SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The side lengths of the triangle are AB = 9 cm, BC = 40 cm, and AC = 41 cm.

We can verify if the triangle is a right-angled triangle using the Pythagorean theorem: AB² + BC² = 9² + 40² = 81 + 1600 = 1681 AC² = 41² = 1681

Since AB² + BC² = AC², the triangle is a right-angled triangle, with AC as the hypotenuse.

For any right-angled triangle, the circum-circle's diameter is equal to the hypotenuse, which means the circum-radius is exactly half of the hypotenuse: Circum-radius = AC / 2 = 41 / 2 = 20½ cm.

Explanation

The formula for the compound amount after n years is: A = P(1 + r/100)ⁿ

Given the Principal (P) is ₹20,000 and the compound interest accrued at the end of 2 years is ₹2,050, the amount after 2 years (A₂) is: A₂ = 20,000 + 2,050 = 22,050

Using the formula for n = 2: 22,050 = 20,000(1 + r/100)² (1 + r/100)² = 22,050 / 20,000 (1 + r/100)² = 441 / 400 1 + r/100 = √(441/400) 1 + r/100 = 21/20

The total amount at the end of the 3rd year (A₃) is: A₃ = P(1 + r/100)³ A₃ = 20,000 × (21/20)³ A₃ = 20,000 × (9261 / 8000) A₃ = 2.5 × 9261 A₃ = 23,152.50

The total compound interest accrued at the end of the 3rd year is: CI = A₃ - P CI = 23,152.50 - 20,000 CI = 3,152.50

Explanation

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

Area = (√3/4) × a²

Given that the side length of the equilateral triangle is a = 22 cm, we substitute this value into the formula:

Area = (√3/4) × 22² Area = (√3/4) × 484 Area = 121√3

The area of the triangle is 121√3 cm².

Explanation

Let the cost price of the motorcycle for Akash be C.

According to the given transaction chain: Navin buys it at a profit of 10%, making his cost price equal to C × 11/10. Vinay buys it at a profit of 20%, making his cost price equal to C × 11/10 × 6/5. Vishal buys it at a loss of 5%, making his cost price equal to C × 11/10 × 6/5 × 19/20.

Given that Vishal paid ₹62,700 for the motorcycle: C × 11/10 × 6/5 × 19/20 = 62700

Simplifying the equation: C × (11 × 6 × 19) / 1000 = 62700 C × 1254 / 1000 = 62700 C = (62700 × 1000) / 1254 C = 50 × 1000 C = 50000

The cost of the motorcycle for Akash was ₹50,000.

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides.

Perimeter(XYZ) / Perimeter(PQR) = XY / PQ

Substituting the given values into this relation:

62 / 42 = XY / 21

Solving for XY:

XY = (62 × 21) / 42

XY = 62 / 2

XY = 31 cm

Explanation

Correct Answer: D

To find the value of (x² + y²)(p² + q²), we can use the given equations: px + qy = 1 qx + py = 2pq / (p² + q²)

Adding these two equations gives: (p + q)(x + y) = 1 + 2pq / (p² + q²) (p + q)(x + y) = (p² + q² + 2pq) / (p² + q²) (p + q)(x + y) = (p + q)² / (p² + q²) x + y = (p + q) / (p² + q²)

Subtracting the second equation from the first gives: (p - q)(x - y) = 1 - 2pq / (p² + q²) (p - q)(x - y) = (p² + q² - 2pq) / (p² + q²) (p - q)(x - y) = (p - q)² / (p² + q²) x - y = (p - q) / (p² + q²)

Using the algebraic identity 2(x² + y²) = (x + y)² + (x - y)²: 2(x² + y²) = [(p + q) / (p² + q²)]² + [(p - q) / (p² + q²)]² 2(x² + y²) = [(p + q)² + (p - q)²] / (p² + q²)² 2(x² + y²) = 2(p² + q²) / (p² + q²)² 2(x² + y²) = 2 / (p² + q²)

Dividing both sides by 2: x² + y² = 1 / (p² + q²)

Multiplying both sides by (p² + q²): (x² + y²)(p² + q²) = 1

Explanation

The length of the direct common tangent (L) between two circles with radii r₁ and r₂ and distance between their centres d is given by the standard geometric relation:

L = √(d² - (r₁ - r₂)²)

Substituting the given values r₁ = 9 cm, r₂ = 4 cm, and d = 13 cm into the formula:

L = √(13² - (9 - 4)²) L = √(169 - 5²) L = √(169 - 25) L = √144 L = 12 cm

Explanation

Let x be the amount of the electricity bill.

