SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The total profit of the business is £10,000.

A receives 20% of this total profit for managing the business as a working partner: Management share for A = 20% of £10,000 = £2,000

The remaining profit to be distributed in proportion to their capitals is: Remaining profit = £10,000 - £2,000 = £8,000

The ratio of capital investments of A and B is: A : B = 30,000 : 50,000 = 3 : 5

B's share of the remaining profit is calculated as: B's share = 8,000 × 5/(3 + 5) = 8,000 × 5/8 = £5,000

Explanation

Correct Answer: C

Let the initial volume of the solution be v. Since the quantity of milk is double the quantity of water, the milk and water are in the ratio 2:1.

The initial volume of water in the solution is v/3, and the initial volume of milk is 2v/3.

Let x be the volume of pure milk to be added to the solution. The total volume of the new solution becomes v + x.

For the solution to contain 90% milk, the remaining 10% of the new solution must be water. Since no water was added, the volume of water remains v/3.

Using the relation for the water content: v/3 = 10% of (v + x) v/3 = (v + x)/10

Multiplying both sides by 30 to clear the denominators: 10v = 3(v + x) 10v = 3v + 3x 7v = 3x x = 7v/3

Thus, the volume of pure milk to be added is 7v/3.

Explanation

Selling Price (SP) of the article is calculated using the formula: SP = Marked Price (MP) × (1 - Discount %)

Substituting the given values: SP = 450 × (1 - 0.20) SP = 450 × 0.80 = 360

The relationship between Selling Price and Cost Price (CP) is given as: SP = 1.5 × CP

Substituting the value of SP to find CP: 360 = 1.5 × CP CP = 360 ÷ 1.5 CP = 240

Explanation

A number is divisible by 6 if it is divisible by both 2 and 3. For a number to be divisible by 2, its last digit must be even. For a number to be divisible by 3, the sum of its digits must be a multiple of 3.

For the number 8,00,552: The units digit is 2, which is even, so the number is divisible by 2. The sum of the digits is 8 + 0 + 0 + 5 + 5 + 2 = 20. Since 20 is not divisible by 3, the number 8,00,552 is not divisible by 3. Consequently, it is not divisible by 6.

Explanation

When a solid object is melted and recast into another shape, the total volume remains conserved. Therefore, the volume of the metallic cone is equal to the total volume of the N solid spheres.

Volume of a cone = (1/3) × π × r_cone² × h_cone Volume of a sphere = (4/3) × π × r_sphere³

Using the conservation of volume: N × (4/3) × π × r_sphere³ = (1/3) × π × r_cone² × h_cone

Simplifying the equation by canceling common terms: 4 × N × r_sphere³ = r_cone² × h_cone N = (r_cone² × h_cone) / (4 × r_sphere³)

Substituting the given values r_cone = 12 cm, h_cone = 24 cm, and r_sphere = 2 cm into the equation: N = (12² × 24) / (4 × 2³) N = (144 × 24) / (4 × 8) N = 3456 / 32 N = 108

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) is given by the formula:

CP / MP = (100 - Discount%) / (100 + Profit%)

Substituting the initial values into the formula:

CP / MP = (100 - 25) / (100 + 80) = 75 / 180 = 5 / 12

This gives the relation:

MP = 2.4 × CP

When the discount is 40%, the new Selling Price (SP) is:

SP = 60% of MP = 0.6 × MP

Substituting MP in terms of CP:

SP = 0.6 × 2.4 × CP = 1.44 × CP

The new profit percentage is:

New Profit% = (SP - CP) / CP × 100 = (1.44 - 1) × 100 = 44%

Explanation

The fraction of the original liquid remaining in a container after successive draw-and-replace operations is given by the formula: Fraction of wine remaining = (1 - x₁/V) × (1 - x₂/V)

In this formula, V represents the initial volume of wine in the cask, x₁ is the first volume drawn and replaced (75 litres), and x₂ is the second volume drawn and replaced (60 litres).

The final ratio of wine to water is 3:2, which means the final fraction of wine in the total mixture is 3 / (3 + 2) = 3/5.

