SSC CGL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

Let the total work be the Least Common Multiple (LCM) of the given time periods (12, 20, and 15), which is 60 units.

The daily efficiency of the pairs is calculated as follows: Efficiency of (A + B) = 60 / 12 = 5 units/day Efficiency of (B + C) = 60 / 20 = 3 units/day Efficiency of (C + A) = 60 / 15 = 4 units/day

Summing these efficiencies gives 2(A + B + C) = 5 + 3 + 4 = 12 units/day. To find the combined efficiency of A, B, and C working together, we divide by 2, resulting in 12 / 2 = 6 units/day.

The total time required for A, B, and C to complete the work together is: Time = Total Work / Combined Efficiency = 60 / 6 = 10 days.

Explanation

Let the total work be 1 unit. Since A and B together can complete the work in 3 days, their combined efficiency is 1/3 units per day.

In the first 2 days, A and B work together and complete 2 × (1/3) = 2/3 of the total work. The remaining work is 1 - 2/3 = 1/3 of the total work.

After B leaves, A completes this remaining 1/3 work in 2 days. Thus, the efficiency of A is (1/3) / 2 = 1/6 units per day. To find the efficiency of B, we subtract A's efficiency from the combined efficiency: 1/3 - 1/6 = 1/6 units per day.

The time taken by B alone to complete the work is the total work divided by B's efficiency, which is 1 / (1/6) = 6 days.

Explanation

Let the efficiency of B be 100%. Since A does 20% less work than B in the same amount of time, the efficiency of A is 80%. The ratio of efficiency of A to B is 80 : 100, which simplifies to 4 : 5. Time taken is inversely proportional to efficiency when the total work is constant, meaning the ratio of time taken by A to the time taken by B is 5 : 4.

Given that A takes 15/2 hours or 7.5 hours to complete the work, we can calculate the time for B by setting up the ratio T_A / T_B = E_B / E_A. Substituting the known values gives 7.5 / T_B = 5 / 4. Solving for T_B, we get T_B = (7.5 × 4) / 5 = 30 / 5 = 6 hours.

Explanation

To find a rational number between 3/4 and 3/8, we first express both fractions with a common denominator. The least common multiple of 4 and 8 is 8, but to match the denominator in the correct option, we can use 16 as the common denominator.

Converting the given fractions: 3/8 = (3 × 2) / (8 × 2) = 6/16 3/4 = (3 × 4) / (4 × 4) = 12/16

A rational number lies between 3/8 and 3/4 if its value is greater than 6/16 and less than 12/16. The fraction 9/16 satisfies this condition because 6/16 < 9/16 < 12/16.

Alternatively, using decimal equivalents, 3/8 = 0.375 and 3/4 = 0.75. The value of 9/16 is 0.5625. Since 0.375 < 0.5625 < 0.75, the rational number 9/16 lies between the two given numbers.

Explanation

Two numbers are defined as coprime if their highest common factor (HCF) is 1. The relationship between any two numbers and their common multiples and factors is governed by the mathematical formula: Product of two numbers = LCM × HCF In this case, the product of the two numbers is given as 117. Since the numbers are coprime, their HCF is 1. Substituting these known values into the equation: 117 = LCM × 1 By simplifying the expression, the LCM is found to be 117. This property holds true for all sets of coprime numbers, where the least common multiple is always equal to the product of the numbers themselves because there are no common prime factors to divide out.

Explanation

Let the side of the square be s. The diameter of the smaller circle is equal to the side of the square, so d₁ = s. The diameter of the larger circle is equal to the diagonal of the square, which is given by the formula d₂ = s√2.

The area of a circle A with diameter d is calculated using the formula A = (π d²)/(4). Since π/4 is a constant, the ratio of the areas of two circles is equal to the ratio of the squares of their diameters.

Areaₛₘₐₗₗₑᵣ{Areaₗₐᵣ_gₑᵣ} = (d₁²)/(d₂²) = (s²)/((s√(2))²) = (s²)/(2s²) = ½

Thus, the ratio of the areas of the smaller circle and the larger circle is 1:2.

Explanation

The total surface area of a sphere is given by the formula A = 4π r². According to the problem, 4π r² = 8π. Dividing both sides by 4π yields r² = 2, which means the radius r = √2.

The volume of a sphere is calculated using the formula V = ⁴⁄₃π r³. By substituting the value of r into this formula, we get: V = ⁴⁄₃π (√2)³ V = ⁴⁄₃π (2√2) V = 8√2{3}π

Thus, the volume of the sphere is 8√2{3}π square units.

