SSC CHSL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

The profit in a partnership is divided in the ratio of the product of the capital and the time duration of investment.

Let the initial capital of A be 4x. Since the capital of B is 3/2 times that of A, the initial capital of B is: 3/2 × 4x = 6x

Now, let us calculate the equivalent capital of A and B for 12 months:

For A: For the first 10 months, the capital is 4x. After 10 months, A withdrew 1/4 of his capital, which is: 1/4 × 4x = x So, the remaining capital for the next 2 months is: 4x - x = 3x Equivalent capital of A = (4x × 10) + (3x × 2) = 40x + 6x = 46x

For B: For the first 8 months, the capital is 6x. After 8 months, B withdrew ½ of his capital, which is: ½ × 6x = 3x So, the remaining capital for the next 4 months is: 6x - 3x = 3x Equivalent capital of B = (6x × 8) + (3x × 4) = 48x + 12x = 60x

The profit-sharing ratio between A and B is: A : B = 46x : 60x = 23 : 30

Total ratio units = 23 + 30 = 53

Given that the total profit incurred is ₹53,000: Amount received by A = (23 / 53) × 53,000 = ₹23,000

Explanation

By the formula for simple interest, Simple Interest = Principal × Rate × Time / 100. Let the common term (Rate × Time) / 100 be represented by k.

Since y is the simple interest on x, we have: y = x × k k = y/x

Since z is the simple interest on y, we have: z = y × k

Substituting the value of k from the first equation into the second equation: z = y × (y/x) z = y²/x y² = zx

Explanation

Let the cost price of the watch be 100%.

Initially, the watch is sold at a 10% loss, making the original selling price 90% of the cost price.

If the watch were sold for Rs. 45 more, it would yield a 5% profit, making the new selling price 105% of the cost price.

The difference between these two selling price percentages represents the Rs. 45 increase: 105% - 90% = 15%

Since 15% of the cost price equals Rs. 45: 1% of the cost price = 45 ÷ 15 = 3

The original selling price of the watch was 90% of the cost price: 90 × 3 = 270

Explanation

Let the selling price of one orange be SP.

The selling price of 36 oranges is 36 × SP. The loss incurred is equal to the selling price of 4 oranges, which is 4 × SP.

The cost price (CP) of the oranges is the sum of the selling price and the loss. CP = 36 × SP + 4 × SP = 40 × SP

The loss percentage is calculated using the formula: Loss percent = (Loss / CP) × 100 Loss percent = (4 × SP / 40 × SP) × 100 = (4 / 40) × 100 = 10%

Explanation

For two successive discount percentages of x% and y%, the single equivalent discount is calculated using the formula:

Equivalent discount = (x + y - xy/100)%

Substituting x = 30 and y = 20 into the formula:

Equivalent discount = (30 + 20 - (30 × 20)/100)% = (50 - 600/100)% = (50 - 6)% = 44%

Explanation

Taking the reciprocals of the given equations: (x + y) / xy = 1/a → 1/y + 1/x = 1/a (x + z) / xz = 1/b → 1/z + 1/x = 1/b (y + z) / yz = 1/c → 1/z + 1/y = 1/c

Adding the first two equations: 2/x + 1/y + 1/z = 1/a + 1/b

Substituting 1/y + 1/z = 1/c into this equation: 2/x + 1/c = 1/a + 1/b 2/x = 1/a + 1/b - 1/c 2/x = (bc + ac - ab) / abc

Solving for x: x = 2abc / (ac + bc - ab)

Explanation

We have the following system of linear equations: 6x - 5y = 13 (Equation 1) 7x + 2y = 23 (Equation 2)

To eliminate y, we multiply Equation 1 by 2 and Equation 2 by 5: 12x - 10y = 26 35x + 10y = 115

Adding these two equations gives: 47x = 141 x = 3

Substituting x = 3 into Equation 2: 7(3) + 2y = 23 21 + 2y = 23 2y = 2 y = 1

Using the values of x and y to evaluate the target expression: 11x + 18y = 11(3) + 18(1) = 33 + 18 = 51

Explanation

Using the standard algebraic identity: (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

The expression to find is: xy + z(x + y) = xy + yz + zx

Substitute the given values x + y + z = 13 and x² + y² + z² = 69 into the identity: 13² = 69 + 2(xy + yz + zx) 169 = 69 + 2(xy + yz + zx) 100 = 2(xy + yz + zx) xy + yz + zx = 50

Explanation

When two trains of equal length L run in the same direction on parallel tracks, the relative speed is the difference between their individual speeds:

Relative Speed = 90 km/h - 60 km/h = 30 km/h

Converting this speed into metres per second:

Relative Speed = 30 × 5/18 = 25/3 m/s

The total distance to be covered for one train to completely cross the other is the sum of their lengths, which is L + L = 2L.

