SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The radius of the circle is half of its diameter, which is 20 / 2 = 10 cm.

Let O be the center of the circle, and A and B be the endpoints of the chord. The line segments OA and OB represent the radii of the circle, so OA = OB = 10 cm.

The length of the chord AB is given as 10 cm. Therefore, in triangle OAB, all three sides are equal: OA = OB = AB = 10 cm. This makes triangle OAB an equilateral triangle. Since every interior angle of an equilateral triangle is 60°, the angle AOB made by the chord at the center of the circle is 60°.

Explanation

The area of an equilateral triangle with side length s is given by the formula:

Area = (√3/4) × s²

Let the side lengths of X, Y, and Z be s_x, s_y, and s_z respectively. According to the problem, the sum of the areas of X and Y is equal to the area of Z:

(√3/4) × s_x² + (√3/4) × s_y² = (√3/4) × s_z²

Dividing the entire equation by the common factor (√3/4) simplifies the relation to:

s_x² + s_y² = s_z²

Substitute the given side lengths s_x = 6 cm and s_y = 8 cm into this equation:

6² + 8² = s_z² 36 + 64 = s_z² 100 = s_z² s_z = √100 = 10 cm

Explanation

Let Jitin's current age be x.

His age 3 years from now will be x + 3, and his age 8 years ago was x - 8.

The average of Jitin's current age and his age 8 years ago is: (x + x - 8) / 2 = x - 4

According to the given condition, his age 3 years from now is double this average: x + 3 = 2 × (x - 4) x + 3 = 2x - 8 x = 11

So, Jitin's current age is 11 years.

His age 5 years ago was: 11 - 5 = 6 years

Explanation

For half-yearly compounding, the annual interest rate is halved and the number of compounding periods is calculated based on six-month intervals.

The rate per half-year R is 20% ÷ 2 = 10%. The number of compounding periods n for 18 months is 18 ÷ 6 = 3.

The formula for the total amount A is: A = P × (1 + R)ⁿ

Substituting the values: A = 6000 × (1 + 0.10)³ A = 6000 × 1.1³ A = 6000 × 1.331 A = 7986

The compound interest is the difference between the final amount and the principal: Compound Interest = A - P Compound Interest = 7986 - 6000 = 1986

Explanation

Using the trigonometric identity tan(90° - θ) = cot θ, we can write tan 70° as tan(90° - 20°) = cot 20°.

Since cot θ = 1/tan θ, we have tan 20° × tan 70° = tan 20° × cot 20° = 1.

Knowing that tan 45° = 1, we substitute these values into the given expression:

3 tan 20° × tan 45° × tan 70° = 3 × 1 × (tan 20° × cot 20°) = 3 × 1 × 1 = 3

Explanation

Using the algebraic identity for the sum of cubes: A³ + B³ = (A + B)³ - 3AB(A + B)

Substitute the given values A + B = -5 and AB = 6 into the identity: A³ + B³ = (-5)³ - 3 × 6 × (-5) A³ + B³ = -125 - 18 × (-5) A³ + B³ = -125 + 90 A³ + B³ = -35

Explanation

According to the BODMAS rule, the operations must be solved in the order of Bracket, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the percentage and "of" terms: 25% of 720 = (25/100) × 720 = 180 15% of 300 = (15/100) × 300 = 45

Next, calculate the compound "of" term: 1/3 of 100/11 of 33/25 = 1/3 × 100/11 × 33/25 = 1/3 × 4 × 3 = 4

Substitute these simplified values back into the main expression: 180 + 57 ÷ 19 × 4 - 45

Perform the division next: 57 ÷ 19 = 3

Now, perform the multiplication: 3 × 4 = 12

Finally, perform the addition and subtraction: 180 + 12 - 45 = 192 - 45 = 147

Explanation

The maximum length of a stick that can be placed inside a cuboid of length l, breadth b, and height h is equal to the length of its body diagonal, which is given by the formula d = √(l² + b² + h²).