Explanation

The volume of water in a conical flask with base radius r and height h is determined by the formula V = ⅓π r²h. When this volume of water is transferred to a cylindrical flask, the volume of the water remains constant.

The volume of water in the cylindrical flask with base radius mr and unknown height H is given by the formula V = π(mr)²H. Setting the two volumes equal to each other, we get:

⅓π r²h = π m²r²H

By dividing both sides of the equation by π r², the expression simplifies to:

⅓h = m²H

To find the height H of the water in the cylindrical flask, we rearrange the formula:

H = (h)/(3m²⁾

Explanation

The selling price of a dozen (12 pairs) of socks after the discount is calculated using the formula SP = MP × (1 - (D)/(100)), where MP is the marked price and D is the discount percentage. Substituting the given values, SP = 180 × (1 - ²⁰⁄₁₀₀) = 180 × 0.8 = 144 rupees.

Since 12 pairs of socks cost 144 rupees, the price of a single pair of socks is ¹⁴⁴⁄₁₂ = 12 rupees. The number of pairs that can be purchased for 48 rupees is determined by dividing the total amount available by the price per pair: ⁴⁸⁄₁₂ = 4 pairs.

Explanation

The amount of discount allowed on an item is the difference between its marked price and its selling price. Using the formula Discount = Marked Price - Selling Price, we get Discount = 12000 - 10500 = 1500.

The rate of discount is calculated as a percentage of the marked price using the formula Discount Rate = ((Discount)/(Marked Price)) × 100. Substituting the values into the formula gives Rate = (¹⁵⁰⁰⁄₁₂₀₀₀) × 100. Simplifying the fraction, we have ¹⁵⁄₁₂₀ × 100, which reduces to ⅛ × 100. Calculating this yields 12.5%.

Explanation

The given equation is (3 / 4)³ (4 / 3)⁻⁷ = (3 / 4)²ˣ. Using the law of exponents (a/b)⁻ⁿ = (b/a)ⁿ, the term (4/3)⁻⁷ can be rewritten as (3/4)⁷. Substituting this back into the equation yields (3 / 4)³ (3 / 4)⁷ = (3 / 4)²ˣ. Applying the product rule for exponents aᵐ · aⁿ = aᵐ⁺ⁿ, the left-hand side becomes (3 / 4)³⁺⁷, which simplifies to (3 / 4)¹⁰. Since the bases on both sides of the equation are equal, the exponents must also be equal, leading to the linear equation 10 = 2x. Dividing both sides by 2 results in x = 5.

Explanation

Let the original three numbers be x, 2x, and 3x based on the ratio 1:2:3. When 5 is added to each of these numbers, the new values become x+5, 2x+5, and 3x+5, which are in the ratio 2:3:4.

Setting up a proportion using the first two terms:

ˣ⁺⁵⁄₂ₓ₊₅ = ⅔

Cross-multiplying to solve for x:

3(x+5) = 2(2x+5) 3x + 15 = 4x + 10 x = 5

The original numbers are x, 2x, and 3x. Substituting x = 5 yields:

1 × 5 = 5 2 × 5 = 10 3 × 5 = 15

Alternatively, observe that each part of the ratio 1:2:3 increases by exactly 1 unit to become 2:3:4 (2-1=1, 3-2=1, and 4-3=1). Since adding 5 to each number results in an increase of 1 ratio unit, then 1 unit is equal to 5. Thus, the numbers are 1 × 5 = 5, 2 × 5 = 10, and 3 × 5 = 15.

Explanation

The given equation a² + b² + c² + 3 = 2(a - b - c) can be rearranged by bringing all terms to one side:

a² - 2a + b² + 2b + c² + 2c + 3 = 0

The constant 3 can be split into 1 + 1 + 1 to complete the squares for each variable:

(a² - 2a + 1) + (b² + 2b + 1) + (c² + 2c + 1) = 0

Using the algebraic identity (x ± y)² = x² ± 2xy + y², the expression simplifies to:

(a - 1)² + (b + 1)² + (c + 1)² = 0

For the sum of squares of real numbers to be zero, each individual term must independently equal zero. This yields:

a - 1 = 0 ⟹ a = 1 b + 1 = 0 ⟹ b = -1 c + 1 = 0 ⟹ c = -1

Substituting these values into the required expression 2a - b + c:

2(1) - (-1) + (-1) = 2 + 1 - 1 = 2

Explanation

Let the cost of one cow be C and the cost of one goat be G. Based on the information provided, we can form two linear equations. The first purchase is 3 cows and 8 goats for 47,200, which gives the equation: 3C + 8G = 47,200

The second scenario states that 8 cows and 3 goats cost 53,000 more than the first purchase. Therefore, the total cost for the second scenario is 47,200 + 53,000 = 100,200. This gives the second equation: 8C + 3G = 100,200

To solve for C efficiently, we can use the sum and difference of these equations. Adding the two equations: 11C + 11G = 147,400 C + G = 13,400

Subtracting the first equation from the second equation: 5C - 5G = 53,000 C - G = 10,600

Now, adding these two simplified equations: (C + G) + (C - G) = 13,400 + 10,600 2C = 24,000 C = 12,000

Explanation

In a circle, a perpendicular line segment drawn from the center to a chord bisects that chord. This construction creates a right-angled triangle where the radius r is the hypotenuse, the distance from the center d is one leg, and half the length of the chord L/2 is the other leg. The radius can be calculated using the Pythagorean theorem:

r² = d² + (L/2)²

Substituting the given values L = 30 cm and d = 8 cm:

r² = 8² + (30/2)² r² = 8² + 15² r² = 64 + 225 r² = 289 r = √(289) = 17

This calculation corresponds to the standard Pythagorean triplet (8, 15, 17).

Explanation

The number of digits in the square root of a number with n digits is given by the formula ⌈ n/2 ⌉. If n is even, the square root has n/2 digits; if n is odd, the square root has (n+1)/2 digits. The number 62478078 contains 8 digits. Since n = 8 is an even number, the number of digits in its square root is 8/2 = 4. This calculation aligns with the mathematical principle that the square of any k-digit number results in a product that has either 2k-1 or 2k digits. Given that 62478078 has 8 digits, its square root must be a 4-digit integer because 1000² = 1,000,000 (7 digits) and 9999² = 99,980,001 (8 digits).

Explanation

Let the side of the square be s. The in-circle of a square touches the midpoints of its four sides, making its diameter equal to the side length of the square. Thus, the in-radius r is half of the side, which is r = ˢ⁄₂. The circum-circle of a square passes through its four vertices, making its diameter equal to the diagonal of the square. Using the Pythagorean theorem, the diagonal of a square with side s is s√2. Therefore, the circum-radius R is half of the diagonal, which is R = s√2{2}. To find the ratio of the in-radius to the circum-radius, we calculate r : R = ˢ⁄₂ : s√2{2}. Simplifying this expression by canceling the common terms s and 2 yields the ratio 1 : √2.

Explanation

The relationship between the ages of the outgoing and incoming members of a group and the change in the average age is given by the formula: Age of Retired Teacher = Age of New Teacher + (Number of Teachers × Decrease in Average Age).

Substituting the given values into the formula, where the number of teachers is 10, the age of the new teacher is 25 years, and the decrease in average age is 3 years:

Age of Retired Teacher = 25 + (10 × 3) Age of Retired Teacher = 25 + 30 Age of Retired Teacher = 55 years

This happens because the total sum of the ages of the 10 teachers must have decreased by 10 × 3 = 30 years to result in an average reduction of 3 years. Since the only change in the group was the replacement of the retired teacher with a 25-year-old teacher, the retired teacher must be 30 years older than the new teacher.

Explanation

The expression is a product of tangent functions where the angles range from 1° to 89°. Using the trigonometric identity tan(90° - θ) = cot θ, we can pair terms from the beginning and the end of the series. For instance, tan 89° can be written as tan(90° - 1°) = cot 1°.

Because tan θ · cot θ = 1, the product of the first and last terms is tan 1° · tan 89° = tan 1° · cot 1° = 1. Following this pattern, tan 2° · tan 88° = 1, tan 3° · tan 87° = 1, and so on, until the pair tan 44° · tan 46° = 1. This accounts for 44 pairs, each resulting in 1. The only term without a pair is the middle term, tan 45°, which is also equal to 1. Multiplying all these results together gives 1 × 1 × … × 1 = 1.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) is given by the formula CP/MP = (100 - Discount%)/(100 + Profit%). Substituting the given values:

CP/MP = (100 - 10)/(100 + 8) = 90/108 = 5/6

If no discount is allowed, the Selling Price (SP) becomes equal to the Marked Price (MP). To find the gain percent when SP = MP, we use the ratio derived above where CP = 5 units and MP = 6 units.

