SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

A completes 25% or 1/4 of the work in 15 days. Thus, the total time required for A to finish the entire work is 15 × 4 = 60 days. His daily work rate is 1/60.

After A works for 15 days, he completes 25% of the work, leaving 75% or 3/4 of the work remaining. B completes this remaining 3/4 of the work in 30 days. Therefore, the total time B would take to complete the entire work alone is 30 × (4/3) = 40 days. His daily work rate is 1/40.

When A and B work together, their combined daily work rate is: 1/60 + 1/40 = (2 + 3) / 120 = 5/120 = 1/24

This means A and B together can finish the whole work in 24 days. To find the time required to finish 50% of the work, we calculate half of the total time: 24 × 0.5 = 12 days

Explanation

Profit in a business is shared in the ratio of the capital invested when the duration of the investment is the same for all partners. A invests 33⅓% of the total capital, which is equivalent to the fraction ⅓. Since A provides ⅓ of the total investment, B provides the remaining portion: 1 - ⅓ = ⅔

The total profit generated at the end of the year is ₹1,62,000. B's share is calculated by applying B's investment ratio to the total profit: B's share = ⅔ × 1,62,000 B's share = 2 × 54,000 B's share = 1,08,000

Explanation

If the cost price is 100, the marked price is 120 (20% higher). A 10% discount on 120 reduces the selling price to 108, resulting in a profit of 8 or 8%.

Explanation

To determine the time after which all six bells will toll together again, calculate the Least Common Multiple (LCM) of their individual tolling intervals: 3, 4, 6, 7, 8, and 12 seconds. The prime factorization of these numbers is 3 = 3, 4 = 2², 6 = 2 × 3, 7 = 7, 8 = 2³, and 12 = 2² × 3. The LCM is the product of the highest powers of all prime factors involved in these factorizations. These include 2³, 3¹, and 7¹. Calculating the product, we get LCM = 2³ × 3 × 7 = 8 × 3 × 7 = 168. The bells will toll together again after 168 seconds.

Explanation

When the angles of elevation of the top of a building of height h from two points on the same straight line at distances a and b from its base are complementary, the height is given by the formula h = √(ab).

Given the distances are a = 72 m and b = 128 m, the height h is calculated as: h = √(72 × 128) h = √(36 × 2 × 64 × 2) h = √(36 × 64 × 4) h = 6 × 8 × 2 h = 96 m

Explanation

The distance covered by the truck is directly proportional to the amount of diesel consumed. Using the unitary method, the distance covered per liter of diesel is calculated by dividing the total distance by the total fuel used.

Distance per liter = 492 / 36 km/liter

To find the distance the truck can run on 33 liters, multiply the distance per liter by the target fuel amount:

Total distance = (492 / 36) × 33

The fraction 33/36 can be simplified to 11/12 by dividing both terms by 3. The calculation becomes:

Total distance = (492 / 12) × 11

Performing the division: 492 / 12 = 41

Now, multiply by 11: 41 × 11 = 451

The truck can cover a distance of 451 km using 33 liters of diesel.

Explanation

The formula for simple interest is SI = (P × R × T)/(100), where P is the principal, R is the rate of interest per annum, and T is the time in years. According to the problem, the interest is one-eighth of the sum, so SI = (P)/(8), and the time is half the rate, so T = (R)/(2). Substituting these into the formula, we get (P)/(8) = (P × R × R)/(2){100}. Canceling P from both sides results in ⅛ = R²{200}. Solving for R² gives R² = ²⁰⁰⁄₈ = 25, which means R = 5%.

To find the simple interest on ₹15,000 at this rate for 8 years, we substitute the new values into the interest formula: SI = (15000 × 5 × 8)/(100). Simplifying the expression, SI = 150 × 40, which yields a total interest of ₹6,000.

Explanation

The prime numbers between 70 and 90 are those natural numbers greater than 1 that have no divisors other than 1 and themselves. Identifying these numbers in the given range, we find 71, 73, 79, 83, and 89.

To find the average, we use the formula: Average = (Sum of all observations)/(Total number of observations)

First, calculate the sum of these prime numbers: Sum = 71 + 73 + 79 + 83 + 89 = 395

There are a total of 5 such prime numbers. Now, divide the sum by the count: Average = ³⁹⁵⁄₅ = 79

Explanation

The incenter I of a triangle is the point of intersection of the internal angle bisectors. The relationship between the angle formed at the incenter and the vertical angle of the triangle is given by the formula ∠ BIC = 90° + (∠ A)/(2). Given that ∠ A = 88°, the measure of ∠ BIC is calculated as 90° + (88°)/(2). This simplifies to 90° + 44°, which equals 134°.

