SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

The given trigonometric expression is (2 tan² 30°)/(1 - tan² 30°) + sec² 45° - sec² 0° = x sec 60°. To solve for x, substitute the standard values for the trigonometric ratios: tan 30° = (1)/(√(3)), sec 45° = √2, sec 0° = 1, and sec 60° = 2.

First, calculate the value of tan² 30°: tan² 30° = ((1)/(√(3)))² = ⅓

Substitute these values into the left-hand side (LHS) of the equation: LHS = (2(1)/(3)){1 - ⅓} + (√2)² - (1)² LHS = (2)/(3){⅔} + 2 - 1 LHS = 1 + 2 - 1 = 2

Now, substitute the value for sec 60° into the right-hand side (RHS): RHS = x(2) = 2x

Equating LHS and RHS: 2 = 2x x = 1

Explanation

The relationship between the sum and difference of sine and cosine functions is given by the identity (sin a + cos a)² + (sin a - cos a)² = 2(sin² a + cos² a). Since sin² a + cos² a = 1, this simplifies to (sin a + cos a)² + (sin a - cos a)² = 2.

Given the expression tan Θ = (sin a - cos a)/(sin a + cos a), let S = sin a + cos a and D = sin a - cos a. We can write D = S tan Θ. Substituting this into the identity:

S² + (S tan Θ)² = 2 S² (1 + tan² Θ) = 2

Using the trigonometric identity 1 + tan² Θ = sec² Θ:

S² sec² Θ = 2 S² = (2)/(sec² Θ) = 2 cos² Θ

Taking the square root of both sides:

S = ± √2 cos Θ

Thus, sin a + cos a = ± √2 cos Θ.

Explanation

The given equation is 7sin²Θ + 3cos²Θ = 4. By splitting the first term, we can rewrite the equation as 4sin²Θ + 3sin²Θ + 3cos²Θ = 4. Factoring out the coefficient 3 from the last two terms gives 4sin²Θ + 3(sin²Θ + cos²Θ) = 4.

Using the fundamental trigonometric identity sin²Θ + cos²Θ = 1, the equation simplifies to 4sin²Θ + 3(1) = 4, which further reduces to 4sin²Θ = 1. Solving for sinΘ in the range 0° < Θ < 90°, we get sinΘ = √(1/4) = 1/2. This corresponds to Θ = 30°. Consequently, the value of tanΘ is tan 30° = 1 / √3. Alternatively, rearranging 4sin²Θ + 3cos²Θ = 4(sin²Θ + cos²Θ) leads directly to 3sin²Θ = cos²Θ, which implies tan²Θ = 1/3 and tanΘ = 1/√3.

Explanation

The expression is sec² 81°{1 + cot² 81°}. Using the trigonometric identity 1 + cot² θ = csc² θ, the denominator can be replaced to give sec² 81°{csc² 81°}. Expressing these in terms of sine and cosine results in 1/cos² 81°{1/sin² 81°}, which simplifies to sin² 81°{cos² 81°} = tan² 81°.

Since tan(90° - θ) = cot θ, we have tan 81° = tan(90° - 9°) = cot 9°. It is given that tan 9° = (p)/(q), so its reciprocal cot 9° = (q)/(p). Therefore, tan² 81° = (cot 9°)² = ((q)/(p))² = q²{p²}.

Explanation

The volume V of a right circular cone is given by the formula V = ⅓ π r² h, where r is the radius of the base and h is the height. If the radius is doubled (rₙₑ_w = 2r) and the height is doubled (hₙₑ_w = 2h), the new volume Vₙₑ_w is calculated as:

Vₙₑ_w = ⅓ π (2r)² (2h) Vₙₑ_w = ⅓ π (4r²) (2h) Vₙₑ_w = 8 × (⅓ π r² h) Vₙₑ_w = 8V

Mathematically, the volume is proportional to the square of the radius and the first power of the height (V ∝ r²h). By doubling both parameters, the volume increases by a factor of 2² × 2¹ = 4 × 2 = 8.

Explanation

The weight W of a sphere is given by the product of its volume V and its density ρ (weight per unit volume). The volume of a sphere with radius r is V = ⁴⁄₃ π r³. Therefore, the weight is proportional to the product of the cube of the radius and the density, expressed as W ∝ r³ · ρ.

