SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The average speed for a journey involving multiple segments is calculated using the formula: Average Speed = Total Distance / Total Time.

First, calculate the total distance covered in the journey: Total Distance = 275 + 315 = 590 km.

Next, calculate the time taken for each segment using the relation Time = Distance / Speed: Time for the first segment t₁ = ²⁷⁵⁄₅₀ = 5.5 hours. Time for the second segment t₂ = ³¹⁵⁄₇₀ = 4.5 hours.

Then, find the total time for the entire journey: Total Time = t₁ + t₂ = 5.5 + 4.5 = 10 hours.

Finally, apply the average speed formula: Average Speed = ⁵⁹⁰⁄₁₀ = 59 km/h.

Explanation

Using the algebraic identity (a + b)³ + (a - b)³ = 2a(a² + 3b²), let a = 4x and b = 2y.

(4x + 2y)³ + (4x - 2y)³ = 2(4x)((4x)² + 3(2y)²) = 8x(16x² + 3 × 4y²) = 8x(16x² + 12y²) = 128x³ + 96xy²

Given that this expression equals 16(Ax³ + Bxy²), we can factor out 16 from the result: 128x³ + 96xy² = 16(8x³ + 6xy²)

By comparing the coefficients of the corresponding terms, we find: A = 8 B = 6

The required value is calculated as follows: ½ √(A² + B²) = ½ √(8² + 6²) = ½ √(64 + 36) = ½ √(100) = ¹⁰⁄₂ = 5

Explanation

To find the greatest number that divides three given numbers a, b, and c while leaving the same remainder in each case, we calculate the Highest Common Factor (HCF) of the absolute differences between the numbers: |a - b|, |b - c|, and |c - a|.

Given the numbers 108, 124, and 156, we first calculate the differences: |124 - 108| = 16 |156 - 124| = 32 |156 - 108| = 48

Next, we determine the HCF of 16, 32, and 48. We can express these numbers in terms of their factors: 16 = 16 × 1 32 = 16 × 2 48 = 16 × 3

Since 16 is a factor of all three differences and is the largest such value, the HCF is 16. Dividing the original numbers by 16 confirms the result: 108 = (16 × 6) + 12 124 = (16 × 7) + 12 156 = (16 × 9) + 12

In each case, the remainder is 12, satisfying the condition of the problem. Thus, 16 is the greatest number that divides 108, 124, and 156 to leave the same remainder.

Explanation

When the profit earned by selling an article at one price is equal to the loss incurred by selling it at another price, the Cost Price (CP) is the arithmetic mean of the two selling prices.

Cost Price (CP) = SP₁ + SP₂{2}

Substituting the given values: CP = (832 + 448)/(2) = ¹²⁸⁰⁄₂ = 640

To find the selling price (SP) at a 10% loss, we use the formula: SP = CP × (1 - (Loss%)/(100))

SP = 640 × (1 - ¹⁰⁄₁₀₀) SP = 640 × 0.9 SP = 576

Explanation

The total runs scored in the first 10 overs are calculated by multiplying the run rate by the number of overs: 10 × 7.2 = 72 runs. Given that the target score is 272 runs, the runs needed in the remaining 40 overs are 272 - 72 = 200 runs. The required average run rate is the ratio of the remaining runs to the remaining overs: ²⁰⁰⁄₄₀ = 5 runs per over.

Explanation

To find the number of integers in the range [500, 650] that are neither divisible by 3 nor by 7, we first calculate the total count of numbers in the range and then subtract those divisible by 3, 7, or both using the principle of inclusion-exclusion.

The total number of integers from 500 to 650 inclusive is N = 650 - 500 + 1 = 151.

First, we find the count of numbers divisible by 3 in this range. The smallest multiple of 3 ≥ 500 is 501, and the largest multiple of 3 ≤ 650 is 648. The number of terms n(3) is calculated as: n(3) = (648 - 501)/(3) + 1 = ¹⁴⁷⁄₃ + 1 = 49 + 1 = 50

Next, we find the count of numbers divisible by 7. The smallest multiple of 7 ≥ 500 is 504 (7 × 72), and the largest multiple ≤ 650 is 644 (7 × 92). The number of terms n(7) is: n(7) = (644 - 504)/(7) + 1 = ¹⁴⁰⁄₇ + 1 = 20 + 1 = 21

Now, we find the count of numbers divisible by both 3 and 7, which means they are divisible by LCM(3, 7) = 21. The smallest multiple of 21 ≥ 500 is 504 (21 × 24), and the largest multiple ≤ 650 is 630 (21 × 30). The number of terms n(21) is: n(21) = (630 - 504)/(21) + 1 = ¹²⁶⁄₂₁ + 1 = 6 + 1 = 7

Using the principle of inclusion-exclusion, the number of integers divisible by 3 or 7 is: n(3 ∪ 7) = n(3) + n(7) - n(3 ∩ 7) = 50 + 21 - 7 = 64

The number of integers neither divisible by 3 nor by 7 is the total count minus those divisible by 3 or 7: Result = 151 - 64 = 87

Explanation

The given expression can be simplified using the algebraic identity: if X + Y + Z = 0, then X³ + Y³ + Z³ = 3XYZ.