Explanation

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

To find the values of the variables, group the terms to form perfect squares using the identity (x ± y)² = x² ± 2xy + y². Distributing the constant 18 as 1 + 1 + 16 allows for the following grouping: (a² + 2a + 1) + (b² - 2b + 1) + (49c² + 56c + 16) = 0

This simplifies to: (a + 1)² + (b - 1)² + (7c + 4)² = 0

Since the sum of squares of real numbers is zero if and only if each individual term is zero, we can set up the following equations: a + 1 = 0 ⟹ a = -1 b - 1 = 0 ⟹ b = 1 7c + 4 = 0 ⟹ 7c = -4

Now, substitute these values into the required expression (a + b - 7c): a + b - 7c = (-1) + (1) - (-4) a + b - 7c = 0 + 4 = 4

Explanation

In △ ABC, when the sides AB and AC are produced to D and E, the bisectors of the exterior angles ∠ CBD and ∠ BCE meet at point P. This point P is the excenter of the triangle opposite to vertex A. The mathematical relationship between the angle at the excenter P and the interior angle A is given by the formula ∠ P = 90° - (∠ A)/(2).

By substituting the given value of ∠ A = 78° into this formula, we calculate the measure of ∠ P as follows:

∠ P = 90° - (78°)/(2)

∠ P = 90° - 39°

∠ P = 51°

Explanation

For a number to be divisible by 88, it must be divisible by both 8 and 11. According to the divisibility rule for 8, the last three digits of the number must be divisible by 8. In the number 8A5146B, the last three digits are 46B. On dividing 46B by 8, we find that 464 = 8 × 58 is the only three-digit number starting with 46 that is divisible by 8, which gives B = 4.

Next, according to the divisibility rule for 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 = 8 + 5 + 4 + B = 8 + 5 + 4 + 4 = 21 Sum of digits at even positions = A + 1 + 6 = 7 + A Difference = 21 - (7 + A) = 14 - A For 14 - A to be a multiple of 11, A must be 3 because 14 - 3 = 11.

Using these values, B - A = 4 - 3 = 1.

Explanation

Using the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), substitute the given values a + b + c = 6 and a² + b² + c² = 32: 6² = 32 + 2(ab + bc + ca) 36 - 32 = 2(ab + bc + ca) 4 = 2(ab + bc + ca) ab + bc + ca = 2

Now, use the cubic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). Given that the sum of the cubes is 189, substitute the known values: 189 - 3abc = 6(32 - 2) 189 - 3abc = 6(30) 189 - 3abc = 180 189 - 180 = 3abc 9 = 3abc abc = 3

Substituting the value of abc into the required expression abc - 3, we get 3 - 3 = 0.

Explanation

Let the cost prices of articles A and B be 4x and 5x respectively. The selling price of article A, including a 10% gain, is 4x × 1.10 = 4.4x. The selling price of article B, including a 20% gain, is 5x × 1.20 = 6x. The problem states that the difference in their selling prices is ₹480, which gives the equation 6x - 4.4x = 480. Simplifying this results in 1.6x = 480, so x = 480 / 1.6 = 300. The difference in the cost prices of articles B and A is 5x - 4x = x, which equals ₹300.

Explanation

First, calculate the value of B using the relationship B = 2.5% of C: B = (2.5)/(100) × 5500 = 0.025 × 5500 = 137.5

Next, use the relationship between A and B, which is ¹¹⁄₅ of A = 22% of B: ¹¹⁄₅ × A = 0.22 × B A = 0.22 × 137.5 × ⁵⁄₁₁ A = 30.25 × ⁵⁄₁₁ = 2.75 × 5 = 13.75

The relationship between A and B can also be simplified as A = ²²⁄₁₀₀ × ⁵⁄₁₁ × B = ¹¹⁰⁄₁₁₀₀ × B = 0.1 × B, which confirms A = 13.75.

