SSC CHSL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

Let the salaries of Disha and Pratima be 17x and 15x, respectively.

The difference between their salaries is: 17x - 15x = 2x

Given that Disha's salary is ₹3,000 more than Pratima's, we can write: 2x = 3000 x = 1500

The salary of Pratima is: 15x = 15 × 1500 = 22,500

Explanation

The relationship between Selling Price (SP) and Profit percentage (P) is given by:

SP₂ = SP₁ × (100 + P₂) / (100 + P₁)

Substituting the given values where SP₁ = 70000, P₁ = 25, and P₂ = 30:

SP₂ = 70000 × (100 + 30) / (100 + 25)

SP₂ = 70000 × 130 / 125

Simplifying the fraction:

SP₂ = 70000 × 26 / 25

SP₂ = 2800 × 26

SP₂ = 72800

Explanation

The combined weighted average formula for two groups is:

Combined Average = (n₁ × A₁ + n₂ × A₂) / (n₁ + n₂)

Here, the initial number of men n₁ is 25 with an average age A₁ of 28 years, and the number of new men n₂ is 5 with an average age A₂ of 25 years.

Substituting these values into the formula:

Combined Average = (25 × 28 + 5 × 25) / (25 + 5)

Combined Average = (700 + 125) / 30

Combined Average = 825 / 30

Combined Average = 27.5 years

Explanation

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any non-negative numbers a and b: (a + b)/2 ≥ √(ab) Which can be rewritten as: a + b ≥ 2√(ab)

Let a = 8 cosec²θ and b = 25 sin²θ. Both terms are positive for real values of θ where cosec θ is defined. Applying the inequality: 8 cosec²θ + 25 sin²θ ≥ 2 × √(8 cosec²θ × 25 sin²θ)

Since cosec²θ = 1/sin²θ, the terms inside the square root simplify as follows: 8 cosec²θ + 25 sin²θ ≥ 2 × √(8 × 25) 8 cosec²θ + 25 sin²θ ≥ 2 × √200 8 cosec²θ + 25 sin²θ ≥ 2 × 10√2 8 cosec²θ + 25 sin²θ ≥ 20√2

This minimum value is achievable because the condition for equality, 8 cosec²θ = 25 sin²θ, yields sin²θ = (2√2)/5. Since (2√2)/5 is approximately 0.566, which lies within the valid range of [0, 1] for sin²θ, real values of θ exist that satisfy this condition.

Explanation

When three equal distances are covered at speeds of s₁, s₂, and s₃, the average speed is calculated using the harmonic mean of the speeds:

Average Speed = 3 / (1/s₁ + 1/s₂ + 1/s₃) = 3s₁s₂s₃ / (s₁s₂ + s₂s₃ + s₃s₁)

Substituting the given values s₁ = 15 km/h, s₂ = 20 km/h, and s₃ = 30 km/h:

Average Speed = 3 × 15 × 20 × 30 / (15 × 20 + 20 × 30 + 30 × 15)

Average Speed = 27000 / (300 + 600 + 450)

Average Speed = 27000 / 1350

Average Speed = 20 km/h

Explanation

In the circumcircle of triangle ABC with circumcentre M and circumradius 15 cm, the distance from the circumcentre M to the vertex B is the circumradius, so MB = 15 cm.

Since ML is perpendicular to the chord BC from the centre M, it bisects the chord BC. Therefore, the length of BL is half of BC: BL = BC / 2 = 24 / 2 = 12 cm

In the right-angled triangle MLB, applying the Pythagorean theorem gives: MB² = ML² + BL² 15² = ML² + 12² 225 = ML² + 144 ML² = 225 - 144 ML² = 81 ML = 9 cm

Explanation

By the Midpoint Theorem of triangles, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. Since L and M are the midpoints of sides XY and XZ in triangle XYZ, we have:

LM = ½ × YZ

The point R lies on the segment LM such that:

LR : RM = 1 : 2

Given that LR = 3 cm, we can find RM as:

RM = 2 × LR = 2 × 3 cm = 6 cm

The total length of LM is the sum of LR and RM:

LM = LR + RM = 3 cm + 6 cm = 9 cm

Substituting this value into the Midpoint Theorem relation:

9 = ½ × YZ YZ = 18 cm

Explanation

Let l and w be the length and width of the rectangle, and d be its diagonal. By the Pythagorean theorem, the relationship between these dimensions is:

d² = l² + w²

Substituting the given values, d = 15 cm and l = 12 cm:

15² = 12² + w² 225 = 144 + w² w² = 225 - 144 w² = 81 w = 9 cm

The area of the rectangle is given by the formula:

Area = length × width Area = 12 × 9 = 108 cm²

Explanation

The given equations are: a + b + c = 2 1/a + 1/b + 1/c = 0 ac = 4/b

Simplifying ac = 4/b gives: abc = 4

Simplifying 1/a + 1/b + 1/c = 0 by taking the common denominator abc gives: (ab + bc + ca) / abc = 0 Since abc = 4, we have: ab + bc + ca = 0

In the question statement, the expression a² + b² + c² = 28 contains a typographical error and represents the sum of the cubes: a³ + b³ + c³ = 28

To find the value of a² + b² + c², we apply the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

Substituting the values of a + b + c = 2, abc = 4, ab + bc + ca = 0, and a³ + b³ + c³ = 28 into the identity: 28 - 3 × 4 = 2 × (a² + b² + c² - 0) 28 - 12 = 2 × (a² + b² + c²) 16 = 2 × (a² + b² + c²) a² + b² + c² = 8

Explanation

To find the share of Y, first combine the ratios X:Y and Y:Z into a single continuous ratio X:Y:Z.

Given: X:Y = 7:5 Y:Z = 4:3

To make the value of Y equal in both ratios, multiply the first ratio by 4 and the second ratio by 5: X:Y = 28:20 Y:Z = 20:15

Thus, the combined ratio is: X:Y:Z = 28:20:15

The sum of the ratio terms is: 28 + 20 + 15 = 63

Now, divide the total amount of 6,300 by the sum of the ratio terms to find the value of one share: 6,300 ÷ 63 = 100

Since Y has 20 shares, the share of Y is: 20 × 100 = 2,000

Explanation

The relationship between the number of men and the number of days required to complete a constant amount of work is inversely proportional, which is expressed by the formula:

M₁ × D₁ = M₂ × D₂

Here, M₁ is the initial number of men, D₁ is the initial number of days, M₂ is the new number of men, and D₂ is the new number of days.

Substituting the given values into the formula:

15 × 10 = 20 × D₂

150 = 20 × D₂

D₂ = 150 / 20 = 7.5 days

Explanation

Let the total working days of the school be x.

Since Kavita's attendance of 216 days represents 90% of the total working days:

90% of x = 216

(90 / 100) × x = 216

x = (216 × 100) / 90

x = 24 × 10

x = 240

Explanation

A number is completely divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.

For the number 963391, the digits at the odd positions (starting from the right) are 1, 3, and 6. The sum of these digits is 1 + 3 + 6 = 10.

The digits at the even positions are 9, 3, and 9. The sum of these digits is 9 + 3 + 9 = 21.

The difference between the two sums is 21 - 10 = 11.

Since the difference is 11, which is a multiple of 11, the number 963391 is completely divisible by 11.

Explanation

In the given question, we are asked to find the value of P when the year 2016 is expressed as having 52 weeks and P days.

The year 2016 is a leap year because it is divisible by 4. A leap year contains exactly 366 days.

To find the number of weeks and extra days, we divide 366 by 7: 366 = (52 × 7) + 2

This division yields exactly 52 weeks and 2 extra days. Comparing this with the given representation of 52 weeks and P days, we find that the value of P is 2.

Explanation

Given the equations: cos(x - y) = √3/2 sin(x + y) = ½

Using standard trigonometric values: cos(30°) = √3/2 sin(30°) = ½

We can set up the system of linear equations: x - y = 30° x + y = 30°

Adding these two equations: (x - y) + (x + y) = 30° + 30° 2x = 60° x = 30°

This satisfies the given condition 0 < x < 90°.

Explanation

Using the fundamental trigonometric identity: cosec² A = 1 + cot² A

We know that cosec A is the reciprocal of sin A, so we can write: 1 / sin² A = 1 + cot² A

Given that cot A = k, substituting this value gives: 1 / sin² A = 1 + k²

Taking the reciprocal of both sides: sin² A = 1 / (1 + k²)

Taking the square root of both sides: sin A = 1 / √(1 + k²)

Since A is given as an acute angle, all trigonometric ratios including sin A are positive. Thus, we take the positive square root.

