SSC CHSL Tier 1 2013 Quantitative Aptitude Questions with Solutions

Explanation

The trigonometric expression is: (2 sin 68°) / cos 22° - (2 cot 15°) / (5 tan 75°) - (3 tan 45° · tan 20° · tan 40° · tan 50° · tan 70°) / 5

Each term can be simplified using complementary angle trigonometric identities:

For the first term: Using cos θ = sin(90° - θ), we get cos 22° = sin(90° - 22°) = sin 68°. (2 sin 68°) / cos 22° = (2 sin 68°) / sin 68° = 2

For the second term: Using tan θ = cot(90° - θ), we get tan 75° = cot(90° - 75°) = cot 15°. (2 cot 15°) / (5 tan 75°) = (2 cot 15°) / (5 cot 15°) = 2/5

For the third term: Using tan θ · tan(90° - θ) = 1: tan 20° · tan 70° = tan 20° · cot 20° = 1 tan 40° · tan 50° = tan 40° · cot 40° = 1 Since tan 45° = 1, the numerator simplifies to: 3 × 1 × 1 × 1 = 3 (3 tan 45° · tan 20° · tan 40° · tan 50° · tan 70°) / 5 = 3/5

Substituting these values back into the expression: 2 - 2/5 - 3/5 = 2 - 5/5 = 2 - 1 = 1

Explanation

Using the complementary angle relationship, we have cot 72° = cot(90° - 18°) = tan 18°.

This gives: 1/cot² 72° = 1/tan² 18° = cot² 18°

Substituting this into the original expression: csc² 18° - 1/cot² 72° = csc² 18° - cot² 18°

Applying the fundamental trigonometric identity csc² θ - cot² θ = 1, we get: csc² 18° - cot² 18° = 1

Explanation

Using trigonometric identities, we can simplify each term of the expression.

First, 1 - sin²α = cos²α and 1 - cos²α = sin²α. Therefore, the first part of the expression becomes cos²α × sin²α.

Second, since α + β = 90°, we have β = 90° - α. This gives cot β = cot(90° - α) = tan α, and tan β = tan(90° - α) = cot α. Substituting these into the second part of the expression gives: 1 + cot²β = 1 + tan²α = sec²α 1 + tan²β = 1 + cot²α = cosec²α So, the second part of the expression simplifies to sec²α × cosec²α.

Multiplying the two parts together: (cos²α × sin²α) × (sec²α × cosec²α) = cos²α × sin²α × (1/cos²α) × (1/sin²α) = 1

Explanation

Given the equation tan θ - cot θ = 0, we can rewrite it as: tan θ = cot θ

Since cot θ is the reciprocal of tan θ, this gives: tan θ = 1/tan θ tan² θ = 1

For the acute angle θ in the first quadrant, this yields tan θ = 1, which means θ = 45°.

To find the value of sin θ + cos θ, we substitute θ = 45°: sin 45° + cos 45° = 1/√2 + 1/√2 = 2/√2 = √2.

Explanation

The sum of an interior angle and an exterior angle of any polygon is always 180°. If the measure of the exterior angle is x, then the interior angle is 2x.

Using this relationship: 2x + x = 180° 3x = 180° x = 60°

So, the exterior angle of the regular polygon is 60°.

The number of sides n of a regular polygon is calculated by dividing 360° by the measure of its exterior angle: n = 360° / 60° n = 6

Explanation

The volume of a right prism is given by the formula: Volume = Base Area × Height

The base of the prism is an equilateral triangle with side length a = 6 cm. The area of an equilateral triangle is given by the formula: Base Area = (√3 / 4) × a²

Substituting a = 6 cm into the formula: Base Area = (√3 / 4) × 6² = (√3 / 4) × 36 = 9√3 cm²

Given that the volume of the prism is 81√3 cm³, we substitute these values into the volume formula: 81√3 = 9√3 × Height

Dividing both sides by 9√3: Height = 9 cm

Explanation

Simplify the square roots in the given equation: √75 = 5√3 √50 = 5√2 √24 = 2√6

Substitute these simplified terms back into the original equation: ((x - 2√6)(5√3 + 5√2)) / (5√3 - 5√2) = 1

Factor out 5 from the numerator and the denominator of the fraction: ((x - 2√6) × 5(√3 + √2)) / (5(√3 - √2)) = 1 ((x - 2√6)(√3 + √2)) / (√3 - √2) = 1