We are given: l + b + h = 20 2(lb + bh + hl) = 256

Using the algebraic identity: (l + b + h)² = l² + b² + h² + 2(lb + bh + hl)

Substitute the given values into the identity: 20² = l² + b² + h² + 256 400 = l² + b² + h² + 256 l² + b² + h² = 400 - 256 l² + b² + h² = 144

Taking the square root on both sides to find the diagonal: d = √144 = 12 cm

Explanation

We are given the trigonometric equation sin 2θ = √3/2. Since sin 60° = √3/2, we can equate the angles:

2θ = 60° θ = 30°

Using this value of θ to find the value of tan θ:

tan θ = tan 30° = 1/√3

Explanation

Let the speed of the boat in still water be 8x and the speed of the stream be 3x.

The downstream speed is the sum of the speed of the boat and the stream: Downstream speed = 8x + 3x = 11x

The boat covers 22 km downstream in 24 minutes. Converting minutes to hours, 24 minutes is 24/60 hours, which simplifies to 2/5 hours.

Using the speed formula, speed = distance / time: Downstream speed = 22 / (2/5) = 22 × 5/2 = 55 km/h

Equating the two expressions for downstream speed: 11x = 55 x = 5

The upstream speed is the difference between the speed of the boat and the stream: Upstream speed = 8x - 3x = 5x = 5 × 5 = 25 km/h

The time taken to cover 50 km upstream is: Time = Distance / Upstream speed = 50 / 25 = 2 hours

Explanation

Using the rate of work formula, the combined rate of the three pipes working together is the sum of their individual rates, where filling is positive and emptying is negative.

Rate of P = 1/5 of the tank per hour Rate of Q = 1/10 of the tank per hour Rate of R = -1/20 of the tank per hour

Combined rate = 1/5 + 1/10 - 1/20

Taking the common denominator of 20:

Combined rate = 4/20 + 2/20 - 1/20 = 5/20 = 1/4 of the tank per hour

The time taken to fill the tank is the reciprocal of the combined rate:

Time = 1 / (1/4) = 4 hours

Explanation

Using the algebraic identity for the square of a binomial: (x + 1/x)² = x² + 1/x² + 2

Substituting the given value x² + 1/x² = 7: (x + 1/x)² = 7 + 2 = 9

Taking the square root: x + 1/x = 3

Now, using the algebraic identity for the sum of cubes: x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

Substituting the value x + 1/x = 3: x³ + 1/x³ = 3³ - 3(3) x³ + 1/x³ = 27 - 9 = 18

Explanation

From the given relationships: 30% of P = 15% of Q 30 × P = 15 × Q Q = 2 P

Similarly: 20% of Q = 10% of R 20 × Q = 10 × R R = 2 Q

Substituting Q = 2 P into the second equation: R = 2 × 2 P = 4 P

To find by what percent R is more than P: Percentage increase = (R - P) / P × 100% Percentage increase = (4 P - P) / P × 100% = 3 P / P × 100% = 300%

Explanation

The total cost price of the milk is calculated by multiplying the quantity of milk by its cost per litre: Cost Price = 30 litres × Rs. 50/litre = Rs. 1500

Since 20 litres of water is added free of cost, the total volume of the mixture becomes: Total Volume = 30 litres + 20 litres = 50 litres

The total selling price is calculated by multiplying this total volume by the selling price per litre: Selling Price = 50 litres × Rs. 48/litre = Rs. 2400

The profit earned by the merchant is: Profit = Selling Price - Cost Price = Rs. 2400 - Rs. 1500 = Rs. 900

The profit percentage is: Profit Percentage = (Profit / Cost Price) × 100 = (900 / 1500) × 100 = 60%

Explanation

The area of an equilateral triangle with side length a is given by the formula (√3/4) × a². Substituting a = 12 cm: Area of the triangle = (√3/4) × 12² = 36√3 cm² Using the approximation √3 ≈ 1.732: Area of the triangle ≈ 36 × 1.732 = 62.352 cm²

The radius r of the incircle of an equilateral triangle with side length a is given by r = a / (2√3). Substituting a = 12 cm: r = 12 / (2√3) = 2√3 cm

The area of the incircle is given by π × r². Substituting the value of r and using π ≈ 22/7: Area of the incircle = π × (2√3)² = 12π ≈ 12 × 22/7 ≈ 37.714 cm²

The area of the triangle not covered by the incircle is the difference between the area of the equilateral triangle and the area of the incircle: Required area = 62.352 - 37.714 = 24.638 cm²

Rounding to two decimal places gives 24.64 sq.cm.