Explanation

Divide both the numerator and the denominator on the left side of the equation by sin θ:

(cot θ + 1) / (cot θ - 1) = 8

Cross-multiply to solve for cot θ:

cot θ + 1 = 8(cot θ - 1)

cot θ + 1 = 8 cot θ - 8

9 = 7 cot θ

cot θ = 9/7

Explanation

Using the successive percentage change formula:

Net change % = x + y + (xy/100)

Here, the markup percentage (x) is +15% and the discount percentage (y) is -20%.

Substituting these values into the formula:

Net change % = 15 - 20 + (15 × (-20) / 100)

Net change % = -5 - (300 / 100)

Net change % = -5 - 3 = -8%

Since the net percentage change is negative, it represents a loss of 8%.

Explanation

By modular arithmetic, 77 can be written as -1 modulo 78 because 77 = 78 - 1.

Substituting this into the given expression: (77⁷⁷ + 77) mod 78 = ((-1)⁷⁷ + (-1)) mod 78

Since 77 is an odd exponent, (-1)⁷⁷ simplifies to -1: = (-1 - 1) mod 78 = -2 mod 78

To obtain the positive remainder, we add the divisor 78 to the negative remainder: 78 - 2 = 76

Explanation

Average speed is defined as the total distance travelled divided by the total time taken.

Total Distance = 18 + 20 + 27 = 65 km

The time taken for each of the three parts of the journey is calculated as distance divided by speed: Time for the first part = 18 / 6 = 3 hours Time for the second part = 20 / 5 = 4 hours Time for the third part = 27 / 9 = 3 hours

Total Time = 3 + 4 + 3 = 10 hours

Average Speed = Total Distance / Total Time = 65 / 10 = 6.5 km/h

Explanation

The standard algebraic identity relating the given terms is: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

Substituting the given values into this identity: 75 - 3 × 15 = 10 × (x² + y² + z² - xy - yz - zx)

Simplifying the left side: 75 - 45 = 10 × (x² + y² + z² - xy - yz - zx) 30 = 10 × (x² + y² + z² - xy - yz - zx)

Dividing both sides by 10 yields: x² + y² + z² - xy - yz - zx = 3

Explanation

The selling price corresponding to a loss of 20% is given as ₹2500. This selling price represents 80% of the cost price CP.

Using the ratio of the two selling prices based on their respective loss percentages:

SP₂ = SP₁ × (100 - Loss₂%) / (100 - Loss₁%)

Substituting the given values:

SP₂ = 2500 × (100 - 30) / (100 - 20) SP₂ = 2500 × 70 / 80 SP₂ = 2500 × 7 / 8 SP₂ = 17500 / 8 = 2187.5

Explanation

Rearranging the given trigonometric equation: 3 sec²x - 4 = 0 3 sec²x = 4 sec²x = 4/3

Taking the positive square root on both sides since x lies in the first quadrant (0 < x < 90): sec x = 2/√3

Since cos x is the reciprocal of sec x: cos x = √3/2

For the interval 0 < x < 90, the angle whose cosine is √3/2 is 30°.

Explanation

In any triangle XYZ with centroid G and midpoints L and M of sides XY and XZ respectively, the line segment LM represents the midsegment parallel to side YZ.

By the midsegment theorem, LM = ½ YZ.

If H is the height of triangle XYZ from vertex X to the base YZ, the distance of the midsegment LM from vertex X is ½ H.

Since G is the centroid of triangle XYZ, its distance from vertex X along the height is ⅔ H.

The perpendicular distance from the centroid G to the line segment LM is: ⅔ H - ½ H = ⅙ H

The area of triangle GLM is given by: Area(ΔGLM) = ½ × LM × (distance from G to LM) = ½ × (½ YZ) × ⅙ H = 1/24 × YZ × H

The area of triangle XYZ is: Area(ΔXYZ) = ½ × YZ × H

Dividing the area of triangle GLM by the area of triangle XYZ gives the ratio: Area(ΔGLM) : Area(ΔXYZ) = (1/24 × YZ × H) : (½ × YZ × H) = 1/12 : ½ = 1:12

Explanation

The perimeter of a rectangle is given by the formula: Perimeter = 2 × (length + breadth)

Let the length and breadth of the rectangular field be 5x and 2x respectively, based on the ratio of 5 : 2.