Using the distance formula:

Distance = Speed × Time

2L = (25/3) × 30

2L = 250

L = 125 metres

Explanation

Let the total work be 1 unit.

Since A and B together can finish the work in 30 days, their combined daily work rate is 1/30 of the work per day.

In 20 days, the work completed by A and B together is: 20 × 1/30 = 2/3 of the work.

The remaining work is: 1 - 2/3 = 1/3 of the work.

A takes another 20 days to finish this remaining 1/3 of the work. Therefore, A's daily work rate is: (1/3) ÷ 20 = 1/60 of the work per day.

To find B's daily work rate, subtract A's daily rate from their combined rate: 1/30 - 1/60 = 2/60 - 1/60 = 1/60 of the work per day.

Since B completes 1/60 of the work in one day, B alone can finish the entire work in 60 days.

Explanation

Let O be the center of the circle and R be the point where the line segment OT intersects the chord PQ. Since the line connecting the center to the external point T perpendicularly bisects the chord PQ, we have:

PR = RQ = 8/2 = 4 cm

In the right-angled triangle PRO, using Pythagoras' theorem:

OP² = PR² + OR² 5² = 4² + OR² OR² = 25 - 16 = 9 OR = 3 cm

Since TP is a tangent to the circle at P, angle OPT is 90°. In the right-angled triangle OPT, PR is perpendicular to the hypotenuse OT. This makes triangle OPT similar to triangle RPO.

Using the ratio of corresponding sides of these similar triangles:

TP/PR = OP/OR TP/4 = 5/3 TP = 20/3 cm

Explanation

The intercept form of a line is x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. The area of the triangle formed by this line and the coordinate axes is given by the formula:

Area = ½ × |a × b|

Dividing the given equation 3x + 4y = 12 by 12 on both sides gives:

x/4 + y/3 = 1

Comparing this with the intercept form, the x-intercept a = 4 and the y-intercept b = 3.

Substituting these values into the area formula:

Area = ½ × 4 × 3 = 6 sq. units

Explanation

Here x is the cube root of (2 + square root of 3), so x cubed equals 2 + square root of 3 directly. Rationalizing, 1 divided by x cubed equals 1 divided by (2 + square root of 3), which simplifies to 2 minus square root of 3. Adding these two results, x cubed plus 1 over x cubed equals (2 + square root of 3) plus (2 - square root of 3), which is 4.

Explanation

In the equilateral triangle PQR, all sides are equal (PQ = QR = PR) and each interior angle measures 60°.

Since the side QR is produced to S such that QR = RS, we have PR = RS.

Because Q, R, and S lie on a straight line, angle PRQ and angle PRS are supplementary angles: angle PRS = 180° - angle PRQ = 180° - 60° = 120°

In triangle PRS, since PR = RS, it is an isosceles triangle where the angles opposite to these sides are equal: angle PSR = angle SPR

The sum of the angles in triangle PRS is 180°: angle PSR + angle SPR + angle PRS = 180° 2 × angle PSR + 120° = 180° 2 × angle PSR = 60° angle PSR = 30°

Explanation

Since AB = AC and AB = AD, we have AB = AC = AD. This implies that the point A is equidistant from points B, C, and D.

Consequently, a circle can be drawn with A as the center and BD as the diameter, passing through the points B, C, and D. Since BD is the diameter of this circle, the angle BCD is an angle in a semicircle.

By the property of circles, the angle subtended by a diameter at any point on the circumference is always a right angle. Therefore, angle BCD must be 90°.