Multiply both sides of the equation by (√3 - √2): (x - 2√6)(√3 + √2) = √3 - √2

Isolate the term containing x by dividing both sides by (√3 + √2): x - 2√6 = (√3 - √2) / (√3 + √2)

Rationalize the denominator on the right-hand side by multiplying both the numerator and the denominator by (√3 - √2): x - 2√6 = (√3 - √2)² / ((√3 + √2)(√3 - √2)) x - 2√6 = (3 + 2 - 2√6) / (3 - 2) x - 2√6 = 5 - 2√6

Add 2√6 to both sides to solve for x: x = 5

Explanation

To find the total number of prime factors, express each composite base as a product of its prime factors:

4¹⁰ = (2²)¹⁰ = 2²⁰ 7³ = 7³ 16² = (2⁴)² = 2⁸ 11 = 11¹ 10² = (2 × 5)² = 2² × 5²

Now, rewrite the entire product using these prime factorizations:

2²⁰ × 7³ × 2⁸ × 11¹ × 2² × 5² = 2³⁰ × 5² × 7³ × 11¹

The total number of prime factors is the sum of the powers of these prime bases:

30 + 2 + 3 + 1 = 36

Explanation

To find the number nearest to the value of √0.9, we calculate the square root directly.

We can rewrite √0.9 as √(90/100) = √90 / 10.

Since 9² = 81 and 10² = 100, √90 lies between 9 and 10, specifically around 9.486.

Dividing this by 10 gives √0.9 ≈ 0.9486.

Comparing this result to 0.9, the absolute difference is |0.9486 - 0.9| = 0.0486, which represents the smallest deviation among the choices, making 0.9 the nearest value.

Explanation

The standard division formula states: Dividend = (Divisor × Quotient) + Remainder

The problem states that the remainder is 36.

The divisor is 5 times the remainder: Divisor = 5 × 36 = 180

The divisor is also 12 times the quotient: Quotient = Divisor ÷ 12 = 180 ÷ 12 = 15

Using these values in the division formula: Dividend = (180 × 15) + 36 Dividend = 2700 + 36 = 2736

Explanation

The expression can be simplified by first converting the mixed fractions into improper fractions: 3 ½ = 7/2 2 ¼ = 9/4 1 ¼ = 5/4 1 ½ = 3/2

Substituting these values into the expression, we get: 7/2 - [ 9/4 ÷ { 5/4 - ½ ( 3/2 - ⅓ - ⅙ ) } ]

Applying the BODMAS rule, we first solve the innermost parentheses ( 3/2 - ⅓ - ⅙ ). Taking the least common multiple of 2, 3, and 6, which is 6, this simplifies to: 9/6 - 2/6 - 1/6 = 6/6 = 1

Next, we evaluate the terms inside the curly braces: 5/4 - ½ × 1 = 5/4 - 2/4 = 3/4

Now, we solve the division inside the square brackets: 9/4 ÷ 3/4 = 9/4 × 4/3 = 3

Finally, we perform the subtraction outside the brackets: 7/2 - 3 = 7/2 - 6/2 = ½

Explanation

Let Arbind's initial income be ₹100. Since he spends 75% of his income, his expenditure is ₹75 and his savings are ₹25.

His income increases by 20%, making his new income ₹120. His expenditure increases by 10%, making his new expenditure 75 × 1.10 = ₹82.5.

His new savings are calculated as 120 - 82.5 = ₹37.5.

The increase in savings is 37.5 - 25 = ₹12.5.

The percentage increase in savings is (12.5 / 25) × 100 = 50%.

Explanation

The fraction of copper in the first alloy is 1 / (1 + 3) = 1/4.

The fraction of copper in the second alloy is 2 / (2 + 5) = 2/7.

In the new alloy, the ratio of tin to copper is 8:3, which means the ratio of copper to tin is 3:8. Thus, the fraction of copper in the new alloy is 3 / (3 + 8) = 3/11.

By applying the rule of alligation on the copper concentrations:

Ratio of first alloy to second alloy = (2/7 - 3/11) : (3/11 - 1/4)

Simplifying the terms:

First term = 2/7 - 3/11 = (22 - 21) / 77 = 1/77

Second term = 3/11 - 1/4 = (12 - 11) / 44 = 1/44

Therefore, the ratio is 1/77 : 1/44. Multiplying both parts of the ratio by 308, we get 4:7.