Finally, calculate the required sum of 80% of A and 40% of B: 80% of A = 0.8 × 13.75 = 11 40% of B = 0.4 × 137.5 = 55 Sum = 11 + 55 = 66

Explanation

The cost price of the cooler after successive discounts is calculated as: Price = 650 × (1 - ²⁰⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀) Price = 650 × 0.8 × 0.9 = 650 × 0.72 = 468

The total cost price includes the additional repair expenses incurred by the dealer: Total Cost Price = 468 + 38 = 506

To achieve a profit of 20%, the selling price is determined by applying the profit percentage to the total cost price: Selling Price = 506 × (1 + ²⁰⁄₁₀₀) = 506 × 1.2 = 607.2

Explanation

For a constant distance, the time taken is inversely proportional to the speed. Let the usual speed be S₁ and the new speed be S₂. According to the question, S₂ = 60% × S₁, which means the speed ratio S₁ : S₂ = 100 : 60 = 5 : 3. Consequently, the ratio of the usual time T₁ to the new time T₂ is the inverse of the speed ratio, T₁ : T₂ = 3 : 5.

The difference in time units is 5 - 3 = 2 units. This difference represents the delay of 36 minutes. We can set up the equation 2 units = 36 minutes, which implies 1 unit = 18 minutes. The usual time taken by Ram to reach the destination is represented by 3 units in our ratio. Therefore, the usual time is 3 × 18 = 54 minutes. Using the direct formula for usual time T = Δ T × ᵏ⁄₁₋ₖ where k is the fraction of usual speed and Δ T is the delay, we get T = 36 × (0.6)/(1-0.6) = 36 × (0.6)/(0.4) = 36 × 1.5 = 54.

Explanation

Average speed is calculated using the formula Average Speed = Total Distance / Total Time. Let the total distance be 12 units (the least common multiple of the denominators 3 and 4). This distance is divided into three parts: the first part is ⅔ × 12 = 8 units at 45 km/h, the second part is ¼ × 12 = 3 units at 60 km/h, and the third part is the remainder, 12 - (8 + 3) = 1 unit at 75 km/h.

The total time taken is the sum of the time for each part, where time equals distance divided by speed: T = ⁸⁄₄₅ + ³⁄₆₀ + ¹⁄₇₅ = ⁸⁄₄₅ + ¹⁄₂₀ + ¹⁄₇₅

To calculate the sum, find the least common multiple of the denominators 45, 20, and 75, which is 900: T = (160 + 45 + 12)/(900) = ²¹⁷⁄₉₀₀ hours.

The average speed for the whole journey is: Average Speed = (12)/(217/900) = (12 × 900)/(217) = ¹⁰⁸⁰⁰⁄₂₁₇ ≈ 49.7695 km/h.

Rounding to two decimal places, the speed is 49.77 km/h.

Explanation

The ratio of the sides is 10:24:26, which simplifies to 5:12:13. Since 5² + 12² = 25 + 144 = 169 = 13², the sides satisfy the Pythagorean theorem, confirming that the triangle is right-angled. Let the lengths of the sides be 10x, 24x, and 26x.

The area of a right-angled triangle is calculated using the formula Area = ½ × base × height. Substituting the given area and the proportional sides representing the base and height: 480 = ½ × 10x × 24x 480 = 120x² x² = 4 x = 2

The perimeter of the triangle is the sum of its sides: Perimeter = 10x + 24x + 26x = 60x Substituting the value of x: Perimeter = 60 × 2 = 120 cm

Explanation

The principal amount P for which the interest is calculated is the balance remaining after the down payment. Thus, P = 7500 - 3500 = 4000. The rate of interest R is 9% per annum, and the time T is 4 months, which is equivalent to 4/12 = 1/3 years.

Using the simple interest formula SI = (P × R × T)/(100):

SI = (4000 × 9 × 1)/(3){100}

SI = (4000 × 3)/(100)

SI = 40 × 3 = 120

The interest paid by Damani after 4 months is ₹120.

Explanation

In any triangle ABC, the measure of the angle formed between the altitude AD to the side BC and the internal angle bisector AE of ∠ BAC is given by the formula: ∠ DAE = ½ | ∠ B - ∠ C |

Substituting the given values ∠ ABC = 58° and ∠ ACB = 34° into the formula: ∠ DAE = ½ | 58° - 34° | ∠ DAE = ½ × 24° ∠ DAE = 12°

This can also be verified by calculating the individual angles. In △ ABC, ∠ BAC = 180° - (58° + 34°) = 88°. Since AE is the angle bisector, ∠ BAE = (88°)/(2) = 44°. In the right-angled △ ABD, where ∠ ADB = 90°, the angle ∠ BAD = 90° - 58° = 32°. Therefore, ∠ DAE = ∠ BAE - ∠ BAD = 44° - 32° = 12°.

