SSC CHSL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The algebraic identity for the sum of cubes of three variables is: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Given that a + b + c = 0, substituting this value into the identity yields: a³ + b³ + c³ - 3abc = 0 × (a² + b² + c² - ab - bc - ca) = 0

This directly gives the value of the expression as 0.

Explanation

The distance travelled by a wheel in one complete revolution is equal to its circumference.

The formula for the circumference of a wheel is: Circumference = 2πr

Given: Radius (r) = 14 cm Number of revolutions = 10

Using π ≈ 22/7, we calculate the circumference: Circumference = 2 × (22/7) × 14 = 88 cm

The total distance travelled in 10 revolutions is: Total Distance = 10 × Circumference Total Distance = 10 × 88 = 880 cm

Explanation

The fundamental trigonometric identity states: sec² θ - tan² θ = 1

Using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), this can be rewritten as: (sec θ - tan θ)(sec θ + tan θ) = 1

Substituting the given value of sec θ + tan θ = 5 into this equation: (sec θ - tan θ) × 5 = 1

Solving for the required expression gives: sec θ - tan θ = 1/5

Explanation

A chord that subtends an angle of 180 degrees at the center of a circle is a diameter of that circle. Since the length of this diameter is 36 cm, the radius is half of this length, which is 18 cm.

Explanation

To simplify the given expression, convert sec²A and cosec²A into terms of cos²A and sin²A:

The first term is: (1 + sec²A) / (1 + cos²A) = (1 + 1/cos²A) / (1 + cos²A) = ((cos²A + 1) / cos²A) / (1 + cos²A) = 1/cos²A

The second term is: (1 + sin²A) / (1 + cosec²A) = (1 + sin²A) / (1 + 1/sin²A) = (1 + sin²A) / ((sin²A + 1) / sin²A) = sin²A

Multiplying the two terms yields: (1/cos²A) × sin²A = sin²A / cos²A = tan²A

Alternatively, substituting A = 45° where sin²45° = ½, cos²45° = ½, sec²45° = 2, and cosec²45° = 2 gives: ((1 + 2) / (1 + ½)) × ((1 + ½) / (1 + 2)) = (3 / (3/2)) × ((3/2) / 3) = 2 × ½ = 1

Since tan²45° = 1, the expression simplifies directly to tan²A.

Explanation

The expression represents a continued fraction that can be simplified step-by-step from the innermost level upward:

First, evaluate the bottom-most fraction: 5 + 3/5 = 28/5

Substitute this value into the next level: 5 + 5/28 = 145/28

Continue the simplification with the next reciprocal term: 5 + 28/145 = 753/145

Proceeding through the remaining levels of the nested fraction: 2 + 290/753 = 1796/753

Evaluating the final division yields the simplified rational value: 25/123

Explanation

Heron's formula is used to find the area of a triangle when all three side lengths are known. The semi-perimeter s is calculated as: s = (a + b + c) / 2

Substituting the given side lengths a = 13, b = 15, and c = 14: s = (13 + 15 + 14) / 2 = 21 inches

Using Heron's formula for the area: Area = √(s(s - a)(s - b)(s - c)) Area = √(21 × (21 - 13) × (21 - 15) × (21 - 14)) Area = √(21 × 8 × 6 × 7) Area = √(1764 × 4) Area = √7056 Area = 84 square inches.

Explanation

According to the given condition, the relationship can be written as:

0.20 × P + 0.40 × Q = 0.80 × Q

Subtracting 0.40 × Q from both sides:

0.20 × P = 0.40 × Q

Dividing both sides by 0.20:

P = (0.40 / 0.20) × Q P = 2 × Q

To find what percentage P is of Q, we calculate:

(P / Q) × 100 = 2 × 100 = 200%

Explanation

Let MP be the marked price and SP be the selling price.

According to the given condition: (2/5) × MP = (5/8) × SP

Rearranging the terms to find the ratio of the selling price to the marked price: SP/MP = (2/5) / (5/8) SP/MP = (2/5) × (8/5) SP/MP = 16/25

The formula for the discount percentage is: Discount Percent = (1 - SP/MP) × 100%

Substituting the ratio of SP to MP into the formula: Discount Percent = (1 - 16/25) × 100% Discount Percent = (9/25) × 100% Discount Percent = 36%

Explanation

A perpendicular drawn from the centre of a circle to a chord bisects the chord. This halves the chord length to 15 cm.