Explanation

In trapezium ABCD, since AB is parallel to DC, the alternate interior angles are equal. This means angle OAB equals angle OCD, and angle OBA equals angle ODC. Additionally, angle AOB and angle COD are vertically opposite angles and thus equal. Consequently, triangle AOB is similar to triangle COD by AAA similarity.

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides:

Area(AOB) / Area(COD) = (AB / CD)²

Substituting the given condition AB = 2 CD into this relation:

Area(AOB) / Area(COD) = (2 CD / CD)² = 2² = 4/1

This directly yields the ratio of 4:1.

Explanation

An isosceles triangle with equal sides a, base b, and altitude h to the base can be divided into two right-angled triangles with hypotenuse a, height h, and base b/2.

By the Pythagorean theorem: a² = h² + (b/2)²

Substituting the given height h = 8: a² = 64 + b²/4

Since the perimeter of the triangle is 64 cm: 2a + b = 64 a = 32 - b/2

Substituting this expression for a into the Pythagorean relation: (32 - b/2)² = 64 + b²/4 1024 - 32b + b²/4 = 64 + b²/4 1024 - 32b = 64 32b = 960 b = 30 cm

The area of the triangle is: Area = ½ × base × height Area = ½ × 30 × 8 = 120 cm²

Explanation

The total volume of the three solid iron cubes is the sum of their individual volumes: Volume = 4³ + 5³ + 6³ = 64 + 125 + 216 = 405 cm³

After losing 62 cm³ of the melted material, the volume of the newly formed cube is: New Volume = 405 - 62 = 343 cm³

Let the edge of the newly formed cube be x. The volume of this cube is given by: x³ = 343 x = 7 cm

The total surface area of a cube with edge x is given by: Surface Area = 6x² = 6 × 7² = 6 × 49 = 294 cm²

Explanation

The volume of a cone is given by the formula V = ⅓πr²h, where r is the radius of the base and h is the height.

Let V₁ and V₂ be the volumes, r₁ and r₂ be the radii, and h₁ and h₂ be the heights of the two cones.

The ratio of their volumes is: V₁/V₂ = (⅓πr₁²h₁) / (⅓πr₂²h₂) V₁/V₂ = (r₁/r₂)² × (h₁/h₂)

Given that V₁/V₂ = 2/3 and r₁/r₂ = 1/2, substituting these values into the relation yields: 2/3 = (1/2)² × (h₁/h₂) 2/3 = 1/4 × (h₁/h₂)

Solving for the ratio of their heights: h₁/h₂ = 2/3 × 4 h₁/h₂ = 8/3

Explanation

The area of the slant surface of a right pyramid with a square base is given by the formula: Slant surface area = 2 × a × l

where a is the side of the square base and l is the slant height.

The slant height l is calculated using the relation: l = √(h² + (a/2)²)

Substituting the given values, side a = 16 cm and height h = 15 cm: l = √(15² + (16/2)²) l = √(225 + 8²) l = √(225 + 64) l = √289 l = 17 cm

Now, substituting a and l into the slant surface area formula: Slant surface area = 2 × 16 × 17 Slant surface area = 544 cm²

Explanation

Let O be the center of the circle and AB be the chord such that the length of AB is equal to the radius of the circle. Since OA and OB are also radii of the circle, triangle OAB is an equilateral triangle where OA = OB = AB.

The angle subtended by the chord AB at the center O is ∠AOB = 60°.

Let C be any point on the major arc of the circle. The angle subtended by the chord at any point on the major arc is half of the angle subtended by it at the center. Therefore, ∠ACB = 60° ÷ 2 = 30°.

Let D be a point on the minor arc of the circle. The quadrilateral ACBD is a cyclic quadrilateral as all four of its vertices lie on the circumference of the circle. The sum of opposite angles in a cyclic quadrilateral is 180°.

This gives ∠ADB + ∠ACB = 180°. Substituting ∠ACB = 30° into the equation yields ∠ADB = 180° - 30° = 150°.

Explanation

The diagonal of a square is related to its side s by the formula diagonal = s√2. Since the diagonal of the square is 40√2 cm, the side of the square s is 40 cm.

The length of each side of the rhombus is equal to the side of the square, so the side of the rhombus a = 40 cm.