In the numerator, let X = a² - 4b², Y = 4(b² - 4c²) = 4b² - 16c², and Z = 16c² - a². The sum X + Y + Z = (a² - 4b²) + (4b² - 16c²) + (16c² - a²) = 0. Thus, the numerator becomes: 3(a² - 4b²)(4b² - 16c²)(16c² - a²) Using the difference of squares formula x² - y² = (x - y)(x + y), this expands to: 3(a - 2b)(a + 2b) · 4(b - 2c)(b + 2c) · (4c - a)(4c + a) 12(a - 2b)(a + 2b)(b - 2c)(b + 2c)(4c - a)(4c + a)

In the denominator, let p = a - 2b, q = 2b - 4c = 2(b - 2c), and r = 4c - a. The sum p + q + r = (a - 2b) + (2b - 4c) + (4c - a) = 0. Thus, the denominator becomes: 3(a - 2b)(2b - 4c)(4c - a) = 3(a - 2b) · 2(b - 2c) · (4c - a) = 6(a - 2b)(b - 2c)(4c - a)

Dividing the simplified numerator by the simplified denominator: (12(a - 2b)(a + 2b)(b - 2c)(b + 2c)(4c - a)(4c + a))/(6(a - 2b)(b - 2c)(4c - a)) Canceling the common terms (a - 2b), (b - 2c), and (4c - a) leaves: ¹²⁄₆(a + 2b)(b + 2c)(4c + a) = 2(a + 2b)(b + 2c)(4c + a)

Explanation

The value of x² + (1)/(x²) is determined using the identity x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2.

Given x = 4 + √(15), the reciprocal ¹⁄ₓ is found by rationalizing the denominator: ¹⁄ₓ = (1)/(4 + √(15)) × 4 - √(15){4 - √(15)} = 4 - √(15){16 - 15} = 4 - √(15)

Calculating the sum of x and its reciprocal: x + ¹⁄ₓ = (4 + √(15)) + (4 - √(15)) = 8

Substituting this value into the square identity: x² + (1)/(x²) = 8² - 2 = 64 - 2 = 62

Explanation

The total percentage difference between an 18% loss and a 12% gain is 18% + 12% = 30%. According to the problem, this 30% gap is equivalent to the price increase of ₹960.

The Cost Price (CP) is calculated as: CP = (Price Difference)/(Percentage Difference) × 100 CP = ⁹⁶⁰⁄₃₀ × 100 = ₹3,200

When the article is sold for ₹3,840, the profit earned is: Profit = 3,840 - 3,200 = ₹640

The profit percentage is determined by the formula: Profit % = (Profit)/(CP) × 100 Profit % = (640)/(3,200) × 100 = 20%

Explanation

In any triangle, the angle formed by the intersection of the internal bisectors of two angles is given by the formula 90° + ½A, where A is the third angle of the triangle. In △ LMN, the bisectors of ∠ L and ∠ N intersect at an angle of 112°. Using the formula:

112° = 90° + (∠ M)/(2)

Subtracting 90° from both sides:

22° = (∠ M)/(2)

Multiplying by 2:

∠ M = 44°

Explanation

Let the speed of the boat in still water be u = 9 km/h and the speed of the stream be v km/h. The downstream speed is u + v and the upstream speed is u - v. Based on the first condition:

(44)/(9 + v) + (35)/(9 - v) = 9

To solve for v, we can test small integer values or solve the quadratic equation derived from 44(9 - v) + 35(9 + v) = 9(81 - v²). Simplifying this gives 711 - 9v = 729 - 9v², which reduces to v² - v - 2 = 0. Factoring the quadratic (v - 2)(v + 1) = 0 gives v = 2 (ignoring the negative value).

With v = 2 km/h, the downstream speed is 9 + 2 = 11 km/h and the upstream speed is 9 - 2 = 7 km/h. The time T required to row 33 km downstream and 28 km upstream is:

T = ³³⁄₁₁ + ²⁸⁄₇ = 3 + 4 = 7 hours

Explanation

In the right-angled triangle formed by the kite's height, the ground, and the string, the height of 90 m represents the perpendicular side opposite to the given angle of elevation, while the string represents the hypotenuse. The trigonometric ratio relating the perpendicular side and the hypotenuse for a given angle θ is the sine function:

sin(θ) = (Perpendicular)/(Hypotenuse)

Substituting the given height h = 90 m and the angle θ = 30° into the formula:

sin(30°) = (90)/(L)

The value of sin(30°) is ½. Applying this to the equation:

½ = (90)/(L)

L = 90 × 2 = 180 m

Explanation

The sum borrowed P in a compound interest installment plan is calculated using the formula for the present value of an annuity. For two equal annual installments I at a rate r, the formula is P = (I)/(1 + r){100} + (I)/((1 + r){100)²}.