The radius of the circle, the half-chord, and the distance of the chord from the centre form a right-angled triangle where the radius is the hypotenuse. Applying the Pythagorean theorem:

r² = d² + (half-chord)²

Substituting the values: 17² = d² + 15² 289 = d² + 225 d² = 64 d = √64 = 8 cm

Explanation

Wages are distributed in proportion to the daily work efficiency of the individuals when they work together for the same number of days.

Efficiency is inversely proportional to the time taken to complete the work.

Efficiency of Tina : Efficiency of Meena = 1/12 : 1/15 = 15 : 12 = 5 : 4

The total ratio is 5 + 4 = 9 parts, which corresponds to the total payment of Rs. 18,000.

Meena's share is calculated as: (4 / 9) × 18,000 = 8,000

Explanation

The length of the direct common tangent to two circles is given by the formula:

Length of direct common tangent = √(d² - (R - r)²)

where d is the distance between the centres of the circles, R is the radius of the larger circle, and r is the radius of the smaller circle.

Substituting the given values: d = 25 cm R = 12 cm r = 5 cm

Length of direct common tangent = √(25² - (12 - 5)²) Length of direct common tangent = √(25² - 7²) Length of direct common tangent = √(625 - 49) Length of direct common tangent = √576 Length of direct common tangent = 24 cm

Explanation

Using the average adjustment relation: Correct Average = Incorrect Average + (Correct Value - Incorrect Value) ÷ Total Number of Students

Substituting the values: Correct Average = 15 + (9 - 15) ÷ 30 Correct Average = 15 - 6 ÷ 30 Correct Average = 15 - 0.2 Correct Average = 14.8 years

This reduces the total sum by 6 years, which translates to a decrease of 0.2 years in the average across all 30 students.

Explanation

The relation between distance, speed, and time difference is given by the formula: Distance = (v₁ × v₂ / (v₂ - v₁)) × Δt

where v₁ is the original speed, v₂ is the increased speed, and Δt is the difference in time.

Substituting the given values into the formula: v₁ = v v₂ = v + 8 Distance = 60 km Δt = 2 hours

60 = (v × (v + 8) / 8) × 2 60 = v(v + 8) / 4 240 = v(v + 8) v² + 8v - 240 = 0

Factoring the quadratic equation: (v - 12)(v + 20) = 0

Since speed must be a positive value, we take v = 12. Thus, the original speed of the car is 12 km/hr.

Explanation

Let the cost price of the wheat be CP.

The difference between a 25% profit and a 25% loss represents a total change of: 25% - (-25%) = 50% of CP

This 50% difference is equal to the reduction in the selling price, which is Rs. 40. 0.50 × CP = 40 CP = 40 / 0.50 = Rs. 80

The initial selling price was at a profit of 25%: Initial Selling Price = CP × (1 + 0.25) = 80 × 1.25 = Rs. 100

Explanation

Using the BODMAS rule, the expression 44 × 12 - (25 + 13 × 10) is evaluated by first solving the terms inside the parentheses.

First, perform the multiplication inside the brackets: 13 × 10 = 130

Next, complete the addition inside the brackets: 25 + 130 = 155

Now, perform the multiplication outside the brackets: 44 × 12 = 528

Finally, subtract the bracket value from the product: 528 - 155 = 373

Explanation

Using the trigonometric identity cot y = base/perpendicular, we have base = 12 and perpendicular = 5.

By the Pythagorean theorem, the hypotenuse is: hypotenuse = √(base² + perpendicular²) = √(12² + 5²) = √(144 + 25) = √169 = 13

Using the definition of cosine: cos y = base/hypotenuse = 12/13

Squaring both sides gives: cos² y = (12/13)² = 144/169

Explanation

By the properties of ratios, if a/b = c/d = e/f = k, then for any non-zero constants x, y, and z: (xa + yc + ze) / (xb + yd + zf) = k

Given that p/q = r/s = t/u = 2/5, we can apply this property with the multipliers x = 4, y = 3, and z = 7: (4p + 3r + 7t) / (4q + 3s + 7u) = 2/5

Thus, the ratio (4p + 3r + 7t) : (4q + 3s + 7u) is equal to 2:5.

Explanation

Using the average formula: Average = Sum of all values ÷ Total number of values

The given values are 32, 34, 38, 21, 26, and x. The number of values is 6, and their average is 30.