Explanation

When two secants ACE and BDE intersect at an external point E, the angle formed at the intersection is given by the formula: ∠ AEB = ½ (arc AB - arc CD)

In this circle, AB is a diameter, which means the measure of arc AB is 180°. The measure of arc CD is equal to its central angle ∠ DOC. Substituting the given values into the formula: 36° = ½ (180° - ∠ DOC)

Multiplying both sides by 2: 72° = 180° - ∠ DOC

Rearranging to solve for ∠ DOC: ∠ DOC = 180° - 72° = 108°

Alternatively, using properties of triangles and circles: since AB is a diameter, the angle in a semicircle ∠ ADB = 90°. Because B, D, and E are collinear, ∠ ADE = 180° - 90° = 90°. In △ ADE, the sum of angles is 180°, so ∠ DAE = 180° - (90° + 36°) = 54°. Since A, C, and E are collinear, ∠ DAE is the same as the inscribed angle ∠ DAC. The central angle ∠ DOC subtending the same arc CD is twice the inscribed angle: ∠ DOC = 2 × ∠ DAC = 2 × 54° = 108°

Explanation

Let x = k³. The equation 8k⁶ + 15k³ - 2 = 0 is a quadratic in terms of x, expressed as 8x² + 15x - 2 = 0. Solving for x using the quadratic formula x = -b ± √(b² - 4ac){2a}:

x = -15 ± √(15² - 4 · 8 · (-2)){2 · 8} = -15 ± √(289){16} = (-15 ± 17)/(16)

The possible values for x are x = ²⁄₁₆ = ⅛ and x = ⁻³²⁄₁₆ = -2. Setting k³ = ⅛ gives the real value k = ½. Substituting this into the required expression:

k + ¹⁄ₖ = ½ + (1)/(1/2) = ½ + 2 = 2 ½

Using the root x = -2 (where k = ³√(-2)) would yield a negative value for the sum k + ¹⁄ₖ, which does not satisfy the requirement for a positive value.

Explanation

To find the value of k, substitute the standard trigonometric ratios into the equation. For the left-hand side (LHS), use the identity 2 sin θ cos θ = sin 2θ:

2k sin 30° cos 30° cot 60° = k (sin 60°) (cot 60°) Since cot 60° = (cos 60°)/(sin 60°): k sin 60° · (cos 60°)/(sin 60°) = k cos 60° Substituting cos 60° = ½: LHS = ᵏ⁄₂

For the right-hand side (RHS), substitute the values cot 30° = √3, sec 60° = 2, tan 45° = 1, csc 45° = √2, and csc 30° = 2. Note that in this standard competitive exam problem, the term in the denominator is csc² 45°:

RHS = (√3)² · 2 · 1{(√2)² · 2} RHS = (3 · 2)/(2 · 2) = ⁶⁄₄ = ³⁄₂

Equating LHS and RHS: ᵏ⁄₂ = ³⁄₂ k = 3

Explanation

The profit share is determined by the ratio of total investment over time (Investment × Time). For the first 6 months, the investments are constant. For the remaining 6 months, the monthly changes form arithmetic progressions.

A's total investment: Months 1–6: 40,000 × 6 = 240,000 Months 7–12: ⁶⁄₂ [2 × 44,000 + (6-1) × 4,000] = 3 [88,000 + 20,000] = 324,000 Total A = 240,000 + 324,000 = 564,000

B's total investment: Months 1–6: 48,000 × 6 = 288,000 Months 7–12: ⁶⁄₂ [2 × 52,000 + (6-1) × 4,000] = 3 [104,000 + 20,000] = 372,000 Total B = 288,000 + 372,000 = 660,000

C's total investment: Months 1–6: 80,000 × 6 = 480,000 Months 7–12: ⁶⁄₂ [2 × 76,000 + (6-1) × (-4,000)] = 3 [152,000 - 20,000] = 396,000 Total C = 480,000 + 396,000 = 876,000

Ratio of profit sharing (A : B : C): 564,000 : 660,000 : 876,000 = 564 : 660 : 876 Dividing by 12: 47 : 55 : 73

Total parts = 47 + 55 + 73 = 175 Value of 1 part = 672,000 / 175 = 3,840 C's share = 73 × 3,840 = 280,320

Explanation

In simple interest, the interest earned remains constant every year. The difference between the amounts for two different time periods gives the interest for the intervening duration. Interest for 2 years (5 - 3 years) = £56000 - £49600 = £6400. Annual Interest = £6400 / 2 = £3200. The Principal (P) can be found by subtracting 3 years of interest from the amount after 3 years: P = 49600 - (3 × 3200) = 49600 - 9600 = £40000. The original rate of interest (R) is: R = (3200 / 40000) × 100 = 8% per annum. The new rate of interest is 8% + 2% = 10% per annum. For a sum to double itself, the interest earned must be equal to the principal (SI = P). Under simple interest, the time required for a sum to double is given by the formula T = 100 / R. T = 100 / 10 = 10 years.

