SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The expression can be simplified using complementary angle relations, as 17° + 73° = 90°. For any angle θ, the co-function identities state that cot θ = tan(90° - θ) and sec θ = csc(90° - θ).

The second term (1)/(tan² 73°) is equal to cot² 73°. Converting this to the complementary angle 17° gives cot² 73° = tan²(90° - 73°) = tan² 17°.

The third term sin 17° sec 73° can be simplified by noting that sec 73° = csc(90° - 73°) = csc 17°. Since csc 17° = (1)/(sin 17°), the term becomes sin 17° · (1)/(sin 17°) = 1.

Substituting these values back into the original expression: sec² 17° - tan² 17° - 1

Using the fundamental trigonometric identity sec² θ - tan² θ = 1, we substitute 1 for the first two terms: 1 - 1 = 0

Explanation

The Simple Interest (SI) is calculated using the formula SI = (P × R × T)/(100), where P is the principal amount, R is the rate of interest per annum, and T is the time period in years. Given the principal P = 4000 and the rate R = 12% per annum, the time T of 18 months must be converted into years. Dividing 18 months by 12 months per year gives T = 1.5 years. Substituting these values into the formula, the calculation becomes SI = (4000 × 12 × 1.5)/(100). This simplifies to 40 × 12 × 1.5, which further simplifies to 40 × 18, resulting in a total simple interest of Rs. 720.

Explanation

To find the value of x²+y²+z², substitute the given trigonometric expressions into the equation:

x² + y² + z² = (acosφ)² + (acosφ)² + (asinθ)²

Expanding the terms: x² + y² + z² = a²cos²²φ + a²cos²²φ + a²sin²θ

Factor out the common term a²cos²θ from the first two terms: x² + y² + z² = a²cos²θ(cos²φ + sin²φ) + a²sin²θ

Using the fundamental trigonometric identity sin²α + cos²α = 1: x² + y² + z² = a²cos²θ(1) + a²sin²θ x² + y² + z² = a²(cos²θ + sin²θ)

Applying the identity again: x² + y² + z² = a²(1) = a²

This expression represents the square of the distance from the origin to a point in spherical coordinates where the radius r is equal to a.

Explanation

The height of the tower can be determined by considering the right-angled triangle formed by the observer's line of sight, the horizontal distance to the tower, and the vertical segment of the tower above the observer's eye level. Let h be the height of the tower above the observer's eye. Using the trigonometric relation for the tangent of the angle of elevation θ:

tan(θ) = (Opposite)/(Adjacent)

Given θ = 30° and the adjacent distance d = 45 m:

h = 45 · tan(30°) = 45 · (1)/(√(3)) = (45)/(√(3))

To simplify, multiply the numerator and denominator by √3:

h = 45√3{3} = 15√3

Substituting the value √3 = 1.732:

h = 15 × 1.732 = 25.98 m

The total height of the tower H is the sum of the height above the observer's eye level and the height of the observer:

H = h + 1.6 = 25.98 + 1.6 = 27.58 m

Explanation

The given equation is a + ¹⁄ₐ = 2. Multiplying the entire equation by a yields a² + 1 = 2a, which can be rearranged into the quadratic form a² - 2a + 1 = 0. This expression is a perfect square, specifically (a - 1)² = 0. Solving for a gives the unique value a = 1.

To find the value of a⁵ + (1)/(a⁵), substitute a = 1 into the expression: 1⁵ + (1)/(1⁵) = 1 + ¹⁄₁ = 1 + 1 = 2.

This result follows a general rule in algebra where if a + ¹⁄ₐ = 2, then a must be 1, and consequently aⁿ + (1)/(aⁿ) will always equal 2 for any integer power n.