Sum of the given values = 32 + 34 + 38 + 21 + 26 = 151

Setting up the equation: 30 = (151 + x) ÷ 6

Multiplying both sides by 6: 180 = 151 + x

Solving for x: x = 180 - 151 x = 29

Explanation

The value of J1 is the average number of pens sold by shopkeeper P in all five months: Sum of pens sold by P = 100 + 80 + 40 + 30 + 10 = 260 J1 = 260 / 5 = 52

The value of J2 is the average number of pens sold by shopkeeper Q in all five months: Sum of pens sold by Q = 10 + 20 + 120 + 60 + 50 = 260 J2 = 260 / 5 = 52

The ratio of the averages can be calculated directly or by comparing their sums since the number of observations is identical: J1 / J2 = 52 / 52 = 1

Explanation

To simplify the given expression, we rewrite the division as multiplication by the reciprocal:

((1 + x²) / (1 - x)) ÷ ((1 - x⁴) / (x - 1)) × (x(1 - x) / (1 + x)) = ((1 + x²) / (1 - x)) × ((x - 1) / (1 - x⁴)) × (x(1 - x) / (1 + x))

Using the algebraic identity 1 - x⁴ = (1 - x²)(1 + x²), we can simplify the first two terms:

((1 + x²) / (1 - x)) × (-(1 - x) / ((1 - x²)(1 + x²))) = -1 / (1 - x²)

Now, multiply this result by the third term:

(-1 / (1 - x²)) × (x(1 - x) / (1 + x))

Since 1 - x² = (1 - x)(1 + x), we substitute this into the expression:

(-1 / ((1 - x)(1 + x))) × (x(1 - x) / (1 + x))

Canceling the common term (1 - x) from the numerator and the denominator gives:

-x / (1 + x)²

Explanation

Correct Answer: D

The effective rate of compound interest for 2 years at a rate of r% per annum is calculated using the formula: Effective Rate = 2r + r²/100

Substituting r = 10: Effective Rate = 2(10) + 10²/100 = 20 + 1 = 21%

Using this effective rate, the compound interest on the principal of Rs. 14,000 is: Compound Interest = 14000 × 21/100 = 140 × 21 = 2940

Explanation

In a right-angled triangle PQR with ∠Q = 90° and altitude QX perpendicular to the hypotenuse PR, the square of each leg is equal to the product of the hypotenuse and the projection of that leg on the hypotenuse.

This gives the relations: PQ² = PX × PR QR² = XR × PR

Dividing these two equations yields the direct ratio: PX / XR = PQ² / QR²

To find QR, apply the Pythagorean theorem on ΔPQR: QR² = PR² - PQ² QR² = 25² - 15² QR² = 625 - 225 = 400 QR = 20 cm

Now, substitute the values of PQ and QR into the ratio formula: PX / XR = 15² / 20² PX / XR = 225 / 400 PX / XR = 9 / 16

Thus, the ratio of PX : XR is 9:16.

Explanation

In an isosceles triangle ABC with AB = AC, the altitude AZ drawn from the vertex A to the base BC (since ∠AZB = 90°) also acts as the median to the base.

As a median, Z bisects the side BC, meaning Z is the midpoint of BC. Therefore, the length of BZ is exactly half of the length of BC.

BZ = BC / 2 = 42 / 2 = 21 cm.

Explanation

To find the last three digits of the multiplication, we only need to consider the last three digits of each of the two numbers and find their product modulo 1000.

The last three digits of 654321 are 321, and the last three digits of 123456 are 456.

Multiplying these two three-digit numbers: 321 × 456 = 146376

Taking the last three digits of this result gives 376.

Explanation

Let the larger number be x and the smaller number be y.

The problem states that the difference between the two numbers is 20% of their sum: x - y = 0.2 × (x + y)

Multiply both sides by 5 to simplify: 5 × (x - y) = x + y 5x - 5y = x + y

Rearranging the terms to group like variables: 5x - x = y + 5y 4x = 6y

To find the ratio of the larger number to the smaller number, divide x by y: x/y = 6/4 = 3/2

This gives the ratio of the larger number to the smaller number as 3:2.

Explanation

The single equivalent discount for two successive discounts of d₁% and d₂% is calculated using the formula d₁ + d₂ - (d₁ × d₂)/100.

Using the given discounts of 40% and 20%: Equivalent discount = 40 + 20 - (40 × 20)/100 = 60 - 8 = 52%

This means the selling price is 100% - 52% = 48% of the marked price.

Selling Price = Marked Price × 48% 4632 = Marked Price × 0.48 Marked Price = 4632 / 0.48 = 9650

The marked price of the article is Rs. 9650.