Explanation

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

The sum of the two numbers is: x + y = 520

Decreasing the bigger number by 4% yields 0.96x, and increasing the smaller number by 12% yields 1.12y. Since these two values are equal: 0.96x = 1.12y

Expressing this as a ratio: x/y = 1.12 / 0.96 = 112 / 96 = 7/6

This gives the ratio of the bigger number to the smaller number as 7:6.

The sum of the ratio terms is 7 + 6 = 13 units, which corresponds to the total sum of 520.

Value of 1 unit = 520 / 13 = 40

The smaller number is represented by 6 units: y = 6 × 40 = 240

Explanation

Let the initial quantities of alcohol and water be 4x and 3x litres respectively.

When 5 litres of water is added, the quantity of alcohol remains unchanged at 4x litres, while the quantity of water becomes 3x + 5 litres.

The new ratio of alcohol to water is given as 4:5. Therefore:

4x / (3x + 5) = 4 / 5

Dividing both sides of the equation by 4:

x / (3x + 5) = 1 / 5

Cross-multiplying:

5x = 3x + 5

2x = 5

x = 2.5

The quantity of alcohol in the mixture is:

4x = 4 × 2.5 = 10 litres.

Explanation

When m consecutive numbers are added to a sequence of consecutive numbers, the average of the series increases by m/2. Since the next 2 numbers are included in the sequence, m = 2. Therefore, the average will increase by 2/2 = 1.

Explanation

The formula for simple interest is SI = (P × R × T) / 100, where P is the principal sum, 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 given as P/16. Since the number of years is numerically equal to the rate percent per annum, T = R.

Substituting these values into the simple interest formula:

P/16 = (P × R × R) / 100

Dividing both sides by P and rearranging the equation:

R² = 100 / 16

Taking the square root on both sides:

R = 10 / 4 = 2.5 = 2 ½

Explanation

Let a + b = 6k, b + c = 7k, and c + a = 8k, where k is a constant.

Adding these three equations: (a + b) + (b + c) + (c + a) = 6k + 7k + 8k 2(a + b + c) = 21k

Substituting the given value a + b + c = 14: 2 × 14 = 21k 28 = 21k k = 28/21 = 4/3

Now, substitute the value of k back into the equation for a + b: a + b = 6k = 6 × (4/3) = 8

To find c, subtract (a + b) from (a + b + c): c = (a + b + c) - (a + b) c = 14 - 8 = 6

Explanation

Evaluating each term of the expression:

√20 = √(4 × 5) = 2√5

√12 = √(4 × 3) = 2√3

∛729 = 9

√81 = 9

Rationalizing the denominator of the term 4/(√5 - √3) by multiplying the numerator and denominator by (√5 + √3): 4/(√5 - √3) = [4(√5 + √3)] / [(√5 - √3)(√5 + √3)] Using the algebraic identity (a - b)(a + b) = a² - b²: 4/(√5 - √3) = 4(√5 + √3) / (5 - 3) 4/(√5 - √3) = 4(√5 + √3) / 2 = 2(√5 + √3) = 2√5 + 2√3

Substituting these values back into the original expression: √20 + √12 + ∛729 - 4/(√5 - √3) - √81 = 2√5 + 2√3 + 9 - (2√5 + 2√3) - 9 = 2√5 + 2√3 + 9 - 2√5 - 2√3 - 9 = 0

Explanation

The list price of the mobile phone is ₹1,500.

First, apply the 10% discount on the list price: Price after first discount = 1500 × (1 - 10/100) = 1500 × 0.9 = ₹1,350

The target net price is ₹1,242.

The additional discount amount required on the intermediate price of ₹1,350 is: Additional discount amount = 1350 - 1242 = ₹108

Calculate the additional discount percentage based on the intermediate price: Additional discount percentage = (108 / 1350) × 100 = 8%

Explanation

Let the initial investments of A, B and C be 5x, 4x and 3x respectively.