Explanation

The expression (2+√3{2-√3} - 4√3)² is solved by first rationalizing the fractional term. Multiplying the numerator and denominator by the conjugate of the denominator, (2+√3), yields:

2+√3{2-√3} × 2+√3{2+√3} = (2+√3)²{2² - (√3)²}

Expanding the numerator using the identity (a+b)² = a² + b² + 2ab and simplifying the denominator gives:

4 + 3 + 4√3{4 - 3} = 7 + 4√3{1} = 7 + 4√3

Substituting this result back into the original expression:

(7 + 4√3 - 4√3)²

The radical terms +4√3 and -4√3 cancel each other out, simplifying the value inside the parentheses to 7. Squaring this result gives 7² = 49.

Explanation

According to the Midpoint Theorem in geometry, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is exactly half the length of that third side. In △ ABC, since P and Q are the midpoints of sides AB and AC respectively, the segment PQ is parallel to BC and its length is given by the relation PQ = ½ BC.

Given that PQ = 6 cm, we can substitute this value into the formula to find the length of the side BC: 6 = ½ BC BC = 6 × 2 BC = 12 cm

Explanation

The sum of the internal angles of a triangle is always 180°. Given the ratio of the angles is 5:3:10, the total number of parts in the ratio is 5 + 3 + 10 = 18 parts.

The value of one part is calculated as: One part = (180°)/(18) = 10°

The largest angle corresponds to the largest ratio value, 10, and the smallest angle corresponds to the smallest ratio value, 3. The difference between these in terms of ratio units is 10 - 3 = 7 parts.

To find the difference in degrees, multiply the difference in parts by the value of one part: Difference = 7 × 10° = 70°

Explanation

Given the trigonometric ratio sinθ = √3{2} within the range 0° < θ < 90°, we identify θ by referencing standard angles. Since sin 60° = √3{2}, it follows that θ = 60°.

The expression to be evaluated is tan(θ - 15°). Substituting the value of θ:

tan(60° - 15°) = tan 45°

According to the standard trigonometric values, tan 45° = 1.

Explanation

To find the greatest number that divides 390, 495, and 300 without leaving a remainder, we calculate the Highest Common Factor (HCF) of these numbers.

Using prime factorization: 390 = 2 × 3 × 5 × 13 495 = 3² × 5 × 11 300 = 2² × 3 × 5²

The common prime factors present in all three numbers are 3 and 5. To find the HCF, we take the product of the lowest power of each common factor: HCF = 3¹ × 5¹ = 15

Alternatively, using the division method: HCF(300, 390): 390 = 300 × 1 + 90 300 = 90 × 3 + 30 90 = 30 × 3 + 0 The HCF of 300 and 390 is 30.

Now, HCF(30, 495): 495 = 30 × 16 + 15 30 = 15 × 2 + 0 The HCF of 30 and 495 is 15.

The greatest number that divides all three numbers is 15.

Explanation

The age of the teacher can be determined by calculating the difference between the total age of the class after including the teacher and the total age of the boys alone. Initially, the total age of the 30 boys is calculated as 30 × 10 = 300 years. When the teacher is included, the total number of people in the group becomes 31 and the average age increases by one year, making the new average 10 + 1 = 11 years. The new total age for the 31 people is 31 × 11 = 341 years. The teacher's age is the difference between these two totals: 341 - 300 = 41 years.

Alternatively, this can be solved using the shortcut formula for inclusion: Teacher's age = New Average + (Original number of people × Increase in average). Substituting the given values, the calculation is 11 + (30 × 1) = 41. This logic follows from the fact that the teacher enters the group with enough years to represent the new average for themselves, plus they provide the extra year needed to increase the average for each of the 30 original boys.

Explanation

To find the value of the given expression, we identify a common denominator for the three fractions, which is (p-n)(n-q)(q-p). We rewrite each fraction so they all share this denominator.

The first term (1)/((p-n)(n-q)) is multiplied by (q-p)/(q-p) to obtain (q-p)/((p-n)(n-q)(q-p)).

The second term (1)/((n-q)(q-p)) is multiplied by ᵖ⁻ⁿ⁄ₚ₋ₙ to obtain (p-n)/((p-n)(n-q)(q-p)).

The third term (1)/((q-p)(p-n)) is multiplied by (n-q)/(n-q) to obtain (n-q)/((p-n)(n-q)(q-p)).