Substituting the values into the formula: 238 = 2 × (5x + 2x) 238 = 2 × 7x 238 = 14x x = 238 / 14 x = 17

Using the value of x, the length of the field is: Length = 5x = 5 × 17 = 85 m

Explanation

The net profit or loss percentage when there is a markup percentage followed by a discount percentage can be directly calculated using the successive percentage change formula:

Net gain percentage = x + y + xy/100

Here, x represents the markup percentage (+35) and y represents the discount percentage (-20).

Substituting these values into the formula:

Net gain percentage = 35 - 20 + (35 × -20)/100 Net gain percentage = 15 - 700/100 Net gain percentage = 15 - 7 = 8

Since the final value is positive, Mohan makes a gain of 8%.

Explanation

Total runs in 20 matches = 20 × 35 = 700

Total runs in the first 12 matches = 12 × 45 = 540

Total runs in the last 8 matches = 700 - 540 = 160

Average of the last 8 matches = 160 / 8 = 20

Explanation

The formula for simple interest is SI = (P × R × T) / 100, where P is the principal, R is the rate of interest per annum, and T is the time in years.

According to the question, the simple interest is equal to the principal, so SI = P.

Substituting SI = P into the formula: P = (P × R × T) / 100 1 = (R × T) / 100 T = 100 / R

The given rate of interest R is 12 2/4% = 12.5%.

Substituting this rate into the equation: T = 100 / 12.5 = 8 years

Explanation

Apply the BODMAS rule to evaluate the expression: 2.1 + 2.25 ÷ [63 - [7.5 × 8 + (13 - 2.5 × 5)]]

First, solve the innermost parentheses: 13 - 2.5 × 5 = 13 - 12.5 = 0.5

Next, evaluate the middle square bracket: 7.5 × 8 + 0.5 = 60 + 0.5 = 60.5

Then, evaluate the outermost square bracket: 63 - 60.5 = 2.5

Now, substitute this value back into the expression: 2.1 + 2.25 ÷ 2.5

Perform division before addition: 2.25 ÷ 2.5 = 0.9

Finally, add the numbers: 2.1 + 0.9 = 3.0

Explanation

Using the algebraic identity for the square of a binomial: (a + b)² = a² + b² + 2ab

We can rearrange this formula to solve for a² + b²: a² + b² = (a + b)² - 2ab

Substituting the given values of a + b = 11 and ab = 15 into the formula: a² + b² = 11² - 2(15) a² + b² = 121 - 30 a² + b² = 91

Explanation

Since the share of Komal is equal to 2 in both ratios, we can directly combine them to get the joint ratio of Seema : Komal : Rita as 1.5 : 2 : 2.5.

Multiplying each part of the ratio by 2 to convert them into whole numbers gives: Seema : Komal : Rita = 3 : 4 : 5

The sum of the ratio parts is 3 + 4 + 5 = 12.

The total sum of ₹ 3,600 is divided into 12 equal parts. Rita's share corresponds to 5 parts out of 12.

Rita's share = (5 ÷ 12) × 3,600 = 5 × 300 = ₹ 1,500.

Explanation

In the right-angled isosceles triangle PQR, the right angle is at Q with PQ = QR = 10 cm.

The length of the hypotenuse PR is calculated using the Pythagorean theorem: PR = √(PQ² + QR²) = √(10² + 10²) = √200 = 10√2 cm

In a right-angled triangle, the length of the perpendicular drawn from the right-angled vertex to the hypotenuse is equal to the product of the two sides containing the right angle divided by the hypotenuse. If QD is the perpendicular from Q on PR: QD = (PQ × QR) / PR QD = (10 × 10) / 10√2 QD = 100 / 10√2 QD = 10 / √2 QD = 5√2 cm

Alternatively, since PQR is an isosceles right-angled triangle, the perpendicular from Q to PR is also the median to the hypotenuse. The length of the median to the hypotenuse in a right-angled triangle is equal to half the length of the hypotenuse: QD = PR / 2 = 10√2 / 2 = 5√2 cm