Let W₁, W₂ be the weights, ₁, ₂ be the densities, and r₁, r₂ be the radii of the two spheres. Based on the given ratios: (W₁)/(W₂) = ⁸⁄₁₇ (₁)/(₂) = ²⁸⁹⁄₆₄

Using the relation (W₁)/(W₂) = (r₁³ · ₁)/(r₂³ · ₂), we can solve for the ratio of the radii: ⁸⁄₁₇ = ((r₁)/(r₂))³ · ²⁸⁹⁄₆₄ ((r₁)/(r₂))³ = ⁸⁄₁₇ · ⁶⁴⁄₂₈₉ ((r₁)/(r₂))³ = (8 · 64)/(17 · 289) = ⁵¹²⁄₄₉₁₃

Taking the cube root of both sides: (r₁)/(r₂) = ³√((8³)/(17³)) = ⁸⁄₁₇

Thus, the ratio of the radii of the two spheres is 8:17.

Explanation

The relationship between the marked price (MP) and the cost price (CP) when given a gain percentage and a discount percentage is expressed by the formula: MP / CP = (100 + Gain%) / (100 - Discount%)

By substituting the given values, where CP is 360, gain is 25%, and discount is 10%, we get: MP / 360 = (100 + 25) / (100 - 10) MP / 360 = 125 / 90

To find the marked price, multiply both sides by 360: MP = (125 / 90) × 360 MP = 125 × 4 MP = 500

Alternatively, the required selling price (SP) to achieve a 25% profit on a cost price of 360 is calculated as: SP = 360 × (1 + 25/100) = 360 × 1.25 = 450

This selling price represents the value after a 10% discount has been applied to the marked price. Therefore: SP = MP × (1 - 10/100) 450 = MP × 0.9 MP = 450 / 0.9 = 500

Explanation

The present value (PV) of a future amount (A) due in n years at a compound interest rate (r) is calculated using the formula PV = A / (1 + r)ⁿ. Here, the future amount A is 600, the interest rate r is 10 percent or 0.10, and the time n is 4 years.

Substituting these values into the formula: PV = 600 / (1 + 0.10)⁴ PV = 600 / (1.1)⁴ PV = 600 / 1.4641 PV = 409.808

The discount allowed by B to A is the difference between the future value and the present value: Discount = 600 - 409.808 = 190.192

To find the discount percent, we calculate the percentage reduction from the future value: Discount Percent = (Discount / Future Value) * 100 Discount Percent = (190.192 / 600) * 100 Discount Percent = 31.6986

Rounding to two decimal places, the discount percent is 31.69.

Explanation

The Marked Price (MP) is determined from the Selling Price (SP) and the discount percentage using the relation SP = MP × (100 - Discount%) / 100. Given SP = 1510 and a discount of 24 ½% or 24.5%, the equation is 1510 = MP × (100 - 24.5) / 100, which simplifies to 1510 = MP × 0.755. Solving for MP gives MP = 1510 / 0.755 = 2000.

The Cost Price (CP) is calculated using the loss percentage: SP = CP × (100 - Loss%) / 100. With a loss of 10%, we have 1510 = CP × (90 / 100), which results in CP = 15100 / 9.

If the item is sold without a discount, the new selling price is the Marked Price of 2000. The net profit is the difference between this price and the cost price: Profit = 2000 - 15100 / 9. Calculating this gives Profit = (18000 - 15100) / 9 = 2900 / 9. Expressing 2900 / 9 as a mixed fraction, we divide 2900 by 9 to get 322 with a remainder of 2, resulting in 322 ²⁄₉.

Explanation

Let the total amount be T = 3000. The problem states that A receives 1/3 as much as B and C together, which can be expressed as A = ⅓(B + C). This implies the ratio A : (B + C) = 1 : 3. The share of A is calculated as ¹⁄₁₊₃ × 3000 = ¼ × 3000 = 750. Similarly, it is given that B receives 2/3 as much as A and C together, which means B = ⅔(A + C), implying the ratio B : (A + C) = 2 : 3. The share of B is calculated as ²⁄₂₊₃ × 3000 = ⅖ × 3000 = 1200. The share of C is found by subtracting the shares of A and B from the total sum: C = T - (A + B) = 3000 - (750 + 1200) = 3000 - 1950 = 1050.

Explanation

To determine the divisibility of the expression 4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴, we factor out the lowest power of 4, which is 4⁶¹: 4⁶¹(4⁰ + 4¹ + 4² + 4³) Calculating the sum inside the parentheses: 1 + 4 + 16 + 64 = 85 The expression is simplified to: 4⁶¹ × 85 By prime factorization, 85 can be written as 5 × 17. Thus, the original expression is: 4⁶¹ × 5 × 17 Since 17 is a factor of the product, the entire expression is divisible by 17.

Explanation

The greatest 4-digit number is 9999. To find the least number that must be added to 9999 to make it exactly divisible by 307, we calculate the remainder when 9999 is divided by 307. Using the relation N = (d × q) + r: 9999 = (307 × 32) + 175 The remainder r is 175. The smallest number to be added to N to make it a multiple of d is given by the formula (d - r). Substituting the divisor and the remainder: 307 - 175 = 132 Adding this value to 9999 results in 10131, which is equal to 307 × 33.