Setting up the equation: (1 - 75/V) × (1 - 60/V) = 3/5

Simplifying this expression: (V - 75)(V - 60) / V² = 3/5 5(V² - 135V + 4500) = 3V² 5V² - 675V + 22500 = 3V² 2V² - 675V + 22500 = 0

Factoring the quadratic equation: 2V² - 600V - 75V + 22500 = 0 2V(V - 300) - 75(V - 300) = 0 (2V - 75)(V - 300) = 0

This yields V = 37.5 or V = 300. Since the volume of the cask must be greater than the individual volumes drawn out (75 litres and 60 litres), the initial volume of wine in the cask is 300 litres.

Explanation

To simplify the given expression, we combine the first two fractional terms:

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

Taking the common denominator for the fractions:

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

Expanding the numerator:

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

Using the fundamental trigonometric identity cos² A + sin² A = 1, the numerator becomes:

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

Factoring out 2 in the numerator:

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

Cancelling the common term (1 + sin A) in the numerator and denominator:

2/cos A + 2

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

2 sec A + 2 = 2(1 + sec A)

Explanation

When two numbers n₁ and n₂ are divided by a divisor d, they leave remainders r₁ and r₂ respectively. The sum of the two numbers is n₁ + n₂ = q₁d + r₁ + q₂d + r₂ = (q₁ + q₂)d + (r₁ + r₂).

When this sum is divided by the same divisor d, the resulting remainder R is given by the relation: r₁ + r₂ = d + R

This relation holds because the sum of the individual remainders r₁ + r₂ is greater than the divisor d, but less than 2d since each individual remainder is less than d.

Substituting the given values into the formula: 57 + 57 = d + 49 114 = d + 49 d = 114 - 49 d = 65

Since the divisor must be greater than the individual remainders (d > 57), the value of 65 is consistent with all conditions.

Explanation

The marked price can be calculated by dividing the final selling price by the remaining percentages after each discount. After a 20 percent discount and a 12 percent discount, the selling price is 792, which gives the equation Marked Price times 0.80 times 0.88 equals 792, resulting in a marked price of 1,125 rupees.

Explanation

The curved surface area (CSA) of a cylinder is given by the formula CSA = 2πrh, and its volume is given by V = πr²h, where r is the radius and h is the height.

Let the initial radius, height, CSA, and volume be r, h, CSA, and V respectively.

According to the given conditions: New curved surface area, CSA' = 1.20 × CSA New height, h' = 0.96 × h

Using the relation for CSA': 2πr'h' = 1.20 × 2πrh r' × 0.96h = 1.20 × rh r' = (1.20 / 0.96) × r r' = 1.25 r

Now, the new volume V' is: V' = π(r')²h' V' = π(1.25 r)²(0.96 h) V' = π(1.5625 r²)(0.96 h) V' = 1.5625 × 0.96 × πr²h V' = 1.5 × V

The net change in volume is: ((V' - V) / V) × 100% = ((1.5 V - V) / V) × 100% = 0.5 × 100% = 50% increase.

Explanation

The loss percentage is calculated using the formula:

Loss Percentage = ((Cost Price - Selling Price) / Cost Price) × 100

Given: Cost Price (CP) = ₹225 Selling Price (SP) = ₹210

Loss = CP - SP = 225 - 210 = ₹15

Loss Percentage = (15 / 225) × 100 = (1 / 15) × 100 = 20 / 3 % ≈ 6.67%

Explanation

Correct Answer: D

In the right-angled triangles LON and LMN, we are given that angle LON = angle LMN = 90° and LN is the common hypotenuse. Since LN is the angle bisector of angle OLM, we have angle OLN = angle MLN.

By the Angle-Angle-Side (AAS) congruence criterion, triangle LON is congruent to triangle LMN. Consequently, the corresponding sides are equal: LO = LM ON = MN

Given ON = 20 cm, we have MN = 20 cm.

