SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The distance Gin jogs each day forms an arithmetic progression for one week starting from Sunday. On Sunday, he jogs a = 3 miles. For the next six days, he increases the distance by d = 1 mile each day. The number of days in a week is n = 7.

The total distance jogged in one week is the sum of this arithmetic progression: Sₙ = ⁿ⁄₂ [2a + (n-1)d] S₇ = ⁷⁄₂ [2(3) + (7-1)1] S₇ = ⁷⁄₂ [6 + 6] S₇ = ⁷⁄₂ × 12 S₇ = 7 × 6 = 42 miles

The question asks for the total distance jogged in 2 weeks. Since the pattern repeats, we multiply the weekly total by 2: Total distance = 42 × 2 = 84 miles

Explanation

The algebraic identity for the difference of two cubes is p³ - q³ = (p - q)(p² + pq + q²). To align this with the form provided in the equation, we manipulate the second factor using the identity (p + q)² = p² + 2pq + q².

By rearranging the square identity, we can express the quadratic part as: p² + pq + q² = (p² + 2pq + q²) - pq = (p + q)² - 1pq

Substituting this expression back into the difference of cubes formula: p³ - q³ = (p - q) {(p + q)² - 1pq}

Comparing this to the given expression (p - q) × {(p + q)² - xpq}, the value of the coefficient x must be 1.

Explanation

To simplify the expression 1 - (1)/(1+√(2)) + (1)/(1-√(2)), we first focus on combining the two fractional terms. We can rewrite the expression as:

1 + ( (1)/(1-√(2)) - (1)/(1+√(2)) )

To combine these fractions, we find a common denominator by multiplying the individual denominators: (1 - √2)(1 + √2) = 1² - (√2)² = 1 - 2 = -1

Next, we calculate the combined numerator: (1 + √2) - (1 - √2) = 1 + √2 - 1 + √2 = 2√2

Substituting these results back into the expression: 1 + 2√2{-1} = 1 - 2√2

Alternatively, rationalizing each fraction individually leads to the same outcome. The term (1)/(1+√(2)) rationalizes to √2 - 1, and the term (1)/(1-√(2)) rationalizes to -(1 + √2). Plugging these back into the original expression gives: 1 - (√2 - 1) + (-1 - √2) = 1 - √2 + 1 - 1 - √2 = 1 - 2√2

Explanation

When a rectangle is inscribed in a circle, its diagonal is equal to the diameter of the circle. Let the sides of the rectangle be a and b, and the diameter of the circle be d. Using the Pythagorean theorem for the right triangle formed by two sides and the diagonal:

d² = a² + b²

Given that the diameter d = 5 cm and one side a = 4 cm, we substitute these values into the formula:

5² = 4² + b² 25 = 16 + b² b² = 25 - 16 b² = 9 b = 3 cm

The area of the rectangle is the product of its two sides:

Area = a × b = 4 cm × 3 cm = 12 cm²

Explanation

In a cyclic quadrilateral, the sum of opposite angles is 180°. Therefore, ∠ A + ∠ C = 180° and ∠ B + ∠ D = 180°. Given ∠ A = 105°, it follows that ∠ C = 180° - 105° = 75°. Additionally, in a trapezium with AD ∥ BC, the consecutive interior angles between the parallel lines are supplementary, which implies ∠ A + ∠ B = 180° and ∠ D + ∠ C = 180°. Substituting the value of ∠ A into the first relation, we find ∠ B = 180° - 105° = 75°. From the cyclic property ∠ B + ∠ D = 180°, we then determine ∠ D = 180° - 75° = 105°. This confirms that a cyclic trapezium is an isosceles trapezium where base angles are equal. The resulting values are ∠ B = 75°, ∠ C = 75°, and ∠ D = 105°.

Explanation

The trigonometric equation sec(15θ) = csc(15θ) can be simplified by using the reciprocal identities sec x = (1)/(cos x) and csc x = (1)/(sin x). Substituting these into the equation gives:

(1)/(cos(15θ)) = (1)/(sin(15θ))

By cross-multiplying the terms, we obtain:

sin(15θ) = cos(15θ)

Dividing both sides by cos(15θ) (where cos(15θ) ≠ 0) yields the tangent form:

tan(15θ) = 1

In trigonometry, the tangent function equals 1 when the angle is 45° in the first quadrant. Setting the argument of the tangent function to 45° allows us to solve for θ:

15θ = 45°

Dividing by 15:

θ = (45°)/(15)

θ = 3°

The calculated value θ = 3° satisfies the given constraint 0° < θ < 10°.