The profit percentage is calculated as: Profit% = [(SP - CP)/CP] × 100 Profit% = [(6 - 5)/5] × 100 Profit% = (1/5) × 100 = 20%

Alternatively, using the actual values, SP = 480 × 0.9 = 432. Since this represents an 8% gain, CP = 432/1.08 = 400. If sold at MP of 480, the profit is 480 - 400 = 80, and the gain percent is (80/400) × 100 = 20%.

Explanation

The equation of a straight line parallel to the x-axis is always of the form y = k, where k represents the constant ordinate for every point on that line. The value of k is determined by the vertical distance and direction from the x-axis. Since the line is positioned 3 units below the x-axis, the y-coordinate for any point on the line is -3. Substituting this value into the general form gives the equation y = -3.

Explanation

In the quadrilateral OPSQ, OP and OQ are radii and SP and SQ are tangents. Since the radius is perpendicular to the tangent at the point of contact, ∠ OPS = ∠ OQS = 90°. The sum of angles in a quadrilateral is 360°, so the angle subtended by the chord PQ at the center O is ∠ POQ = 180° - ∠ PSQ = 180° - 20° = 160°.

The angle subtended by the major arc at the center is the reflex ∠ POQ = 360° - 160° = 200°. According to the theorem that the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle, the angle ∠ PRQ subtended by the major arc at point R on the minor arc is half of the reflex angle at the center.

∠ PRQ = ½ × (reflex ∠ POQ) = ½ × 200° = 100°

Explanation

The value of a machine depreciating at a constant annual rate is calculated using the formula V = P(1 - ʳ⁄₁₀₀)ⁿ, where P is the present value, r is the rate of depreciation, and n is the number of years. Substituting the given values P = 50,000, r = 10, and n = 2, the calculation is 50,000 × (1 - ¹⁰⁄₁₀₀)². This simplifies to 50,000 × (0.9)², which equals 50,000 × 0.81 = 40,500. Alternatively, using successive reduction, the value after the first year is 50,000 - 10%(50,000) = 45,000, and the value after the second year is 45,000 - 10%(45,000) = 40,500.

Explanation

In △ ABC, since AB = AC, the triangle is isosceles, meaning the angles opposite to these sides are equal, so ∠ ABC = ∠ ACB. The sum of the interior angles of any triangle is 180°. Given ∠ BAC = 40°, the sum of the remaining two angles is 180° - 40° = 140°. Since ∠ ABC = ∠ ACB, each of these angles is 140° / 2 = 70°. The external angle at vertex B is supplementary to the internal angle ABC because they form a linear pair on a straight line. Thus, the external angle at B = 180° - 70° = 110°. Alternatively, using the exterior angle theorem, the external angle at B is the sum of the interior opposite angles, ∠ BAC + ∠ ACB = 40° + 70° = 110°.

Explanation

Let the total work be represented by the least common multiple of the individual completion times. The LCM of 18 and 5 is 90 units. Therefore, A's work rate is 90 / 18 = 5 units per day, and B's work rate is 90 / 5 = 18 units per day. B worked for 2 days, completing 18 × 2 = 36 units of work. The remaining work to be completed by A is 90 - 36 = 54 units. Since A completes 5 units per day, the time required for A to finish the remaining work is 54 / 5 = 10.8 days.

Explanation

The speed of the man in still water is u = 3 km/hr and the velocity of the stream is v = 2 km/hr. When swimming upstream, the effective speed is the difference between the man's speed and the stream's speed, which is u - v = 3 - 2 = 1 km/hr. When swimming downstream, the effective speed is the sum of the man's speed and the stream's speed, which is u + v = 3 + 2 = 5 km/hr.

The total time taken for a round trip of distance d = 10 km is given by the sum of the time taken upstream and the time taken downstream: T = (d)/(u - v) + (d)/(u + v) T = ¹⁰⁄₁ + ¹⁰⁄₅ T = 10 + 2 = 12 hours

Explanation

From the given efficiency ratio, 10 men = 20 women = 40 children, we establish that 1 man = 4 children and 1 woman = 2 children. The total work can be calculated in terms of child-months as 40 × 7 = 280 units. The target is to complete half of this work, which is ²⁸⁰⁄₂ = 140 units.