According to the given condition, a 13% reduction on the bill amount equals ₹260.39.

We can express this relationship using the formula: Reduction = Bill Amount × Reduction Percentage

Substituting the given values into the formula: 260.39 = x × 13/100

Rearranging the terms to solve for x: x = 260.39 × 100 / 13 x = 26039 / 13 x = 2003

Thus, the total amount of his electricity bill was ₹2,003.

Explanation

If the cost price of the goods is 100, the marked price is 130 (30% above cost price). A 10% discount on the marked price of 130 is 13, making the selling price 117, which results in a gain of 17%.

Explanation

The marked price (MP) of an item is related to its selling price (SP) and the discount percentage (d) by the formula:

MP = SP ÷ (1 - d / 100)

Substituting the given values, where SP = 765 and d = 15:

MP = 765 ÷ (1 - 15 / 100) MP = 765 ÷ 0.85 MP = 76500 ÷ 85 MP = 900

Explanation

Let V be the total number of valid votes cast in the election.

The winning candidate received 55% of the total valid votes, which equals 42,680. This relationship can be expressed as:

0.55 × V = 42,680

Solving for V:

V = 42,680 / 0.55 V = 77,600

The candidate who stood third received 10% of the total valid votes. The number of valid votes received by this candidate is:

10% of V = 0.10 × 77,600 = 7,760

Explanation

Let x = √(a/b). The given equation can be written in terms of x as: x = 8/3 + 1/x

To solve for x, multiply the entire equation by 3x: 3x² - 8x - 3 = 0

Factoring the quadratic equation: (3x + 1)(x - 3) = 0

Since x represents the principal square root of a real positive ratio, x must be positive. Therefore, x = 3.

This gives: √(a/b) = 3 a/b = 9 a = 9b

Substitute a = 9b into the second given equation, a + b = 30: 9b + b = 30 10b = 30 b = 3

Using b = 3, we find a: a = 9 × 3 = 27

Now, calculate the value of ab: ab = 27 × 3 = 81

Explanation

The first train starts from station P at 6 p.m. and travels for 2 hours before the second train starts from station Q at 8 p.m.

Distance covered by the first train in these 2 hours = 75 km/h × 2 hours = 150 km

Remaining distance between the two trains at 8 p.m. = 690 km - 150 km = 540 km

Since the two trains are traveling towards each other, their relative speed is the sum of their individual speeds: Relative speed = 75 km/h + 105 km/h = 180 km/h

Time taken to meet after 8 p.m. = Remaining distance / Relative speed = 540 km / 180 km/h = 3 hours

Adding 3 hours to 8 p.m. gives the meeting time as 11 p.m.

Explanation

To find the year with the highest enrolment, the total enrolment for each year is calculated by summing the number of boys and girls across all three sports (Basketball, Football, and Cricket).

Total Enrolment for 2018 = 46 + 34 + 42 + 37 + 45 + 35 = 239 Total Enrolment for 2019 = 40 + 41 + 44 + 34 + 42 + 32 = 233 Total Enrolment for 2020 = 43 + 40 + 40 + 32 + 38 + 36 = 229 Total Enrolment for 2021 = 37 + 44 + 43 + 25 + 34 + 28 = 211 Total Enrolment for 2022 = 34 + 41 + 41 + 32 + 41 + 39 = 228

Comparing these values, the maximum total enrolment is 239, which was achieved in the year 2018.

Explanation

The perimeter of a parallelogram is given by the formula 2(a + b), where a and b are the lengths of the adjacent sides.

Given that the perimeter is 48 cm and the length of one adjacent side a is 8 cm: 2(8 + b) = 48 8 + b = 24 b = 16 cm

The area of a parallelogram is calculated using the formula: Area = base × height

Using the base b = 16 cm and the given height of 6 cm: Area = 16 × 6 = 96 cm²

Explanation

Using the work rate formula, let the total work be the Least Common Multiple (LCM) of the individual days, which is LCM(8, 20, 24) = 120 units.

The combined daily work rate of Ali, Avtar Singh, and Ashok working together is 120 / 8 = 15 units per day.

The daily work rate of Ali alone is 120 / 20 = 6 units per day.

The daily work rate of Avtar Singh alone is 120 / 24 = 5 units per day.

The daily work rate of Ashok alone is the combined rate minus the individual rates of Ali and Avtar Singh: 15 - (6 + 5) = 4 units per day.

The number of days Ashok alone takes to complete the work is 120 / 4 = 30 days.