Explanation

To determine the relationship between the three variables, we simplify each expression to its decimal form:

x = (0.25)¹^/² = √(0.25) = 0.5

y = 0.4

z = (0.216)¹^/³ = ³√(0.216) = 0.6

Comparing these values, we have 0.6 > 0.5 > 0.4. Substituting the variables back into this inequality gives z > x > y.

Explanation

The total runs scored by the cricket team are calculated by multiplying the number of players by their average score. Total runs = 11 × 23 = 253

To find the runs scored by the remaining 10 players, subtract the first player's score from the total: Runs of remaining players = 253 - 113 = 140

The average of the remaining 10 players is then calculated by dividing their total runs by the number of players: Average = ¹⁴⁰⁄₁₀ = 14 runs

Explanation

To evaluate the expression ˣ⁄ₐ₊ₓ + (y)/(b+y) + (z)/(c+z), we transform each term to utilize the given equations a² = by + cz, b² = cz + ax, and c² = ax + by.

First, multiply the numerator and denominator of the first term by a: ˣ⁄ₐ₊ₓ = (ax)/(a²+ax) Substituting a² = by+cz into the denominator: (ax)/(by+cz+ax)

Next, multiply the numerator and denominator of the second term by b: (y)/(b+y) = (by)/(b²+by) Substituting b² = cz+ax into the denominator: (by)/(cz+ax+by)

Finally, multiply the numerator and denominator of the third term by c: (z)/(c+z) = (cz)/(c²+cz) Substituting c² = ax+by into the denominator: (cz)/(ax+by+cz)

Since all three terms now share the common denominator ax+by+cz, we can add them directly: (ax)/(ax+by+cz) + (by)/(ax+by+cz) + (cz)/(ax+by+cz) = (ax+by+cz)/(ax+by+cz)

The expression simplifies to 1.

Explanation

To find the value of (x+y)/(x-y), we use the algebraic identities (x+y)² = x² + y² + 2xy and (x-y)² = x² + y² - 2xy.

Substituting the given values x²+y²=29 and xy=10 into the first identity: (x+y)² = 29 + 2(10) = 29 + 20 = 49. Since x>0 and y>0, their sum must be positive, so x+y = √(49) = 7.

Next, substituting the values into the second identity: (x-y)² = 29 - 2(10) = 29 - 20 = 9. Since x>y, the difference x-y must be positive, so x-y = √9 = 3.

The ratio is then calculated as (x+y)/(x-y) = ⁷⁄₃.

Explanation

In an equilateral triangle with side length a, the altitude h is given by h = √3{2}a. The centroid of the triangle, which also serves as the circumcenter and incenter, divides the altitude in a 2:1 ratio. The circumradius R corresponds to the distance from a vertex to the centroid, while the inradius r corresponds to the distance from the centroid to the midpoint of a side. Mathematically, R = ⅔h and r = ⅓h. The ratio of the circumradius to the inradius is therefore R : r = ⅔h : ⅓h, which simplifies directly to 2:1. Alternatively, using the side formulas, R = (a)/(√(3)) and r = (a)/(2√(3)), so R : r = (a)/(√(3)) : (a)/(2√(3)) = 1 : ½ = 2:1.

Explanation

Since AB is a diameter and O is the center, AOB is a straight line. The angle BOC is given as 120°. The angles AOC and BOC form a linear pair, so ∠ AOC = 180° - ∠ BOC = 180° - 120° = 60°.

According to the circle theorem, the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circle. Here, arc AC subtends ∠ AOC at the center and ∠ ADC at the circumference. Therefore, ∠ AOC = 2 × ∠ ADC. Substituting the value, we get 60° = 2 × ∠ ADC, which gives ∠ ADC = 30°.

Explanation

Let the vertical standing part of the tree be h₁ and the broken part that bends to touch the ground be h₂. The total height of the tree is H = h₁ + h₂. These parts form a right-angled triangle with the ground where the distance from the foot to the top (base) is 10 m and the angle of elevation is 30°.