The combined rate of 5 men, 5 women, and 5 children expressed in children units is 5(4) + 5(2) + 5 = 20 + 10 + 5 = 35 units per month. Using the work formula Time = (Work)/(Rate), the time required is:

Time = ¹⁴⁰⁄₃₅ = 4 months

Explanation

The expression is a product of cosine functions for every integer degree from 1° to 179°. The sequence of terms can be written as: cos 1° · cos 2° · cos 3° … cos 89° · cos 90° · cos 91° … cos 179° In any product of terms, if one of the factors is zero, the entire product is zero. In this specific sequence, the term cos 90° is present because the angles increase by 1° increments starting from 1° and ending at 179°. By trigonometric identity, the value of cos 90° is exactly 0. Therefore, substituting this value into the product gives: (cos 1° · cos 2° … cos 89°) · 0 · (cos 91° … cos 179°) = 0

Explanation

The algebraic identity for the cube of a binomial is (a - b)³ = a³ - b³ - 3ab(a - b). By substituting a = p and b = 2q, the expansion is (p - 2q)³ = p³ - (2q)³ - 3(p)(2q)(p - 2q), which simplifies to p³ - 8q³ - 6pq(p - 2q). Given the condition p - 2q = 4, we substitute this value into the equation to get 4³ = p³ - 8q³ - 6pq(4). This calculation leads to 64 = p³ - 8q³ - 24pq. Subtracting 64 from both sides of the equation results in p³ - 8q³ - 24pq - 64 = 0. Alternatively, using the method of substitution by choosing values that satisfy p - 2q = 4, such as q = 0 and p = 4, we can evaluate the target expression directly. Substituting these values into p³ - 8q³ - 24pq - 64 yields 4³ - 8(0)³ - 24(4)(0) - 64, which evaluates to 64 - 0 - 0 - 64 = 0.

Explanation

Given the equation sin θ + csc θ = 2, we can square both sides to find the required value. Using the algebraic identity (a + b)² = a² + b² + 2ab, we get:

(sin θ + csc θ)² = 2²

sin²θ + csc²θ + 2(sin θ)(csc θ) = 4

Since csc θ = 1/sin θ, the product sin θ · csc θ = 1. Substituting this into the equation:

sin²θ + csc²θ + 2(1) = 4

sin²θ + csc²θ = 4 - 2

sin²θ + csc²θ = 2

This result is consistent with the property that if x + ¹⁄ₓ = 2 for a positive real number x, then x must be 1. Here, sin θ = 1 and csc θ = 1, so any power sinⁿθ + cscⁿθ will also equal 2.

Explanation

Let the prices of the school bag and the shoe be represented by 7x and 5x respectively, based on the given ratio of 7:5. The problem states that the price of the school bag is Rs. 200 more than the shoe. This difference in terms of the ratio is:

7x - 5x = 200 2x = 200 x = 100

The price of the shoe is represented by 5x. Substituting the value of x into this expression:

Price of shoe = 5 × 100 = 500

Using the unitary method, the difference between the ratio parts is 7 - 5 = 2 units. These 2 units correspond to a value of Rs. 200. Therefore, 1 unit is equal to Rs. 100. The price of the shoe, which is 5 units, is calculated as 5 × 100 = 500.

Explanation

The selling price after two successive discounts of d₁% and d₂% is calculated using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)). Given the marked price MP = 80 and successive discounts of 5% each, the first discount reduces the price to 80 × 0.95 = 76. The second 5% discount is then applied to this new price, resulting in 76 × 0.95 = 72.2.

Alternatively, the effective single discount for two successive discounts x and y is given by (x + y - ˣʸ⁄₁₀₀)%. For two 5% discounts, the net discount is (5 + 5 - (5 × 5)/(100))% = 9.75%. The selling price is then 80 × (1 - 0.0975) = 80 × 0.9025 = 72.2.

Explanation

To determine which successive discount is better for the customer, we calculate the final price after applying each set of discounts to an initial value of 100. For a set of successive discounts d₁, d₂, d₃, the final price P is given by the formula P = 100 × (1 - d₁) × (1 - d₂) × (1 - d₃).

For sequence (a) with discounts of 20%, 15%, and 10%, the final price calculation is 100 × (1 - 0.20) × (1 - 0.15) × (1 - 0.10) = 100 × 0.80 × 0.85 × 0.90 = 61.2. This results in a total effective discount of 100 - 61.2 = 38.8%.

For sequence (b) with discounts of 25%, 12%, and 8%, the final price calculation is 100 × (1 - 0.25) × (1 - 0.12) × (1 - 0.08) = 100 × 0.75 × 0.88 × 0.92. Simplifying the first part, 100 × 0.75 × 0.88 = 75 × 0.88 = 66. Then, 66 × 0.92 = 60.72. This results in a total effective discount of 100 - 60.72 = 39.28%. Because 39.28% is greater than 38.8%, sequence (b) provides a higher overall discount and is better for the customer.