Summing these terms together over the common denominator gives the following numerator: (q-p) + (p-n) + (n-q)

Combining like terms in the numerator, we find: q - p + p - n + n - q = 0

Because the numerator is zero, the value of the entire expression is: (0)/((p-n)(n-q)(q-p)) = 0

Explanation

To find the time C takes to complete the work alone, we use the efficiency method. Let the total work be the Least Common Multiple (LCM) of the time taken by A, B, and the group (A + B + C). The LCM of 20, 15, and 5 is 60 units.

The efficiency (work done per day) of each component is calculated as: Efficiency of A = 60 / 20 = 3 units/day Efficiency of B = 60 / 15 = 4 units/day Efficiency of A + B + C = 60 / 5 = 12 units/day

The efficiency of C is obtained by subtracting the individual efficiencies of A and B from the combined efficiency of all three: Efficiency of C = (A + B + C) - (A + B) = 12 - (3 + 4) = 12 - 7 = 5 units/day

The time taken by C to complete the total work alone is the total work divided by C's efficiency: Time taken by C = 60 / 5 = 12 days

Alternatively, using the work rate formula 1/T_A + 1/T_B + 1/T_C = 1/Tₜₒₜₐₗ: 1/20 + 1/15 + 1/T_C = 1/5 1/T_C = 1/5 - (1/20 + 1/15) 1/T_C = 1/5 - 7/60 1/T_C = (12 - 7) / 60 = 5 / 60 = 1 / 12 T_C = 12 days

Explanation

When a solid object is melted and recast into smaller objects, the total volume remains constant. The volume of a sphere is calculated using the formula V = ⁴⁄₃ π r³. To find the total number of small lead balls, the volume of the large lead ball is divided by the volume of a single small lead ball.

First, the units of measurement must be consistent. The radius of the large ball is R = 6 cm, which is equivalent to 60 mm. The radius of each small ball is r = 3 mm. The number of balls n is given by the relation n = (4)/(3) π R³{⁴⁄₃ π r³}, which simplifies to n = ((R)/(r))³ as the constant terms cancel out.

Substituting the specific values into the formula: n = (⁶⁰⁄₃)³ n = 20³ n = 20 × 20 × 20 n = 8000

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when both the gain percentage and discount percentage are known is given by the formula: (MP)/(CP) = (100 + Gain%)/(100 - Discount%)

Given that the gain is 33% and the discount is 5%, substitute these values into the formula: (MP)/(CP) = (100 + 33)/(100 - 5) = ¹³³⁄₉₅

Simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor, 19: (MP)/(CP) = ⁷⁄₅ = 1.4

The markup percentage represents how much the Marked Price is above the Cost Price. It is calculated as: Markup% = ((MP)/(CP) - 1) × 100 Markup% = (1.4 - 1) × 100 Markup% = 0.4 × 100 = 40%

Explanation

The perimeter of a quadrilateral is the sum of its four sides. Given that the sides are in the ratio 2:3:4:5, let the lengths of the sides be 2x, 3x, 4x, and 5x. The sum of these sides must equal the total perimeter of 280 m:

2x + 3x + 4x + 5x = 280 14x = 280 x = ²⁸⁰⁄₁₄ = 20

The longest side corresponds to the largest part of the ratio, which is 5x. By substituting the value of x, the length of the longest side is calculated as:

Longest side = 5 × 20 = 100 m

Explanation

Let the number of apples Kamal ate be x. The problem states that he sold 40% more than he ate, which means the number of apples sold is 140% of the number of apples eaten. This relationship is expressed by the formula S = x × (1 + ⁴⁰⁄₁₀₀).

Given that the number of apples sold S is 70, we can substitute the values into the equation: 1.4x = 70

To find x, we divide 70 by 1.4: x = (70)/(1.4) x = ⁷⁰⁰⁄₁₄ x = 50

Explanation

To find the time taken by a train to pass a stationary person, we use the formula t = ᵈ⁄ᵥ, where d is the distance covered and v is the speed. When a train passes a person, the distance to be covered is equal to the length of the train, which is 100 meters.