Explanation

Let the capacity of the cistern be the LCM of 20 and 30, which is 60 units. The efficiency of filling pipe A is 60 / 20 = 3 units per minute, and the efficiency of emptying pipe B is 60 / 30 = -2 units per minute.

In a single cycle of 2 minutes (where A is open for the first minute and B for the second), the net work done is 3 - 2 = 1 unit. To avoid overcounting the time once the cistern is already full, we subtract the work of the final filling stroke of pipe A from the total capacity: 60 - 3 = 57 units.

The time taken to fill these 57 units at the cycle rate of 1 unit per 2 minutes is 57 × 2 = 114 minutes. At the end of the 114th minute, the cistern contains exactly 57 units. In the 115th minute, it is pipe A's turn to operate. Since pipe A fills 3 units per minute, it will fill the remaining 60 - 57 = 3 units in exactly 1 minute. The total time taken is 114 + 1 = 115 minutes.

Explanation

The rate of power consumption for a single machine is determined by dividing the amount consumed by the time taken, which is (k/5)/(t) = ᵏ⁄₅ₜ units per hour. The total consumption for multiple machines over a specific duration is the product of the rate of one machine, the total number of machines, and the time elapsed.

Using the formula: Total Consumption = Rate × Number of Machines × Time

Substituting the given values: Total Consumption = ᵏ⁄₅ₜ × 3 × 10 Total Consumption = ³⁰ᵏ⁄₅ₜ Total Consumption = ⁶ᵏ⁄ₜ

By simplifying the fraction, the constant factor 30 divided by 5 results in 6, leaving the final expression as 6k/t.

Explanation

When multiple cubes are melted to form a new cube, the total volume remains conserved. The volumes of the three cubes are 6³ = 216 cubic cm, 8³ = 512 cubic cm, and 1³ = 1 cubic cm. Summing these values gives the volume of the new cube as 216 + 512 + 1 = 729 cubic cm.

Let s be the side length of the new cube. Since the volume of a cube is s³, we have s³ = 729, which gives s = ³√(729) = 9 cm. The total surface area of a cube is given by the formula 6s². Substituting the side length of the new cube, the surface area is 6 × 9² = 6 × 81 = 486 sq.cm.

Explanation

Let the time taken to walk one way be W and the time taken to ride one way be R. Based on the problem, the time for walking one way and riding back is W + R = 37 minutes. The time for walking both ways is 2W = 55 minutes.

To find the time taken to ride both ways (2R), we can use the following relationship: 2 × (W + R) = 2W + 2R Substituting the known values: 2 × 37 = 55 + 2R 74 = 55 + 2R 2R = 74 - 55 2R = 19

Alternatively, since walking both ways takes 55 minutes, walking one way takes 55 / 2 = 27.5 minutes. Substituting this into the first equation, 27.5 + R = 37, which gives R = 9.5 minutes. Riding both ways thus takes 9.5 × 2 = 19 minutes.

Explanation

Simple Interest (SI) is the same for every year. Given the SI for 3 years is 3,000, the SI for 1 year is 3,000 / 3 = 1,000. Therefore, the SI for 2 years is 1,000 × 2 = 2,000.

The Compound Interest (CI) for 2 years is 2,050. The difference between the CI and SI for two years represents the interest earned on the first year's interest. This difference is 2,050 - 2,000 = 50.

To find the rate of interest (R), we calculate what percentage 50 is of the first year's interest (1,000): R = (50)/(1,000) × 100 = 5% per annum.

Now, using the formula for Simple Interest SI = (P × R × T)/(100), where SI for one year is 1,000 and T = 1: 1,000 = (P × 5 × 1)/(100) 1,000 = (5P)/(100) P = (1,000 × 100)/(5) = 20,000.

Explanation

Given the condition ab + bc + ca = 0, we can express the terms in the denominators as follows: Since -bc = ab + ca, the first denominator becomes a² - bc = a² + ab + ca = a(a + b + c). Similarly, since -ca = ab + bc, the second denominator becomes b² - ac = b² + ab + bc = b(a + b + c). And since -ab = bc + ca, the third denominator becomes c² - ab = c² + bc + ca = c(a + b + c).