Explanation

The perpendicular from the centre of a circle to a chord bisects the chord, forming a right-angled triangle with the radius and the distance of the chord from the centre. By the Pythagorean theorem, the relation between the radius r, the distance of the chord from the centre d, and the length of the chord L is:

r² = d² + (L/2)²

For the first chord of length 48 cm at a distance of 7 cm:

r² = 7² + (48/2)² r² = 7² + 24² r² = 49 + 576 r² = 625 r = 25 cm

For the second chord at a distance of 15 cm, using the radius of 25 cm:

25² = 15² + (L/2)² 625 = 225 + (L/2)² (L/2)² = 400 L/2 = 20 L = 40 cm

Explanation

Correct Answer: D

Let the initial cost price be CP. Since the initial profit is 20%, the initial selling price is 1.2CP.

When both the cost price and selling price are reduced by Rs. 20: New cost price = CP - 20 New selling price = 1.2CP - 20

The new profit is 40%, which gives the relation: New selling price = 1.4 × New cost price

Substituting the expressions into this relation: 1.2CP - 20 = 1.4 × (CP - 20) 1.2CP - 20 = 1.4CP - 28 1.4CP - 1.2CP = 28 - 20 0.2CP = 8 CP = 8 ÷ 0.2 CP = 40

Explanation

Let the efficiency of B be 1 unit of work per day. Since A is thrice as good a workman as B, the efficiency of A is 3 units of work per day.

Together, their combined efficiency is 3 + 1 = 4 units of work per day.

They can complete the entire work together in 32 days. Total work = Combined efficiency × Number of days = 4 × 32 = 128 units.

The number of days B alone will take to complete this total work is: Time taken by B = Total work / Efficiency of B = 128 / 1 = 128 days.

Explanation

Let the amounts invested in the three schemes at 4%, 6%, and 10% be P₁, P₂, and P₃ respectively. The total sum is P₁ + P₂ + P₃ = 15500.

The simple interest earned from each scheme at the end of one year is equal. Using the formula SI = P × R × T / 100 with T = 1:

P₁ × 4 / 100 = P₂ × 6 / 100 = P₃ × 10 / 100

This simplifies to: 4P₁ = 6P₂ = 10P₃

To find the ratio of P₁ : P₂ : P₃, divide each term by the least common multiple of 4, 6, and 10, which is 60: P₁/15 = P₂/10 = P₃/6

Therefore, the ratio of the investments is P₁ : P₂ : P₃ = 15 : 10 : 6.

The sum of the ratio terms is 15 + 10 + 6 = 31.

The investment at the rate of 6% corresponds to P₂: P₂ = (10/31) × 15500 = 10 × 500 = 5000

The sum invested at the rate of 6% is Rs. 5000.

Explanation

To find the value of the given expression, we combine the fractions by finding a common denominator:

(sin θ + cos θ)/(sin θ - cos θ) + (sin θ - cos θ)/(sin θ + cos θ)

We cross-multiply to obtain a single fraction:

[(sin θ + cos θ)² + (sin θ - cos θ)²] / [(sin θ - cos θ)(sin θ + cos θ)]

Using the algebraic identity (a + b)² + (a - b)² = 2(a² + b²) for the numerator:

(sin θ + cos θ)² + (sin θ - cos θ)² = 2(sin² θ + cos² θ)

Since sin² θ + cos² θ = 1, the numerator simplifies to:

2 × 1 = 2

Using the difference of squares identity (a - b)(a + b) = a² - b² for the denominator:

(sin θ - cos θ)(sin θ + cos θ) = sin² θ - cos² θ

Substituting the simplified numerator and denominator back into the expression yields:

2 / (sin² θ - cos² θ)

Explanation

The sum of the averages is equivalent to the average of the total sum of male and female teachers across all schools.

A1 + A2 = (Sum of female teachers / 5) + (Sum of male teachers / 5) = (Total teachers) / 5

First, find the total number of teachers in each school: School J = 180 + 240 = 420 School K = 240 + 210 = 450 School L = 210 + 60 = 270 School M = 150 + 30 = 180 School N = 90 + 90 = 180

Total number of teachers across all 5 schools = 420 + 450 + 270 + 180 + 180 = 1500

Now, calculate (A1 + A2): A1 + A2 = 1500 / 5 = 300

Explanation

According to the inscribed angle theorem, an angle subtended by a chord on the circumference of a circle is 90° if and only if the chord is the diameter of the circle.

The relationship between the diameter d and the radius r of a circle is given by the formula: d = 2r

Substituting the given radius r = 12 cm into the formula: d = 2 × 12 = 24 cm

The chord MN is the diameter of the circle, making its length 24 cm.

