180° - 78°, which results in 2x = 102°. Dividing this result by 2 yields x = 51°.

Explanation

The present value (PV) of a future sum is calculated using the formula PV = FV / (1 + r/n)ⁿᵗ, where FV is the future value, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years.

Given values are FV = 10,000, r = 0.12 (12%), n = 2 (semi-annual compounding), and t = 2 years.

First, calculate the periodic interest rate: i = r/n = 0.12 / 2 = 0.06. Next, calculate the total number of compounding periods: N = n × t = 2 × 2 = 4.

Substituting these into the formula: PV = 10000 / (1 + 0.06)⁴ PV = 10000 / (1.06)⁴ PV = 10000 / 1.26247696 PV = 7920.9366...

Rounding to two decimal places, the present value is Rs. 7,920.94.

Explanation

The volume V of a solid conical vessel is calculated using the formula V = ⅓ π r² h, where r is the base radius and h is the vertical height. Substituting the given values r = 7 cm and h = 15 cm, we have V = ⅓ × ²²⁄₇ × 7 × 7 × 15. By simplifying the expression, the volume becomes 22 × 7 × 5 = 770 cm³.

The weight of the vessel is determined by multiplying the total volume by the density of the iron. Given that the iron weighs 15 g/cm³, the weight in grams is 770 × 15 = 11550 g. Since there are 1000 grams in a kilogram, the weight of the vessel is 11550 / 1000 = 11.55 kg.

Explanation

The value of the expression p³ + q³ can be determined using the algebraic identity p³ + q³ = (p + q)³ - 3pq(p + q).

Given that p + q = 6 and pq = 4, substituting these values into the identity yields: p³ + q³ = (6)³ - 3(4)(6) p³ + q³ = 216 - 72 p³ + q³ = 144

The question asks for the value of 3(p³ + q³). By substituting the calculated value of p³ + q³: 3(144) = 432