Using trigonometric ratios: h₁ = 10 tan 30° = 10 × (1)/(√(3)) = (10)/(√(3)) m h₂ = (10)/(cos 30°) = (10)/(√(3)){2} = (20)/(√(3)) m

The total height of the tree is the sum of these two parts: H = h₁ + h₂ = (10)/(√(3)) + (20)/(√(3)) = (30)/(√(3)) m

Rationalizing the denominator by multiplying the numerator and denominator by √3: H = 30√3{3} = 10√3 m

Explanation

The effective compound interest rate for 2 years at 10% per annum is calculated using the formula r + r + (r²)/(100), which results in 10 + 10 + (10 × 10)/(100) = 21%. Since the compound interest is given as Rs. 525, the principal P is determined by the relation: 21% of P = 525 P = ⁵²⁵⁄₂₁ × 100 = 2500

For the simple interest calculation, the time is doubled to 4 years (2 × 2) and the rate is halved to 5% per annum (10% ÷ 2). Using the simple interest formula SI = (P × R × T)/(100): SI = (2500 × 5 × 4)/(100) SI = 25 × 20 = 500

Explanation

The trigonometric values for the specified angles are tan 30° = (1)/(√(3)) and tan 60° = √3. Substituting these values into the given expression tan θ = tan 30° · tan 60° yields:

tan θ = (1)/(√(3)) · √3

tan θ = 1

For an acute angle θ, the equation tan θ = 1 is satisfied when θ = 45°. To find the value of 2θ, we multiply this angle by 2:

2θ = 2 × 45° = 90°

Explanation

The total work can be assumed as the Least Common Multiple (LCM) of the individual time taken by A, B, and C. The LCM of 10, 12, and 15 is 60 units.

Efficiency of A = ⁶⁰⁄₁₀ = 6 units per day Efficiency of B = ⁶⁰⁄₁₂ = 5 units per day Efficiency of C = ⁶⁰⁄₁₅ = 4 units per day

Combined efficiency of A, B, and C = 6 + 5 + 4 = 15 units per day.

The time taken by A, B, and C working together is calculated by dividing the total work by their combined efficiency: Time = ⁶⁰⁄₁₅ = 4 days.

Alternatively, using the work rate formula, the combined rate is ⅒ + ¹⁄₁₂ + ¹⁄₁₅ = (6 + 5 + 4)/(60) = ¹⁵⁄₆₀ = ¼ of the work per day. Thus, the entire work is completed in 4 days.

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. Since the radius is half of the diameter, the ratio of the radii r₁ : r₂ is equal to the ratio of the diameters d₁ : d₂, which is given as 4:5.

The ratio of the volumes of the two cones can be expressed as: (V₁)/(V₂) = (1)/(3) π r₁² h₁{⅓ π r₂² h₂} = ((r₁)/(r₂))² × (h₁)/(h₂)

Substituting the given volume ratio 1:4 and the radius ratio 4:5 into the formula: ¼ = (⅘)² × (h₁)/(h₂) ¼ = ¹⁶⁄₂₅ × (h₁)/(h₂)

To solve for the ratio of the heights h₁ : h₂, rearrange the equation: (h₁)/(h₂) = ¼ × ²⁵⁄₁₆ (h₁)/(h₂) = ²⁵⁄₆₄

Explanation

The selling price after successive discounts is calculated by applying each discount rate to the decreasing balance or by finding a single equivalent discount. Using the sequential method, the selling price is given by the formula SP = MP × (1 - d₁/100) × (1 - d₂/100). Substituting the given values, SP = 2300 × (1 - 0.25) × (1 - 0.10). This simplifies to SP = 2300 × 0.75 × 0.90. Multiplying 2300 by 0.75 gives 1725, and subsequently multiplying 1725 by 0.90 results in 1552.5. Alternatively, the effective discount rate can be found using the formula d₁ + d₂ - (d₁ × d₂ / 100), which is 25 + 10 - (250 / 100) = 32.5%. The selling price is 2300 × (100 - 32.5)% = 2300 × 0.675 = 1552.5.

Explanation

The age difference between two individuals remains constant over time. Since Ram is 20 years younger than Shyam today, he was also 20 years younger five years ago. Let the ages of Ram and Shyam five years ago be 3x and 5x respectively, based on the given ratio 3:5.