Explanation

In an isosceles triangle ABC where AB = AC, the angles opposite the equal sides are equal, thus ∠ C = ∠ B = 35°. The sum of the interior angles of triangle ABC is 180°, so the vertex angle ∠ BAC = 180° - (∠ B + ∠ C) = 180° - (35° + 35°) = 110°.

In an isosceles triangle, the median AD drawn to the base BC also acts as the angle bisector of the vertex angle ∠ BAC. Therefore, the measure of ∠ BAD is exactly half of the vertex angle ∠ BAC. Calculating this gives ∠ BAD = 110° / 2 = 55°. Alternatively, since the median to the base of an isosceles triangle is also the altitude, △ ABD is a right-angled triangle with ∠ ADB = 90°. In △ ABD, ∠ BAD = 180° - (90° + 35°) = 180° - 125° = 55°.

Explanation

The relationship between radians and degrees is defined by the identity π radians = 180°. Therefore, 1 radian = (180)/(π) degrees. Using the given value π = ²²⁄₇, the calculation is as follows:

1 radian = (180)/(22/7) = (180 × 7)/(22) = (90 × 7)/(11) = ⁶³⁰⁄₁₁ degrees

Dividing 630 by 11 gives 57 degrees with a remainder of 3. To convert the remaining ³⁄₁₁ degrees into minutes, we multiply by 60 (since 1° = 60^′):

³⁄₁₁ × 60 = ¹⁸⁰⁄₁₁ = 16 ⁴⁄₁₁ minutes

To convert the remaining ⁴⁄₁₁ minutes into seconds, we multiply by 60 (since 1^′ = 60^′^′):

⁴⁄₁₁ × 60 = ²⁴⁰⁄₁₁ ≈ 21.81 seconds

Rounding the seconds to the nearest whole number gives 22^′^′. Combining these parts, the measure of 1 radian is approximately 57° 16^′ 22^′^′.

Explanation

The speed of the train in metres per second is found by multiplying the speed in km/hr by ⁵⁄₁₈. v = 36 × ⁵⁄₁₈ = 10 m/s

When a train crosses a standing man, the distance covered is equal to the length of the train (Lₜ). Using the formula Distance = Speed × Time: Lₜ = 10 m/s × 10 s = 100 m

To cross a platform, the train must travel a total distance equal to the sum of its own length and the length of the platform (Lₚ). Given Lₚ = 55 m, the total distance is: D = Lₜ + Lₚ = 100 + 55 = 155 m

The time required to cross the platform is calculated as: t = (Total Distance)/(Speed) = ¹⁵⁵⁄₁₀ = 15.5 seconds

Converting the decimal to a fraction, 15.5 seconds is equal to 15 ½ seconds.

Explanation

A quadratic expression of the form ax² + bx + c is a perfect square if its discriminant D = b² - 4ac is equal to zero. In the given expression n² - tn + 1/4, the coefficients are a = 1, b = -t, and c = 1/4.

Setting the discriminant to zero: (-t)² - 4(1)(1/4) = 0 t² - 1 = 0 t² = 1 t = ± 1

Alternatively, we can compare the expression to the standard identity for a perfect square binomial (A ± B)² = A² ± 2AB + B². By mapping n² - tn + 1/4 to this form, we identify A² = n², which gives A = n, and B² = 1/4, which gives B = 1/2. The middle term -tn must then correspond to ± 2AB: -tn = ± 2(n)(1/2) -tn = ± n -t = ± 1 t = ∓ 1

Both values, t = 1 and t = -1, result in the expression being a perfect square, specifically (n - 1/2)² and (n + 1/2)² respectively.

Explanation

The given equation is x / a = 1 / a - 1 / x. To find the value of x - x², first multiply the entire equation by the common denominator ax to eliminate the fractions:

ax · ˣ⁄ₐ = ax · (¹⁄ₐ - ¹⁄ₓ)

x² = x - a

To obtain the expression x - x², rearrange the terms by subtracting x² from both sides and adding a to both sides:

x² + a = x

a = x - x²

By comparing the resulting equation to the required expression, the value is determined to be a.