In the right-angled triangle LON, we apply the Pythagorean theorem to find LO: LN² = LO² + ON² 29² = LO² + 20² 841 = LO² + 400 LO² = 441 LO = 21 cm

Since LM = LO, we have LM = 21 cm.

The perimeter of triangle LMN is the sum of its three sides: Perimeter = LM + MN + LN Perimeter = 21 + 20 + 29 = 70 cm

Explanation

Using the algebraic identity (x² + 1/x²)² = x⁴ + 1/x⁴ + 2, substitute the given value:

(x² + 1/x²)² = 194 + 2 = 196 x² + 1/x² = √196 = 14

Similarly, using the identity (x + 1/x)² = x² + 1/x² + 2:

(x + 1/x)² = 14 + 2 = 16 x + 1/x = √16 = 4

Now, find the value of x³ + 1/x³ using the formula x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x):

x³ + 1/x³ = 4³ - 3(4) = 64 - 12 = 52

The required expression is x³ + 1/x³ + x + 1/x:

52 + 4 = 56

Explanation

When the profit earned by selling an article at one price is equal to the loss incurred by selling it at another price, the Cost Price (CP) is the arithmetic mean of the two Selling Prices (SP₁ and SP₂).

CP = (SP₁ + SP₂) / 2

Substituting the given values: CP = (1600 + 1400) / 2 = 3000 / 2 = 1500

To obtain a profit of 20%, the required selling price is: Required SP = CP × (1 + Profit%) Required SP = 1500 × (1 + 0.20) = 1500 × 1.2 = 1800

Explanation

For a period of 2 years, the simple interest is distributed equally across both years. Since the simple interest for 2 years is ₹500, the simple interest for the first year is: 500 / 2 = ₹250

The difference between the compound interest and the simple interest over the 2-year period is: CI - SI = 550 - 500 = ₹50

This difference represents the interest earned during the second year on the first year's interest of ₹250. The rate of interest (r) is calculated as: r = (Difference / SI of first year) × 100 r = (50 / 250) × 100 = 20%

Explanation

In triangle ABC, since PQ is parallel to BC, by the Basic Proportionality Theorem, we have:

AP / AB = AQ / AC

Let the length of QC be x cm. Since AP = QC, we have AP = x cm. The length of AC is AQ + QC = (4 + x) cm.

Substituting the given values into the proportion:

x / 16 = 4 / (4 + x)

Cross-multiplying the terms gives:

x(4 + x) = 64 x² + 4x - 64 = 0

Using the quadratic formula to solve for x:

x = (-4 ± √(4² - 4 × 1 × (-64))) / 2 x = (-4 ± √(16 + 256)) / 2 x = (-4 ± √272) / 2 x = (-4 ± 4√17) / 2 x = -2 ± 2√17

Since the length must be a positive value, we take the positive root:

x = 2√17 - 2

Explanation

For two circles touching each other externally, the length of their direct common tangent t is given by the formula:

t = 2 × √(R × r)

where R is the radius of the larger circle and r is the radius of the smaller circle.

Given that the length of the common tangent PQ is 16 cm and the radius of the larger circle R is 20 cm, we substitute these values into the formula:

16 = 2 × √(20 × r)

Dividing both sides by 2:

8 = √(20 × r)

Squaring both sides of the equation:

64 = 20 × r

Solving for r:

r = 64 / 20 = 3.2 cm

Explanation

Using the algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values into the identity: 5² = 15 + 2(ab + bc + ca) 25 = 15 + 2(ab + bc + ca) 2(ab + bc + ca) = 10 ab + bc + ca = 5

Now, use the identity for the sum of cubes: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

Substitute the known values into this equation: a³ + b³ + c³ - 3abc = 5 × (15 - 5) a³ + b³ + c³ - 3abc = 5 × 10 = 50

Subtract 27 from both sides to find the required value: a³ + b³ + c³ - 3abc - 27 = 50 - 27 = 23

Explanation

To find the city that recorded the maximum temperature in each of the five months, we compare the temperatures across all cities for each month:

In January, the maximum temperature is 26°C, recorded by Hyderabad. In February, the maximum temperature is 27°C, recorded by Hyderabad. In March, the maximum temperature is 34°C, recorded by Hyderabad. In April, the maximum temperature is 37°C, recorded by Hyderabad. In May, the maximum temperature is 42°C, recorded by Hyderabad.