The diagonals of the rhombus, d₁ and d₂, are in the ratio 3:4, so they can be written as d₁ = 3x and d₂ = 4x.

Using the relationship between the side and the diagonals of a rhombus: 4a² = d₁² + d₂²

Substitute the values into the equation: 4 × 40² = (3x)² + (4x)² 4 × 1600 = 9x² + 16x² 6400 = 25x² x² = 256 x = 16

Thus, the diagonals of the rhombus are: d₁ = 3 × 16 = 48 cm d₂ = 4 × 16 = 64 cm

The area of the rhombus is given by: Area = ½ × d₁ × d₂ Area = ½ × 48 × 64 = 1536 cm²

Explanation

Using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b) with a = sin² θ and b = cos² θ, we can write:

sin⁶ θ + cos⁶ θ = (sin² θ + cos² θ)³ - 3 sin² θ cos² θ (sin² θ + cos² θ)

Since sin² θ + cos² θ = 1, this simplifies to:

sin⁶ θ + cos⁶ θ = 1 - 3 sin² θ cos² θ

We can rewrite the term sin² θ cos² θ using the double-angle formula sin 2θ = 2 sin θ cos θ:

sin² θ cos² θ = (sin² 2θ) / 4

Substituting this back into the expression gives:

sin⁶ θ + cos⁶ θ = 1 - (3/4) sin² 2θ

For any real angle θ, the value of sin² 2θ lies in the interval [0, 1]. To maximize the expression, we must subtract the minimum possible value. The minimum value of sin² 2θ is 0, which occurs when θ = 0, π/2, etc.

Substituting sin² 2θ = 0 gives the greatest value:

1 - (3/4) × 0 = 1

Explanation

For any real number θ, the expressions of the form a sinθ + b cosθ and b sinθ - a cosθ satisfy the algebraic identity:

(a sinθ + b cosθ)² + (b sinθ - a cosθ)² = a² + b²

This is derived by expanding both squared terms: (a² sin²θ + b² cos²θ + 2ab sinθ cosθ) + (b² sin²θ + a² cos²θ - 2ab sinθ cosθ) = (a² + b²)(sin²θ + cos²θ) = a² + b²

Given that 3 sinθ + 5 cosθ = 5, we can set a = 3 and b = 5. Let the required expression 5 sinθ - 3 cosθ be y. Substituting these values into the identity gives:

5² + y² = 3² + 5²

25 + y² = 9 + 25

y² = 9

y = ±3

Explanation

Given the trigonometric relation: sin θ + sin²θ = 1

We can rewrite this as: sin θ = 1 - sin²θ

Using the standard identity sin²θ + cos²θ = 1, we know that 1 - sin²θ = cos²θ. Therefore: sin θ = cos²θ

Squaring both sides of this equation gives: sin²θ = cos⁴θ

Now, substitute cos⁴θ with sin²θ in the required expression: cos²θ + cos⁴θ = cos²θ + sin²θ

Since cos²θ + sin²θ = 1, the value of the expression is 1.

Explanation

Correct Answer: D

The height h of a pillar, when the angles of elevation from two points on the same side at distances a and b from its base are complementary, is given by the standard formula h = √(ab).

In this problem, the distances are a = 9 m and b = 16 m.

Substituting these values directly into the formula: h = √(9 × 16) h = √144 h = 12 m

Explanation

The sum of the interior angles of a polygon with n sides is calculated using the formula (n - 2) × 180°.

Substituting the given sum of 1440° into the formula:

(n - 2) × 180 = 1440

Dividing both sides by 180:

n - 2 = 8

n = 10

Explanation

The total surface area of a right prism is given by the formula: Total Surface Area = 2 × Base Area + Perimeter of Base × Height

The base of the prism is a right-angled triangle with sides 5 cm, 12 cm, and 13 cm. Base Area = ½ × 5 × 12 = 30 cm² Perimeter of Base = 5 + 12 + 13 = 30 cm

Substituting these values into the formula: 360 = 2 × 30 + 30 × Height 360 = 60 + 30 × Height 300 = 30 × Height Height = 10 cm

Explanation

The given expression is the product of cosine values of integer angles from 1° to 100°:

cos 1° × cos 2° × cos 3° × ... × cos 90° × ... × cos 100°

This sequence contains the term cos 90°. Since the trigonometric value of cos 90° is 0, multiplying any real numbers by 0 results in 0.