Substituting these into the given expression: (1)/(a(a + b + c)) + (1)/(b(a + b + c)) + (1)/(c(a + b + c))

Factoring out (1)/(a + b + c) gives: (1)/(a + b + c) ( ¹⁄ₐ + (1)/(b) + (1)/(c) )

Taking the LCM inside the parentheses: (1)/(a + b + c) ( (bc + ac + ab)/(abc) )

Using the given condition ab + bc + ca = 0, the numerator of the term in parentheses becomes 0: (1)/(a + b + c) × (0)/(abc) = 0

Explanation

For the quadratic equation 2x² - 7x + 12 = 0, the sum of roots α + β = -b/a = -(-7)/2 = 7/2 and the product of roots α β = c/a = 12/2 = 6. The required expression α/β + β/α is simplified by taking the common denominator as (α² + β²) / α β. Using the algebraic identity α² + β² = (α + β)² - 2α β, we calculate the numerator: (7/2)² - 2(6) = 49/4 - 12 = 1/4. Dividing this numerator by the product of roots gives (1/4) / 6 = 1/24.

Explanation

For a quadratic expression of the form f(x) = ax² + bx + c, the maximum or minimum value occurs at the vertex of the parabola. If the coefficient of the squared term a is negative, the parabola opens downwards, and the vertex represents the greatest value of the expression.

In the given expression 2 - 3x - 4x², we can identify the coefficients as a = -4, b = -3, and c = 2. Since a < 0, the expression reaches its maximum value at x = -b / (2a).

Substituting the values into the formula: x = -(-3) / (2 × -4) x = 3 / -8 x = -3/8

Explanation

To make the expression x⁴ - 2x² + k a perfect square, it must conform to the algebraic identity (a - b)² = a² - 2ab + b². By comparing the given expression to this identity, we can set a² = x⁴, which implies a = x². The middle term of the identity is -2ab, which corresponds to the term -2x² in the given expression. Substituting a = x² into the middle term equation, we have -2(x²)b = -2x². Solving for b by dividing both sides by -2x² gives b = 1. The constant term k must be equal to b² for the expression to be a perfect square. Therefore, k = 1² = 1. When k = 1, the expression becomes x⁴ - 2x² + 1, which is equivalent to (x² - 1)².

Explanation

If (x - 1) and (x + 3) are factors of the quadratic expression x² + k₁x + k₂, then the expression must be identical to the product of these factors. Expanding the product of the given factors:

(x - 1)(x + 3) = x(x + 3) - 1(x + 3) = x² + 3x - x - 3 = x² + 2x - 3

By comparing the coefficients of the expanded form x² + 2x - 3 with the given expression x² + k₁x + k₂, we find that the coefficient of x corresponds to k₁ = 2 and the constant term corresponds to k₂ = -3.

This can also be verified using the Factor Theorem. If (x - 1) and (x + 3) are factors, then x = 1 and x = -3 are the zeros of the polynomial. Substituting these values into x² + k₁x + k₂ = 0 provides two equations: 1 + k₁ + k₂ = 0 and 9 - 3k₁ + k₂ = 0. Subtracting the second equation from the first yields 4k₁ - 8 = 0, which gives k₁ = 2. Substituting k₁ = 2 back into 1 + k₁ + k₂ = 0 results in 1 + 2 + k₂ = 0, giving k₂ = -3.

Explanation

Let the number of boys be 200 and the number of girls be 300, making the total number of students 500. If 25% of boys hold scholarships, then 75% do not. Similarly, if 30% of girls hold scholarships, then 70% do not.

Number of boys without scholarships = 200 × ⁷⁵⁄₁₀₀ = 150 Number of girls without scholarships = 300 × ⁷⁰⁄₁₀₀ = 210

The total number of students who are not scholarship holders is 150 + 210 = 360. To find the percentage, we divide this by the total student population and multiply by 100:

Percentage = ³⁶⁰⁄₅₀₀ × 100 = 72%

Explanation

The sum of the original 20 items is calculated using the formula Sum = Mean × Number of items, which gives 55 × 20 = 1100. The total value of the two items to be removed is 45 + 30 = 75. After removing these items, the sum of the remaining 18 items is 1100 - 75 = 1025. The new mean is found by dividing the remaining sum by the remaining number of items: Meanₙₑ_w = ¹⁰²⁵⁄₁₈. Performing the division, 1025 ÷ 18 ≈ 56.944, which approximates to 56.9.

Explanation

Let the cost price (CP) be 100. Since the profit is 20% of the cost price, the profit is 20. The selling price (SP) is the sum of the cost price and the profit, so SP = 100 + 20 = 120. To find the percentage of profit on the sale price, we use the formula Profit % on SP = (Profit / SP) × 100. Substituting the values, we get (20 / 120) × 100 = (1 / 6) × 100 = 50 / 3 = 16 ⅔%.

Explanation

The expression is calculated by evaluating the left-hand side and the right-hand side of the equation. First, 25% of 120 is equivalent to ¼ × 120, which equals 30. Next, 40% of 380 is calculated as 0.4 × 380, which equals 152. Summing these two values gives 30 + 152 = 182.