Given I = 980 and r = 4%, the value of the factor 1 + ʳ⁄₁₀₀ = 1 + ⁴⁄₁₀₀ = ²⁶⁄₂₅. Substituting these values into the formula, we get P = (980)/(26){25} + (980)/((26){25)²}. This simplifies to P = 980 × ²⁵⁄₂₆ + 980 × ⁶²⁵⁄₆₇₆.

Factoring out 980 × ²⁵⁄₂₆, we have P = 980 × ²⁵⁄₂₆ × (1 + ²⁵⁄₂₆) = 980 × ²⁵⁄₂₆ × ⁵¹⁄₂₆. Calculating the numerator gives 980 × 25 × 51 = 1,249,500, and the denominator is 26 × 26 = 676. Dividing these results in P = (1,249,500)/(676) ≈ 1848.37. Rounding 1848.37 to the nearest tens yields 1,850.

Explanation

When two circles touch each other externally at point T and have a direct common tangent PQ (where P and Q are points of contact on the circles), the angle subtended by the segment PQ at the point of contact T is always 90°. This can be shown by drawing a common internal tangent through T that intersects PQ at a point M. Since the lengths of tangents from an external point to a circle are equal, MP = MT and MQ = MT. This makes M the midpoint of PQ such that MP = MQ = MT, which implies that T lies on a semicircle with PQ as the diameter. Therefore, ∠ PTQ = 90°. In Δ PTQ, the sum of internal angles is 180°. Given ∠ TPQ = 42° and ∠ PTQ = 90°, the remaining angle ∠ PQT = 180° - (90° + 42°) = 48°.

Explanation

Given sec A = ¹⁷⁄₈, we identify the components of the right-angled triangle as Hypotenuse (H) = 17 and Base (B) = 8. Using the Pythagorean triplet (8, 15, 17), the Perpendicular (P) is calculated as: P = √(17² - 8²) = √(289 - 64) = √(225) = 15

From these values, we determine the required trigonometric ratios: sin A = (P)/(H) = ¹⁵⁄₁₇ cos A = (B)/(H) = ⁸⁄₁₇ tan A = (P)/(B) = ¹⁵⁄₈ cot A = (B)/(P) = ⁸⁄₁₅

Substituting these ratios into the expression (34 sin A + 15 cot A)/(68 cos A - 16 tan A): Numerator: 34 (¹⁵⁄₁₇) + 15 (⁸⁄₁₅) = 2(15) + 8 = 30 + 8 = 38 Denominator: 68 (⁸⁄₁₇) - 16 (¹⁵⁄₈) = 4(8) - 2(15) = 32 - 30 = 2

The value of the expression is ³⁸⁄₂ = 19.

Explanation

Since AB is the diameter of the circle, the angles subtended by it at any point on the circumference are right angles. Therefore, ∠ ACB = 90° and ∠ ADB = 90°. By applying the triangle angle sum property to △ ACB, where ∠ CAB = 42°, we calculate ∠ CBA = 180° - (90° + 42°) = 48°. Similarly, in △ ADB, where ∠ ABD = 57°, we calculate ∠ DAB = 180° - (90° + 57°) = 33°.

Given that C and D lie on opposite sides of the diameter AB, the measure of ∠ CAD is the sum of ∠ CAB and ∠ DAB. This gives ∠ CAD = 42° + 33° = 75°. The measure of ∠ CBD is the sum of ∠ CBA and ∠ ABD, resulting in ∠ CBD = 48° + 57° = 105°. The calculations yield 75° and 105° as the respective measures for the two angles.

Explanation

The problem utilizes the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) By substituting the given values a² + b² + c² = 6.25 and the sum of products (ab + bc + ca) = 0.52 into the identity, we get: (a + b + c)² = 6.25 + 2(0.52) (a + b + c)² = 6.25 + 1.04 = 7.29 Taking the square root of both sides: a + b + c = ± √(7.29) = ± 2.7 The question specifies the constraint (a + b + c) < 0. Therefore, we discard the positive value and conclude: a + b + c = -2.7

Explanation

To find the greatest number that leaves a remainder of 6 when dividing 156, 181, and 331, we find the Highest Common Factor (HCF) of the numbers after subtracting the remainder from each. This is because if a number x leaves a remainder r when dividing n, then x must be a factor of (n - r).