The profit of partners in a business is shared in the ratio of their equivalent investments over the period of one year (12 months):

Equivalent investment of A = 5x × 12 = 60x

Equivalent investment of B = (4x × 4) + (4x + 1000) × 8 = 16x + 32x + 8000 = 48x + 8000

Equivalent investment of C = (3x × 8) + (3x + 2000) × 4 = 24x + 12x + 8000 = 36x + 8000

The profit sharing ratio of A to B is given as 15:14, so: 60x / (48x + 8000) = 15 / 14

Dividing the numerator on both sides by 15: 4x / (48x + 8000) = 1 / 14

Cross-multiplying: 56x = 48x + 8000 8x = 8000 x = 1000

The investment of C at the beginning was 3x: 3x = 3 × 1000 = ₹3,000

Explanation

When two trains run in opposite directions, their relative speed is the sum of their individual speeds, and the total distance to be covered to completely cross each other is the sum of their lengths.

Total distance = 125 + 125 = 250 metres Time taken = 6 seconds

Relative speed = Distance / Time = 250 / 6 metres/second

To convert the speed from metres/second to km/hour, multiply by 18/5: Relative speed = (250 / 6) × (18 / 5) = 150 km/hour

Since the trains are moving in opposite directions: Relative speed = Speed of first train + Speed of second train 150 = 65 + Speed of second train Speed of second train = 150 - 65 = 85 km/hour

Explanation

Let the original cost price be CP.

The article is initially sold at a profit of 12%, so the initial selling price is: SP₁ = 1.12 × CP

According to the new condition, if the cost price is 10% less, the new cost price becomes: CP₂ = 0.90 × CP

If the selling price is ₹5.75 more, the new selling price becomes: SP₂ = 1.12 × CP + 5.75

Under these conditions, the profit is 30%. Therefore: SP₂ = 1.30 × CP₂ 1.12 × CP + 5.75 = 1.30 × 0.90 × CP 1.12 × CP + 5.75 = 1.17 × CP 1.17 × CP - 1.12 × CP = 5.75 0.05 × CP = 5.75 CP = 5.75 / 0.05 = 115

The original cost price is ₹115.

To make a profit of 20%, the selling price must be: Selling Price = 1.20 × CP = 1.20 × 115 = 138

Explanation

Let the cost price of one pen be CP and the selling price of one pen be SP.

The relation given is that the cost price of 12 pens equals the selling price of 8 pens: 12 × CP = 8 × SP

Rearranging this equation gives the ratio of the selling price to the cost price: SP / CP = 12 / 8 = 3/2

The formula for the gain percentage is: Gain % = ((SP - CP) / CP) × 100 = (SP/CP - 1) × 100

Substituting the ratio into the formula: Gain % = (3/2 - 1) × 100 = ½ × 100 = 50%

Explanation

Using the algebraic identity for the difference of cubes: a³ - b³ = (a - b)³ + 3ab(a - b)

Substitute the given values a - b = 3 and a³ - b³ = 117 into the formula: 117 = 3³ + 3ab(3) 117 = 27 + 9ab 90 = 9ab ab = 10

Now, use the identity relating the sum and difference of two numbers: (a + b)² = (a - b)² + 4ab

Substitute the values of (a - b) and ab: (a + b)² = 3² + 4(10) (a + b)² = 9 + 40 (a + b)² = 49

Taking the square root on both sides yields the absolute value: |a + b| = √49 = 7

Explanation

The algebraic identity for the given expression is: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca))

To find the value of ab + bc + ca, we use the expansion of (a + b + c)²: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values into this relation: 15² = 83 + 2(ab + bc + ca) 225 = 83 + 2(ab + bc + ca) 2(ab + bc + ca) = 225 - 83 2(ab + bc + ca) = 142 ab + bc + ca = 71

Now, substituting these values back into the identity: a³ + b³ + c³ - 3abc = 15 × (83 - 71) a³ + b³ + c³ - 3abc = 15 × 12 a³ + b³ + c³ - 3abc = 180

Explanation

To find the intersection point (p, q) of the two equations, we solve the system of linear equations:

3x + 2y = 18 -2x + 3y = 1

Multiplying the first equation by 2 and the second equation by 3:

6x + 4y = 36 -6x + 9y = 3

Adding these two equations eliminates x:

13y = 39 y = 3

Substituting y = 3 back into the first equation:

3x + 2(3) = 18 3x + 6 = 18 3x = 12 x = 4

Thus, the point of intersection (p, q) is (4, 3), which gives p = 4 and q = 3.

The value of p + q is:

4 + 3 = 7

Explanation

The given equation is x = 4√15 / (√5 + √3).

We need to find the value of (x + √20)/(x - √20) + (x + √12)/(x - √12).