Hyderabad recorded the highest temperature among all the cities in every single month from January to May 2022.

Explanation

Using the algebraic identity (x + y)² = x² + y² + 2xy, we can substitute the given values: (x + y)² = 427 + 2 × 202 (x + y)² = 427 + 404 (x + y)² = 831 Taking the square root of both sides gives: x + y = √831

Similarly, using the algebraic identity (x - y)² = x² + y² - 2xy, we substitute the given values: (x - y)² = 427 - 2 × 202 (x - y)² = 427 - 404 (x - y)² = 23 Taking the square root of both sides gives: x - y = √23

To find the value of (x + y)/(x - y), we divide the two equations: (x + y)/(x - y) = √831 / √23

Explanation

Number of girls studying English = 50% of 70 = 35 Number of girls studying Computers = 40% of 90 = 36

Required ratio = 35 : 36

Explanation

For half-yearly compounding, the annual interest rate is halved and the number of compounding periods is doubled. Here, the principal P = 50000, the annual rate R = 8%, and the time period is 1 year.

The half-yearly rate r = 8% / 2 = 4% per half-year, and the number of periods n = 2.

Using the compound interest formula: CI = P × ((1 + r)ⁿ - 1)

Substituting the values into the formula: CI = 50000 × ((1 + 0.04)² - 1) CI = 50000 × (1.04² - 1) CI = 50000 × (1.0816 - 1) CI = 50000 × 0.0816 CI = 4080

Explanation

The formula for the arc length of a sector is: s = (θ / 360) × 2πr

Given: s = 16.5 cm θ = 22.5° π = 22 / 7

Substituting the values into the formula: 16.5 = (22.5 / 360) × 2 × (22 / 7) × r

Simplifying the angle fraction: 22.5 / 360 = 1 / 16

Substituting this back into the equation: 16.5 = (1 / 16) × 2 × (22 / 7) × r 16.5 = (1 / 8) × (22 / 7) × r 16.5 = (11 / 28) × r

Solving for the radius r: r = (16.5 × 28) / 11 r = 1.5 × 28 r = 42

Explanation

The overall selling price multiplier from A to C is calculated by combining A's profit and B's loss:

Multiplier = (1 + Profit% of A) × (1 - Loss% of B) Multiplier = (1 + 33/100) × (1 - 25/100) = 133/100 × 3/4 = 399/400

Let the cost price of the cycle for A be x. The price paid by C is:

x × 399/400 = 3059 x = 3059 × 400 / 399

Dividing both 3059 and 399 by their common factor of 133:

3059 ÷ 133 = 23 399 ÷ 133 = 3

Substituting these values back into the equation:

x = 23 × 400 / 3 = 9200 / 3 = 3066 ⅔

Explanation

The given equation is 4y - 4/y = 11.

Factoring out 4 on the left side gives: 4(y - 1/y) = 11 y - 1/y = 11/4

Squaring both sides of this equation: (y - 1/y)² = (11/4)² y² + 1/y² - 2 = 121/16

Adding 2 to both sides to find y² + 1/y²: y² + 1/y² = 121/16 + 2 y² + 1/y² = (121 + 32)/16 y² + 1/y² = 153/16

Converting the improper fraction 153/16 into a mixed fraction: 153/16 = 9 9/16

Explanation

To find the value of y⁵ - 1/y⁵, we can use the algebraic relation: y⁵ - 1/y⁵ = (y² + 1/y²)(y³ - 1/y³) - (y - 1/y)

First, we determine the values of y² + 1/y² and y³ - 1/y³ using the given value of y - 1/y = -9.

y² + 1/y² = (y - 1/y)² + 2 y² + 1/y² = (-9)² + 2 = 81 + 2 = 83

y³ - 1/y³ = (y - 1/y)³ + 3(y - 1/y) y³ - 1/y³ = (-9)³ + 3(-9) = -729 - 27 = -756

Now, substitute these values into the main equation: y⁵ - 1/y⁵ = 83 × (-756) - (-9) y⁵ - 1/y⁵ = -62748 + 9 = -62739

Explanation

Let the cost price of the mobile be ₹100. Since the marked price is 25% above the cost price, the marked price is ₹125.