Explanation

To find the percentage decrease in the speed of the runner, use the percentage change formula:

Percentage Decrease = (Original Speed - New Speed) / Original Speed × 100

Substituting the given values into the formula: Original Speed = 40 km/h New Speed = 30 km/h

Decrease in Speed = 40 - 30 = 10 km/h

Percentage Decrease = (10 / 40) × 100 = 25%

Explanation

Using the rule of alligation, the ratio of the number of items sold at a profit of 25% to the number of items sold at a loss of 10% (represented as -10%) to obtain a net profit of 4% is calculated as follows:

Ratio of Quantity 1 to Quantity 2 = (Net Profit Percentage - Loss Percentage) / (Profit Percentage - Net Profit Percentage)

Substituting the values into the formula: Ratio = (4 - (-10)) / (25 - 4) = 14 / 21 = 2 / 3

The items sold at a profit and those sold at a loss are in the ratio of 2 : 3.

Total parts in the ratio = 2 + 3 = 5

Given that the total number of items is 40, we find the value of each part: 40 / 5 = 8

The number of items sold at a loss represents 3 parts of the ratio: 3 × 8 = 24

Explanation

To find the minimum number of spherical metal balls needed to make the cylindrical vessel, the total volume of the spherical balls melted must equal the volume of the cylindrical vessel.

The volume of a cylinder is given by the formula V = π R² H, where R is the radius and H is the height of the cylinder. Substituting the given values where R = 6 cm and H = 5 cm: Volume of cylinder = π × 6² × 5 = 180π cm³

The volume of a sphere is given by the formula V = 4/3 π r³, where r is the radius of the sphere. Given that the diameter of each metal ball is 2 cm, the radius r is 1 cm: Volume of one sphere = 4/3 × π × 1³ = 4/3 π cm³

Let N be the number of spherical balls required. Equating the total volume of the balls to the volume of the cylinder: N × 4/3 π = 180π N × 4/3 = 180 N = 180 × 3/4 N = 135

Explanation

For a consecutive sequence of n numbers starting at x, the average is the arithmetic mean of the first and last terms: Average = (first term + last term) / 2

Let the 35 consecutive natural numbers start at x. The first term is x and the last term is x + 34. N = (x + x + 34) / 2 = x + 17

When we drop the first 10 numbers (from x to x + 9) and include the next 10 numbers (from x + 35 to x + 44), the new sequence starts at x + 10 and ends at x + 44. M = (x + 10 + x + 44) / 2 = x + 27

Thus, the difference between the averages is: M - N = (x + 27) - (x + 17) = 10

Using the algebraic identity: M² - N² = (M - N)(M + N)

Substituting the given values: 600 = 10 × (M + N) M + N = 60

Solving this system of linear equations: M - N = 10 M + N = 60

Adding the two equations: 2M = 70 M = 35

Subtracting the first equation from the second: 2N = 50 N = 25

The average of 3M and 5N is: Average = (3M + 5N) / 2 Average = (3 × 35 + 5 × 25) / 2 Average = (105 + 125) / 2 Average = 230 / 2 Average = 115

Explanation

Using the standard algebraic identities, we can rewrite the terms containing θ:

sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² - 2sin²θ cos²θ = 1 - 2sin²θ cos²θ

sin⁶θ + cos⁶θ = (sin²θ + cos²θ)³ - 3sin²θ cos²θ (sin²θ + cos²θ) = 1 - 3sin²θ cos²θ

Substituting these identities into the expression for A:

A = 2(1 - 3sin²θ cos²θ) - 3(1 - 2sin²θ cos²θ)

A = 2 - 6sin²θ cos²θ - 3 + 6sin²θ cos²θ

A = -1

Substituting the constant value A = -1 into the equation for cos α:

cos α = √((3 + A) / (5 + A))

cos α = √((3 - 1) / (5 - 1))

cos α = √(2 / 4)

cos α = √(1 / 2)

cos α = 1 / √2

Since cos α = 1 / √2, the angle α is equal to 45°.