The difference in their ages five years ago was 5x - 3x = 2x. Setting this equal to the constant difference of 20 years gives 2x = 20, so x = 10. Consequently, five years ago, Ram was 3 × 10 = 30 years old and Shyam was 5 × 10 = 50 years old. To find their present ages, we add 5 years to each, resulting in 35 years for Ram and 55 years for Shyam. The sum of their present ages is 35 + 55 = 90 years.

Explanation

The relationship between Cost Price (CP) and Selling Price (SP) during a loss is given by the formula SP = CP - Loss. Given a loss of 15%, we can assume the Cost Price to be 100 units. The loss amount is then 15% of 100, which is 15 units.

Using the formula, the Selling Price is 100 - 15 = 85 units. The ratio of Cost Price to Selling Price is expressed as CP : SP = 100 : 85. By dividing both terms by their greatest common divisor, 5, the ratio simplifies to 20 : 17.

Alternatively, expressing the loss percentage as a fraction gives 15% = ¹⁵⁄₁₀₀ = ³⁄₂₀. In this fractional representation, the denominator represents the Cost Price (20 units) and the numerator represents the loss (3 units). The Selling Price is CP - Loss = 20 - 3 = 17 units. Thus, the ratio of CP to SP is 20 : 17.

Explanation

Let B's salary be 100. Since A's salary is 40% less than B's, A's salary is calculated as 100 - 40 = 60. To find how much percent B's salary is more than A's, we use the percentage increase formula relative to A's salary:

Percentage More = (Difference)/(A's Salary) × 100

Substituting the values: (100 - 60)/(60) × 100 = ⁴⁰⁄₆₀ × 100 = ⅔ × 100 = 66.66...%

Alternatively, using the standard comparison formula where if one value is r% less than another, the second is more than the first by: (r)/(100 - r) × 100 = (40)/(100 - 40) × 100 = ⁴⁰⁄₆₀ × 100 = 66.66...%

Rounding this value to the nearest integer as presented in the options yields 66%.

Explanation

When an object travels equal distances at two different speeds, the average speed for the entire journey is the harmonic mean of those speeds. Since the distance from home to office is the same as the distance from office back home, we use the formula:

Vₐᵥ_g = (2v₁v₂)/(v₁ + v₂)

Given that v₁ = 12 kmph and v₂ = 10 kmph, substituting these values into the formula gives:

Vₐᵥ_g = (2 × 12 × 10)/(12 + 10)

Vₐᵥ_g = ²⁴⁰⁄₂₂

Vₐᵥ_g = ¹²⁰⁄₁₁ ≈ 10.909 kmph

Rounding to one decimal place yields an average speed of 10.9 kmph.

Explanation

The given expression for x represents 3¹^/³ - 3⁻¹^/³. Using the algebraic identity (a - b)³ = a³ - b³ - 3ab(a - b), we can evaluate the required expression.

Let a = 3¹^/³ and b = 3⁻¹^/³. Here, ab = 3¹^/³ · 3⁻¹^/³ = 1. Given x = a - b, cubing both sides gives: x³ = (a - b)³ = a³ - b³ - 3ab(a - b)

Substituting the values of a³, b³, and ab: x³ = (3¹^/³)³ - (3⁻¹^/³)³ - 3(1)(x) x³ = 3 - ⅓ - 3x x³ = (9 - 1)/(3) - 3x x³ = ⁸⁄₃ - 3x

To find the value of 3x³ + 9x, multiply the entire equation by 3: 3x³ = 8 - 9x 3x³ + 9x = 8

Explanation

The algebraic identity for the cube of a binomial is given by (x + y)³ = x³ + y³ + 3xy(x + y). According to the problem, xy(x + y) = m. Substituting this value into the identity, we get:

x³ + y³ + 3m = (x + y)³

To express the result in terms of m, x, and y, we rearrange the given equation xy(x + y) = m to solve for (x + y):

x + y = m/(xy)

Now, substitute this back into the cubic expression:

x³ + y³ + 3m = (m/(xy))³ = m³/(x³y³)

Thus, the value of the expression is m³/(x³y³).