Explanation

The area theorem for similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given that Δ ABC ∼ Δ DEF, we apply the formula:

(Area(Δ ABC))/(Area(Δ DEF)) = ((BC)/(EF))²

Substituting the provided values BC = 3 cm, EF = 4 cm, and Area(Δ ABC) = 54 cm² into the equation:

(54)/(Area(Δ DEF)) = (¾)²

(54)/(Area(Δ DEF)) = ⁹⁄₁₆

Solving for the Area(Δ DEF):

Area(Δ DEF) = (54 × 16)/(9)

By simplifying the fraction, 54 ÷ 9 = 6. Therefore:

Area(Δ DEF) = 6 × 16 = 96 cm²

Explanation

The relationship between selling prices and their respective gain or loss percentages can be expressed using the formula: SP₂ = (SP₁ × (100 + Gain%))/((100 - Loss%))

Given that the initial selling price SP₁ is 170 with a loss of 15%, and the desired gain is 20%, we substitute these values into the formula: SP₂ = (170 × (100 + 20))/((100 - 15)) SP₂ = (170 × 120)/(85)

Simplifying the fraction, 170 ÷ 85 = 2: SP₂ = 2 × 120 = 240

Explanation

When interest is compounded semi-annually, the rate of interest is halved and the number of compounding periods is doubled. The effective rate per half-year is r = 10% / 2 = 5%, and the number of periods is n = 2T, where T is the time in years. Using the compound interest formula A = P(1 + r/100)ⁿ:

92,610 = 80,000 (1 + ⁵⁄₁₀₀)ⁿ

(92,610)/(80,000) = (¹⁰⁵⁄₁₀₀)ⁿ

(9,261)/(8,000) = (²¹⁄₂₀)ⁿ

Recognizing that 9,261 is the cube of 21 and 8,000 is the cube of 20, we can rewrite the equation as:

(²¹⁄₂₀)³ = (²¹⁄₂₀)ⁿ

Comparing the powers, we find n = 3. Since n represents the number of semi-annual periods, the time in years is T = n/2 = 3/2 = 1.5 years.

Explanation

Let the share of P be x. Based on the problem, Q gets Rs. 30 more than P, so Q's share is x + 30. R gets Rs. 60 more than Q, making R's share (x + 30) + 60 = x + 90. The sum of their shares is Rs. 300, expressed by the equation x + (x + 30) + (x + 90) = 300.

Simplifying this gives 3x + 120 = 300, which leads to 3x = 180 and x = 60. Substituting x back into the shares, P receives Rs. 60, Q receives 60 + 30 = 90, and R receives 90 + 60 = 150. The ratio of their shares P:Q:R is 60:90:150. Dividing each value by 30 results in the simplified ratio of 2:3:5.

Explanation

Let O be the center of the circle and r be its radius. Let P and Q be the midpoints of chords AB and CD respectively. Since the perpendicular from the center of a circle to a chord bisects the chord, we have AP = ½ AB = 5 cm and CQ = ½ CD = 12 cm. Let the distance from the center to chord AB be OP = x. Since the chords are on opposite sides of the center and the total distance between them is 17 cm, the distance from the center to chord CD is OQ = 17 - x.

In the right-angled triangles OPA and OQC, we apply the Pythagoras theorem: r² = OP² + AP² = x² + 5² r² = OQ² + CQ² = (17 - x)² + 12²

Equating the two expressions for r²: x² + 25 = (17 - x)² + 144 x² + 25 = 289 - 34x + x² + 144 34x = 289 + 144 - 25 34x = 408 x = 12

Substituting x = 12 into the first equation: r² = 12² + 5² r² = 144 + 25 r² = 169 r = √(169) = 13 cm

Explanation

The sum of the initial 50 numbers is calculated as 50 × 38 = 1900. When the numbers 45 and 55 are discarded, the sum of the remaining 48 numbers becomes 1900 - (45 + 55) = 1800. The new average is found by dividing this remaining sum by the remaining count of numbers: ¹⁸⁰⁰⁄₄₈ = 37.5.

Alternatively, using the deviation method, the two discarded numbers 45 and 55 are 7 and 17 units higher than the original average of 38, respectively. The total excess removed from the group is 7 + 17 = 24. This total reduction is shared among the remaining 48 numbers, resulting in a decrease in the average by ²⁴⁄₄₈ = 0.5. Consequently, the new average is 38 - 0.5 = 37.5.