Explanation

We simplify the given equation by taking the common denominator:

cos θ / (1 + sin θ) + cos θ / (1 - sin θ) = 4

Factoring out cos θ in the numerator gives:

cos θ [ (1 - sin θ + 1 + sin θ) / (1 - sin² θ) ] = 4

Using the trigonometric identity 1 - sin² θ = cos² θ:

cos θ [ 2 / cos² θ ] = 4

2 / cos θ = 4

cos θ = ½

Since θ is an acute angle, we have θ = 60°.

Explanation

To solve the expression, simplify the two bracketed terms separately.

For the first bracket, find the lowest common multiple of the denominators 6, 8, and 12, which is 24. 1 + 5/6 + 7/8 + 11/12 = 24/24 + 20/24 + 21/24 + 22/24 = 87/24

For the second bracket, find the lowest common multiple of 4 and 8, which is 8. 3/4 - 5/8 = 6/8 - 5/8 = 1/8

Now, divide the first result by the second result: (87/24) ÷ (1/8) = (87/24) × 8 = 87/3 = 29

Explanation

Under a Buy X, Get Y Free offer, the customer receives a total of X + Y items while paying only for X items. The effective discount percentage is calculated using the formula:

Discount % = (Free Items ÷ Total Items) × 100

In a BUY 2 GET 1 FREE offer, the customer pays for 2 items and gets 1 item free, making the total number of items received 3.

Discount % = (1 ÷ 3) × 100 = 33.33%

Explanation

In a circle, the tangent at any point is perpendicular to the radius through the point of contact. Let O be the centre of the circle and P be the point of contact of the tangent from X. This forms a right-angled triangle OPX, with the right angle at P.

Using the Pythagorean theorem: OX² = OP² + PX²

Given that the radius OP = 5 cm and the tangent length PX = 12 cm, substituting these values gives: OX² = 5² + 12² OX² = 25 + 144 OX² = 169 OX = √169 = 13 cm

Explanation

Since OA, OB, and OC are radii of the same circle, we have OA = OB = OC.

In triangle OAB, since OA = OB, the angles opposite to these sides are equal, which gives angle OAB = angle ABO = 40°.

In triangle OAC, since OA = OC, the angles opposite to these sides are equal, which gives angle OAC = angle ACO = 30°.

The measure of angle BAC is the sum of these two angles: angle BAC = angle OAB + angle OAC = 40° + 30° = 70°

By the circle theorem, the angle subtended by an arc at the centre is twice the angle subtended by it at any point on the circumference: angle BOC = 2 × angle BAC = 2 × 70° = 140°

Explanation

The relationship between two different selling prices for the same cost price is given by the formula:

SP₂ / SP₁ = (100 + Gain%) / (100 - Loss%)

Given: First selling price (SP₁) = 1300 Loss% = 35% Desired Gain% = 30%

Substituting these values into the formula:

SP₂ / 1300 = (100 + 30) / (100 - 35)

SP₂ / 1300 = 130 / 65

SP₂ / 1300 = 2

SP₂ = 1300 × 2 = 2600

To gain 30%, the craftsman must sell the table for 2,600.

Explanation

The difference between the simple interest on the same sum for two different time periods at the same rate is given by the formula:

Difference in Simple Interest = P × R × (T₂ - T₁) / 100

Here, the difference in simple interest is ₹9120, the rate of interest R is 8% per annum, the first time period T₁ is 3 years, and the second time period T₂ is 7 years. Let P be the principal sum.

Substituting these values into the formula:

9120 = P × 8 × (7 - 3) / 100

9120 = P × 8 × 4 / 100

9120 = 32P / 100

P = (9120 × 100) / 32

P = 2850000 / 100 (or simply 912000 / 32)

P = 28500

Explanation

Using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b), let a = 2x and b = 3y.

This yields the relation: (2x + 3y)³ = 8x³ + 27y³ + 3(2x)(3y)(2x + 3y) (2x + 3y)³ = 8x³ + 27y³ + 18xy(2x + 3y)

Substitute the given values 2x + 3y = 9 and xy = 3 into the identity: 9³ = 8x³ + 27y³ + 18 × 3 × 9 729 = 8x³ + 27y³ + 486

Subtract 486 from both sides to find the required value: 8x³ + 27y³ = 729 - 486 8x³ + 27y³ = 243

Explanation

Using the rule of alligation, the ratio of the number of technicians to non-technicians is calculated from their respective average salaries and the overall average salary:

Technicians average salary: 12,000 Non-technicians average salary: 4,000 Overall average salary: 6,000

The difference between the technicians average and the overall average is 12,000 - 6,000 = 6,000. The difference between the overall average and the non-technicians average is 6,000 - 4,000 = 2,000.