Subtracting the remainder 6 from each: 156 - 6 = 150 181 - 6 = 175 331 - 6 = 325

Next, we find the HCF of 150, 175, and 325. Using the prime factorization method: 150 = 2 × 3 × 5² 175 = 5² × 7 325 = 5² × 13

The common factor among these three numbers is 5², which equals 25. Therefore, 25 is the greatest common divisor for 150, 175, and 325. When this divisor is applied to the original numbers 156, 181, and 331, it leaves a remainder of 6 in each instance.

Explanation

Let the total work be the Least Common Multiple of 48 and 60, which is 240 units. The efficiency of A is 240 / 48 = 5 units per day, and the efficiency of B is 240 / 60 = 4 units per day. Working together, their combined efficiency is 5 + 4 = 9 units per day.

In 12 days, A and B complete 12 × 9 = 108 units of work. The remaining work is 240 - 108 = 132 units. B is required to finish 25% of this remaining work, which is calculated as 132 × 0.25 = 33 units. The time taken by B to complete this portion is 33 / 4 = 8.25 days. To convert the decimal part into hours, 0.25 × 24 = 6 hours. This results in a total time of 8 days 6 hours.

Explanation

The volume of water in the frustum-shaped bucket is transferred to the cylindrical drum, meaning the volumes are equal. The volume of a frustum is given by the formula V_f = ⅓ π h (R² + r² + Rr), where h = 35 cm, R = 18 cm, and r = 12 cm. Substituting these values:

V_f = ⅓ π × 35 × (18² + 12² + 18 × 12) V_f = (35π)/(3) × (324 + 144 + 216) V_f = (35π)/(3) × 684 = 35π × 228

The volume of water in the cylindrical drum is V_c = π R_d² h_d, where R_d = 20 cm and h_d is the height of the water. Setting V_c = V_f:

π × 20² × h_d = 35π × 228 400 × h_d = 35 × 228 h_d = (35 × 228)/(400) h_d = (7 × 228)/(80) h_d = (7 × 57)/(20) = ³⁹⁹⁄₂₀ h_d = 19.95 cm

Explanation

The given quadratic equation is tan² A + 2 tan A - 63 = 0. Let tan A = x. The equation becomes x² + 2x - 63 = 0. Factoring the quadratic expression, we get:

x² + 9x - 7x - 63 = 0 (x + 9)(x - 7) = 0

Thus, tan A = 7 or tan A = -9. Since it is given that 0 < A < (π)/(2), the angle A lies in the first quadrant where all trigonometric ratios are positive. Therefore, tan A = 7.

In a right-angled triangle, if we take the perpendicular p = 7 and the base b = 1, then tan A = p/b = 7/1 = 7. The hypotenuse h is calculated using the Pythagorean theorem:

h = √(p² + b²) = √(7² + 1²) = √(49 + 1) = √(50)

Using the definitions of sine and cosine: sin A = ᵖ⁄ₕ = (7)/(√(50)) cos A = ᵇ⁄ₕ = (1)/(√(50))

Substituting these values into the required expression: 2 sin A + 5 cos A = 2((7)/(√(50))) + 5((1)/(√(50))) = (14)/(√(50)) + (5)/(√(50)) = (19)/(√(50))

Explanation

For a number to be divisible by 56, it must be divisible by both 7 and 8, as these are co-prime factors of 56.

A number is divisible by 8 if its last three digits are divisible by 8. For the number x8942y4, the last three digits are 2y4. Testing possible values for the digit y: If y = 2, 224 / 8 = 28. If y = 6, 264 / 8 = 33. For all other values of y from 0 to 9, 2y4 is not divisible by 8. The problem specifies the largest value of y, so we take y = 6.

Now, for the number x894264 to be divisible by 7, we apply the divisibility rule for 7. One method is to check the alternating sum of blocks of three digits from right to left: 264 - 894 + x = x - 630. For the number to be divisible by 7, this result must be a multiple of 7. Since 630 is exactly divisible by 7 (7 × 90 = 630), the value of x must also be a multiple of 7. As x is the leading digit of a 7-digit natural number, the only possible value for x is 7.

With x = 7 and y = 6, we calculate the required value: x² + y = 7² + 6 = 49 + 6 = 55

Explanation

To simplify the expression, we follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction). The expression can be evaluated by calculating the two segments separately and then finding their product:

Segment 1: ³⁄₂ + ⁵⁄₃ ÷ ⅓ × ⁵⁄₄ - ⁵⁄₂ First, perform the division: ⁵⁄₃ ÷ ⅓ = ⁵⁄₃ × 3 = 5. Next, perform the multiplication: 5 × ⁵⁄₄ = ²⁵⁄₄. Then, perform addition and subtraction: ³⁄₂ + ²⁵⁄₄ - ⁵⁄₂ = ⁶⁄₄ + ²⁵⁄₄ - ¹⁰⁄₄ = ²¹⁄₄.

Segment 2: ½ × ⅓ - ⁶⁄₂ First, perform the multiplication: ½ × ⅓ = ⅙. Then, perform the subtraction: ⅙ - 3 = (1 - 18)/(6) = -¹⁷⁄₆.