Setting up the full equation where x represents the unknown value, we have 182 = x × 637. Solving for x gives the fraction ¹⁸²⁄₆₃₇. By dividing both the numerator and the denominator by their common factor of 91, we simplify the fraction: 182 ÷ 91 = 2 and 637 ÷ 91 = 7. Thus, the value is ²⁄₇.

Explanation

The thief travels for 30 minutes (from 1:30 p.m. to 2:00 p.m.) before the theft is discovered. The distance covered by the thief during this head start is calculated as: Distance = Speed × Time = 40 km/hr × 0.5 hr = 20 km

The owner starts the chase at 2:00 p.m. at a speed of 50 km/hr. Since both the thief and the owner are moving in the same direction, their relative speed is: Relative Speed = 50 km/hr - 40 km/hr = 10 km/hr

The time taken by the owner to close the 20 km gap and overtake the thief is: Time = (Distance)/(Relative Speed) = (20 km)/(10 km/hr) = 2 hours

Adding these 2 hours to the owner's start time of 2:00 p.m., the owner will overtake the thief at 4:00 p.m.

Explanation

In Δ ABC, the condition AB = BC implies that the triangle is isosceles. By the property of isosceles triangles, the angles opposite to equal sides must be equal, so ∠ A = ∠ C. Given that ∠ A = (2x - 20)°, we have ∠ C = (2x - 20)° as well.

The sum of the interior angles of a triangle is always 180°. Applying this property: ∠ A + ∠ B + ∠ C = 180° (2x - 20) + x + (2x - 20) = 180 5x - 40 = 180 5x = 220 x = ²²⁰⁄₅ x = 44

Since ∠ B is defined as x°, the measure of ∠ B is 44°.

Explanation

For ∠ ABD = 45° to be a valid result in an isosceles triangle where D is a point on AC such that BD = DC, the triangle must be an isosceles right-angled triangle with the right angle at B.

In △ ABC, if we assume AB = BC and ∠ ABC = 90°: The base angles are ∠ BAC = ∠ BCA = 45°. The point D lies on AC such that BD = DC. In a right-angled triangle, the median to the hypotenuse is equal to half the hypotenuse (BD = AD = DC). This confirms that D is the midpoint of AC. In an isosceles triangle (AB = BC), the median drawn from the vertex (B) to the base (AC) also serves as the angle bisector of the vertex angle. Therefore, ∠ ABD = ½ ∠ ABC = 90°/2 = 45°.

This configuration satisfies the condition that the triangle is isosceles and BD = DC, yielding the value provided in the options.

Explanation

When two circles with radii r₁ and r₂ touch each other externally, the distance between their centers is the sum of their radii, r₁ + r₂. For three circles with radii r₁, r₂, and r₃ touching each other externally, the side lengths of the triangle formed by joining their centers are d₁₂ = r₁ + r₂, d₂₃ = r₂ + r₃, and d₃₁ = r₃ + r₁. The perimeter P of this triangle is the sum of these three sides:

P = (r₁ + r₂) + (r₂ + r₃) + (r₃ + r₁) = 2(r₁ + r₂ + r₃)

Substituting the given radii r₁ = 3.5 cm, r₂ = 4.5 cm, and r₃ = 5.5 cm into the formula:

P = 2(3.5 + 4.5 + 5.5) P = 2(13.5) P = 27 cm

Explanation

Using the trigonometric identity sec² Θ - tan² Θ = 1, we can factor this as (sec Θ + tan Θ)(sec Θ - tan Θ) = 1. Given that sec Θ + tan Θ = 5, we substitute this value to find sec Θ - tan Θ = 1/5.

To find tan Θ, we subtract the second equation from the first: (sec Θ + tan Θ) - (sec Θ - tan Θ) = 5 - 1/5 2 tan Θ = 24/5 tan Θ = 12/5

Now, we substitute the value of tan Θ into the required expression: (tan Θ + 1)/(tan Θ - 1) = (12/5 + 1)/(12/5 - 1) ((12 + 5)/5)/((12 - 5)/5) = (17/5)/(7/5) = 17/7

Explanation

The given expression is E = sec Θ + tan³ Θ csc Θ. By simplifying the second term using trigonometric identities, we have:

E = sec Θ + (sin³ Θ)/(cos³ Θ) · (1)/(sin Θ) E = sec Θ + (sin² Θ)/(cos³ Θ) E = sec Θ + (sin² Θ)/(cos² Θ) · (1)/(cos Θ) E = sec Θ + tan² Θ sec Θ

Factoring out sec Θ and applying the fundamental identity 1 + tan² Θ = sec² Θ:

E = sec Θ (1 + tan² Θ) E = sec Θ · sec² Θ = sec³ Θ E = (sec² Θ)³^/²

Using the identity sec² Θ = 1 + tan² Θ and substituting the given value tan² Θ = 1 - e²:

sec² Θ = 1 + (1 - e²) = 2 - e²

Substituting this back into the expression for E:

E = (2 - e²)³^/²

Explanation

For the interval 0° < Θ < 90°, the value of cos Θ lies strictly between 0 and 1, meaning 0 < cos Θ < 1. For any real number x in the range 0 < x < 1, the square of the number is always smaller than the number itself (x² < x). This occurs because multiplying the inequality x < 1 by the positive value x yields x² < x. Substituting x = cos Θ into this relation, we get cos² Θ < cos Θ, which is equivalent to cos Θ > cos² Θ.

Explanation

To solve for x, substitute the standard values of the trigonometric ratios into the given equation: x(sin 60°)tan² 30° - tan 45° = csc 60°cot 30° - sec² 45°.

The values are: sin 60° = √3/2 tan 30° = 1/√3 ⟹ tan² 30° = ⅓ tan 45° = 1 csc 60° = 2/√3 cot 30° = √3 sec 45° = √2 ⟹ sec² 45° = 2

Substituting these values into the equation: x · √3/2 · ⅓ - 1 = 2/√3 · √3 - 2

Simplify the right-hand side (RHS): RHS = 2 - 2 = 0

Simplify the left-hand side (LHS): LHS = x√3/6 - 1

Equating LHS and RHS: x√3/6 - 1 = 0 x√3/6 = 1 x = 6/√3

By rationalizing the denominator: x = 6√3/3 = 2√3

Explanation

To find the value of x + ¹⁄₂ₓ from the equation (5x)/(2x² + 5x + 1) = ⅓, begin by cross-multiplying to eliminate the fractions:

3(5x) = 1(2x² + 5x + 1)

15x = 2x² + 5x + 1

Subtract 5x from both sides of the equation to isolate the quadratic and constant terms:

10x = 2x² + 1

To obtain the required expression x + ¹⁄₂ₓ, divide every term in the simplified equation by 2x:

¹⁰ˣ⁄₂ₓ = (2x²)/(2x) + ¹⁄₂ₓ

5 = x + ¹⁄₂ₓ

Explanation

To find the value of a² - b², we first add and subtract the given equations to simplify the expressions for a and b.

Adding the two equations: (x³ + ³⁄ₓ) + (3x + (1)/(x³)) = 4(a³ + b³) + 4(a³ - b³) x³ + 3x + ³⁄ₓ + (1)/(x³) = 8a³ Recognizing the expansion of (x + ¹⁄ₓ)³ = x³ + 3x²(¹⁄ₓ) + 3x((1)/(x²)) + (1)/(x³), we get: (x + ¹⁄ₓ)³ = 8a³ Taking the cube root on both sides: x + ¹⁄ₓ = 2a

Subtracting the second equation from the first: (x³ + ³⁄ₓ) - (3x + (1)/(x³)) = 4(a³ + b³) - 4(a³ - b³) x³ - 3x + ³⁄ₓ - (1)/(x³) = 8b³ Recognizing the expansion of (x - ¹⁄ₓ)³ = x³ - 3x²(¹⁄ₓ) + 3x((1)/(x²)) - (1)/(x³), we get: (x - ¹⁄ₓ)³ = 8b³ Taking the cube root on both sides: x - ¹⁄ₓ = 2b

We now have a = ½(x + ¹⁄ₓ) and b = ½(x - ¹⁄ₓ). Squaring both terms to find a² - b²: a² - b² = [½(x + ¹⁄ₓ)]² - [½(x - ¹⁄ₓ)]² a² - b² = ¼ [(x + ¹⁄ₓ)² - (x - ¹⁄ₓ)²] Using the algebraic identity (m + n)² - (m - n)² = 4mn: a² - b² = ¼ [4 · x · ¹⁄ₓ] a² - b² = ¼ [4] a² - b² = 1

Explanation

The value is calculated using algebraic identities starting from the relation x - ¹⁄ₓ = 6. Squaring both sides of this equation yields (x - ¹⁄ₓ)² = 6², which simplifies to x² + (1)/(x²) - 2 = 36, or x² + (1)/(x²) = 38.

To find the value of x⁴ + (1)/(x⁴), square this result again: (x² + (1)/(x²))² = 38², which gives x⁴ + (1)/(x⁴) + 2 = 1444. Solving for the required expression, we get x⁴ + (1)/(x⁴) = 1444 - 2 = 1442.

This follows the general shortcut formula: if x - ¹⁄ₓ = k, then x⁴ + (1)/(x⁴) = (k² + 2)² - 2. Substituting k = 6 results in (6² + 2)² - 2 = 38² - 2 = 1444 - 2 = 1442.