Explanation

The SSS (Side-Side-Side) similarity criterion for triangles states that if the corresponding sides of two triangles are proportional, then the triangles are similar. In Euclidean geometry, two triangles △ ABC and △ DEF are similar if the ratios of their corresponding sides are equal, such that AB/DE = BC/EF = AC/DF = k, where k is a constant scale factor. This proportionality of sides is a sufficient condition for triangles to be similar because it mathematically implies that their corresponding interior angles are also equal (∠A = ∠D, ∠B = ∠E, and ∠C = ∠F), thereby fulfilling the formal definition of geometric similarity.

Explanation

In a triangle, the angle formed by the intersection of the internal bisectors of two base angles is calculated using the formula ∠BOC = 90° + ½∠BAC. This point O is the incenter of △ABC.

Given the vertex angle ∠BAC = 60°, we substitute this value into the formula:

∠BOC = 90° + 60°/2

∠BOC = 90° + 30°

∠BOC = 120°

This property is derived from the sum of angles in △ABC and △OBC. In △ABC, ∠B + ∠C = 180° - ∠A. In △OBC, the sum of angles is ∠BOC + ½∠B + ½∠C = 180°. Substituting ½(∠B + ∠C) with ½(180° - ∠A) leads directly to the standard relation used above.

Explanation

To find the value of the expression (sinθ + sinφ)/(cosθ + cosφ) + (cosθ - cosφ)/(sinθ - sinφ), we combine the two fractions over a common denominator. The resulting numerator is (sinθ + sinφ)(sinθ - sinφ) + (cosθ - cosφ)(cosθ + cosφ). Applying the algebraic identity (a + b)(a - b) = a² - b² to both terms in the numerator, it simplifies to (sin²θ - sin²φ) + (cos²θ - cos²φ). By regrouping these terms, we can utilize the fundamental trigonometric identity sin² x + cos² x = 1. The numerator becomes (sin²θ + cos²θ) - (sin²φ + cos²φ), which evaluates to 1 - 1 = 0. Since the numerator of the combined fraction is zero, the value of the entire expression is 0.

Explanation

To find the least number that must be subtracted from 63520 to make it a perfect square, we use the long division method to find its square root and determine the remainder.

First, group the digits from right to left in pairs: 6, 35, 20. The largest square number less than or equal to the first period 6 is 2² = 4. The first digit of the root is 2. Subtracting 4 from 6 gives 2, and bringing down the next pair 35 gives 235.

Next, double the current quotient (2) to get 4 as the divisor's first digit. We find a digit x such that (40 + x) · x ≤ 235. For x = 5, 45 · 5 = 225. Subtracting 225 from 235 gives 10, and bringing down the final pair 20 gives 1020.

Now, double the current quotient 25 to get 50 as the new divisor's prefix. We find a digit y such that (500 + y) · y ≤ 1020. For y = 2, 502 · 2 = 1004. The remainder is 1020 - 1004 = 16.

Since 16 is the remainder after calculating the square of 252, it is the excess value that prevents 63520 from being a perfect square. Thus, subtracting 16 from 63520 results in 63504, which is 252².

Explanation

The total age of the 5 family members three years ago is calculated by multiplying the number of members by the average age: 5 × 17 = 85 years. Over the next three years, each of the 5 members ages by 3 years, adding a total of 5 × 3 = 15 years to the family sum. Thus, the current total age of the original 5 members is 85 + 15 = 100 years.

With the birth of the baby, the family now consists of 6 members. The problem states that the current average age remains 17 years. Therefore, the total current age of all 6 members is 6 × 17 = 102 years. The age of the baby is the difference between the total age of the 6-member family and the current total age of the 5 original members: 102 - 100 = 2 years.

Explanation

Given the equation p + ¹⁄ₚ₊₂ = 1, we add 2 to both sides to make the terms consistent with the target expression: (p+2) + ¹⁄ₚ₊₂ = 1 + 2 (p+2) + ¹⁄ₚ₊₂ = 3

Let x = p+2. The equation simplifies to x + ¹⁄ₓ = 3, and the value to be found is x³ + (1)/(x³) - 3.