To achieve a profit of 5%, the selling price must be ₹105.

The discount given is the difference between the marked price and the selling price: Discount = 125 - 105 = ₹20

The discount percentage is calculated on the marked price: Discount % = (Discount / Marked Price) × 100 = (20 / 125) × 100 = 16%

Explanation

The compound interest earned specifically in the nth year can be calculated using the formula:

Iₙ = P × r × (1 + r)ⁿ⁻¹

Where: P = Principal amount = ₹50,000 r = Annual interest rate = 10% = 0.1 n = The specific year = 3

Substituting these values into the formula:

I₃ = 50,000 × 0.1 × (1 + 0.1)³⁻¹ I₃ = 5,000 × 1.1² I₃ = 5,000 × 1.21 I₃ = 6,050

Thus, the interest earned in the 3rd year is ₹6,050.

Explanation

Using the simple interest formula, the difference in simple interest is given by: ΔSI = (ΔP × R × T) / 100

Here, the difference in principal amounts (ΔP) is 2500 - 1500 = ₹1000, the time (T) is 3 years, and the difference in interest (ΔSI) is ₹360.

Substituting these values into the formula: 360 = (1000 × R × 3) / 100 360 = 30 × R R = 360 / 30 R = 12%

This gives the rate of interest per annum.

Explanation

The perimeter of the triangular flower bed is 20 metres and two of its sides are 5 metres and 6 metres.

The third side c is calculated as: Perimeter = a + b + c 20 = 5 + 6 + c c = 9 metres

The semi-perimeter s of the triangle is: s = Perimeter / 2 = 20 / 2 = 10 metres

Using Heron's formula, the area of the triangle is: Area = √(s(s - a)(s - b)(s - c)) Area = √(10 × (10 - 5) × (10 - 6) × (10 - 9)) Area = √(10 × 5 × 4 × 1) Area = √200 Area = 10√2 m²

Explanation

The standard equation of a circle centered at the origin is given by the formula x² + y² = r², where r represents the radius of the circle.

Comparing the given equation x² + y² = 25 with this standard form yields r² = 25. Taking the square root of both sides gives the radius r = 5.

Explanation

Let the work rate of B be 1 unit per day. Since A is twice as good a worker as B, the work rate of A is 2 units per day.

Together, their combined work rate is 2 + 1 = 3 units per day.

They complete the entire work in 20 days, which means the total work is: Total work = Combined work rate × Time = 3 × 20 = 60 units

The number of days A alone will take to complete the work is: Time taken by A = Total work / Work rate of A = 60 / 2 = 30 days

Explanation

According to the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles.

Exterior angle = 35° + 65° = 100°

Explanation

Average speed is given by the formula:

Average Speed = Total Distance / Total Time

First, find the time taken for each part of the journey.

Time taken for the first 210 km at 70 km/h: t₁ = 210 / 70 = 3 hours

Time taken for the next 170 km at 85 km/h: t₂ = 170 / 85 = 2 hours

Now, calculate the total distance and the total time: Total Distance = 210 + 170 = 380 km Total Time = 3 + 2 = 5 hours

Substitute these values into the average speed formula: Average Speed = 380 / 5 = 76 km/h

Explanation

Using the complementary angle identities sec θ = cosec(90° - θ) and tan θ = cot(90° - θ), the terms of the expression can be simplified.

For the first term, sec 54° = sec(90° - 36°) = cosec 36°. Dividing this by cosec 36° gives 1.

For the second term, tan 70° = tan(90° - 20°) = cot 20°. Dividing this by cot 20° gives 1.

For the third term, the standard value of tan 45° is 1, so 2 tan 45° = 2 × 1 = 2.