Explanation

In any triangle, the centroid G is the point where the three medians intersect. A fundamental property of the centroid is that it divides each median in the ratio 2:1 from the vertex to the midpoint of the opposite side. Given that AD is the median from vertex A to side BC, the centroid G lies on AD such that AG:GD = 2:1. To find the ratio AG:AD, we express the total length of the median as AD = AG + GD. By assigning AG = 2k and GD = k for some constant k, the total length becomes AD = 2k + k = 3k. Therefore, the ratio AG:AD is 2k:3k, which simplifies to 2:3.

Explanation

In a right-angled triangle, the circumcentre always lies at the midpoint of the hypotenuse. This implies that the hypotenuse of the triangle is the diameter of its circumcircle. Therefore, the circumradius R is equal to half the length of the hypotenuse.

In Δ PQR, since ∠ RPQ = 90°, the side QR is the hypotenuse. Using the Pythagoras theorem:

QR = √(PR² + PQ²)

Substituting the given values PR = 6 cm and PQ = 8 cm:

QR = √(6² + 8²) = √(36 + 64) = √(100) = 10 cm

The radius of the circumcircle is given by the formula:

R = (Hypotenuse)/(2) = (QR)/(2)

R = ¹⁰⁄₂ = 5 cm

Explanation

Two angles are defined as supplementary if the sum of their measures is equal to 180°. Given that the angles are in the ratio 2:3, let the first angle be 2x and the second angle be 3x. According to the definition of supplementary angles:

2x + 3x = 180° 5x = 180° x = (180°)/(5) = 36°

By substituting the value of x back into the expressions for the angles, the first angle is 2 × 36° = 72° and the second angle is 3 × 36° = 108°.

Explanation

In any triangle, the centroid O divides the median AD in the ratio 2:1, measuring from the vertex to the midpoint of the opposite side. This mathematical property implies that the length of the segment AO is twice the length of the segment OD. Given that the length of AO is 10 cm, we can set up the ratio AO : OD = 2 : 1, which translates to the equation AO = 2 × OD. Substituting the known value, we have 10 = 2 × OD. Dividing both sides by 2 gives OD = 5 cm.

Explanation

In a parallelogram ABCD, the diagonal AC divides it into two triangles of equal area. Since the area of Δ ABC is given as 12 cm², the total area of the parallelogram ABCD = 2 × 12 = 24 cm².

Let the area of the parallelogram be S = 24. Using the properties of mid-points in a parallelogram:

Area of Δ ABP: Since P is the mid-point of BC, the area of Δ ABP = ½ × AB × BP sin(∠ B) = ½ × AB × ½ BC sin(∠ B) = ¼ S. Area(Δ ABP) = ¼ × 24 = 6 cm².

Area of Δ ADQ: Since Q is the mid-point of CD, the area of Δ ADQ = ½ × AD × DQ sin(∠ D) = ½ × AD × ½ CD sin(∠ D) = ¼ S. Area(Δ ADQ) = ¼ × 24 = 6 cm².

Area of Δ PCQ: Since P and Q are mid-points of BC and CD respectively, the area of Δ PCQ = ½ × PC × CQ sin(∠ C) = ½ × ½ BC × ½ CD sin(∠ C) = ⅛ S. Area(Δ PCQ) = ⅛ × 24 = 3 cm².

The area of Δ APQ is obtained by subtracting the areas of these three triangles from the total area of the parallelogram: Area(Δ APQ) = Area(ABCD) - [Area(Δ ABP) + Area(Δ ADQ) + Area(Δ PCQ)] Area(Δ APQ) = 24 - [6 + 6 + 3] = 24 - 15 = 9 cm².

Explanation

Given the equations x = a sec α cos β, y = b sec α sin β, and z = c tan α, we can rewrite them as ˣ⁄ₐ = sec α cos β, (y)/(b) = sec α sin β, and (z)/(c) = tan α. Substituting these into the required expression (x²)/(a²) + (y²)/(b²) - (z²)/(c²), we obtain the following:

(sec α cos β)² + (sec α sin β)² - (tan α)² sec² α cos² β + sec² α sin² β - tan² α

By factoring out sec² α from the first two terms, we get: sec² α (cos² β + sin² β) - tan² α

Using the fundamental trigonometric identity cos² θ + sin² θ = 1, the expression simplifies to: sec² α (1) - tan² α = sec² α - tan² α

Applying the identity sec² θ - tan² θ = 1, we find the final value to be 1.