Explanation

Given the equation (x)/(x² - 2x + 1) = ⅓, we can take the reciprocal of both sides to simplify the expression:

(x² - 2x + 1)/(x) = 3

By dividing each term in the numerator by x, we get:

x - 2 + ¹⁄ₓ = 3

Rearranging the terms gives the sum of the variable and its reciprocal:

x + ¹⁄ₓ = 5

To find the value of x³ + (1)/(x³), we use the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting a = x and b = ¹⁄ₓ, we apply the shortcut formula where if x + ¹⁄ₓ = k, then x³ + (1)/(x³) = k³ - 3k:

x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x)(¹⁄ₓ)(x + ¹⁄ₓ)

x³ + (1)/(x³) = 5³ - 3(5)

x³ + (1)/(x³) = 125 - 15 = 110

Explanation

The first discount of 10% is applied to the marked price of 1600. The price after this initial discount is calculated as: Price after first discount = 1600 × (1 - ¹⁰⁄₁₀₀) = 1600 × 0.9 = 1440

The final price paid by the customer is 1224. The second discount of x% is applied to the intermediate price of 1440. The amount of this second discount is the difference between the intermediate price and the final selling price: Second discount amount = 1440 - 1224 = 216

The rate of the second discount, x, is the percentage of this discount amount relative to the intermediate price: x = (²¹⁶⁄₁₄₄₀) × 100 x = ²¹⁶⁰⁰⁄₁₄₄₀ = ²¹⁶⁰⁄₁₄₄ x = 15

Alternatively, using the successive discount formula: 1224 = 1600 × (1 - ¹⁰⁄₁₀₀) × (1 - ˣ⁄₁₀₀) 1224 = 1600 × 0.9 × (1 - ˣ⁄₁₀₀) 1224 = 1440 × (1 - ˣ⁄₁₀₀) 1 - ˣ⁄₁₀₀ = ¹²²⁴⁄₁₄₄₀ = 0.85 ˣ⁄₁₀₀ = 0.15 x = 15

Explanation

Express the values of A and B as ratios of C based on the given percentages. Since A is 50% larger than C, A = 1.5C. Since B is 25% larger than C, B = 1.25C.

The percentage by which A is larger than B is calculated by comparing their relative values: (A)/(B) = (1.5C)/(1.25C) = (1.5)/(1.25)

To simplify the calculation, remove the decimals by multiplying both the numerator and denominator by 100: ¹⁵⁰⁄₁₂₅ = ⁶⁄₅

The ratio ⁶⁄₅ implies that for every 5 parts of B, A has 6 parts. The increase is 1 part over the base of 5 parts. Converting this fractional increase to a percentage: ⅕ × 100 = 20%

Explanation

For any expression of the form x + 1/x = k, the value of x² + 1/x² is derived using the identity (a + b)² = a² + b² + 2ab. Squaring both sides gives (x + 1/x)² = k², which simplifies to x² + 1/x² + 2 = k², or x² + 1/x² = k² - 2.

Given x + 1/x = 4: x² + 1/x² = 4² - 2 = 16 - 2 = 14

To find the value of x⁴ + 1/x⁴, we repeat the process by squaring the expression for x² + 1/x²: (x² + 1/x²)² = 14² x⁴ + 1/x⁴ + 2(x²)(1/x²) = 196 x⁴ + 1/x⁴ + 2 = 196 x⁴ + 1/x⁴ = 196 - 2 = 194

Explanation

The base of a right circular cone is a circle. The perimeter (circumference) of the base is given by C = 2π r, where r is the radius. Given C = 8cm, we find the radius: 2π r = 8 r = (8)/(2π) = (4)/(π)cm

The volume V of a right circular cone is calculated using the formula V = ⅓π r² h. Substituting the radius r = (4)/(π) and the height h = 21cm: V = ⅓ × π × ((4)/(π))² × 21 V = ⅓ × π × (16)/(π²) × 21 V = (16)/(π) × 7 V = (112)/(π)cm³

Explanation

The circumference of a circle is determined by the formula 2 π r, where r is the radius. Let R be the radius of the outer circle and r be the radius of the inner ground. The width of the road is the constant distance between these two concentric circles, defined by the expression R - r. According to the problem, the difference between the circumferences is 66 metres, which is expressed as 2 π R - 2 π r = 66.

By factoring out 2 π, the equation becomes 2 π (R - r) = 66. Using the approximation π = ²²⁄₇, we can solve for the width (R - r):

2 × ²²⁄₇ × (R - r) = 66 ⁴⁴⁄₇ × (R - r) = 66 R - r = (66 × 7)/(44) R - r = (3 × 7)/(2) R - r = 10.5

The value of R - r is 10.5, which corresponds to the width of the road in metres.