Explanation

Let the marked prices of articles A and B be MP_A and MP_B respectively.

According to the given condition, the value of the 20% discount on article A is equal to the value of the 15% discount on article B: 20% of MP_A = 15% of MP_B

This can be written as: 20/100 × MP_A = 15/100 × MP_B

Simplifying this relation gives: 20 × MP_A = 15 × MP_B

To find the ratio of the marked price of article A to that of article B: MP_A / MP_B = 15 / 20

Reducing the fraction to its simplest form by dividing both the numerator and the denominator by 5: MP_A / MP_B = 3 / 4

The ratio of the marked prices of articles A and B is 3 : 4.

Explanation

We can solve this quickly by substituting a convenient value for θ. Let θ = 45°.

The trigonometric values are: sec 45° = √2 tan 45° = 1 cosec 45° = √2 cot 45° = 1

This gives: m = √2 - 1 n = √2 + 1

Substitute these values into the given expression m + n(m - 1): E = (√2 - 1) + (√2 + 1)((√2 - 1) - 1) E = √2 - 1 + (√2 + 1)(√2 - 2) E = √2 - 1 + (2 - 2√2 + √2 - 2) E = √2 - 1 - √2 E = -1

Alternatively, using algebraic simplification: m = (1 - sin θ)/cos θ n = (1 + cos θ)/sin θ

Rearranging the expression: m + n(m - 1) = m(n + 1) - n

Now calculate m(n + 1): m(n + 1) = ((1 - sin θ)/cos θ) × ((1 + cos θ + sin θ)/sin θ) m(n + 1) = (1 + cos θ + sin θ - sin θ - sin θ cos θ - sin² θ) / (sin θ cos θ) m(n + 1) = (1 - sin² θ + cos θ - sin θ cos θ) / (sin θ cos θ) m(n + 1) = (cos² θ + cos θ - sin θ cos θ) / (sin θ cos θ) m(n + 1) = cos θ (cos θ + 1 - sin θ) / (sin θ cos θ) m(n + 1) = (cos θ + 1 - sin θ) / sin θ m(n + 1) = (1 + cos θ)/sin θ - 1 m(n + 1) = n - 1

Substituting this back: m(n + 1) - n = (n - 1) - n = -1

Explanation

Jitendra's income is given as Rs.2,800. Since the ratio of the incomes of Jitendra and Narendra is 4 : 7, Narendra's income is calculated as 2800 × 7/4 = Rs.4,900.

Jitendra saves Rs.400, which means his expenditure is Rs.2,800 - Rs.400 = Rs.2,400.

Since the ratio of their expenditures is 6 : 5, Narendra's expenditure is calculated as 2400 × 5/6 = Rs.2,000.

Narendra's savings is the difference between his income and his expenditure, which is Rs.4,900 - Rs.2,000 = Rs.2,900.

Explanation

Using the algebraic identity (a + b)² = a² + 2ab + b², we can rewrite the given expression by distributing the denominator:

(4x² + 9y² + 12xy)/144 = (4/144)x² + (9/144)y² + (12/144)xy

Simplifying each fraction gives:

(1/36)x² + (1/16)y² + (1/12)xy

We can express the terms as perfect squares:

(1/36)x² = (x/6)² (1/16)y² = (y/4)²

The middle term can be written as:

2 × (x/6) × (y/4) = xy/12

Therefore, the expression is a perfect square of the form:

(x/6 + y/4)²

Explanation

Let CP be the cost price of one pen.