Since √20 = 2√5 and √12 = 2√3, the expression can be rewritten as: (x + 2√5)/(x - 2√5) + (x + 2√3)/(x - 2√3)

To solve this, we can express x in terms of 2√5 and 2√3 to apply the componendo and dividendo rule.

Dividing both sides of the original equation by 2√5: x / (2√5) = 2√3 / (√5 + √3)

Applying componendo and dividendo: (x + 2√5) / (x - 2√5) = (2√3 + √5 + √3) / (2√3 - √5 - √3) (x + 2√5) / (x - 2√5) = (3√3 + √5) / (√3 - √5)

Now, dividing both sides of the original equation by 2√3: x / (2√3) = 2√5 / (√5 + √3)

Applying componendo and dividendo: (x + 2√3) / (x - 2√3) = (2√5 + √5 + √3) / (2√5 - √5 - √3) (x + 2√3) / (x - 2√3) = (3√5 + √3) / (√5 - √3) = -(3√5 + √3) / (√3 - √5)

Adding the two resulting expressions together: (x + 2√5) / (x - 2√5) + (x + 2√3) / (x - 2√3) = (3√3 + √5) / (√3 - √5) - (3√5 + √3) / (√3 - √5) = (3√3 + √5 - 3√5 - √3) / (√3 - √5) = (2√3 - 2√5) / (√3 - √5) = 2(√3 - √5) / (√3 - √5) = 2

Explanation

The work done by 4 men is equal to that of 6 women, which gives the efficiency relation: 4M = 6W M/W = 3/2

Taking the efficiency of a man as 3 units/hour and a woman as 2 units/hour, the efficiency of 4 men is 4 × 3 = 12 units/hour.

The initial work, W₁, is calculated as: W₁ = Efficiency × Days × Hours = 12 × 12 × 7 = 1008 units

The new work is twice as large: W₂ = 2 × 1008 = 2016 units

The combined efficiency of 10 men and 3 women working together is: E = (10 × 3) + (3 × 2) = 36 units/hour

Using the work formula: W₂ = E × Days × Hours 2016 = 36 × Days × 8 2016 = 288 × Days Days = 2016 / 288 = 7 days

Explanation

Given the equation x² - 3x + 1 = 0. Dividing both sides by x, we get: x - 3 + 1/x = 0 x + 1/x = 3

Using algebraic identities, we find: x² + 1/x² = (x + 1/x)² - 2 = 3² - 2 = 7 x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) = 3³ - 3(3) = 18

Using these values, we determine x⁵ + 1/x⁵: x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) - (x + 1/x) x⁵ + 1/x⁵ = 7 × 18 - 3 = 126 - 3 = 123

Explanation

Let the equal sides of the isosceles right-angled triangle ABC be AB = BC = x. Using Pythagoras' theorem, the hypotenuse AC is given by AC² = AB² + BC² = x² + x² = 2x², which means AC = x√2.

Since ΔABE and ΔACD are both equilateral triangles, they are similar to each other. The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Area(ΔABE) / Area(ΔACD) = AB² / AC² = x² / (x√2)² = x² / 2x² = 1/2, which simplifies to the ratio 1:2.

Explanation

Let O be the center of the concentric circles. Since AB is a diameter of the bigger circle and E lies on its circumference, ∠AEB = 90° because the angle subtended by a diameter in a semicircle is a right angle. Thus, AE is perpendicular to BE.

Since BD is tangent to the smaller circle at D, the radius OD is perpendicular to BE. Because both AE and OD are perpendicular to BE, OD is parallel to AE.

In ΔABE, O is the midpoint of AB. Since OD is parallel to AE, by the Midpoint Theorem, D is the midpoint of BE, which gives: AE = 2 × OD = 2 × 8 = 16 cm

In right-angled ΔODB, using the Pythagorean theorem: BD = √(OB² - OD²) = √(13² - 8²) = √(169 - 64) = √105 cm

Since D is the midpoint of BE, we have DE = BD = √105 cm.

In right-angled ΔAED, using the Pythagorean theorem: AD = √(AE² + DE²) = √(16² + (√105)²) = √(256 + 105) = √361 = 19 cm

Explanation

Let O be the common center of the two concentric circles. Draw a perpendicular line segment OM from O to the line ABCD, meeting it at M.

Since a perpendicular drawn from the center of a circle to a chord bisects the chord, M is the midpoint of the chord BC of the smaller circle, and it is also the midpoint of the chord AD of the larger circle.