Explanation

Using the trigonometric co-function identity: cot θ = tan (90° - θ)

By substituting θ = 3A, the given equation cot 3A = tan (A - 36°) becomes: tan (90° - 3A) = tan (A - 36°)

Equating the angles: 90° - 3A = A - 36° 4A = 126° A = 31.5°

Explanation

For any set of numbers in an arithmetic progression, such as consecutive even numbers, the average of the series is always equal to the average of the first and the last terms.

Average = (Smallest number + Largest number) / 2

Given that the average of the six consecutive even numbers is 35:

35 = (Smallest number + Largest number) / 2

Sum of the smallest and largest number = 35 × 2 = 70

Explanation

Let the length of the rectangle be l and its breadth be b.

The perimeter of a rectangle is given by 2(l + b). Therefore, half of the perimeter is: l + b = 30

The length of the rectangle is 6 cm more than its breadth: l = b + 6

Substituting this relation into the first equation: b + 6 + b = 30 2b + 6 = 30 2b = 24 b = 12 cm

Using the value of b to find l: l = 12 + 6 = 18 cm

The area of the rectangle is the product of its length and breadth: Area = l × b Area = 18 × 12 = 216 cm²

Explanation

This simplification problem contains typographical errors where decimal points were omitted. Restoring the correct decimal points and replacing the equal sign with a addition sign, the expression is evaluated as follows:

First Term: (7 × 0.4 + 0.8) ÷ (5 × 0.25 + 0.19) = (2.8 + 0.8) ÷ (1.25 + 0.19) = 3.6 ÷ 1.44 = 2.5

Second Term: (5 × 0.4 + 1.6) ÷ (8 × 0.4 + 1.6) = (2 + 1.6) ÷ (3.2 + 1.6) = 3.6 ÷ 4.8 = 0.75

Third Term: (7 × 0.4 + 0.2) ÷ (8 × 0.7 + 0.4) = (2.8 + 0.2) ÷ (5.6 + 0.4) = 3 ÷ 6 = 0.5

Summing the three terms: 2.5 + 0.75 + 0.5 = 3.75 = 15/4

Explanation

For a journey consisting of equal distance intervals, the average speed is given by the harmonic mean of the individual speeds.

Let the distance of each of the four trips be d.

The speeds for the four trips are: v₁ = 720 km/hr v₂ = 720/2 = 360 km/hr v₃ = 360/2 = 180 km/hr v₄ = 180/2 = 90 km/hr

The total distance traveled is 4d.

The total time taken is the sum of the times for each individual trip: Total Time = d/720 + d/360 + d/180 + d/90 Total Time = (d + 2d + 4d + 8d) / 720 Total Time = 15d / 720 Total Time = d / 48

The average speed is the total distance divided by the total time: Average Speed = 4d / (d / 48) = 4 × 48 = 192 km/hr

Explanation

Correct Answer: D

The algebraic identity relating the sum of three terms, their squares, and their pairwise products is: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

First, simplify the given equations by dividing both sides by their common factors: From 9a + 9b + 9c = 81, we divide by 9 to get: a + b + c = 9

From 4ab + 4bc + 4ca = 160, we divide by 4 to get: ab + bc + ca = 40

Substitute these simplified values into the algebraic identity: 9² = a² + b² + c² + 2(40) 81 = a² + b² + c² + 80 a² + b² + c² = 81 - 80 = 1

Multiply both sides by 6 to find the required expression: 6a² + 6b² + 6c² = 6(a² + b² + c²) = 6 × 1 = 6

Explanation

For any real number a, if a + 1/a = k, then squaring both sides gives: a² + 1/a² = k² - 2

Using this relation for the given value k = 8: a² + 1/a² = 8² - 2 = 64 - 2 = 62

Applying the same relation again to find the value of a⁴ + 1/a⁴: a⁴ + 1/a⁴ = (a² + 1/a²)² - 2 a⁴ + 1/a⁴ = 62² - 2 a⁴ + 1/a⁴ = 3844 - 2 = 3842

Explanation

By definition, in a right-angled triangle, the sine of an angle P is the ratio of the opposite side to the hypotenuse:

sin P = opposite / hypotenuse = 15 / 22

We can determine the adjacent side using the Pythagorean theorem:

adjacent = √(hypotenuse² - opposite²) adjacent = √(22² - 15²) adjacent = √(484 - 225) adjacent = √259

Since the tangent of an angle P is the ratio of the opposite side to the adjacent side:

tan P = opposite / adjacent = 15 / √259

Explanation

Let the values of a, b, and c be represented in terms of a constant multiplier k as a = 2k, b = 3k, and c = 5k.