Loss is defined as Total Cost Price minus Total Selling Price: Loss = 18 × CP - 18 × SP

Given that the loss is equal to the cost price of 3 pens (3 × CP) and the selling price of 18 pens (18 × SP) is Rs. 360: 3 × CP = 18 × CP - 360 18 × CP - 3 × CP = 360 15 × CP = 360 CP = 360 ÷ 15 = 24

The cost price of one pen is Rs. 24.

Explanation

Correct Answer: D

When two identical circles of radius r intersect such that each passes through the center of the other, the distance between their centers is equal to the radius r.

The length of the common chord of two such intersecting circles is given by the standard formula: Length of common chord = r√3

Given that the radius of each circle r = 21 cm, we substitute this value into the formula: Length of common chord = 21√3 cm

This relation is derived using Pythagoras' theorem. The line joining the centers and the common chord bisect each other perpendicularly. This forms a right-angled triangle where the hypotenuse is the radius r, the base is half the distance between the centers (r/2), and the perpendicular is half the length of the common chord.

Applying Pythagoras' theorem: (Half of common chord)² = r² - (r/2)² = 3r²/4 Half of common chord = (r√3)/2

Multiplying by 2 yields the total length of the common chord as r√3.

Explanation

Let the speed of the car be V km/hr. Since the speed of the train is 30% more than the speed of the car, the speed of the train is 1.3V km/hr.

The distance between point P and point Q is 130 km. The time taken by the car to cover this distance is 130/V hours, and the time taken by the train is 130/(1.3V) hours.

Because both start and arrive at the same time but the train loses 30 minutes (0.5 hours) at a station, the difference in their travel times is 0.5 hours. This relation is expressed as:

130/V - 130/(1.3V) = 0.5

Simplify the equation by factoring out 130/V:

(130/V) × (1 - 1/1.3) = 0.5 (130/V) × (0.3/1.3) = 0.5 (130/V) × (3/13) = 0.5 30/V = 0.5 V = 60

The speed of the car is 60 km/hr. The speed of the train is:

1.3 × 60 = 78 km/hr

Explanation

The algebraic identity for the sum of cubes is: a³ + b³ = (a + b)³ - 3ab(a + b)

By substituting a = 5x and b = y, the expression becomes: 125x³ + y³ = (5x + y)³ - 15xy(5x + y)

Now, substitute the given values 5x + y = 17 and xy = 6 into the equation: 125x³ + y³ = 17³ - 15 × 6 × 17 125x³ + y³ = 4913 - 1530 125x³ + y³ = 3383

Explanation

We can simplify the given expression by grouping the terms: ((sec A - tan A) + 1)((sec A - tan A) - 1)

This is in the form (x + y)(x - y) = x² - y², where x = sec A - tan A and y = 1: (sec A - tan A)² - 1²

Expanding the squared term: sec² A + tan² A - 2 sec A tan A - 1

Using the fundamental trigonometric identity sec² A - 1 = tan² A, we can rewrite the expression as: (sec² A - 1) + tan² A - 2 sec A tan A = tan² A + tan² A - 2 sec A tan A = 2 tan² A - 2 sec A tan A

Factoring out 2 tan A: 2 tan A (tan A - sec A)

Explanation

Using the algebraic identity for the square of a trinomial: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Substituting x = a, y = -b, and z = c into this identity gives: (a - b + c)² = a² + (-b)² + c² + 2(a)(-b) + 2(-b)(c) + 2(c)(a)

Simplifying the terms: (a - b + c)² = a² + b² + c² - 2ab - 2bc + 2ca

Explanation

The number of integers strictly between 23 and 43 is calculated as 43 - 23 - 1 = 19. These integers are 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, and 42.

Among these 19 integers, the prime numbers are 29, 31, 37, and 41, which gives a total of 4 prime numbers.

Subtracting the number of prime numbers from the total integers gives the number of composite numbers: 19 - 4 = 15. These 15 composite numbers are 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, and 42.