Using the algebraic identity x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ), we substitute the known sum: x³ + (1)/(x³) = (3)³ - 3(3) x³ + (1)/(x³) = 27 - 9 = 18

The final value of the expression is obtained by subtracting 3 from this result: 18 - 3 = 15

Explanation

The given equation is (3a+4b)/(3c+4d) = (3a-4b)/(3c-4d). By applying the property of alternendo, we can rearrange the terms as:

(3a+4b)/(3a-4b) = (3c+4d)/(3c-4d)

Applying the rule of Componendo and Dividendo, where if (x)/(y) = (p)/(q) then (x+y)/(x-y) = (p+q)/(p-q), we get:

((3a+4b) + (3a-4b))/((3a+4b) - (3a-4b)) = ((3c+4d) + (3c-4d))/((3c+4d) - (3c-4d))

(6a)/(8b) = (6c)/(8d)

Simplifying the constants by dividing both sides by ⁶⁄₈:

(a)/(b) = (c)/(d)

By cross-multiplying, we obtain the relationship:

ad = bc

Explanation

In a right-angled triangle, one angle is 90°. Since the sum of all internal angles in any triangle is 180°, the sum of the two smaller (acute) angles must be 180° - 90° = 90°. Let the two smaller angles be x and y, where y represents the smallest angle. Based on the given information, the difference between these two angles is x - y = 8°, and their sum is x + y = 90°. To find the value of the smallest angle y, we subtract the difference equation from the sum equation: (x + y) - (x - y) = 90° - 8°. This simplifies to 2y = 82°. Solving for y gives y = 41°.

Explanation

The orthocenter O of a triangle ABC is the point of intersection of its altitudes. Let BE be the altitude from vertex B to side AC, and CF be the altitude from vertex C to side AB. These altitudes intersect at the orthocenter O.

In the quadrilateral AFOE, the angles at F (∠ AFO) and E (∠ AEO) are both 90° because CF and BE are perpendicular to the respective sides. The sum of the interior angles of any quadrilateral is 360°. Therefore, in quadrilateral AFOE:

∠ FAE + ∠ AFO + ∠ FOE + ∠ AEO = 360°

Substituting the known values:

∠ BAC + 90° + ∠ FOE + 90° = 360° ∠ BAC + ∠ FOE = 180°

The angles ∠ FOE and ∠ BOC are vertically opposite angles, which means ∠ FOE = ∠ BOC. Substituting this into the equation:

∠ BAC + ∠ BOC = 180°

Given that ∠ BOC = 150°, we can solve for ∠ BAC:

∠ BAC + 150° = 180° ∠ BAC = 180° - 150° ∠ BAC = 30°

Explanation

In simple interest, the interest accrued remains constant for every equal interval of time. The interest earned in the third year is the difference between the amount after 3 years and the amount after 2 years.

Interest for 1 year = Rs. 5832 - Rs. 5182 = Rs. 650

The amount after 2 years (A₂) is the sum of the principal (P) and the interest for 2 years (I₂). We calculate the total interest for the first two years as:

I₂ = 2 × Rs. 650 = Rs. 1300

To find the principal, we subtract this 2-year interest from the total amount at the end of 2 years:

P = A₂ - I₂ P = Rs. 5182 - Rs. 1300 = Rs. 3882

Explanation

Let the height of the helicopter be h = 1500 m. When two objects are on the same side of a vertical line and their angles of depression are ₁ and ₂ (where ₁ < ₂), the distance d between them is given by the formula d = h(cot ₁ - cot ₂).

Substituting the given values ₁ = 30° and ₂ = 60°: d = 1500(cot 30° - cot 60°) d = 1500(√3 - (1)/(√(3)))

To solve the expression in the parentheses, find a common denominator: d = 1500((3 - 1)/(√(3))) d = 1500((2)/(√(3))) d = (3000)/(√(3))

Rationalizing the denominator by multiplying the numerator and denominator by √3: d = 3000 × √3{3} d = 1000√3

The distance between the two ships is 1000√3 metres.

Explanation

Let the efficiency (work done per day) of B be x. Since A does thrice the work of B in the same amount of time, the efficiency of A is 3x. The combined efficiency of A and B working together is 3x + x = 4x.

The total work is calculated by multiplying the combined efficiency of A and B by the total time they take to finish the work: Total Work = 4x × 9 = 36x units.