We calculate the value of 3α:

3α = 3 × 45° = 135°

Explanation

Equating the powers of identical bases on both sides of the three equations:

From the first equation, since 8 is 2³, we equate the exponents of base 2: x + y - 2z = 3(8z - 5 - y) x + y - 2z = 24z - 15 - 3y x + 4y - 26z = -15

From the second equation, since 25 is 5², we equate the exponents of base 5: 4y - 6z = 2(y + z) 4y - 6z = 2y + 2z 2y = 8z y = 4z

From the third equation, since 9 is 3², we equate the exponents of base 3: 4x - 3z = 2(x + z) 4x - 3z = 2x + 2z 2x = 5z x = 2.5z

Substituting the expressions for x and y into the first simplified equation: 2.5z + 4(4z) - 26z = -15 2.5z + 16z - 26z = -15 -7.5z = -15 z = 2

Now, solving for x and y: y = 4(2) = 8 x = 2.5(2) = 5

Substituting these values to find the required expression: 2x + 3y + 5z = 2(5) + 3(8) + 5(2) 2x + 3y + 5z = 10 + 24 + 10 = 44

Explanation

From the first equation: 117 cos² A + 129 sin² A = 120

Using the trigonometric identity sin² A = 1 - cos² A: 117 cos² A + 129(1 - cos² A) = 120 117 cos² A + 129 - 129 cos² A = 120 129 - 12 cos² A = 120 12 cos² A = 9 cos² A = 9/12 = 3/4

From the second equation: 170 cos² B + 159 sin² B = 161

Using the trigonometric identity sin² B = 1 - cos² B: 170 cos² B + 159(1 - cos² B) = 161 170 cos² B + 159 - 159 cos² B = 161 11 cos² B + 159 = 161 11 cos² B = 2 cos² B = 2/11

Since sec² B = 1/cos² B: sec² B = 11/2

Now, substituting these values to find cos² A sec² B: cos² A sec² B = (3/4) × (11/2) = 33/8 = 4.125 ≈ 4

Explanation

By simplifying the expression using BODMAS rules:

First, simplify the terms inside the first bracket: [3/7 of 4 1/5 ÷ 18/25 + 17/24]

Convert the mixed fraction: 4 1/5 = 21/5

Solve the "of" operation: 3/7 of 21/5 = 3/7 × 21/5 = 9/5

Perform the division: 9/5 ÷ 18/25 = 9/5 × 25/18 = 5/2

Perform the addition by finding a common denominator: 5/2 + 17/24 = 60/24 + 17/24 = 77/24

Next, simplify the terms inside the second bracket: [289/16 ÷ (3/4 + 2/3)²]

Find the sum inside the parenthesis: 3/4 + 2/3 = (9 + 8)/12 = 17/12

Square the sum: (17/12)² = 289/144

Perform the division: 289/16 ÷ 289/144 = 289/16 × 144/289 = 9

Now, multiply the two simplified bracket values to find A: A = 77/24 of 9 = 77/24 × 9 = 231/8

Calculate the value of 8A: 8A = 8 × 231/8 = 231

Explanation

Let the cost price of the item for A be x.

A sells the item to B at a 20% profit, so the cost price for B is 1.20x.

B sells it to C at a 10% profit, so the selling price for B is: 1.20x × 1.10 = 1.32x

Since B receives Rs. 1,32,000 from C: 1.32x = 1,32,000 x = 1,00,000

Thus, the cost price of the item for A is Rs. 1,00,000.

If C had purchased the item directly from A, he would have spent 5% less than what he spent with B: New selling price for A = 1,32,000 × (1 - 0.05) = 1,32,000 - 6,600 = Rs. 1,25,400

The profit A would have made under this scenario is: Profit = New selling price for A - Cost price for A Profit = 1,25,400 - 1,00,000 = Rs. 25,400

Explanation

Let t be the time taken by the passenger train to complete the journey. Using the formula Speed = Distance / Time, the speed of the passenger train is 1176 / t km/h.

The express train takes 5 hours less than the passenger train, making its travel time t - 5 hours, and its speed 1176 / (t - 5) km/h.