For the smaller circle, the radius OB = 10 cm, and the chord BC = 12 cm. BM = BC / 2 = 12 / 2 = 6 cm

In the right-angled triangle OMB, applying the Pythagorean theorem: OM² = OB² - BM² OM² = 10² - 6² OM² = 100 - 36 OM² = 64 OM = 8 cm

For the larger circle, the radius OA = 17 cm. In the right-angled triangle OMA, applying the Pythagorean theorem: AM² = OA² - OM² AM² = 17² - 8² AM² = 289 - 64 AM² = 225 AM = 15 cm

Since M is the midpoint of AD: AD = 2 × AM = 2 × 15 = 30 cm

Explanation

By the linear pair axiom, the sum of an interior angle of a triangle and its adjacent exterior angle is 180°.

The interior angle at B is: ∠B = 180° - 120° = 60°

The interior angle at C is: ∠C = 180° - 105° = 75°

Using the angle sum property of a triangle, the sum of all interior angles in triangle ABC is 180°: ∠A + ∠B + ∠C = 180°

Substituting the calculated values: ∠A + 60° + 75° = 180° ∠A + 135° = 180° ∠A = 180° - 135° = 45°

Explanation

In any triangle, the sum of the lengths of the medians is always less than the perimeter of the triangle.

To prove this, extend the median AD to a point G such that AD = DG, and join CG. In triangles ABD and GCD, BD = CD because AD is the median, AD = GD by construction, and angle ADB equals angle GDC as vertically opposite angles. By SAS congruence, triangle ABD is congruent to triangle GCD, which gives AB = CG.

In triangle ACG, by the triangle inequality, the sum of two sides is greater than the third side, so AC + CG > AG. Substituting CG = AB and AG = 2AD, we get AB + AC > 2AD, or AD < ½(AB + AC).

Applying the same inequality to the other two medians: BE < ½(AB + BC) CF < ½(BC + AC)

Adding these three inequalities together: AD + BE + CF < ½(AB + AC + AB + BC + BC + AC) AD + BE + CF < ½(2AB + 2BC + 2CA) AD + BE + CF < AB + BC + CA

Explanation

Let the two numbers be 12x and 12y, where x and y are co-prime integers whose highest common factor is 1.

The product of the two numbers is given as 2160: 12x × 12y = 2160 144xy = 2160 xy = 15

Since x and y must be co-prime, we find the pairs of co-prime factors of 15. The possible pairs of (x, y) are: (1, 15) (3, 5)

This gives exactly 2 such possible pairs of numbers, which are (12, 180) and (36, 60).

Explanation

Let the time taken by the faster pipe alone to fill the tank be x hours. Since the other pipe fills the tank 10 hours slower, the time taken by the slower pipe is x + 10 hours.

When both pipes function simultaneously, they fill the tank in 12 hours. Expressing their work rates per hour gives the equation:

1/x + 1/(x + 10) = 1/12

Combining the terms on the left side:

(x + 10 + x) / (x(x + 10)) = 1/12

(2x + 10) / (x² + 10x) = 1/12

Cross-multiplying to solve the equation:

12(2x + 10) = x² + 10x

24x + 120 = x² + 10x

x² - 14x - 120 = 0

Factoring the quadratic equation:

(x - 20)(x + 6) = 0

Since the time taken to fill a tank must be a positive value, we discard x = -6, which leaves x = 20.

Explanation

The volume of a right circular cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height.

The sum of the volumes of the two cones is: V_cones = (1/3)π(3)²h + (1/3)π(4)²h V_cones = (1/3)πh(9 + 16) V_cones = (25/3)πh

The volume of a solid sphere is given by the formula V = (4/3)πR³, where R is the radius. For a sphere of radius 5 cm: V_sphere = (4/3)π(5)³ V_sphere = (500/3)π

Since the two cones are melted to form the sphere, their combined volume equals the volume of the sphere: (25/3)πh = (500/3)π

Canceling π/3 from both sides: 25h = 500 h = 20 cm

Explanation

Let h be the height of the tower in metres, and x be the initial distance of the point on the ground from the foot of the tower.