Substituting these simplified terms back into the expression yields: 1 + 1 - 2 = 0

Explanation

Given the ratio a:b = 5:7, we can substitute a = 5 and b = 7 directly into the expression because the terms in the ratio are homogeneous of degree 2.

Substituting these values into the ratio (6a² - 2b²) : (b² - a²):

Numerator term: 6a² - 2b² = 6(5²) - 2(7²) = 6(25) - 2(49) = 150 - 98 = 52

Denominator term: b² - a² = 7² - 5² = 49 - 25 = 24

Simplifying the ratio 52:24 by dividing both terms by their greatest common divisor, which is 4: 52 ÷ 4 = 13 24 ÷ 4 = 6

This gives the simplified ratio of 13:6.

Explanation

To simplify the given expression, rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (√3 + √2):

((√3 + √2) × (√3 + √2)) / ((√3 - √2) × (√3 + √2))

Applying the algebraic identities (a + b)² = a² + b² + 2ab for the numerator and (a - b)(a + b) = a² - b² for the denominator:

((√3)² + (√2)² + 2√(3 × 2)) / ((√3)² - (√2)²) = (3 + 2 + 2√6) / (3 - 2) = 5 + 2√6

Substituting the given value of √6 = 2.45:

5 + 2 × 2.45 = 5 + 4.9 = 9.9

Explanation

By using the successive percentage change formula, the net profit or loss percentage when there is a markup of x% followed by a discount of y% is given by the relation:

Net change = x - y - xy/100

Substituting the given values of markup, x = 25, and discount, y = 10:

Net profit percentage = 25 - 10 - (25 × 10)/100 Net profit percentage = 15 - 2.5 = 12.5%

This is equivalent to 12½%.

Alternatively, assuming the cost price of the item is ₹100: Marked price = ₹100 + 25% of ₹100 = ₹125 Discount offered = 10% of ₹125 = ₹12.50 Selling price = ₹125 - ₹12.50 = ₹112.50 Gain = Selling price - Cost price = ₹112.50 - ₹100 = ₹12.50 Gain percentage = (Gain / Cost price) × 100 = (12.50 / 100) × 100 = 12.5% = 12½%

Explanation

In triangle ABC, AP is the bisector of angle A and is perpendicular to BC at point P. This means that in triangles ABP and ACP: angle BAP = angle CAP (since AP is the angle bisector) AP is a common side angle APB = angle APC = 90° (since AP is perpendicular to BC)

Therefore, triangle ABP is congruent to triangle ACP by the Angle-Side-Angle congruence criterion. From this congruence, the corresponding sides are equal: BP = PC AB = AC

Using the given measures, BP = PC gives: x = 3y

And AB = AC gives: 4x = 5y + 21

Substituting x = 3y into the equation 4x = 5y + 21: 4(3y) = 5y + 21 12y = 5y + 21 7y = 21 y = 3

Using y = 3 to find x: x = 3(3) = 9

Calculating the value of (x + y): 9 + 3 = 12

Explanation

Correct Answer: D

Using the trigonometric definitions in a right-angled triangle, sec A is the ratio of the hypotenuse to the base:

sec A = hypotenuse/base = 17/15

Let the hypotenuse be 17 and the base be 15. By Pythagoras' theorem, the relation between the hypotenuse (h), base (b), and perpendicular (p) is:

h² = p² + b²

Substituting the values:

17² = p² + 15² 289 = p² + 225 p² = 289 - 225 p² = 64 p = √64 = 8

Since cot A is the ratio of the base to the perpendicular:

cot A = base/perpendicular = 15/8

Explanation

When two objects move in the same direction, their relative speed is the difference of their individual speeds.

Relative speed = 80 km/h - 26 km/h = 54 km/h

To convert this speed from km/h to m/s, multiply by 5/18:

Relative speed = 54 × 5/18 = 15 m/s

The distance covered when one train completely crosses another is equal to the sum of their lengths.