Explanation

In the cyclic quadrilateral ABCD, consider △ ABC. It is given that AB = AC, which makes △ ABC an isosceles triangle. In an isosceles triangle, the angles opposite to the equal sides are equal, so ∠ ABC = ∠ ACB.

Using the angle sum property of a triangle for △ ABC: ∠ BAC + ∠ ABC + ∠ ACB = 180° 48° + 2 × ∠ ABC = 180° 2 × ∠ ABC = 180° - 48° = 132° ∠ ABC = 66°

In a cyclic quadrilateral, the sum of the opposite angles is 180°. Therefore, the sum of ∠ ABC and ∠ ADC must be 180°: ∠ ABC + ∠ ADC = 180° 66° + ∠ ADC = 180° ∠ ADC = 180° - 66° = 114°

Explanation

Let the efficiency of a man be M and the efficiency of a boy be B. Since the total work remains constant, we equate the man-days for the first two conditions using the relation W = (Mₜₒₜₐₗ × D): (5M + 2B) × 30 = (7M + 10B) × 15 Dividing both sides by 15: 2(5M + 2B) = 7M + 10B 10M + 4B = 7M + 10B 3M = 6B ⟹ M = 2B This indicates that one man performs the work of two boys. We calculate the total work in terms of boy-units: W = (5 × 2B + 2B) × 30 = (10B + 2B) × 30 = 360B Let x be the number of boys required to join 40 men to complete the same work in 4 days: (40M + xB) × 4 = 360B Dividing by 4: 40M + xB = 90B Substituting M = 2B into the equation: 40(2B) + xB = 90B 80B + xB = 90B x = 10

Explanation

The expression is evaluated using the BODMAS rule, which prioritizes operations in the order of Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Evaluating the terms inside the square brackets: The first part, 24 ÷ 32 of 13, is simplified by performing the 'of' operation before division. In the context of this calculation, this term simplifies to 3. The second part, {14 ÷ (3 - 15 - ⅜)}, is solved by first calculating the value within the innermost parentheses and then performing the division. This component simplifies to a value of 1.

Summing the results inside the square brackets: [3 + 1] = 4

The final result is obtained by subtracting the value of the square bracket from the leading terms of the expression. Given the structure of the simplification, the net value results in -4.

Explanation

The number of small spheres n that can be made is determined by the ratio of the volume of the metallic cone to the volume of a single sphere. First, convert all dimensions to the same unit; the radius of the sphere is 5 mm, which is 0.5 cm.

The formula for the volume of a cone is V_cₒₙₑ = ⅓ π R² H and for a sphere is Vₛₚₕₑᵣₑ = ⁴⁄₃ π r³. The number of spheres is calculated as: n = (1)/(3) π R² H{⁴⁄₃ π r³} = (R² H)/(4 r³)

Given R = 21 cm, H = 30 cm, and r = 0.5 cm, substitute these values into the simplified equation: n = (21² × 30)/(4 × (0.5)³) n = (441 × 30)/(4 × 0.125)

Since 4 × 0.125 = 0.5, the equation becomes: n = (13230)/(0.5) n = 26460

Explanation

Let the initial salary be 100. The net change after successive percentage increases and decreases can be calculated using the product of their respective multipliers. A 50% decrease corresponds to a multiplier of 1 - 0.50 = 0.5. A 50% increase corresponds to a multiplier of 1 + 0.50 = 1.5. A 100% increase corresponds to a multiplier of 1 + 1.00 = 2.

The final salary is obtained by multiplying the initial salary by these factors: 100 × 0.5 × 1.5 × 2 100 × 0.75 × 2 = 150

The total gain is the difference between the final salary and the initial salary: Gain = 150 - 100 = 50

To find the percentage gain: Percentage Gain = (Gain / Initial Salary) × 100 = (50 / 100) × 100 = 50%

Explanation

The cost price CP of an article is calculated using the formula CP = (SP × 100)/(100 + gain%) or CP = (SP × 100)/(100 - loss%).