The ratio of technicians to non-technicians is 2,000 : 6,000, which simplifies to 1 : 3.

Since 1 part represents the 100 technicians, the total number of parts representing all workers is 1 + 3 = 4 parts.

The total number of workers is 100 × 4 = 400.

Explanation

To find the minimum production of a medicine by any company, we compare the production values of all the companies for each medicine from the given table.

For Company A, the production of Medicine P is 1.125 million. For Company B, the production values are 2.225 million for P, 1.175 million for Q, and 2.500 million for R. For Company C, the production values are 1.550 million for P, 1.750 million for Q, and 1.950 million for R.

Comparing all the values in the table: 1.125 < 1.175 < 1.550 < 1.750 < 1.950 < 2.100 < 2.220 < 2.225 < 2.500

The value of 1.125 million is the absolute lowest production value in the entire dataset. This minimum production corresponds to Medicine P manufactured by Company A.

Explanation

In the isosceles triangle SQR with base QR, the sides SQ and SR are equal. Since the sum of angles in a triangle is 180° and the vertex angle ∠QSR = 60°, the base angles are equal: ∠SRQ = ∠SQR = (180° - 60°) ÷ 2 = 60°

In the isosceles triangle PQR with base QR, the sides PQ and PR are equal. With the vertex angle ∠QPR = 100°, the base angles are: ∠PRQ = ∠PQR = (180° - 100°) ÷ 2 = 40°

Since both P and S lie on the same side of the base QR, and the angle ∠SRQ is greater than ∠PRQ, the ray PR lies within the angle SRQ. Therefore, the angle ∠SRP is the difference between these two angles: ∠SRP = ∠SRQ - ∠PRQ = 60° - 40° = 20°

Explanation

Let the shares of A, B, and C be 3x, 2x, and 5x, respectively.

The share of C is 300 more than B, which gives: 5x - 2x = 300 3x = 300 x = 100

The difference between the shares of A and B is: 3x - 2x = x

Substituting the value of x: x = 100

Explanation

The given expression can be simplified using the algebraic identity: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Let a = 0.1, b = 0.68, and c = 0.22.

The individual terms of the identity match the terms in the expression: a² = 0.1² = 0.01 b² = 0.68 × 0.68 c² = 0.22 × 0.22 2ca = 2 × 0.22 × 0.1 = 0.044 2ab = 2 × 0.1 × 0.68 = 0.136 2bc = 2 × 0.22 × 0.68 = 0.44 × 0.68

Thus, the expression is equivalent to: (0.1 + 0.68 + 0.22)² = 1² = 1

Explanation

The division formula relates the dividend, divisor, quotient, and remainder as: Dividend = Divisor × Quotient + Remainder

Given that the remainder is 46, we can determine the divisor: Divisor = 5 × Remainder Divisor = 5 × 46 = 230

Now, we can find the quotient using the divisor: Divisor = 10 × Quotient 230 = 10 × Quotient Quotient = 23

Substituting these values into the division formula gives: Dividend = 230 × 23 + 46 Dividend = 5290 + 46 Dividend = 5336

Explanation

Under compound interest, the amount after n years is given by the formula Aₙ = P(1 + r)ⁿ, where P is the principal sum and r is the annual rate of interest.

The amount in 2 years is: P(1 + r)² = 6120

The amount in 3 years is: P(1 + r)³ = 7340

Dividing the amount of the third year by the amount of the second year gives the growth factor for one year: 1 + r = 7340 / 6120 = 367 / 306

Substituting this growth factor back into the equation for the second year: P(367 / 306)² = 6120 P = 6120 × (306 / 367)² P = 6120 × 93636 / 134689 P = 573052320 / 134689 P ≈ 4254.63

Explanation

Using standard trigonometric identities, we have: csc²B - 1 = cot²B sec²B - 1 = tan²B

Substituting these into the given expression: (csc²B - 1)(sec²B - 1) = cot²B × tan²B

Since tangent and cotangent are reciprocals of each other, cot B × tan B = 1. Therefore: cot²B × tan²B = (cot B × tan B)² = 1² = 1

Explanation

Using the standard trigonometric identity: (sin θ + cos θ)² + (sin θ - cos θ)² = 2

Given that sin θ + cos θ = 1/29, substitute this value into the identity: (1/29)² + (sin θ - cos θ)² = 2 1/841 + (sin θ - cos θ)² = 2 (sin θ - cos θ)² = 2 - 1/841 (sin θ - cos θ)² = 1681/841

Taking the square root on both sides: sin θ - cos θ = ± 41/29

Now, evaluate the required expression: (sin θ + cos θ) / (sin θ - cos θ) = (1/29) / (± 41/29) = ± 1/41

Considering the positive value, the result is 1/41.