To find the time taken by A alone to finish this work, divide the total work by A's individual efficiency: Time taken by A = ³⁶ˣ⁄₃ₓ = 12 days.

Explanation

The volume of a cylinder is calculated using the formula V = π r² h. Since the ratio of the diameters is 3:2, the ratio of the radii r₁:r₂ is also 3:2. For two cylinders with equal volumes, V₁ = V₂, the relationship is expressed as:

π r₁² h₁ = π r₂² h₂

By canceling π from both sides and rearranging the equation to solve for the ratio of the heights, we get:

h₁ / h₂ = r₂² / r₁² = (r₂ / r₁)²

Substituting the inverse of the radius ratio, which is 2/3, into the equation:

h₁ / h₂ = (2 / 3)² = 4 / 9

The resulting ratio of their heights is 4:9.

Explanation

The transition from a 10% loss to a 6% gain represents a total change of 16% in the value of the cycle relative to its cost price. Given that increasing the selling price by Rs. 200 accounts for this entire shift, we can establish that 16% of the cost price is equal to Rs. 200.

Let the cost price be CP. The difference in percentage is 6% - (-10%) = 16%. The equation is: 16% of CP = 200 ¹⁶⁄₁₀₀ × CP = 200 CP = (200 × 100)/(16) CP = 12.5 × 100 CP = 1250

This can also be solved using the direct formula for cost price when there is a change from loss to gain: CP = (Increase in Price)/(Loss% + Gain%) × 100 CP = (200)/(10 + 6) × 100 CP = ²⁰⁰⁄₁₆ × 100 = 1250

Explanation

Let the total population of the city be 100. The poor population is given as 60%, which means there are 60 poor people and 100 - 60 = 40 rich people. The total illiterate population is 40% of the total, which is 40 people.

The illiterate population among the rich is 10% of the rich population, calculated as: 10% of 40 = 0.10 × 40 = 4

The illiterate poor population is the difference between the total illiterate population and the illiterate rich population: 40 - 4 = 36

To find the percentage of the poor population that is illiterate, divide the number of illiterate poor by the total poor population and multiply by 100: Percentage = (36 / 60) × 100 = 60%

Explanation

When a train crosses a stationary object of negligible length like a pillar, the total distance traveled by the train is equal to its own length. Here, the distance is 100 metres and the speed is 50 km/hr. To calculate the time in seconds, the speed must be converted from km/hr to m/sec using the conversion factor 1 km/hr = ⁵⁄₁₈ m/sec.

Speed in m/sec = 50 × ⁵⁄₁₈ = ²⁵⁰⁄₁₈ m/sec

Using the relation Time = (Distance)/(Speed):

Time = (100)/(250){18}

Time = (100 × 18)/(250)

Time = (2 × 18)/(5)

Time = ³⁶⁄₅ = 7.2 seconds

Explanation

To find the value of p + ¹⁄ₚ, we start by taking the reciprocal of the given equation (2p)/(p² - 2p + 1) = ¼, which results in:

(p² - 2p + 1)/(2p) = 4

By distributing the denominator 2p to each term in the numerator, we obtain:

(p²)/(2p) - ²ᵖ⁄₂ₚ + ¹⁄₂ₚ = 4

ᵖ⁄₂ - 1 + ¹⁄₂ₚ = 4

Adding 1 to both sides of the equation gives:

ᵖ⁄₂ + ¹⁄₂ₚ = 5

Factoring out ½ from the terms on the left side leads to:

½ (p + ¹⁄ₚ) = 5

Multiplying both sides by 2 provides the final value:

p + ¹⁄ₚ = 10

This algebraic manipulation efficiently isolates the desired expression p + ¹⁄ₚ by transforming the original rational equation into a linear form in terms of the sum.

Explanation

The algebraic identity for the square of a trinomial is (l + m + n)² = l² + m² + n² + 2(lm + mn + nl). By substituting the given values l + m + n = 9 and l² + m² + n² = 31 into this formula, we get 9² = 31 + 2(lm + mn + nl). This simplifies to 81 = 31 + 2(lm + mn + nl). Subtracting 31 from both sides gives 50 = 2(lm + mn + nl). Dividing both sides by 2 results in lm + mn + nl = 25.