Final Calculation: Assuming the two segments are multiplied (as is standard when expressions are placed adjacent to each other): ²¹⁄₄ × (-¹⁷⁄₆) = ⁷⁄₄ × (-¹⁷⁄₂) = -¹¹⁹⁄₈

Given the options provided, the value -¹¹⁹⁄₈ (which is -14.875) is the intended result, often represented with typographical variations in such exam formats.

Explanation

The expression 16cos³(π)/(6) - 12cos (π)/(6) can be simplified by factoring out 4 to get 4(4cos³(π)/(6) - 3cos (π)/(6)). Using the triple angle identity cos 3θ = 4cos³θ - 3cosθ where θ = (π)/(6), the expression becomes 4cos(3 × (π)/(6)). This simplifies to 4cos (π)/(2). Since cos (π)/(2) = 0, the value of the expression is 4 × 0 = 0.

Alternatively, substituting the standard value cos (π)/(6) = √3{2} into the expression gives 16(√3{2})³ - 12(√3{2}). This calculates to 16(3√3{8}) - 6√3 = 6√3 - 6√3 = 0.

Explanation

The number of bicycles sold by company B is 90 and the number of bicycles sold by company D is 165. The ratio of bicycles sold by B to D is calculated as follows:

Ratio = (B)/(D)

Substituting the given values:

Ratio = ⁹⁰⁄₁₆₅

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 15:

90 ÷ 15 = 6 165 ÷ 15 = 11

The simplified ratio is 6:11.

Explanation

The radius r of the circle is half the diameter, so r = 58/2 = 29 cm. The perpendicular distance h of a chord of length L from the center of a circle is calculated using the formula h = √(r² - (L/2)²).

For the chord of length 42 cm, the distance from the center is h₁ = √(29² - (42/2)²) = √(29² - 21²). Using the identity a² - b² = (a-b)(a+b), we get h₁ = √((29-21)(29+21)) = √(8 × 50) = √(400) = 20 cm.

For the chord of length 40 cm, the distance from the center is h₂ = √(29² - (40/2)²) = √(29² - 20²) = √((29-20)(29+20)) = √(9 × 49) = √(441) = 21 cm.

The minimum distance between two parallel chords occurs when both chords are located on the same side of the center. In this case, the distance between them is the difference of their distances from the center, which is |h₂ - h₁| = 21 - 20 = 1 cm.

Explanation

The relationship between the efficiency of worker groups and the time taken is governed by the formula M₁ D₁ = M₂ D₂, where M is the group efficiency and D is the number of days. Comparing the first group (6 men and 8 women) with the target group (15 men and 20 women), the target group is a direct scalar multiple: 15 men + 20 women = 2.5 × (6 men + 8 women)

Substituting this into the work-equivalence equation: (6M + 8W) × 10 = 2.5(6M + 8W) × D₂ 10 = 2.5 × D₂ D₂ = 4

This result is verified by calculating the man-to-woman efficiency ratio from the given conditions: 10(6M + 8W) = 2(26M + 48W) 60M + 80W = 52M + 96W 8M = 16W ⟹ 1M = 2W

Total work in terms of women units is 10(6(2W) + 8W) = 200W. The collective efficiency of the target group is 15(2W) + 20W = 50W. The time taken is calculated as (200W)/(50W) = 4 days.

Explanation

The expression 70³ + 20³ - 90³ follows the algebraic identity where if a + b + c = 0, then a³ + b³ + c³ = 3abc. By defining a = 70, b = 20, and c = -90, we check the sum of the terms:

a + b + c = 70 + 20 + (-90) = 90 - 90 = 0

Since the condition a + b + c = 0 is satisfied, the value of the expression is calculated as:

3abc = 3 × 70 × 20 × (-90) = 3 × (70 × 20) × (-90) = 3 × 1400 × (-90) = 4200 × (-90) = -378000

Explanation

The interest earned over a period of 4 years is the difference between the amounts at 10 years and 6 years. Interest for 4 years = ₹72,000 - ₹59,200 = ₹12,800. Therefore, the simple interest for one year is ₹12,800 / 4 = ₹3,200.

The interest for 6 years is 6 × ₹3,200 = ₹19,200. Subtracting this from the amount after 6 years gives the principal P = ₹59,200 - ₹19,200 = ₹40,000. The original rate of interest R can be calculated using the formula SI = (P × R × T) / 100. Substituting the values for one year, we get 3,200 = (40,000 × R × 1) / 100, which simplifies to R = 8%.

If the rate of interest is increased by 2%, the new rate R' becomes 8% + 2% = 10%. To find the time T required for the sum to become ₹76,000, we first determine the total interest needed: SI' = ₹76,000 - ₹40,000 = ₹36,000. Using the simple interest formula again, 36,000 = (40,000 × 10 × T) / 100, which simplifies to 36,000 = 4,000 × T. Solving for T gives T = 36,000 / 4,000 = 9 years.