Explanation

The profit sharing ratio in a partnership is equal to the ratio of the product of investment and time period.

Let the investment duration of Q be t months.

Profit of P / Profit of Q = (Investment of P × Time of P) / (Investment of Q × Time of Q)

Using the given values: 4/3 = (4000 × 12) / (6000 × t) 4/3 = 48000 / 6000t 4/3 = 8 / t 4 × t = 24 t = 6 months

Since the total investment period is one year (12 months), the number of months after which Q joined is: 12 - t = 12 - 6 = 6 months.

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.

Let R be the common annual rate of interest.

The simple interest received from B is: SI = (12000 × R × 3) / 100 = 360R

The simple interest received from C is: SI = (10000 × R × 4) / 100 = 400R

The total interest received from both is ₹6,080: 360R + 400R = 6080 760R = 6080 R = 6080 / 760 R = 8

The annual rate of interest is 8 percent.

Explanation

Using the work-rate formula, (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂, where M represents the number of women, D represents the number of days, and W represents the fraction of work completed.

Given that M₁ = 10 women, D₁ = 20 days, and W₁ = ½. For the remaining portion, the work to be done is W₂ = ½ and the time allowed is D₂ = 10 days. Let M₂ be the total number of women required for this stage.

Substituting these values into the formula: (10 × 20) / ½ = (M₂ × 10) / ½

Since the work fraction is identical on both sides, it cancels out: 10 × 20 = M₂ × 10 M₂ = 20 women

To find how many more women are required, subtract the initial number of women from the total needed: 20 - 10 = 10 women

Explanation

Using the rule of allegation, the ratio of the number of passed students (P) to failed students (F) is: P : F = (50 - 35) : (55 - 50) = 15 : 5 = 3 : 1

This means the ratio of the number of passed students to failed students is 3 : 1.

The total marks obtained by any group is the product of the number of students and their average marks.

Total marks of passed students = 3 × 55 = 165 Total marks of failed students = 1 × 35 = 35

The respective ratio of the total marks of passed students to failed students is: 165 : 35 = 33 : 7

Explanation

In a right-angled triangle PQR where angle Q = 90°, the trigonometric ratio cot P is defined as the ratio of the side adjacent to angle P to the side opposite to angle P.

cot P = Adjacent side / Opposite side = PQ / QR

Given that PQ = 9 cm and QR = 11 cm, substituting these values into the formula gives:

cot P = 9/11

Explanation

Volume of a sphere with radius R is given by V₁ = 4/3 π R³.

Volume of a hemisphere with radius r is given by V₂ = 2/3 π r³.

According to the given condition, the radius of the sphere is thrice the radius of the hemisphere, which means R = 3r.

The ratio of the volume of the sphere to the volume of the hemisphere is: V₁ / V₂ = (4/3 π R³) / (2/3 π r³) = 2 × (R / r)³

Substituting R = 3r into this formula: V₁ / V₂ = 2 × (3r / r)³ = 2 × 3³ = 2 × 27 = 54

This yields the ratio of 54:1.

Explanation

According to the degree measure theorem in geometry, the angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Using this relation: Angle at the centre = 2 × Angle on the major arc

Substituting the given value of 65°: Angle at the centre = 2 × 65° = 130°

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, let a = tan θ and b = cot θ.

Since tan θ and cot θ are reciprocal to each other, their product is tan θ × cot θ = 1.

Substituting these values into the identity: (tan θ + cot θ)² = tan² θ + cot² θ + 2(tan θ × cot θ)

Given that tan θ + cot θ = 8, we substitute this into the equation: 8² = tan² θ + cot² θ + 2(1) 64 = tan² θ + cot² θ + 2 tan² θ + cot² θ = 64 - 2 tan² θ + cot² θ = 62

Explanation

The discount percentage is calculated using the standard formula:

Discount Percentage = (Discount / List Price) × 100

First, determine the discount amount by subtracting the selling price from the listed price:

Discount = List Price - Selling Price Discount = Rs.20,000 - Rs.14,000 = Rs.6,000