From the first position of observation, the angle of elevation is 45°. Applying the trigonometric ratio: tan 45° = h/x

Since tan 45° = 1, we get: 1 = h/x x = h

On travelling 60 m towards the tower, the new distance from the foot of the tower is x - 60, and the angle of elevation becomes 60°. Applying the trigonometric ratio: tan 60° = h/(x - 60)

Since tan 60° = √3, we get: √3 = h/(h - 60) h = √3(h - 60) h = h√3 - 60√3 h(√3 - 1) = 60√3 h = 60√3 / (√3 - 1)

Multiplying the numerator and the denominator by (√3 + 1) to rationalize: h = 60√3(√3 + 1) / ((√3 - 1)(√3 + 1)) h = 60(3 + √3) / (3 - 1) h = 60(3 + √3) / 2 h = 30(3 + √3)

Explanation

Let x = a - b, y = b - c, and z = c - a.

The sum of these variables is: x + y + z = (a - b) + (b - c) + (c - a) = 0

Since (a - c)² = (c - a)², the given expression can be written as: (a - b)²/((b - c)(c - a)) + (b - c)²/((a - b)(c - a)) + (c - a)²/((a - b)(b - c))

Taking the common denominator as (a - b)(b - c)(c - a), the expression becomes: ((a - b)³ + (b - c)³ + (c - a)³) / ((a - b)(b - c)(c - a))

According to the algebraic identity, if x + y + z = 0, then x³ + y³ + z³ = 3xyz. Applying this to the numerator: (a - b)³ + (b - c)³ + (c - a)³ = 3(a - b)(b - c)(c - a)

Substituting this value back into the expression: 3(a - b)(b - c)(c - a) / ((a - b)(b - c)(c - a)) = 3

Explanation

The number of candidates qualified in the Science discipline from 2006 to 2008 is 542, 654, and 583 respectively, which gives an average of 593. The number of candidates qualified from 2009 to 2011 is 612, 648, and 660 respectively, which gives an average of 640. The difference between these two averages is 640 - 593 = 47.

Explanation

By using the complementary angle trigonometric identities, we know that for any angle θ, sin(90° - θ) = cos θ and cot(90° - θ) = tan θ.

For the angle 67°, we can write: cot 67° = cot(90° - 23°) = tan 23° sin 67° = sin(90° - 23°) = cos 23°

Substituting these relationships into the given expression: 2 cos²23° cot²67° − sin²23° − sin²67° cot²67°

We obtain: 2 cos²23° tan²23° − sin²23° − cos²23° tan²23°

Since tan θ = sin θ / cos θ, we can simplify cos²23° tan²23° to sin²23°. Replacing this back into the expression gives: 2 sin²23° − sin²23° − sin²23°

Combining the like terms: (2 − 1 − 1) sin²23° = 0

Explanation

The base of the pyramid is an equilateral triangle of side a = 10√3 cm.

The area of the base is calculated using the formula: Base Area = (√3/4) × a² = (√3/4) × (10√3)² = 75√3 sq cm

The total surface area of the pyramid is 270√3 sq cm. The lateral surface area (LSA) is: LSA = Total Surface Area - Base Area = 270√3 - 75√3 = 195√3 sq cm

The lateral surface area is also expressed as: LSA = ½ × Perimeter of base × slant height (l) Perimeter of the base = 3 × 10√3 = 30√3 cm 195√3 = ½ × 30√3 × l 195√3 = 15√3 × l l = 13 cm

The inradius (r) of the equilateral triangular base is: r = a / (2√3) = (10√3) / (2√3) = 5 cm

The relation between the height (h), slant height (l), and inradius (r) of the right pyramid is: l² = h² + r² 13² = h² + 5² 169 = h² + 25 h² = 144 h = 12 cm

Explanation

Let the radius of the cylinder be r₁ and its height be h₁. The volume of the cylinder V₁ is given by the formula V₁ = π r₁² h₁.

Let the radius of the cone be r₂ and its height be h₂. The volume of the cone V₂ is given by the formula V₂ = ⅓ π r₂² h₂.

Given that the heights of the cylinder and the cone are equal, we have h₁ = h₂ = h. The ratio of their volumes is:

V₁ / V₂ = 3 / 1

Substituting the volume formulas into the ratio gives:

(π r₁² h) / (⅓ π r₂² h) = 3

r₁² / (⅓ r₂²) = 3

3 r₁² / r₂² = 3

r₁² / r₂² = 1

r₁ = r₂

Since the radius of the cylinder is equal to the radius of the cone, their diameters are also equal.

Explanation

The surface area of the open rectangular box is equal to the area of the original rectangular sheet minus the total area of the four square corners that were cut off.