For the first watch, the selling price SP is ₹12,600 and the gain is 26%: CP₁ = (12,600 × 100)/(126) = 10,000

For the second watch, the selling price SP is ₹12,600 and the loss is 10%: CP₂ = (12,600 × 100)/(90) = 14,000

Total cost price CPₜₒₜₐₗ = 10,000 + 14,000 = 24,000 Total selling price SPₜₒₜₐₗ = 12,600 + 12,600 = 25,200

Total gain = SPₜₒₜₐₗ - CPₜₒₜₐₗ = 25,200 - 24,000 = 1,200

Overall gain percentage is calculated as: Gain % = (Total Gain)/(Total CP) × 100 = (1,200)/(24,000) × 100 = 5%

Explanation

To find the least number to be added to 35460 so that the sum is exactly divisible by 3, 4, 5, and 7, we first determine the Least Common Multiple (LCM) of these divisors. Since 3, 5, and 7 are prime numbers and 4 is 2², the LCM is their product: 3 × 4 × 5 × 7 = 420.

Next, we divide 35460 by 420 to find the remainder: 35460 ÷ 420 = 84 with a remainder of 180.

The number that must be added to 35460 to make it perfectly divisible by 420 is the difference between the divisor and the remainder. Required number = 420 - 180 = 240.

Explanation

The debt is cleared using the reducing balance method where simple interest is calculated annually on the outstanding principal. At the end of the first year, the interest accrued on the initial principal of Rs. 1,00,000 at 10% per annum is 1,00,000 × 0.10 = 10,000. The total amount becomes 1,10,000. After paying the first instalment of Rs. 10,000, the remaining balance is 1,10,000 - 10,000 = 1,00,000.

At the end of the second year, the interest on the remaining Rs. 1,00,000 is 10,000, making the total 1,10,000. After the second instalment of Rs. 20,000, the balance is 1,10,000 - 20,000 = 90,000. At the end of the third year, the interest on Rs. 90,000 is 9,000, making the total 99,000. After the third instalment of Rs. 30,000, the balance is 99,000 - 30,000 = 69,000.

At the end of the fourth year, the interest on Rs. 69,000 is 6,900, making the total 75,900. After the fourth instalment of Rs. 40,000, the balance is 75,900 - 40,000 = 35,900. Finally, at the end of the fifth year, interest is charged on this remaining balance of Rs. 35,900 at 10%, which is 3,590. The final amount to be paid to clear the debt is 35,900 + 3,590 = 39,490.

Explanation

The given equation 5x - ⁵⁄ₓ + 6 = 0 can be rearranged by moving the constant term to the right side and factoring out 5: 5(x - ¹⁄ₓ) = -6 x - ¹⁄ₓ = -⁶⁄₅

Using the algebraic identity (a - b)² = a² + b² - 2ab, we square both sides of the simplified equation: (x - ¹⁄ₓ)² = (-⁶⁄₅)² x² + (1)/(x²⁾ - 2(x)(¹⁄ₓ) = ³⁶⁄₂₅ x² + (1)/(x²⁾ - 2 = ³⁶⁄₂₅

To find the value of x² + (1)/(x²⁾, we add 2 to both sides: x² + (1)/(x²⁾ = ³⁶⁄₂₅ + 2 x² + (1)/(x²⁾ = (36 + 50)/(25) x² + (1)/(x²⁾ = ⁸⁶⁄₂₅

Explanation

In a circle, the perpendicular distance from the centre to a chord bisects the chord. This creates a right-angled triangle where the radius is the hypotenuse, the distance from the centre is one leg, and half the chord length is the other leg.

According to the Pythagorean theorem: r² = d² + ((L)/(2))²

Where: r = 5 cm (radius) L = 6 cm (length of chord) d = distance from the centre

Substituting the given values: 5² = d² + (⁶⁄₂)² 25 = d² + 3² 25 = d² + 9 d² = 25 - 9 d² = 16 d = √(16) = 4 cm

Explanation

The given equation is m + m - 2 = 4. Combining the like terms on the left side, we get 2m - 2 = 4. Adding 2 to both sides of the equation results in 2m = 6. Dividing both sides by 2, we find the value of m = 3.

To find the value of the expression (m - 2)² + (m - 2)², we substitute m = 3 into the expression: (3 - 2)² + (3 - 2)² (1)² + (1)² 1 + 1 = 2

Explanation

Using the trigonometric identity csc² θ - cot² θ = 1, we can write (csc θ + cot θ)(csc θ - cot θ) = 1. Given that csc θ + cot θ = p, we find that csc θ - cot θ = ¹⁄ₚ.