Next, substitute the values into the formula to find the percentage:

Discount Percentage = (6,000 / 20,000) × 100 = 3/10 × 100 = 30%

Explanation

The inradius r of an equilateral triangle with side length a is given by the formula:

r = a / 2√3

Substituting the given inradius r = 9√3:

9√3 = a / 2√3

Solving for a:

a = 9√3 × 2√3 a = 18 × 3 a = 54 cm

The perimeter of an equilateral triangle is three times its side length:

Perimeter = 3a Perimeter = 3 × 54 Perimeter = 162 cm

Explanation

Using the algebraic identity for the difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)

We are given that a³ - b³ = 988 and a - b = 4. Substituting these values into the formula: 988 = 4(a² + ab + b²)

Dividing both sides by 4: a² + ab + b² = 247

Multiplying this equation by 2 gives the required expression: 2a² + 2ab + 2b² = 2 × 247 = 494

Explanation

According to the circle geometry theorem, the angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the major segment of the circle.

By this relation: Angle at the center = 2 × Angle in the major segment

Given that the angle subtended at the center O is ∠AOB = 130°. Substituting this value into the relation: 130° = 2 × Angle in the major segment Angle in the major segment = 130° / 2 Angle in the major segment = 65°

Explanation

Let T be the total marks of the exam. The difference between the two percentage scores is: 50% - 20% = 30% of T

This percentage difference corresponds to the total marks difference between being 25 marks short of passing and 20 marks above passing: 25 + 20 = 45 marks

Therefore: 30% of T = 45 T = (45 × 100) / 30 = 150 marks

The passing marks can be calculated using the first case, where Rajiv scored 20% of the total marks and failed by 25 marks: Passing Marks = 20% of 150 + 25 Passing Marks = (20 / 100) × 150 + 25 = 30 + 25 = 55

Explanation

Let O be the center of the circle, and let r be the radius of the circle. The line segments OA and OB represent the radii from the center to the endpoints of the chord AB, so OA = OB = r. Since the length of the chord AB is also equal to the radius of the circle, AB = r.

In triangle OAB, all three sides are equal in length (OA = OB = AB = r), which makes it an equilateral triangle. Every interior angle of an equilateral triangle is equal to 60°. Therefore, the angle subtended by the chord AB at the center of the circle, which is angle AOB, is 60°.

Explanation

We are given the equation (x² - 1)/x = 8, which simplifies to: x - 1/x = 8

We need to find the value of (x⁶ - 1)/x³, which simplifies to: x³ - 1/x³

Using the standard algebraic identity: a³ - b³ = (a - b)³ + 3ab(a - b)

For a = x and b = 1/x, the identity becomes: x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

Substituting the given value x - 1/x = 8 into the equation: x³ - 1/x³ = 8³ + 3 × 8 x³ - 1/x³ = 512 + 24 = 536

Explanation

The radius of the circle is r = 10 cm. The diameter of the circle is d = 2r = 2 × 10 = 20 cm.

Since the length of the chord is 20 cm, which is equal to the diameter, this chord is the diameter of the circle. According to the geometric properties of a circle, the angle subtended by a diameter at any point on the circumference of the circle is a right angle, which measures 90°.

Explanation

Using the work-efficiency relationship, Work = Number of workers × Efficiency × Time.

Let the daily work efficiency of a man be M and that of a boy be B. Since the total work is constant: 12M × 8 = 4B × 40 96M = 160B M/B = 160/96 = 5/3

Assigned efficiencies are M = 5 and B = 3. Total Work = 12 × 5 × 8 = 480 units

The combined daily efficiency of 3 men and 9 boys is: 3M + 9B = 3(5) + 9(3) = 15 + 27 = 42 units/day

The time required for 3 men and 9 boys to complete the work is: Total Work ÷ Combined efficiency = 480 ÷ 42 = 80/7 days

Explanation

To find the value of (6cosθ + sinθ) / (6cosθ - 4sinθ), divide both the numerator and the denominator by sinθ:

(6cosθ/sinθ + sinθ/sinθ) / (6cosθ/sinθ - 4sinθ/sinθ)

Since cotθ = cosθ / sinθ, the expression simplifies to:

(6cotθ + 1) / (6cotθ - 4)

Given that 6cotθ = 5, substitute this value directly into the expression:

(5 + 1) / (5 - 4) = 6 / 1 = 6

Explanation

Using the relation Distance = Speed × Time, the distance x covered after repairing is: x = 108 × 3 = 324 km

The new distance to be covered before repairing is 5x: 5x = 5 × 324 = 1620 km

Using the relation Time = Distance / Speed, the time taken to cover this distance before repairing at a speed of 72 km/h is: Time = 1620 ÷ 72 = 22.5 hours

Converting 22.5 hours into hours and minutes yields 22 hours and 30 minutes.

Alternatively, this can be solved directly using ratios: Time = 5x / 72 Since x = 108 × 3, substituting x gives: Time = 5 × 108 × 3 ÷ 72 = 15 × 108 ÷ 72 = 15 × 1.5 = 22.5 hours, which is 22 hours 30 minutes.

Explanation

Let P be the income of P. According to the given condition, one-third of P's income is Rs. 4400 less than half of P's income:

½ P - ⅓ P = 4400

Subtracting the fractions on the left side:

(3P - 2P) / 6 = 4400

P/6 = 4400

P = 4400 × 6 = 26400

The ratio of the incomes of P and Q is given as P:Q = 1:2. This means Q's income is twice that of P's income:

Q = 2 × P = 2 × 26400 = 52800

Explanation

To simplify the expression, let E = (2² + 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1) ... (2¹²⁸ + 1).

Multiply and divide the entire expression by (2² - 1):

E = [(2² - 1)(2² + 1)(2⁴ + 1)(2⁸ + 1) ... (2¹²⁸ + 1)] / (2² - 1)

Using the algebraic identity (a - b)(a + b) = a² - b²:

(2² - 1)(2² + 1) = (2²)² - 1 = 2⁴ - 1

Substituting this back into the numerator:

E = [(2⁴ - 1)(2⁴ + 1)(2⁸ + 1) ... (2¹²⁸ + 1)] / (2² - 1)

Applying the identity successively:

(2⁴ - 1)(2⁴ + 1) = 2⁸ - 1 (2⁸ - 1)(2⁸ + 1) = 2¹⁶ - 1

This pattern continues until the last term is multiplied:

(2¹²⁸ - 1)(2¹²⁸ + 1) = 2²⁵⁶ - 1

Since the denominator (2² - 1) is equal to 3, the expression simplifies directly to:

(2²⁵⁶ - 1) / 3

Explanation

The student got 64 marks and failed by 20% of the passing marks, which means the student's marks are 80% of the passing marks. Therefore, the passing marks are 64 / 0.80 = 80. Since the passing percentage is 40%, the maximum marks of the exam are 80 / 0.40 = 200.

Explanation

In an equilateral triangle, the centroid, circumcentre, orthocentre, and incentre coincide. The centroid divides each median, which also serves as the altitude, in the ratio 2:1.

The distance from the vertex to the centroid is the circumradius (R), which represents two-thirds of the total length of the median (h). This relationship is expressed by the formula:

R = 2/3 × h

Rearranging the formula to find the median:

h = 3/2 × R

Substituting the given circumradius of 14 cm into the equation:

h = 3/2 × 14 = 21 cm

The length of the median in this triangle is 21 cm.

Explanation

The given series is 1 - 7 + 2 - 8 + 3 - 9 + 4 - 10 + ... to 100 terms.

By grouping the terms of the series into pairs of two consecutive terms, we get: (1 - 7) + (2 - 8) + (3 - 9) + (4 - 10) + ... to 50 pairs

Each pair simplifies to -6: -6 - 6 - 6 - 6 ... to 50 terms

Using the multiplication formula for repeated addition, the sum of these 50 equal terms is: Sum = 50 × (-6) = -300