Original length of the sheet = 24 cm Original breadth of the sheet = 18 cm Original area of the sheet = 24 cm × 18 cm = 432 cm²

Side of each square cut off from the corners = 3 cm Area of one square cut-off = 3 cm × 3 cm = 9 cm² Area of four squares cut-off = 4 × 9 cm² = 36 cm²

Surface area of the remaining sheet = Original area - Total cut-off area Surface area = 432 cm² - 36 cm² = 396 cm²

This remaining sheet is folded to form the open rectangular box, meaning the surface area of the box is also 396 cm².

Explanation

The sides of the triangle are 12 cm, 16 cm, and 20 cm. These lengths satisfy the Pythagorean theorem since 12² + 16² = 144 + 256 = 400, which is equal to 20². This confirms that the triangle is a right-angled triangle where the two shorter sides, 12 cm and 16 cm, represent the base and height, and the longest side, 20 cm, is the hypotenuse.

The area of a right-angled triangle is given by the formula: Area = ½ × base × height

Substituting the values: Area = ½ × 12 × 16 = 96 cm²

Explanation

The radius r of the sphere is half of its diameter, which is 6 cm. Since the cylinder has the same radius, its radius is also 6 cm.

The formula for the volume of a sphere is V = (4/3) π r³ and the formula for the volume of a cylinder is V = π r² h.

Equating the two volumes gives: π r² h = (4/3) π r³

Dividing both sides by π r² simplifies the relation to: h = (4/3) r

Substituting r = 6 cm into the equation: h = (4/3) × 6 = 8 cm

Explanation

Let h be the height of the tower. When the angle of elevation of the sun is 45°, let the length of the shadow be x. In the right-angled triangle formed, tan 45° = h/x. Since tan 45° = 1, we get x = h.

When the angle of elevation changes to 30°, the shadow length increases by 10 metres, becoming x + 10. In this right-angled triangle, tan 30° = h/(x + 10).

Since tan 30° = 1/√3 and x = h, we substitute x to get 1/ | radic;3 = h/(h + 10).

Cross-multiplying yields h + 10 = h√3.

Rearranging the terms gives h√3 - h = 10, which simplifies to h(√3 - 1) = 10.

Solving for h: h = 10 / (√3 - 1)

Multiplying the numerator and the denominator by the conjugate (√3 + 1) to rationalize: h = 10(√3 + 1) / ((√3 - 1)(√3 + 1)) h = 10(√3 + 1) / (3 - 1) h = 10(√3 + 1) / 2 h = 5(√3 + 1) metres

Explanation

Divide both the numerator and the denominator of the given expression by cos θ to express it in terms of tan θ:

(5 sin θ - 3 cos θ) / (5 sin θ + 2 cos θ) = (5 tan θ - 3) / (5 tan θ + 2)

Substitute the given value 5 tan θ = 4 directly into the equation:

(4 - 3) / (4 + 2) = 1/6

Explanation

From the first relation, we can write: (x cos θ − y sin θ)² = x² + y²

Expanding the left side: x² cos² θ + y² sin² θ − 2xy sin θ cos θ = x² + y²

Using the identity x² + y² = (x² + y²)(cos² θ + sin² θ) = x² cos² θ + x² sin² θ + y² cos² θ + y² sin² θ, we can substitute this on the right side: x² cos² θ + y² sin² θ − 2xy sin θ cos θ = x² cos² θ + x² sin² θ + y² cos² θ + y² sin² θ

Subtracting x² cos² θ + y² sin² θ from both sides yields: x² sin² θ + 2xy sin θ cos θ + y² cos² θ = 0 (x sin θ + y cos θ)² = 0

This gives: x sin θ + y cos θ = 0 x sin θ = −y cos θ tan θ = −y/x

From the tangent relation, we can express the sine and cosine functions as: sin θ = −y / √(x² + y²) cos θ = x / √(x² + y²)

Squaring both trigonometric functions: sin² θ = y² / (x² + y²) cos² θ = x² / (x² + y²)

Substituting these values into the second given equation: cos² θ / a² + sin² θ / b² = 1 / (x² + y²)

We obtain: x² / [a²(x² + y²)] + y² / [b²(x² + y²)] = 1 / (x² + y²)

Multiplying the entire equation by (x² + y²) simplifies directly to: x² / a² + y² / b² = 1