Explanation

Let the total capacity of the tank be 6 units. Since pipes A, B, and C can fill the tank in 6 hours, their combined rate of work is 6 units / 6 hours = 1 unit per hour.

In 2 hours, pipes A, B, and C working together fill: 2 hours × 1 unit per hour = 2 units of the tank.

The remaining part of the tank to be filled is: 6 units - 2 units = 4 units.

This remaining 4 units is filled by pipes A and B together in 8 hours. Therefore, the combined rate of work of A and B is: 4 units / 8 hours = 0.5 units per hour.

The individual rate of work of pipe C is the difference between the combined rate of all three pipes and the combined rate of A and B: 1 - 0.5 = 0.5 units per hour.

The time taken by pipe C alone to fill the entire tank of 6 units is: 6 units / 0.5 units per hour = 12 hours.

Explanation

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

Let r₁ = 10 cm and r₂ = 24 cm be the radii of the two given circles. The area of the new circle is the sum of the areas of the two smaller circles. If R is the radius of this new circle, the relation is:

πR² = πr₁² + πr₂²

Dividing both sides of the equation by π gives:

R² = r₁² + r₂²

Substituting the given values into this relation:

R² = 10² + 24²

R² = 100 + 576

R² = 676

R = √676 = 26 cm

Explanation

Using the law of exponents (xᵐ)ⁿ = xᵐⁿ and the algebraic identity (u+v)(u-v) = u² - v², the terms in the expression simplify as follows:

(xᵇ⁺ᶜ)ᵇ⁻ᶜ = x⁽b² - c²)

(xᶜ⁺ᵃ)ᶜ⁻ᵃ = x⁽c² - a²)

(xᵃ⁺ᵇ)ᵃ⁻ᵇ = x⁽a² - b²)

Multiplying these expressions together by adding their exponents:

x⁽b² - c²) × x⁽c² - a²) × x⁽a² - b²) = x⁽b² - c² + c² - a² + a² - b²)

Simplifying the exponent in the power:

b² - c² + c² - a² + a² - b² = 0

Therefore, the entire expression simplifies to:

x⁰ = 1

Explanation

The product of two numbers is equal to the product of their LCM and HCF.

First Number × Second Number = LCM × HCF

Let the other number be x. Substituting the given values:

189 × x = 2079 × 27

x = (2079 × 27) / 189

Since 189 divided by 27 is equal to 7, the equation simplifies to:

x = 2079 / 7

x = 297

Explanation

The total percentage of people below 36 years is the sum of the percentages of the age groups up to 15, 16-25, and 26-35: Percentage below 36 years = 20.00% + 18.25% + 16.75% = 55%

Let the total population of the city be P. 55% of P = 22 million P = (22 × 100) / 55 = 40 million

The percentage of people in the age group 56-65 is 12.50%. Number of people in the age group 56-65 = 12.50% of 40 million = 0.125 × 40 = 5 million.

Explanation

The percentage of the population in the age group 16-25 is 18.25%, and the percentage of the population in the age group 46-55 is 15.00%.

The difference in percentage between these two age groups is: 18.25% - 15.00% = 3.25%

Since this difference represents 0.975 million people, the total population (100%) can be calculated directly: Total Population = (0.975 ÷ 3.25) × 100 = 30 million

Explanation

Using the standard trigonometric identities:

1 + cot²α = cosec²α 1 + tan²α = sec²α

Substituting these identities into the given expression yields:

1/cosec²α + 1/sec²α

Using the reciprocal relations where 1/cosec²α = sin²α and 1/sec²α = cos²α, the expression simplifies to:

sin²α + cos²α

According to the fundamental Pythagorean identity, sin²α + cos²α = 1.

Explanation

Let the monument be represented by AB of height H, and the building by CD of height h. The distance between their bases is BD = a.

Draw a horizontal line CE perpendicular to AB. Thus, AE = AB - BE = H - h, and CE = BD = a.