Given that the average speed of the passenger train is 70 km/h less than that of the express train:

1176 / (t - 5) - 1176 / t = 70

Simplifying the equation by dividing all terms by 70:

16.8 / (t - 5) - 16.8 / t = 1

16.8 × (t - (t - 5)) = t(t - 5)

16.8 × 5 = t² - 5t

84 = t² - 5t

t² - 5t - 84 = 0

Factoring this quadratic equation gives:

(t - 12)(t + 7) = 0

Since time must be a positive value, we have t = 12.

Explanation

For any four terms in a proportion a : b :: c : d, the product of extremes equals the product of means, which gives the relation a × d = b × c.

From the first proportion 2145 : x :: 3003 : 42: x = (2145 × 42) / 3003

Simplifying by dividing 42 and 3003 by 21, we get: x = (2145 × 2) / 143 x = 15 × 2 = 30

Now, substituting x = 30 into the second proportion x : 2508 :: y : 11704: y = (30 × 11704) / 2508

Dividing 30 and 2508 by 6: y = (5 × 11704) / 418

Since 11704 / 418 = 28: y = 5 × 28 = 140

Explanation

A number is divisible by 6 if it is simultaneously divisible by both 2 and 3.

The number 3,49,722 ends in the digit 2, which is an even number, meaning it is divisible by 2.

To test divisibility by 3, the sum of the digits of 3,49,722 is calculated as: 3 + 4 + 9 + 7 + 2 + 2 = 27

Since the sum 27 is divisible by 3, the number itself is divisible by 3.

Because 3,49,722 satisfies the divisibility rules for both 2 and 3, it is divisible by 6.

Explanation

To evaluate the expression, we can use the value substitution method by choosing a suitable value for P. Let P = 30°.

For P = 30°, we have sin 30° = ½ and cos 30° = (√3)/2.

Substituting these values into the first term inside the bracket: (√3 + 2 sin P) / (1 - 2 cos P) = (√3 + 2(½)) / (1 - 2((√3)/2)) = (√3 + 1) / (1 - √3)

Substituting these values into the second term inside the bracket: (1 + 2 cos P) / (√3 - 2 sin P) = (1 + 2((√3)/2)) / (√3 - 2(½)) = (1 + √3) / (√3 - 1)

Since (√3 + 1) / (1 - √3) = - (1 + √3) / (√3 - 1), let x = (1 + √3) / (√3 - 1).

The given expression becomes: (-x)³ + x³ = -x³ + x³ = 0

Explanation

We can simplify the given algebraic expression by using a substitution.

Let x = a⁽1/6). Using this substitution, we can express each fractional power of a as an integer power of x: a⁽1/6) = x a⁽1/3) = (a⁽1/6))² = x² a⁽1/2) = (a⁽1/6))³ = x³ a⁽2/3) = (a⁽1/6))⁴ = x⁴

Now, substitute these terms into the given expression: (x⁴ + 2x³ + 3x² + 2x + 1)(x² - 2x + 1) - x³(x³ - 2)

We can simplify the algebraic terms in the parentheses: x⁴ + 2x³ + 3x² + 2x + 1 = (x² + x + 1)² x² - 2x + 1 = (x - 1)²

Multiplying the two factored terms together: (x² + x + 1)²(x - 1)² = [ (x - 1)(x² + x + 1) ]²

Using the standard algebraic cubic identity (x - 1)(x² + x + 1) = x³ - 1, we get: [ x³ - 1 ]² = x⁶ - 2x³ + 1

Now, substitute this result back into the main expression: (x⁶ - 2x³ + 1) - x³(x³ - 2) = x⁶ - 2x³ + 1 - x⁶ + 2x³ = 1

Since the entire expression simplifies to a constant value of 1, it is completely independent of the value of a. Therefore, for a = 7, the value remains 1.

Explanation

Let the daily work done by a man be M and a woman be W. Since the total work remains constant, the work equations can be equated as follows:

(5M + 8W) × 34 = (4M + 18W) × 28

Dividing both sides by 2:

(5M + 8W) × 17 = (4M + 18W) × 14

85M + 136W = 56M + 252W

85M - 56M = 252W - 136W

29M = 116W

M = 4W

This gives the efficiency relation where 1 man is equivalent to 4 women.