Explanation

The radius of a circle is perpendicular to the tangent at the point of contact. This forms a right-angled triangle where the hypotenuse is the distance from the external point to the center of the circle, one leg is the radius, and the other leg is the length of the tangent.

Using the Pythagorean theorem: Length of tangent = √(d² - r²)

Substitute the given values d = 29 cm and r = 20 cm: Length of tangent = √(29² - 20²) Length of tangent = √((29 - 20)(29 + 20)) Length of tangent = √(9 × 49) Length of tangent = 3 × 7 = 21 cm

Explanation

Correct Answer: C

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.

CP = (1032 + 648) / 2 CP = 1680 / 2 CP = 840

To earn a 30% profit, the selling price (SP) must be 130% of the cost price.

SP = 840 × 1.30 SP = 1092

Explanation

For the largest sphere circumscribed by a cube, the sphere is inscribed inside the cube. The diameter of this sphere is equal to the edge length of the cube.

Given the edge length of the cube is 7 cm, the diameter of the sphere is d = 7 cm, which gives a radius of r = 3.5 cm or 7/2 cm.

Using the formula for the volume of a sphere: V = 4/3 × π × r³

Substituting the values: V = 4/3 × 22/7 × (7/2)³ V = 4/3 × 22/7 × 343/8 V = 539/3 V ≈ 179.67 cm³

Explanation

The area of a circle is given by the formula A = πr², where r is the radius.

Using the given area of 154 cm²: 154 = (22/7) × r² r² = 154 × (7/22) r² = 49 r = 7 cm

Since two parallel tangents to a circle touch the circle at the opposite endpoints of a diameter, the line segment PQ connecting the points of contact P and Q must be a diameter of the circle.

Length of PQ = 2r = 2 × 7 = 14 cm.

Explanation

Correct Answer: A

According to the triangle inequality theorem, three line segments can form a triangle if and only if the sum of the lengths of any two sides is strictly greater than the length of the third side. For any three side lengths a, b, and c, this means the three inequalities a + b > c, a + c > b, and b + c > a must all be true.

For the side lengths 8, 6, and 9, we apply this theorem: 8 + 6 = 14, which is greater than 9 8 + 9 = 17, which is greater than 6 6 + 9 = 15, which is greater than 8

Since the sum of any two sides is greater than the third side in every case, these side lengths satisfy the triangle inequality theorem and can successfully form a triangle.

Explanation

The single equivalent discount for two successive discounts of d₁% and d₂% is calculated using the formula:

Equivalent Discount = d₁ + d₂ - (d₁ × d₂)/100 %

Substituting the given values, where d₁ = 10% and d₂ = 10%:

Equivalent Discount = 10 + 10 - (10 × 10)/100 %

Equivalent Discount = 20 - 1 % = 19%

Explanation

When a journey is divided into two equal halves of distance, covered at speeds v₁ and v₂, the average speed V for the entire journey is given by the harmonic mean formula:

V = 2v₁v₂ / (v₁ + v₂)

Here, the speed for the first half of the distance, v₁, is 40 km per hour and the speed for the second half, v₂, is 60 km per hour.

Substituting these values into the formula:

V = 2 × 40 × 60 / (40 + 60) V = 4800 / 100 V = 48 km per hour

This formula yields the average speed directly, making the total travel time of 7 hours redundant for this calculation.

Explanation

Let the efficiency of Deepak be 1 unit per day. Since Babita is 1.25 times as efficient as Deepak, her efficiency is 1.25 units per day.

Their combined efficiency is 1 + 1.25 = 2.25 units per day.