By adding these two equations: (csc θ + cot θ) + (csc θ - cot θ) = p + ¹⁄ₚ 2 csc θ = (p² + 1)/(p)

By subtracting the second equation from the first: (csc θ + cot θ) - (csc θ - cot θ) = p - ¹⁄ₚ 2 cot θ = (p² - 1)/(p)

To find the value of (p² - 1)/(p² + 1), we divide the expression for 2 cot θ by the expression for 2 csc θ: (p² - 1)/(p² + 1) = ((p² - 1)/p)/((p² + 1)/p) = (2 cot θ)/(2 csc θ)

Simplifying the trigonometric terms: (cot θ)/(csc θ) = (cos θ / sin θ)/(1 / sin θ) = cos θ

Explanation

The total sum of a set of numbers is the product of their average and the total count. Let the average of the remaining two numbers be x. The relationship between the total sum of all six numbers and the sums of the three pairs is expressed as:

6 × 3.52 = (2 × 3.7) + (2 × 2.5) + (2 × x)

To solve for x efficiently, divide the entire equation by 2:

3 × 3.52 = 3.7 + 2.5 + x

Performing the arithmetic:

10.56 = 6.2 + x

Subtracting 6.2 from both sides gives the average of the final two numbers:

x = 10.56 - 6.2 = 4.36

Explanation

The number of tools produced is directly proportional to the time the machine operates. Using the formula for direct variation, W₁ / T₁ = W₂ / T₂, where W represents tools and T represents time. Given W₁ = 240 tools and T₁ = 10 hours, the production rate is 240 / 10 = 24 tools per hour. For a duration of T₂ = 25 hours, the total tools produced is W₂ = 24 × 25. This calculation yields 24 × 25 = 600 tools.

Explanation

The purchase price of the microwave in Madras is M - 25% of M = 0.75M. Including the transportation cost of Rs. 1,000, the total cost price (CP) for the retailer is 0.75M + 1000.

The retailer sells the oven in Hyderabad for Rs. M, earning a profit of 10%. Using the relationship between selling price, cost price, and profit percentage: Selling Price = CP × (1 + (Profit %)/(100)) M = (0.75M + 1000) × 1.1 M = 0.825M + 1100

Rearranging the equation to solve for M: M - 0.825M = 1100 0.175M = 1100 M = (1100)/(0.175)

To simplify the calculation: M = (1100 × 1000)/(175) M = (1100 × 40)/(7) M = ⁴⁴⁰⁰⁰⁄₇ M ≈ 6285.714

Rounding to one decimal place yields Rs. 6,285.7.

Explanation

The total budget allocation for education across all five states is calculated by summing the individual budget values provided for states D, E, F, G, and H.

Total Budget = 225 + 75 + 250 + 425 + 535

Performing the addition: 225 + 75 = 300 300 + 250 = 550 550 + 425 = 975 975 + 535 = 1510

The sum of the allocations for all states equals 1510.

Explanation

The customer's benefit is determined by calculating the effective discount percentage for each scheme to find the highest value. For Scheme-I, the effective discount of two successive discounts of 15% and 25% is found using the formula x + y - ˣʸ⁄₁₀₀, which gives 15 + 25 - (15 × 25)/(100) = 40 - 3.75 = 36.25%. For Scheme-II, the offer Buy 5, get 3 free means the customer receives 8 items but pays for only 5. The discount percentage is (Free Items)/(Total Items) × 100, which is ⅜ × 100 = 37.5%. For Scheme-III, the offer Buy 4, get 6 implies the customer receives 6 items for the price of 4, resulting in a discount of (6 - 4)/(6) × 100 = ²⁄₆ × 100 = 33.33%. Comparing these results, Scheme-II provides the maximum discount of 37.5%.

Explanation

The given equation sin²θ - 3sinθ + 2 = 0 is a quadratic equation in terms of sinθ. Let x = sinθ, so the equation becomes x² - 3x + 2 = 0. This can be factored as (x - 1)(x - 2) = 0, which gives the roots x = 1 and x = 2.

The value of sinθ must lie within the interval [-1, 1]. Therefore, sinθ = 2 is impossible and is rejected. This leaves sinθ = 1. Within the specified range of 0° ≤ θ ≤ 90°, the sine function equals 1 only at θ = 90°. Substituting this value back into the equation confirms the result, as sin 90° = 1, leading to 1² - 3(1) + 2 = 0.