First, the speed must be converted from km/hr to m/s: v = 120 × ⁵⁄₁₈ = (20 × 5)/(3) = ¹⁰⁰⁄₃ m/s

Now, substitute the values into the time formula: t = (100)/(100/3) = 100 × ³⁄₁₀₀ = 3 seconds

Explanation

The given equation is x² + 5x + 6 = 0. From this, we can derive the relation x² + 6 = -5x.

We need to find the value of the expression: (2x)/(x² - 7x + 6)

By rearranging the terms in the denominator, we can rewrite the expression as: (2x)/((x² + 6) - 7x)

Substituting the value x² + 6 = -5x into the denominator: (2x)/(-5x - 7x) = ²ˣ⁄₋₁₂ₓ

Canceling x from the numerator and the denominator, we get: ²⁄₋₁₂ = -⅙

Explanation

The value of a² + b² can be directly determined using the algebraic identity (a+b)² + (a-b)² = 2(a² + b²). Substituting the given values a+b = 5 and a-b = 3 into this identity yields 5² + 3² = 2(a² + b²). This simplifies to 25 + 9 = 2(a² + b²), or 34 = 2(a² + b²). Dividing by 2, we get a² + b² = 17. Alternatively, solving the simultaneous equations a+b=5 and a-b=3 by addition gives 2a=8, which means a=4. Substituting a=4 into the first equation gives 4+b=5, which means b=1. Therefore, a² + b² = 4² + 1² = 16 + 1 = 17.

Explanation

A perpendicular drawn from the centre of a circle to a chord bisects the chord. Given the length of the chord is 12 cm, the perpendicular divides it into two segments of 6 cm each. The perpendicular distance from the centre to the chord is 8 cm. The radius r, the perpendicular distance d, and the half-chord a form a right-angled triangle where the radius is the hypotenuse.

Using the Pythagorean theorem: r² = a² + d² r² = 6² + 8² r² = 36 + 64 r² = 100 r = 10 cm

The diameter of the circle is twice the radius: Diameter = 2 × r = 2 × 10 = 20 cm

Explanation

From the given condition (a²)/(b+c) = 1, we have a² = b + c. Adding a to both sides yields a² + a = a + b + c, which simplifies to a(1 + a) = a + b + c. This provides the relation ¹⁄₁₊ₐ = (a)/(a + b + c). Due to the cyclic symmetry of the equations, the same logic applied to (b²)/(c+a) = 1 and (c²)/(a+b) = 1 yields (1)/(1+b) = (b)/(a + b + c) and (1)/(1+c) = (c)/(a + b + c) respectively. Summing these three equations results in:

¹⁄₁₊ₐ + (1)/(1+b) + (1)/(1+c) = (a)/(a + b + c) + (b)/(a + b + c) + (c)/(a + b + c)

= (a + b + c)/(a + b + c) = 1

For a quicker competitive approach, use the method of substitution by assuming a = b = c because the system is symmetric. The condition (a²)/(2a) = 1 simplifies to ᵃ⁄₂ = 1, which gives a = 2. Substituting a = b = c = 2 into the required expression gives ¹⁄₁₊₂ + ¹⁄₁₊₂ + ¹⁄₁₊₂ = ⅓ + ⅓ + ⅓ = 1.

Explanation

The Incentre of a triangle is the point of concurrency where the three internal angle bisectors of the triangle meet. By the Angle Bisector Theorem, any point lying on the bisector of an angle is equidistant from the two rays forming that angle. Consequently, the point where all three angle bisectors intersect must be at an equal perpendicular distance from all three sides of the triangle. This common distance is known as the inradius r, and the Incentre serves as the center of the incircle, which is the unique circle that is tangent to each of the triangle's sides.