The total work expressed in terms of women's daily work is:

Total Work = (5 × 4W + 8W) × 34 = 28W × 34 = 952W

The efficiency of 3 men and 5 women is:

3M + 5W = 3 × 4W + 5W = 17W

The number of days required for 3 men and 5 women to complete the task is:

Days = Total Work ÷ Efficiency = 952W ÷ 17W = 56 days

Explanation

Given Sec A = √11/3, we can find Cos A: Cos A = 3/√11

Using the identity Sin²A = 1 - Cos²A: Sin²A = 1 - (3/√11)² = 1 - 9/11 = 2/11

From Sin²A and Cos²A, we can determine the remaining values: Cosec²A = 1/Sin²A = 11/2 Tan²A = Sin²A/Cos²A = (2/11)/(9/11) = 2/9 Cot²A = 1/Tan²A = 9/2

Substitute these values into the numerator and denominator of the given expression: Numerator = Cosec²A + Tan²A = 11/2 + 2/9 = (99 + 4)/18 = 103/18 Denominator = Sin²A + Cot²A = 2/11 + 9/2 = (4 + 99)/22 = 103/22

Divide the numerator by the denominator: (103/18) / (103/22) = (103/18) × (22/103) = 22/18 = 11/9

Explanation

Let a = 0.1, b = 0.2, and c = 0.3.

Substituting these values into the numerator of the expression: a³ + b³ + c³ = (0.1)³ + (0.2)³ + (0.3)³ = 0.001 + 0.008 + 0.027 = 0.036

The next term in the numerator can be expressed as: 3 × (0.005 + 0.016 + 0.027) = 3 × 0.048 = 0.144 Here, the terms inside the parentheses represent: 0.005 = a²(b + c) = (0.1)² × (0.2 + 0.3) 0.016 = b²(c + a) = (0.2)² × (0.3 + 0.1) 0.027 = c²(a + b) = (0.3)² × (0.1 + 0.2)

The last term of the numerator is: 0.036 = 6abc = 6 × 0.1 × 0.2 × 0.3

Combining these terms gives the standard algebraic expansion of a perfect cube: Numerator = a³ + b³ + c³ + 3a²(b + c) + 3b²(c + a) + 3c²(a + b) + 6abc = (a + b + c)³ Numerator = (0.1 + 0.2 + 0.3)³ = (0.6)³ = 0.216

Now, substituting the values into the denominator: a² + b² + c² = (0.1)² + (0.2)² + (0.3)² = 0.01 + 0.04 + 0.09 = 0.14 The remaining terms are: 0.04 = 2ab = 2 × 0.1 × 0.2 0.06 = 2ca = 2 × 0.3 × 0.1 0.12 = 2bc = 2 × 0.2 × 0.3

Combining these gives the standard expansion of a perfect square: Denominator = a² + b² + c² + 2ab + 2bc + 2ca = (a + b + c)² Denominator = (0.1 + 0.2 + 0.3)² = (0.6)² = 0.36

Thus, the value of A is: A = (a + b + c)³ / (a + b + c)² = a + b + c = 0.6

Using this value of A, we calculate 60A: 60A = 60 × 0.6 = 36

Explanation

Let M, F, and S be the present ages of the man, his father (the grandfather), and his son (the grandson), respectively.

From the given present ratios, we can express F and S in terms of M: F = (5/4)M S = (1/6)M

Four years ago, their ages were M - 4, F - 4, and S - 4. Using the ratio of the man's age to his father's age four years ago: (M - 4) / (F - 4) = 11/14

Substitute F = (5/4)M into this equation: (M - 4) / ((5/4)M - 4) = 11/14 14(M - 4) = 11((5/4)M - 4) 14M - 56 = (55/4)M - 44 14M - (55/4)M = 56 - 44 (1/4)M = 12 M = 48

Now, calculate the present ages of the grandfather and grandson: Present age of grandfather (F) = (5/4) × 48 = 60 Present age of grandson (S) = (1/6) × 48 = 8

We can verify this with the second ratio from four years ago: (M - 4) / (S - 4) = (48 - 4) / (8 - 4) = 44 / 4 = 11/1, which matches the given condition.

To find the ratio of their ages 12 years from now: Grandfather's age 12 years from now = 60 + 12 = 72 Grandson's age 12 years from now = 8 + 12 = 20

The required ratio is: 72 / 20 = 18 / 5, or 18:5.