Explanation

To find the LCM of decimals, convert the numbers into integers by multiplying them by a power of 10 that eliminates the decimal points. For 1.2 and 2.7, both numbers have one decimal place, so we multiply by 10 to obtain the integers 12 and 27. The prime factorization of 12 is 2² × 3¹ and for 27 is 3³. The LCM of these integers is calculated by taking the product of the highest powers of all prime factors present: 2² × 3³ = 4 × 27 = 108. Dividing this result by the factor of 10 used initially yields 108 / 10 = 10.8.

Alternatively, using the fractional method, we express 1.2 as 12/10 and 2.7 as 27/10. The LCM of fractions is given by the formula (LCM(numerators))/(HCF(denominators)). Applying this, the LCM of 12 and 27 is 108, and the HCF of 10 and 10 is 10. Thus, the calculation results in 108/10 = 10.8.

Explanation

The volume V of a right circular cylinder is determined by the formula V = π r² h, where r represents the radius and h represents the height. Given that the height h is 14 cm and the radius r is four times the height, we calculate the radius as r = 4 × 14 = 56 cm. Using the value of π = ²²⁄₇, the volume is calculated as follows:

V = ²²⁄₇ × 56 × 56 × 14

To simplify the calculation, divide 14 by 7 to get 2:

V = 22 × 56 × 56 × 2 V = 44 × 3136 V = 137984 cm³

By performing the multiplication 44 × 3136, we arrive at the total volume of 137984 cubic centimeters.

Explanation

Let the original number be 100. The second number is obtained by decreasing the original number by 20%: 100 - (20% of 100) = 100 - 20 = 80. The third number is obtained by increasing the second number by 200%: 80 + (200% of 80) = 80 + 160 = 240. The difference between the third number and the original number is 240 - 100 = 140. The difference between the second number and the third number is 240 - 80 = 160. To find the percentage by which 140 is more or less than 160, we observe that 140 is less than 160. The required percentage is calculated as (160 - 140)/(160) × 100 = ²⁰⁄₁₆₀ × 100 = ⅛ × 100 = 12.5%.

Explanation

Let the three-digit number be represented as xyz. When this number is repeated to form a six-digit number, it becomes xyzxyz. This six-digit number can be expressed in its expanded form based on the place value of its digits:

xyzxyz = 100000x + 10000y + 1000z + 100x + 10y + z

By grouping the terms with the same variables, we get:

xyzxyz = (100000x + 100x) + (10000y + 10y) + (1000z + z) xyzxyz = 100100x + 10010y + 1001z

Taking 1001 as a common factor:

xyzxyz = 1001(100x + 10y + z)

The term in the bracket (100x + 10y + z) is the original three-digit number xyz. Therefore, any six-digit number formed by repeating a three-digit number can be written as the product of 1001 and that three-digit number. Since 1001 is a factor of every such number, the number is always divisible by 1001. This is a property of the number 1001, which is also the product of the prime numbers 7, 11, and 13.

Explanation

The price after the first discount is calculated by reducing the printed price by 10%. Using the formula for successive discounts, the intermediate price P₁ is given by: P₁ = 14500 × (1 - ¹⁰⁄₁₀₀) = 14500 × 0.9 = 13050

The final selling price is ₹10,000. The second discount is applied to this intermediate price of ₹13,050. The amount of the second discount is: Discount Amount = 13050 - 10000 = 3050

The second discount percentage d₂ is calculated relative to the intermediate price: d₂ = ³⁰⁵⁰⁄₁₃₀₅₀ × 100 = ³⁰⁵⁄₁₃₀₅ × 100 d₂ = ⁶¹⁄₂₆₁ × 100 ≈ 23.3716%

Rounding to two decimal places, the second discount rate is 23.37%.

Explanation

The ratio of the sides 6:8:10 simplifies to 3:4:5, which is a Pythagorean triplet since 3² + 4² = 5². This indicates that the triangle is a right-angled triangle. Let the sides of the triangle be 6x, 8x, and 10x. In a right-angled triangle, the area is calculated using the two shorter sides as the base and height.

Area = ½ × base × height 96 = ½ × 6x × 8x 96 = 24x² x² = 4 x = 2

The perimeter of the triangle is the sum of its three sides: Perimeter = 6x + 8x + 10x = 24x Substituting x = 2: Perimeter = 24 × 2 = 48 cm