Working together, they complete the work in 40 days, so the total work is: Total Work = Combined Efficiency × Total Days = 2.25 × 40 = 90 units

The number of days Deepak alone will take to complete this work is: Days = Total Work / Deepak's Efficiency = 90 / 1 = 90 days

Explanation

The fourth proportion x to three numbers a, b, and c is defined by the relation a / b = c / x, which gives the formula:

x = (b × c) / a

Substituting the given values, where a = 0.24, b = 0.6, and c = 20:

x = (0.6 × 20) / 0.24

x = 12 / 0.24

Multiplying both the numerator and the denominator by 100 to simplify the calculation:

x = 1200 / 24

x = 50

Explanation

The given expression can be simplified using the algebraic identity for the sum of cubes:

a³ + b³ = (a + b)(a² - ab + b²)

By identifying the numbers in the expression: a³ = 4913, which means a = 17 (since 17³ = 4913) b³ = 343, which means b = 7 (since 7³ = 343) a² = 17² = 289 b² = 7² = 49 ab = 17 × 7 = 119

Substituting these values into the expression:

(4913 + 343) / (289 + 49 - 119) = (17³ + 7³) / (17² + 7² - 17 × 7)

Applying the identity:

(17 + 7)(17² - 17 × 7 + 7²) / (17² + 7² - 17 × 7) = 17 + 7 = 24

Explanation

Using the work-equivalence formula:

M₁ × D₁ = M₂ × D₂

Here, M₁ is the initial number of workers (69), D₁ is the initial number of days (20), M₂ is the final number of workers required, and D₂ is the final number of days (12).

Substituting the given values into the formula:

69 × 20 = M₂ × 12

1380 = 12 × M₂

M₂ = 1380 / 12 = 115 workers

The number of additional workers that need to be employed is the difference between the final number of workers and the initial number of workers:

Additional workers = M₂ - M₁ = 115 - 69 = 46 workers

Explanation

Given the condition p + q + r = 0, we can express r as -(p + q). Substituting this into each denominator:

For the first term: p² - qr = p² - q(-(p + q)) = p² + pq + q²

For the second term: q² - pr = q² - p(-(p + q)) = p² + pq + q²

For the third term: r² - pq = (-(p + q))² - pq = p² + 2pq + q² - pq = p² + pq + q²

Since all three denominators are identical, the expression can be written as a single fraction: (p² + q² + r²) / (p² + pq + q²)

Using r² = (-(p + q))² = p² + 2pq + q², the numerator becomes: p² + q² + (p² + 2pq + q²) = 2p² + 2pq + 2q² = 2(p² + pq + q²)

Thus, the simplified value of the expression is: 2(p² + pq + q²) / (p² + pq + q²) = 2

Alternatively, by choosing convenient values that satisfy p + q + r = 0, such as p = 1, q = 1, and r = -2, the expression evaluates directly to: 1² / (1² - 1 × (-2)) + 1² / (1² - 1 × (-2)) + (-2)² / ((-2)² - 1 × 1) = 1/3 + 1/3 + 4/3 = 6/3 = 2

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, we square both sides of the given equation:

(z + 1/z)² = 4²

z² + 1/z² + 2(z)(1/z) = 16

z² + 1/z² + 2 = 16

z² + 1/z² = 14

Substituting this value into the required expression:

½(z² + 1/z²) = ½ × 14 = 7

Explanation

For any three numbers a, b, and c, their fourth proportional x is defined by the relation:

a/b = c/x

This can be rewritten to solve for x as:

x = (b × c) / a

Given the numbers a = 1.4, b = 8.4, and c = 4.2, substituting these values into the formula gives:

x = (8.4 × 4.2) / 1.4

Simplifying the division first:

8.4 / 1.4 = 6

Now, multiplying the terms:

x = 6 × 4.2 = 25.2

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, we can rewrite sin 55° as sin(90° - 35°) = cos 35°.

Substituting this into the given expression: sin² 35° + sin² 55° = sin² 35° + cos² 35°

Using the fundamental Pythagorean trigonometric identity sin² θ + cos² θ = 1, we get: sin² 35° + cos² 35° = 1

Explanation

Q's speed is calculated by dividing the total race distance of 1000 metres by Q's total time: Speed of Q = 1000 m / 125 s = 8 m/s.

P completes the race in 120 seconds. At this moment, Q needs another 5 seconds (125 s - 120 s) to reach the finish line.

The distance Q covers in these remaining 5 seconds represents the distance by which P beats Q: Distance = Speed of Q × Time difference = 8 m/s × 5 s = 40 m.

Explanation

The length L of the direct common tangent to two circles of radii r₁ and r₂ that touch each other externally is given by the formula:

L = 2√(r₁r₂)

Given the radii of the two circles: r₁ = 15 cm r₂ = 18 cm

Substitute these values into the formula:

L = 2 × √(15 × 18) L = 2 × √270 L = 2 × √(9 × 30) L = 2 × 3 × √30 L = 6√30 cm