Explanation

The given conditions are based on the trigonometric relation (cos² α)/(a²) + (sin² α)/(b²) = 1 and the ratio (sin α)/(sin β) = b. From the second equation, we can express sin α in terms of sin β: sin α = b sin β

Squaring both sides gives: sin² α = b² sin² β

Substituting this into the first equation, and using the identity cos² α = 1 - sin² α: (1 - sin² α)/(a²) + (sin² α)/(b²) = 1

Now, substitute sin² α = b² sin² β into the equation: (1 - b² sin² β)/(a²) + (b² sin² β)/(b²) = 1 (1 - b² sin² β)/(a²) + sin² β = 1

To solve for sin² β, multiply the entire equation by a²: 1 - b² sin² β + a² sin² β = a² sin² β (a² - b²) = a² - 1

Dividing by (a² - b²), we get the value: sin² β = (a² - 1)/(a² - b²)

Explanation

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the denominator sin² 30° + cos² 30° simplifies directly to 1.

Now, substituting the standard values for the trigonometric ratios in the numerator: cos 60° = 1/2, sec 30° = 2/√3, and tan 45° = 1.

The expression becomes: (1/2)² + 4(2/√3)² - (1)²{1} = 1/4 + 4(4/3) - 1 = 1/4 + 16/3 - 1

To add these fractions, find the common denominator, which is 12: = (3/12) + (64/12) - (12/12) = (3 + 64 - 12) / 12 = 55/12

Explanation

The equation 3ˣ - 3ˣ⁻¹ = 486 can be solved by factoring the left side. Using the property of exponents aᵐ - aᵐ⁻ⁿ = aᵐ⁻ⁿ(aⁿ - 1), we can rewrite the expression as follows:

3ˣ⁻¹(3¹ - 1) = 486

3ˣ⁻¹(3 - 1) = 486

3ˣ⁻¹(2) = 486

Dividing both sides by 2, we obtain:

3ˣ⁻¹ = 243

Since 243 can be expressed as a power of 3, specifically 3⁵ = 3 × 3 × 3 × 3 × 3 = 243, we can set the exponents equal to each other:

3ˣ⁻¹ = 3⁵

x - 1 = 5

x = 6

Explanation

When two individuals A and B can complete a piece of work in x and y days respectively, the time taken for them to finish the work together is calculated using the formula: Time = (x × y)/(x + y)

Substituting the given values where x = 4 days and y = 12 days: Time = (4 × 12)/(4 + 12) Time = ⁴⁸⁄₁₆ Time = 3 days

Alternatively, using the efficiency method, let the total work be the Least Common Multiple (LCM) of 4 and 12, which is 12 units. The daily work efficiency of A is 12 / 4 = 3 units, and the daily work efficiency of B is 12 / 12 = 1 unit. Their combined efficiency when working together is 3 + 1 = 4 units per day. To complete the total work of 12 units at a rate of 4 units per day, they will take 12 / 4 = 3 days.

Explanation

The time taken by A to complete the entire work is 10 × 4 = 40 days. Similarly, the time taken by B to complete the entire work is 20 × 3 = 60 days. To find the time taken by both A and B working together, we use the combined work formula T = (xy)/(x+y), where x and y represent the days required by each individual. Substituting the values, we get T = (40 × 60)/(40 + 60) = ²⁴⁰⁰⁄₁₀₀ = 24 days. This can also be solved by calculating their combined one-day work efficiency, which is ¹⁄₄₀ + ¹⁄₆₀ = ³⁺²⁄₁₂₀ = ⁵⁄₁₂₀ = ¹⁄₂₄, meaning they complete the task in 24 days.

Explanation

Let the total work be 1 unit. Since A and B together can finish the work in 30 days, their combined work rate is 1/30 of the work per day. In 20 days, they completed 20 × (1/30) = 2/3 of the total work.

The remaining work to be completed is 1 - 2/3 = 1/3. The problem states that A finished this remaining 1/3 of the work in 20 days. To find the time A takes to complete the entire work alone, we divide the total work by A's daily work rate. If A completes 1/3 of the work in 20 days, A's one-day work rate is (1/3) / 20 = 1/60. Therefore, A alone can finish the entire work in 1 / (1/60) = 60 days.

Explanation

The circumference of the circular wire is calculated using the formula C = π d. Given the diameter d = 112 cm and using π = ²²⁄₇, the circumference is C = ²²⁄₇ × 112 = 22 × 16 = 352 cm. When the wire is bent into a rectangle, its perimeter remains equal to the circumference of the original circle. Let the dimensions of the rectangle be 9x and 7x based on the given ratio. The perimeter P of the rectangle is 2(length + breadth) = 2(9x + 7x) = 32x. Setting the perimeter equal to the circumference, we have 32x = 352, which simplifies to x = 11. The length of the smaller side is 7x = 7 × 11 = 77 cm.