Explanation

Let the lengths of the two perpendicular sides of the right-angled triangle be a and b, and the hypotenuse be c. According to the problem, a - b = 17 and the area is ½ ab = 84. This implies ab = 168.

Substituting a = b + 17 into the product equation gives (b + 17)b = 168, which simplifies to the quadratic equation b² + 17b - 168 = 0. Factoring the quadratic, we get (b + 24)(b - 7) = 0. Since length must be positive, b = 7 cm and a = 24 cm.

To find the perimeter, we first calculate the hypotenuse using the Pythagorean theorem: c = √(a² + b²) = √(24² + 7²) = √(576 + 49) = √(625) = 25 cm.

The dimensions of the triangle form the standard Pythagorean triplet (7, 24, 25), which satisfies both the difference condition 24 - 7 = 17 and the area condition ½ × 7 × 24 = 84. The perimeter is the sum of the three sides: Perimeter = a + b + c = 24 + 7 + 25 = 56 cm.

Explanation

The expression is evaluated by treating the two parts as a numerator and a denominator. Following the BODMAS rule, the numerator is simplified as follows:

Numerator: 5 - 35 ÷ 5 × 15 + 5 First, perform division: 35 ÷ 5 = 7 Next, perform multiplication: 7 × 15 = 105 Then, perform addition and subtraction: 5 - 105 + 5 = -95

Denominator: 12 - 2 Perform subtraction: 12 - 2 = 10

The final value is obtained by dividing the numerator by the denominator: -95 ÷ 10 = -9.5

Explanation

For a sequence of n consecutive numbers where n is odd, the arithmetic mean is equal to the middle term of the series. With n = 9 consecutive numbers, the middle term is the ⁹⁺¹⁄₂ = 5ᵗʰ term. Given that the average is 20, the 5ᵗʰ term is 20. Since the numbers are consecutive, the smallest number (the 1ˢᵗ term) is found by subtracting 4 from the 5ᵗʰ term.

Smallest number = Average - ⁿ⁻¹⁄₂ Smallest number = 20 - ⁹⁻¹⁄₂ Smallest number = 20 - 4 = 16

Explanation

To find the increase in monthly expenditure, first calculate the total quantity of petrol consumed per month using the formula Consumption = Total Distance / Mileage. Given a distance of 3045 km and a mileage of 15 km per litre, the monthly consumption is 3045 / 15 = 203 litres.

The increase in price per litre is calculated as 5% of the original price of 82. This results in an increase of 82 × 0.05 = 4.1 per litre. The total increase in monthly expenditure is the product of the total monthly consumption and the price hike per litre, which is 203 × 4.1 = 832.3. Rounding to the nearest whole number gives 832.

Explanation

The selling price (SP) required to achieve a desired profit after an initial loss can be calculated using the following relationship:

SPₙₑ_w = SPₒₗ_d × (100 + Profit%)/(100 - Loss%)

In this case, the initial selling price is 425, the loss percentage is 15%, and the target profit percentage is 5%. Substituting these values into the formula:

SPₙₑ_w = 425 × (100 + 5)/(100 - 15)

SPₙₑ_w = 425 × ¹⁰⁵⁄₈₅

Dividing 425 by 85 results in 5. Therefore:

SPₙₑ_w = 5 × 105 = 525

Alternatively, the Cost Price (CP) can be determined first. Since a 15% loss means the selling price is 85% of the CP:

CP = (425)/(0.85) = 500

To earn a 5% profit, the article must be sold for 105% of the CP:

SPₙₑ_w = 500 × 1.05 = 525

Explanation

According to the subscription table provided for the Mutual Fund Company, the subscription amount for scheme X in the month of December was 40. In the month of February, the subscription for the same scheme X was 30. To find the difference in the subscription of scheme X between these two months, we calculate the absolute difference between the two values:

Difference = 40 - 30 = 10

Based on the units provided in the options, this numerical difference corresponds to 10 crores.

Explanation

The share of wages for individuals working together is distributed in the ratio of their efficiencies. Efficiency is inversely proportional to the time taken to complete the work.

Given the time taken by A, B, and C as 8, 10, and 12 days respectively, their efficiency ratio is: A : B : C = ⅛ : ⅒ : ¹⁄₁₂

To simplify this ratio, multiply by the LCM of 8, 10, and 12, which is 120: A = ¹²⁰⁄₈ = 15 B = ¹²⁰⁄₁₀ = 12 C = ¹²⁰⁄₁₂ = 10

The ratio of their shares is 15 : 12 : 10. The total number of parts is 15 + 12 + 10 = 37.