Explanation

To find the value of the expression csc² 60° + sec² 60° - cot² 60° + tan² 30°, we substitute the standard trigonometric values:

sin 60° = (√3)/2 ⟹ csc 60° = (2)/(√3) cos 60° = ½ ⟹ sec 60° = 2 tan 60° = √3 ⟹ cot 60° = (1)/(√3) tan 30° = (1)/(√3)

Substituting these values into the expression: ((2)/(√3))² + 2² - ((1)/(√3))² + ((1)/(√3))²

The terms - ((1)/(√3))² and + ((1)/(√3))² cancel each other out: 4/3 + 4 - ⅓ + ⅓ = 4/3 + 4

Alternatively, using the identity csc² θ - cot² θ = 1: (csc² 60° - cot² 60°) + sec² 60° + tan² 30° 1 + 2² + ((1)/(√3))² 1 + 4 + ⅓ = 5 + ⅓ = 16/3

Therefore, the value of the expression is 16/3 = 5 1/3.

Explanation

The overall gain percentage on the transaction is (10 / 500) × 100 = 2%. Using the rule of alligation to determine the ratio of the cost price of the table (sold at 10% loss) to the chair (sold at 10% gain) relative to the mean gain of 2%, the ratio is calculated as (10 - 2) : (2 - (-10)) = 8 : 12, which simplifies to 2 : 3. The cost price of the chair is determined by apportioning the total cost of Rs. 500 according to this ratio: (3 / (2 + 3)) × 500 = (3 / 5) × 500 = 300.

Explanation

To find the least number that must be added to 8961 to make it exactly divisible by 84, we first divide 8961 by 84 to find the remainder.

Using long division: 8961 = (84 × 100) + 561 561 = (84 × 6) + 57

The total quotient is 106 and the remainder R is 57. The amount to be added to a number N to make it perfectly divisible by a divisor D is given by the formula D - R.

Required number = 84 - 57 = 27

Adding 27 to 8961 results in 8988, which is exactly divisible by 84 (84 × 107 = 8988).

Explanation

The age of the teacher can be determined using the formula for adding a new member to a group: Teacher age = Old Average + (New Total Strength × Increase in Average).

The initial average age of the 39 students is 15 years. When the teacher is included, the total number of people becomes 40 (39 students + 1 teacher). The average age increases by 3 months, which is equivalent to 3/12 = 0.25 years.

Substituting the values into the formula: Teacher age = 15 + (40 × 0.25) Teacher age = 15 + 10 = 25 years

Alternatively, calculating the difference between the total age of the group before and after including the teacher: Total age of 40 people = 40 × 15.25 = 610 years Total age of 39 students = 39 × 15 = 585 years Teacher age = 610 - 585 = 25 years

Explanation

The expression x⁴+y⁴-2x²y² follows the algebraic identity (A-B)² = A²+B²-2AB, where A = x² and B = y². Therefore, the expression simplifies to (x²-y²)². Using the difference of squares identity, this can be further written as ((x+y)(x-y))².

Given x = a + ¹⁄ₐ and y = a - ¹⁄ₐ, we calculate the sum and difference of x and y: x + y = (a + ¹⁄ₐ) + (a - ¹⁄ₐ) = 2a x - y = (a + ¹⁄ₐ) - (a - ¹⁄ₐ) = ²⁄ₐ

Multiplying these results gives (x + y)(x - y) = 2a · ²⁄ₐ = 4. Substituting this value back into the simplified identity, we get 4² = 16.

Explanation

Two triangles are similar by the AAA (Angle-Angle-Angle) criterion if their corresponding angles are equal. To express this similarity correctly, the vertices of the triangles must be written in an order that reflects the matching pairs of equal angles. Based on the given values, we see that ∠ A = ∠ F = 50°, ∠ B = ∠ E = 70°, and ∠ C = ∠ D = 60°. This establishes the one-to-one correspondence between the vertices as A ↔ F, B ↔ E, and C ↔ D. Therefore, the triangle ABC is similar to the triangle FED, which is written as △ ABC ∼ △ FED.