Explanation

The centroid of a triangle is the point of concurrency where the three medians of the triangle intersect. A median is defined as a line segment that joins a vertex of the triangle to the midpoint of the side opposite to that vertex. In every triangle, the three medians always meet at a single point that lies in the interior of the triangle. Geometrically, the centroid serves as the center of mass or the arithmetic mean of all the points in the triangle. A key property of the centroid is that it divides each median into two parts in a 2:1 ratio, where the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

Explanation

According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In triangle PQR, the side QR is extended to S, making ∠ PRS the exterior angle at vertex R. The remote interior angles are ∠ QPR and ∠ PQR.

The relationship is expressed as: ∠ PRS = ∠ QPR + ∠ PQR

Substituting the given values: 110° = 72° + ∠ PQR

Solving for ∠ PQR: ∠ PQR = 110° - 72° ∠ PQR = 38°

Explanation

The area of a trapezium is calculated using the formula Area = ½ × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them. In this problem, the parallel sides are AB and CD, the height h is 4 cm, the length of CD is 6 cm, and the area is given as 16 cm².

Substituting these values into the formula: 16 = ½ × (AB + 6) × 4

Simplifying the equation by dividing 4 by 2: 16 = (AB + 6) × 2

Dividing both sides by 2: 8 = AB + 6

Solving for AB: AB = 8 - 6 = 2 cm

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, let a = tanθ and b = cotθ. Squaring both sides of the given equation tanθ + cotθ = 5 yields:

(tanθ + cotθ)² = 5²

tan²θ + cot²θ + 2tanθ = 25

Since tanθ and cotθ are reciprocal trigonometric functions, their product tanθ · cotθ is equal to 1. Substituting this value into the expanded equation results in:

tan²θ + cot²θ + 2(1) = 25

tan²θ + cot²θ = 25 - 2

tan²θ + cot²θ = 23

Explanation

A number N can be expressed using the division algorithm as N = (Divisor × Quotient) + Remainder. In this case, N = 56k + 29. When the same number is divided by 8, we substitute the expression: N/8 = (56k + 29)/8. Since 56 is a multiple of 8 (56 = 8 × 7), the term 56k is perfectly divisible by 8 and leaves a remainder of 0. Therefore, the required remainder is simply the remainder obtained by dividing the original remainder (29) by the new divisor (8). Dividing 29 by 8 gives 8 × 3 = 24 with a remainder of 29 - 24 = 5.

Explanation

The relationship between the Marked Price (MP) and the Cost Price (CP) when the profit (gain) and discount percentages are known is expressed by the formula: (MP)/(CP) = (100 + Gain %)/(100 - Discount %)

Assuming a standard base Cost Price of Rs. 100 to determine the absolute value matching the options, we substitute the given percentages into the formula: MP = 100 × (100 + 25)/(100 - 20) MP = 100 × ¹²⁵⁄₈₀ MP = 100 × 1.5625 MP = 156.25

In this scenario, a marked price of Rs. 156.25 allows the shopkeeper to reduce the price by 20% (a reduction of Rs. 31.25), resulting in a selling price of Rs. 125, which yields the required 25% gain over the cost of Rs. 100.

Explanation

The correct average can be calculated by determining the net change in the total marks and applying it to the initial average using the formula:

Correct Average = Initial Average + (Σ Correct Marks - Σ Incorrect Marks)/(n)

The sum of the correctly entered marks is 56 + 50 = 106, while the sum of the marks wrongly entered is 65 + 24 = 89. The net change in the total sum of marks is:

Net Change = 106 - 89 = +17

Since there are 17 students, the change in the average is:

Change in Average = ⁺¹⁷⁄₁₇ = +1

Adding this change to the initial average of 71 results in the correct average:

Correct Average = 71 + 1 = 72

Explanation

The formula for simple interest is SI = P × r × t, where SI is the interest earned, P is the principal sum, r is the annual interest rate expressed as a decimal, and t is the time in years. According to the problem, the simple interest is two-fifth of the principal, which can be written as SI = ⅖P. The time period t is given as 5 years. Substituting these values into the formula, we get the equation ⅖P = P × r × 5. By dividing both sides of the equation by P, we are left with ⅖ = 5r. To solve for r, we divide both sides by 5, resulting in r = ²⁄₂₅. Converting this fraction into a decimal, we find that r = 0.08.