Substituting these values into the given equation: 5b - a + 2c = 115 5(3k) - 2k + 2(5k) = 115 15k - 2k + 10k = 115 23k = 115 k = 5

Using the value of k, the value of b is calculated as: b = 3k = 3 × 5 = 15

Explanation

The weight of the 50th student is determined by subtracting the combined weight of the first 49 students and the last 50 students from the total weight of all 100 students.

Using the relation: Total Weight = Average Weight × Number of Students

Total weight of all 100 students = 100 × 34 = 3400 kg

Total weight of the first 49 students = 49 × 34 = 1666 kg

Total weight of the last 50 students = 50 × 32 = 1600 kg

Weight of the 50th student = Total weight of 100 students - (Total weight of first 49 students + Total weight of last 50 students) Weight of the 50th student = 3400 - (1666 + 1600) Weight of the 50th student = 3400 - 3266 = 134 kg

This can also be solved using the deviation method: Taking the overall average of 34 kg as the reference: Deviation of the first 49 students = 49 × (34 - 34) = 0 kg Deviation of the last 50 students = 50 × (32 - 34) = -100 kg Since the sum of all deviations from the average must equal zero, the deviation of the 50th student must be +100 kg. Weight of the 50th student = 34 + 100 = 134 kg

Explanation

The given expression is based on the algebraic identity: (a³ + b³) / (a² - ab + b²) = a + b

By setting the values of a and b as: a = 0.91 b = 0.09

We get: ab = 0.91 × 0.09 = 0.0819 a² = (0.91)² b² = (0.09)²

Substituting these values into the identity: [(0.91)³ + (0.09)³] / [(0.91)² - 0.0819 + (0.09)²] = 0.91 + 0.09 = 1

Explanation

The two-digit even numbers form an arithmetic progression starting from 10 and ending at 98, with a common difference of 2. The number of terms is 45, and using the sum formula of an arithmetic progression, the sum is (45 / 2) * (10 + 98) = 2430.

Explanation

According to the BODMAS rule, the operations are performed in the order of Brackets, Of, Division, Multiplication, Addition, and Subtraction.

Simplifying the term containing of: 45 of (1/5) = 45 × 1/5 = 9

Simplifying the multiplication term: (12/5) × (20/3) = (12 × 20) / (5 × 3) = 240 / 15 = 16

The term 117 is a typographical representation of 11 - 7, which simplifies to: 11 - 7 = 4

Substituting these values back into the expression: 4 + 9 + 16 = 29

Explanation

To simplify the given expression, find the common denominator of the two fractions:

(1 - sin A)/cos A + cos A/(1 - sin A)

= ((1 - sin A)² + cos² A) / (cos A (1 - sin A))

Expanding the squared term in the numerator:

= (1 - 2 sin A + sin² A + cos² A) / (cos A (1 - sin A))

Using the trigonometric identity sin² A + cos² A = 1, substitute this into the numerator:

= (1 - 2 sin A + 1) / (cos A (1 - sin A))

= (2 - 2 sin A) / (cos A (1 - sin A))

Factor out 2 from the numerator:

= 2 (1 - sin A) / (cos A (1 - sin A))

Cancel the common term (1 - sin A) in both the numerator and denominator:

= 2 / cos A

Since 1/cos A = sec A, the expression simplifies to:

= 2 sec A

Explanation

The sum of the ages of the 5 children is 75 years (5 multiplied by 15), and the sum of the ages of all 7 family members is 175 years (7 multiplied by 25). The combined age of the father and mother is 100 years (175 minus 75), and since the father is 4 years older than the mother, the mother's age is calculated as (100 - 4) divided by 2, which equals 48 years.

Explanation

Given the algebraic identity (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x), substitute x = N and N + 1/N = √3 into the formula:

(√3)³ = N³ + 1/N³ + 3√3 3√3 = N³ + 1/N³ + 3√3 N³ + 1/N³ = 0

Squaring both sides of this equation using the identity (a + b)² = a² + b² + 2ab:

(N³ + 1/N³)² = 0² N⁶ + 1/N⁶ + 2 = 0 N⁶ + 1/N⁶ = -2

Substitute this value into the required expression:

N⁶ + 1/N⁶ + 11 = -2 + 11 = 9