Explanation

In △ ABC, the sum of angles is ∠ A + ∠ B + ∠ C = 180°. Given the ratio 4 ∠ A = 3 ∠ B = 12 ∠ C, we can equate each term to a constant k. This gives ∠ A = k/4, ∠ B = k/3, and ∠ C = k/12. Substituting these values into the angle sum equation:

k/4 + k/3 + k/12 = 180°

To solve for k, find the common denominator, which is 12:

(3k + 4k + k)/12 = 180°

(8k)/12 = 180°

(2k)/3 = 180°

2k = 540°

k = 270°

Since ∠ A = k/4, we calculate its value as:

∠ A = 270° / 4 = 67.5°

Explanation

The trigonometric functions sinθ and cosθ both have a maximum possible value of 1 for any real angle θ. Given the condition sinθ + cosθ = 2, this equality can only be satisfied if both terms reach their maximum values simultaneously, meaning sinθ = 1 and cosθ = 1.

By substituting these specific values into the required expression: sin²θ + cos²θ = (1)² + (1)² = 1 + 1 = 2

Explanation

The equation is 2ˣ⁺⁴ - 2ˣ⁺² = 3. By applying the exponent rule aᵐ⁺ⁿ = aᵐ · aⁿ, the terms can be expanded to factor out the common expression 2ˣ⁺²:

2ˣ⁺²(2² - 1) = 3

Simplifying the term within the parentheses: 2ˣ⁺²(4 - 1) = 3 2ˣ⁺²(3) = 3

Dividing both sides of the equation by 3 yields: 2ˣ⁺² = 1

Since any base raised to the power of zero equals one, we express the right side as a power of 2: 2ˣ⁺² = 2⁰

Comparing the exponents of the identical bases: x + 2 = 0 x = -2

Explanation

The three medians of a triangle divide it into six smaller triangles of equal area. In triangle ABC, the medians AD, BE, and CF intersect at the centroid O, creating six constituent triangles: BOD, COD, COE, AOE, AOF, and BOF.

Since the total area of triangle ABC is 96 square cm, the area of each individual small triangle is calculated as:

Area of triangle BOD = ⅙ × Area of triangle ABC

Substituting the given value:

Area of triangle BOD = 96/6 = 16 cm²

Explanation

Let the height of the temple be h = 75 m. The two persons are on opposite sides of the temple. Let their distances from the base of the temple be d₁ and d₂ respectively.

For the first person with an angle of elevation of 30°, we use the relation tan θ = Perpendicular / Base: tan 30° = 75 / d₁ 1 / √3 = 75 / d₁ d₁ = 75√3 m

For the second person with an angle of elevation of 60°: tan 60° = 75 / d₂ √3 = 75 / d₂ d₂ = 75 / √3 = 25√3 m

The total distance between the two persons is the sum of d₁ and d₂ since they are on opposite sides: D = d₁ + d₂ = 75√3 + 25√3 = 100√3 m

Using the value √3 ≈ 1.732: D = 100 × 1.732 = 173.2 m

Explanation

The difference D between compound interest and simple interest for a duration of 2 years is expressed by the formula D = P × (ʳ⁄₁₀₀)², where P is the principal sum and r is the rate of interest per annum.

Substituting the given difference D = 20 and the rate r = 5 into the formula: 20 = P × (⁵⁄₁₀₀)² 20 = P × (¹⁄₂₀)² 20 = P × ¹⁄₄₀₀

By solving for P: P = 20 × 400 P = 8000

This mathematical relationship arises because the difference between the two types of interest over 2 years is simply the interest earned on the first year's simple interest. Since the first year's interest is ⁵⁄₁₀₀P, the interest on that amount in the second year is ⁵⁄₁₀₀ × (⁵⁄₁₀₀P), which is equated to 20 to find the principal.

Explanation

According to the Intersecting Chords Theorem, when two chords AB and PQ of a circle intersect at a point D inside the circle, the product of the segments of one chord is equal to the product of the segments of the other chord. The relationship is given by the formula AD × DB = PD × QD. Substituting the given values AD = 4 cm, DB = 6 cm, and QD = 3 cm into this equation, we get 4 × 6 = PD × 3, which results in 24 = 3 × PD. Dividing both sides by 3 yields PD = 8 cm. The total length of the chord PQ is the sum of its two segments, calculated as PQ = PD + QD = 8 + 3 = 11 cm.