B's share is calculated as: B = ¹²⁄₃₇ × 5550 B = 12 × 150 B = 1800

Explanation

The volume of a sphere is calculated using the formula Vₛ = ⁴⁄₃ π R³ and the volume of a cone is calculated using V_c = ⅓ π r² h. Given that both the sphere and the cone have a radius R = r = 7 cm, and the cone has a height h = 10 cm, the difference in their volumes can be expressed as:

Difference = ⁴⁄₃ π R³ - ⅓ π R² h = ⅓ π R² (4R - h)

Substituting the given values R = 7, h = 10, and π = ²²⁄₇ into the equation:

Difference = ⅓ × ²²⁄₇ × 7² × (4 × 7 - 10) Difference = ⅓ × 22 × 7 × (28 - 10) Difference = ¹⁵⁴⁄₃ × 18 Difference = 154 × 6 = 924

The difference in volumes is exactly 924 cm³.

Explanation

According to the Angle Bisector Theorem, the ratio of AD to DC is equal to the ratio of AB to BC, which is 12:18 or 2:3. Since the total length of AC is 15 cm, the length of AD is calculated as (2 / (2 + 3)) * 15 = 6 cm.

Explanation

The given expression follows the algebraic form (a³ + b³)/(a² - ab + b²), where a = 428 and b = 348. Using the standard algebraic identity for the sum of two cubes, we know that a³ + b³ = (a + b)(a² - ab + b²).

When we substitute this identity into the numerator of the fraction, the term (a² - ab + b²) appears in both the numerator and the denominator, allowing them to cancel each other out:

((a + b)(a² - ab + b²))/((a² - ab + b²)) = a + b

Substituting the numerical values back into the simplified expression, we get:

428 + 348 = 776

Explanation

The speed of the first train is determined by the formula Speed = (Distance)/(Time). Substituting the given values, the speed of the first train is (250 km)/(5 h) = 50 km/h.

The ratio of the speeds of the two trains is 2:7. In this ratio, the first train's speed corresponds to 2 parts. Therefore, 2 units = 50 km/h, which means 1 unit = 25 km/h.

To find the sum of the speeds, we add the ratio components: 2 + 7 = 9 units. Multiplying the total units by the value per unit, we get 9 × 25 km/h = 225 km/h.

Explanation

The algebraic identity relating these terms is x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx)). First, the value of x² + y² + z² is determined using the expansion (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). Substituting the given values, 11² = x² + y² + z² + 2(-6), which gives 121 = x² + y² + z² - 12. Solving this, x² + y² + z² = 133.

Now, substitute the known values into the cubic identity: 1604 - 3xyz = 11[133 - (-6)]. This simplifies to 1604 - 3xyz = 11(139). Multiplying 11 by 139 gives 1529. Thus, the equation becomes 1604 - 3xyz = 1529. Rearranging to solve for the product term results in 3xyz = 1604 - 1529 = 75. Dividing by 3 yields xyz = 25.

Explanation

Let h = 24 m be the height of the pole and H be the height of the tower. Let d be the horizontal distance between the pole and the tower on the level ground. From the top of the pole, the angle of depression to the base of the tower is 30°, which establishes the relationship tan 30° = (h)/(d). Substituting the known height of the pole, we have (1)/(√(3)) = (24)/(d), which yields d = 24√3 m.

The total height of the tower H can be divided into two segments: the portion equal to the height of the pole (24 m) and the additional height h' above the level of the pole's top. For the upper segment, the angle of elevation is 60°, giving tan 60° = (h')/(d). Substituting the value of d, we get √3 = (h')/(24√(3)), which simplifies to h' = 24 × 3 = 72 m. The total height of the tower is the sum of these two segments: H = 24 + 72 = 96 m.

Explanation

If a number is divisible by 3, 7, and 11, it must be divisible by their least common multiple. Since 3, 7, and 11 are prime numbers, their LCM is 3 × 7 × 11 = 231.

To find the values of P and Q, we divide the smallest possible 5-digit number starting with 750, which is 75000, by 231.

75000 = 231 × 324 + 156

To find the next multiple of 231 that starts with 750, we add the difference between the divisor and the remainder to 75000: 75000 + (231 - 156) = 75000 + 75 = 75075

By comparing 75075 with the given form 750PQ, we identify that P = 7 and Q = 5. Any subsequent multiple, such as 75075 + 231 = 75306, would change the third digit from 0 to 3, failing to match the 750PQ format.

Using the values P = 7 and Q = 5, the required value is: P + 2Q = 7 + 2(5) = 7 + 10 = 17

Explanation

Let V represent the total number of votes polled. The problem states that Raju received 30% of V and Ravi received 32% of V. Additionally, 5% of the total votes were invalid. To find Ashok's share, we subtract the percentages for Raju, Ravi, and the invalid votes from the total 100%: Ashok's votes = 100% - (30% + 32% + 5%) = 100% - 67% = 33% of V.