Let the angle of depression of the top of the building C from the top of the monument A be θ. In right-angled triangle AEC, the angle ACE is θ. tan θ = AE / CE = (H - h) / a

The angle of depression of the bottom of the building D from the top of the monument A is complementary to θ, which is 90° - θ. In right-angled triangle ABD, the angle ADB is 90° - θ. tan(90° - θ) = AB / BD = H / a

Since tan(90° - θ) = cot θ, we have: cot θ = H / a tan θ = a / H

Equating the two expressions for tan θ: (H - h) / a = a / H

Taking the reciprocal on both sides gives: a / (H - h) = H / a

Rearranging the terms: H / a = a / (H - h)

Explanation

Applying the componendo and dividendo rule to the given equation:

((secθ + tanθ) + (secθ - tanθ)) / ((secθ + tanθ) - (secθ - tanθ)) = ((2 + √3) + (2 - √3)) / ((2 + √3) - (2 - √3))

This simplifies to:

2 secθ / 2 tanθ = 4 / 2√3

secθ / tanθ = 2 / √3

Since secθ = 1/cosθ and tanθ = sinθ/cosθ, the LHS simplifies to:

(1/cosθ) / (sinθ/cosθ) = 2 / √3

1/sinθ = 2 / √3

sinθ = √3 / 2

In circular measure, the angle in the first quadrant whose sine is √3/2 is π/3.

Explanation

Using the trigonometric identity cos θ = sin(90° - θ), we have: sin 47° = cos 43° = x/√(x² + y²)

In a right-angled triangle with angle 47°, sin 47° is the ratio of the perpendicular to the hypotenuse. Therefore: perpendicular = x hypotenuse = √(x² + y²)

Using Pythagoras' theorem: base = √(hypotenuse² - perpendicular²) base = √((x² + y²) - x²) base = √y² = y

Since tan θ is the ratio of the perpendicular to the base, we obtain: tan 47° = x/y

Explanation

First, convert all the dimensions of the wall into centimeters to match the unit of the brick dimensions. The length of the wall is 9 m, which equals 900 cm, and the height is 6 m, which equals 600 cm. The thickness of the wall is already given as 20 cm.

The total volume of the wall is calculated by multiplying its length, height, and thickness: Volume of the wall = 900 cm × 600 cm × 20 cm = 10,800,000 cm³

The volume of one brick is calculated using its given dimensions: Volume of one brick = 30 cm × 15 cm × 10 cm = 4,500 cm³

The number of bricks required is determined by dividing the total volume of the wall by the volume of a single brick: Number of bricks = 10,800,000 ÷ 4,500 = 2,400

Explanation

The volume of a sphere is given by the formula: V = (4/3) × π × R³

where R is the radius of the sphere. Since the diameter of the sphere is 6 cm, its radius R is 3 cm. V = (4/3) × π × 3³ = 36π cm³

The wire is in the shape of a cylinder. Its volume is given by the formula: V = π × r² × h

where r is the radius and h is the length of the wire. For a wire with a diameter of 2 mm (equivalent to 0.2 cm), the radius r is 0.1 cm. V = π × (0.1)² × h = 0.01π × h cm³

Since the sphere is melted and recast into the wire, their volumes must be equal: 36π = 0.01π × h h = 36 / 0.01 = 3600 cm

Converting the length of the wire to meters: h = 3600 / 100 = 36 m

This value is represented in the options as 36 cm due to a typographical unit error in the original exam paper.

Explanation

The equation contains a typographical error where cot θ is written instead of cos θ. The correct formulation of this equation is:

sec θ - cos θ = 3/2

Since cos θ = 1/sec θ, we can substitute this relation into the equation:

sec θ - 1/sec θ = 3/2

Let x = sec θ. Substituting x into the equation gives:

x - 1/x = 3/2

Multiplying the entire equation by 2x to eliminate the fractions:

2x² - 2 = 3x 2x² - 3x - 2 = 0

Now, factor the quadratic equation:

2x² - 4x + x - 2 = 0 2x(x - 2) + 1(x - 2) = 0 (2x + 1)(x - 2) = 0

This yields two solutions for x:

x = 2 or x = -1/2

Since θ is a positive acute angle (0 < θ < 90°), sec θ must be positive and greater than 1. Therefore, we discard the negative root -1/2.

sec θ = 2

Explanation

The volume of a pyramid is given by the formula V = 1/3 × Base Area × h, where h is the height of the pyramid.