Explanation

The rise in the water level is determined by the principle that the volume of the displaced water equals the volume of the submerged object. Let h be the rise in the water level in the cylindrical vessel.

The volume of the solid sphere is calculated using the formula Vₛₚₕₑᵣₑ = ⁴⁄₃ π r³. Given the radius of the sphere is 3 cm: Vₛₚₕₑᵣₑ = ⁴⁄₃ × π × 3³ = ⁴⁄₃ × π × 27 = 36π cubic cm.

The volume of the displaced water in the cylinder is given by V_c_yₗᵢₙ_dₑᵣ = π R² h, where R is the radius of the vessel. Given the radius of the vessel is 4 cm: V_dᵢₛₚₗₐ_cₑ_d = π × 4² × h = 16π h cubic cm.

Equating the volume of the sphere to the volume of the displaced water: 16π h = 36π 16h = 36 h = ³⁶⁄₁₆ = ⁹⁄₄ = 2.25 cm.

Explanation

Let h be the height of the tower. In the right-angled triangle formed with the first point at a distance of 25 meters, the relationship is given by tan(x) = ʰ⁄₂₅. For the second point at a distance of 64 meters, the angle of elevation is 90° - x. In this right-angled triangle, tan(90° - x) = ʰ⁄₆₄.

Using the trigonometric property tan(90° - x) = cot(x), the second equation becomes cot(x) = ʰ⁄₆₄. Multiplying these two equations together gives tan(x) × cot(x) = ʰ⁄₂₅ × ʰ⁄₆₄. Since tan(x) × cot(x) = 1, the expression simplifies to 1 = (h²)/(1600).

Solving for h results in h² = 1600, which gives h = √(1600) = 40 meters. This follows the general formula for a tower of height h observed from two points at distances a and b with complementary angles of elevation: h = √(ab). Substituting the given values, h = √(25 × 64) = 5 × 8 = 40.

Explanation

Let the total work be 72 units, which is the least common multiple of 24 and 36. Since A and B together complete the work in 24 days, their combined efficiency is 72 / 24 = 3 units per day. It is given that B completes 1/3 of the work in 12 days, which implies B would take 12 × 3 = 36 days to complete the full work. Therefore, the efficiency of B is 72 / 36 = 2 units per day.

The efficiency of A is the difference between the combined efficiency and B's efficiency, calculated as 3 - 2 = 1 unit per day. The remaining work after B has completed 1/3 of the total is 2/3 of the total work, which is 2/3 × 72 = 48 units. The time taken by A to complete this remaining work is 48 / 1 = 48 days.

Explanation

The volume of a cone is calculated using the formula V_cₒₙₑ = ⅓ π r₁² h₁ and the volume of a cylinder is calculated using V_c_yₗᵢₙ_dₑᵣ = π r₂² h₂. According to the problem, the heights of both figures are equal (h₁ = h₂ = h) and the ratio of the radius of the cone to the radius of the cylinder is 2:1 (r₁ = 2x and r₂ = x).

To find the ratio of their volumes, we substitute these values into the ratio expression: V_cₒₙₑ{V_c_yₗᵢₙ_dₑᵣ} = (1)/(3) π (2x)² h{π x² h}

By canceling the common terms π, x², and h, the expression simplifies to: V_cₒₙₑ{V_c_yₗᵢₙ_dₑᵣ} = (1)/(3) · 4{1} = ⁴⁄₃

The ratio of the volume of the cone to the volume of the cylinder is 4:3.

Explanation

The net profit or loss percentage resulting from a markup followed by a discount can be calculated using the successive percentage change formula: Net% = x + y + ˣʸ⁄₁₀₀. In this scenario, the markup x is +50 and the discount y is -25.

Applying the values: Net% = 50 - 25 + (50 × (-25))/(100) Net% = 25 - ¹²⁵⁰⁄₁₀₀ Net% = 25 - 12.5 = 12.5%

Since the result is positive, it indicates a profit of 12.5%. This can also be verified by assuming a Cost Price (CP) of 100. A 50% markup makes the Marked Price (MP) 150. A 25% discount on this MP is 150 × 0.25 = 37.5. Subtracting the discount from the MP gives a Selling Price (SP) of 150 - 37.5 = 112.5. The profit is 112.5 - 100 = 12.5, which corresponds to a 12.5% profit on the original cost.