Comparing the shares of the three candidates, Ashok is the winner with 33% of the total votes, and Raju received the least number of votes with 30%. The difference between the winner and the person with the least votes is given as 5136. We can set up the following equation: 33% of V - 30% of V = 5136 3% of V = 5136 0.03V = 5136

To find the total number of votes polled, we solve for V: V = (5136)/(0.03) V = ⁵¹³⁶⁰⁰⁄₃ V = 171200

Explanation

The relationship between the selling price (SP), cost price (CP), and profit percentage (P) is given by the formula SP = CP × (100 + P)/(100). To find the new selling price for a different profit margin without calculating the cost price separately, we can use the ratio of the two selling prices:

SPₙₑ_w = SPₒₗ_d × 100 + Pₙₑ_w{100 + Pₒₗ_d}

Substituting the given values where SPₒₗ_d = 60,000, Pₒₗ_d = 20%, and Pₙₑ_w = 30%:

SPₙₑ_w = 60,000 × (100 + 30)/(100 + 20) SPₙₑ_w = 60,000 × ¹³⁰⁄₁₂₀ SPₙₑ_w = 60,000 × ¹³⁄₁₂

Dividing 60,000 by 12 gives 5,000. Multiplying this by 13:

SPₙₑ_w = 5,000 × 13 = 65,000

Alternatively, calculating the cost price first: CP = (60,000 × 100)/(120) = 50,000 The new selling price for a 30% profit is: SPₙₑ_w = 50,000 × ¹³⁰⁄₁₀₀ = 65,000

Explanation

The value of p³ + q³ is determined using the algebraic identity p³ + q³ = (p + q)³ - 3pq(p + q). By substituting the given values p + q = 6 and pq = 4 into this expression:

p³ + q³ = (6)³ - 3(4)(6)

Evaluating the powers and products, 6³ = 216 and 3 × 4 × 6 = 72. Substituting these values back into the equation:

p³ + q³ = 216 - 72 = 144

This identity is derived from the expansion of (p + q)³ = p³ + q³ + 3pq(p + q). Rearranging the terms to isolate the sum of cubes allows for a direct calculation using the known sum and product of the variables.

Explanation

The two numbers can be calculated by multiplying their ratios by their HCF, giving 5 × 4 = 20 and 4 × 4 = 16. The LCM of 20 and 16 is 80. Alternatively, LCM = HCF × product of the ratios in simplest form, which is 4 × 5 × 4 = 80.

Explanation

In a right-angled triangle, the cosine of an angle A is defined as the ratio of the base (b) to the hypotenuse (h). Given cos A = ⁹⁄₄₁, we assign b = 9 and h = 41. According to the Pythagorean theorem, the perpendicular (p) is calculated as p = √(h²⁾⁻ᵇ^^^². Substituting the values, p = √(41²⁾⁻⁹^^^² = √(1681 - 81) = √(1600) = 40. The cotangent of angle A is defined as the ratio of the base to the perpendicular, cot A = ᵇ⁄ₚ. Therefore, cot A = ⁹⁄₄₀.

Explanation

In any quadrilateral ABCD, the angle formed by the intersection of the bisectors of two adjacent angles is equal to half the sum of the remaining two angles. This property can be expressed by the formula: ∠ CED = (∠ A + ∠ B)/(2)

This relationship exists because the sum of the internal angles of quadrilateral ABCD is 360°, meaning ∠ A + ∠ B + ∠ C + ∠ D = 360°. In triangle CED, the sum of angles is 180°, so ∠ CED + ½∠ C + ½∠ D = 180°. Multiplying the triangle equation by 2 gives 2∠ CED + ∠ C + ∠ D = 360°. Equating the two expressions for 360° results in ∠ A + ∠ B = 2∠ CED.

Substituting the given values into the formula: 57° = (47° + ∠ B)/(2) 114° = 47° + ∠ B ∠ B = 114° - 47° = 67°

Explanation

The expression is evaluated using the VBODMAS rule, which specifies the sequence of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, resolve the operations inside the square bracket. Evaluating the terms in the nested parentheses gives 12 - 9 = 3 and 3 - 8 = -5. Next, the of operation takes priority inside the bracket, resulting in 3 of 5 = 15. The expression inside the square bracket then becomes 18 ÷ 6 × 15 - (-5). Following the order of operations, we perform the division 18 ÷ 6 = 3, then the multiplication 3 × 15 = 45, and finally the subtraction 45 - (-5) = 45 + 5 = 50.

Next, evaluate the of operation outside the square bracket, which is 5 of 2 = 10. The entire expression simplifies to 40 ÷ 10 × 50 ÷ 25. Following the order of operations from left to right for division and multiplication, we first calculate 40 ÷ 10 = 4. Then, we multiply this result by the value of the square bracket to get 4 × 50 = 200. Finally, dividing 200 by 25 yields the result 8.