Combined length of both trains = Relative speed × Time = 15 m/s × 30 s = 450 m

Explanation

For student S, the marks obtained in Physics are 60 and in Mathematics are 78.

The ratio of marks in Physics to Mathematics is: 60 : 78 = (60 ÷ 6) : (78 ÷ 6) = 10 : 13

Explanation

Using the fundamental trigonometric identity: sec²θ - tan²θ = 1 (sec θ - tan θ)(sec θ + tan θ) = 1

Given that sec θ + tan θ = 1/√3, we have: sec θ - tan θ = √3

Solving these equations: 2 sec θ = √3 + 1/√3 = 4/√3 → sec θ = 2/√3 2 tan θ = 1/√3 - √3 = -2/√3 → tan θ = -1/√3

This gives: cos θ = √3/2 cot θ = -√3

Taking the positive values of these trigonometric ratios to find the positive value of their sum: cos θ = √3/2 cot θ = √3

Adding these values: cot θ + cos θ = √3 + √3/2 = 3√3/2

Explanation

The volume V of a solid sphere with radius r is given by the formula V = 4/3 π r³.

Using the given volume of 4500 π cm³:

4/3 π r³ = 4500 π

Dividing both sides by π:

4/3 r³ = 4500

Multiplying both sides by 3/4:

r³ = 4500 × 3/4 r³ = 3375

Taking the cube root of both sides:

r = 15 cm

The surface area S of a solid sphere is given by the formula S = 4 π r². Substituting r = 15 cm into this formula:

S = 4 π × 15² S = 4 π × 225 S = 900 π cm²

Explanation

To find the fourth proportional, first calculate the values of the three given numbers:

First number (a) = 1/3 of 15 = 1/3 × 15 = 5 Second number (b) = 4/5 of 25 = 4/5 × 25 = 20 Third number (c) = 3/7 of 35 = 3/7 × 35 = 15

Let the fourth proportional be x.

The relationship between four proportional numbers a, b, c, and x is given by the formula: a/b = c/x

Substitute the calculated values into the formula: 5/20 = 15/x

Simplify the fraction: 1/4 = 15/x

Solve for x: x = 15 × 4 = 60

Explanation

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

Perimeter(ΔABC) / Perimeter(ΔPQR) = BC / QR

Substituting the given values:

36 / 23 = BC / 3.8

BC = (36 × 3.8) / 23 BC = 136.8 / 23 BC = 1368 / 230 BC = 684 / 115

Converting this improper fraction into a mixed fraction:

684 / 115 = 5 109/115 cm

Explanation

The percentage reduction in serious accidents from the year 2016 to 2017 is calculated using the formula:

Percentage Reduction = ((Initial Value - Final Value) / Initial Value) × 100%

From the given data, the number of serious accidents in the year 2016 is 50, and the number of serious accidents in the year 2017 is 30.

Substituting these values into the formula:

Percentage Reduction = ((50 - 30) / 50) × 100% Percentage Reduction = (20 / 50) × 100% Percentage Reduction = 40%

Explanation

Given the quadratic equation: x² - 4x - 1 = 0

Dividing the entire equation by x: x - 4 - 1/x = 0 x - 1/x = 4

Squaring both sides of this equation using the algebraic identity (a - b)² = a² + b² - 2ab: (x - 1/x)² = 4² x² + 1/x² - 2 = 16 x² + 1/x² = 18

To find the value of the given expression: x² + 1/x² - 5

Substitute the value of x² + 1/x²: 18 - 5 = 13

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 the principal sum be P. The simple interest is P/4, and the rate of interest R is given as 4T. Substituting these into the formula:

P/4 = (P × 4T × T) / 100 1/4 = 4T² / 100 16T² = 100 T² = 25/4 T = 5/2 = 2.5 years

Using T = 2.5, the original rate of interest is R = 4 × 2.5 = 10%.

The rate of interest increases by 2%, so the new rate of interest is R' = 10% + 2% = 12%.

The new simple interest on ₹3,600 for 6 years is calculated as:

SI' = (3600 × 12 × 6) / 100 SI' = 36 × 72 = 2592