The base is an equilateral triangle. Let each side of the triangle be s. The area of an equilateral triangle is (√3 / 4) × s².

Substituting this base area and the height into the volume formula gives:

Volume = 1/3 × (√3 / 4) × s² × h

Using the given values of volume (48√3 cc) and height (4 cm):

48√3 = 1/3 × (√3 / 4) × s² × 4

Simplifying the equation by canceling 4 in the numerator and denominator:

48√3 = (√3 / 3) × s²

Dividing both sides by √3:

48 = s² / 3

s² = 144

s = 12 cm

Explanation

The volume of a cone formed by rotating a right-angled triangle about one of its perpendicular sides is given by the formula V = ⅓πr²h, where the side about which the triangle is rotated acts as the height (h) and the other perpendicular side acts as the radius (r).

When the triangle is rotated around the side AB: Height h₁ = AB = 3 cm Radius r₁ = BC = 4 cm Volume V₁ = ⅓ × π × 4² × 3 = 16π cm³

When the triangle is rotated around the side BC: Height h₂ = BC = 4 cm Radius r₂ = AB = 3 cm Volume V₂ = ⅓ × π × 3² × 4 = 12π cm³

The percentage by which the volume of the first cone is more than the second cone is given by: x = ((V₁ - V₂) / V₂) × 100 x = ((16π - 12π) / 12π) × 100 x = (4/12) × 100 = ⅓ × 100 = 33 ⅓ %

Explanation

By the Pythagorean theorem, the perpendicular height p of the right-angled triangle is:

p = √(hypotenuse² - base²) p = √(13² - 5²) p = √(169 - 25) p = √144 p = 12 units

The area of a right-angled triangle is calculated using the formula:

Area = ½ × base × height Area = ½ × 5 × 12 Area = 30 sq. units

Explanation

In any triangle, the angle opposite to the longer side is always greater in magnitude. Here, the given side lengths of triangle PQR are QR = 5 cm, PR = 4 cm, and PQ = 3 cm. Comparing these lengths, we have QR > PR > PQ. Since ∠P is opposite to side QR, ∠Q is opposite to side PR, and ∠R is opposite to side PQ, the angles follow the same descending order as their opposite sides, which gives ∠P > ∠Q > ∠R.

Explanation

Two linear equations in two variables representing straight lines do not intersect if they are parallel. For the general system of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the lines are parallel and have no points of intersection when a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

In this system: x + y = 4 3x + 3y = 6

The ratio of the coefficients of x is 1/3, and the ratio of the coefficients of y is 1/3. However, the ratio of the constant terms is 4/6, which simplifies to 2/3.

Since 1/3 = 1/3 ≠ 2/3, the lines are parallel and run in the same direction with different y-intercepts. Consequently, they will never cross each other, meaning they do not intersect.

Explanation

To find the value of (x/yz) ÷ (y/zx), we simplify the expression: (x/yz) ÷ (y/zx) = (x/yz) × (zx/y) = (x²z) / (y²z) = x² / y²

From the given ratio: yz : zx : xy = 1 : 2 : 3

Comparing the second and the first terms of the ratio: zx / yz = 2 / 1 x / y = 2

Squaring both sides of this equation: x² / y² = 2² = 4

Explanation

Squaring both sides of the given equation x - 1/x = 2: (x - 1/x)² = 2² x² - 2 + 1/x² = 4 x² + 1/x² = 6

Squaring both sides of this resulting equation: (x² + 1/x²)² = 6² x⁴ + 2 + 1/x⁴ = 36 x⁴ + 1/x⁴ = 34

Explanation

Rationalizing the denominator of the given expression for x: x = 1 / (√2 + 1) x = (√2 - 1) / ((√2 + 1)(√2 - 1)) x = (√2 - 1) / (2 - 1) x = √2 - 1

Rearranging the terms gives: x + 1 = √2

Squaring both sides of the equation: (x + 1)² = (√2)² x² + 2x + 1 = 2 x² + 2x = 1

Adding 2 to both sides of the equation yields: x² + 2x + 3 = 4