Explanation

The total working time of the school remains constant regardless of the number of periods. This can be calculated using the formula: Total time = Number of periods × Duration of each period.

In the first case, total working time = 8 × 45 = 360 minutes.

In the second case, the school has 9 periods for the same total time. Let the duration of each new period be x minutes. 9 × x = 360 x = ³⁶⁰⁄₉ x = 40 minutes.

Explanation

Let the cost price (CP) be 4x and the selling price (SP) be 5x based on the given ratio of 4:5. The profit is calculated by the difference between the selling price and the cost price:

Profit = SP - CP = 5x - 4x = x

The percentage of profit is calculated using the formula:

Profit Percentage = (Profit)/(CP) × 100

Substituting the known values into the formula:

Profit Percentage = ˣ⁄₄ₓ × 100 = ¼ × 100 = 25

Explanation

The ratio of gold to copper is given as 3:2. To find the percentage of gold, we first calculate the total number of parts in the ornament by summing the ratio components: 3 + 2 = 5 parts. Gold accounts for 3 out of these 5 parts. The percentage of gold is determined by the formula:

Percentage of gold = (Ratio of gold)/(Total ratio parts) × 100

Substituting the values: Percentage = ⅗ × 100 = 3 × 20 = 60

Explanation

The speed of a boat in still water is calculated using the formula u = (D + U)/(2), where D represents the downstream speed and U represents the upstream speed. This relationship exists because the downstream speed is the sum of the boat's speed and the stream's speed (u + v), while the upstream speed is the difference between the boat's speed and the stream's speed (u - v).

Given the downstream rate D = 8 km/hr and the upstream rate U = 4 km/hr, substituting these values into the formula gives:

Speed in still water = (8 + 4)/(2)

Speed in still water = ¹²⁄₂ = 6 km/hr

By adding the two rates, the speed of the stream is eliminated, allowing for the direct calculation of the boat's intrinsic speed.

Explanation

The given equation is √(3ⁿ) = 2187. Using the property of radicals where √x = x¹^/², the equation can be rewritten as (3ⁿ)¹^/² = 2187. Applying the power of a power rule (aᵐ)ᵖ = aᵐᵖ, this simplifies to 3ⁿ^/² = 2187.

To solve for n, express 2187 as a power of 3. By prime factorization, 2187 = 3 × 729 = 3 × 3⁶ = 3⁷. Substituting this back into the equation gives 3ⁿ^/² = 3⁷. Since the bases are identical, the exponents must be equal, leading to the linear equation n/2 = 7. Solving for n results in n = 7 × 2 = 14.

Explanation

Divide the numerator and the denominator of the expression (6x)/(x²⁾⁺ˣ⁺¹ by x:

(6x / x)/((x²⁾⁺ˣ⁺¹⁾^/ˣ = (6)/(x + 1 + 1){x}

By rearranging the terms in the denominator, the expression becomes:

(6)/((x + 1){x) + 1}

Substitute the given value x + ¹⁄ₓ = 5 into this expression:

(6)/(5 + 1) = ⁶⁄₆ = 1

Alternatively, the given equation x + ¹⁄ₓ = 5 can be written as x² + 1 = 5x by multiplying both sides by x. Substituting x² + 1 = 5x into the denominator of the original expression gives:

(6x)/((x²⁾⁺¹⁾⁺ˣ = (6x)/(5x + x) = ⁶ˣ⁄₆ₓ = 1

Explanation

The Triangle Inequality Theorem states that for any three line segments to form a triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. A practical application of this rule is to verify if the sum of the two shorter sides exceeds the length of the longest side. In the case of the measurements 11 cm, 3 cm, and 12 cm, the two smaller sides are 11 cm and 3 cm. Adding these together yields 11 + 3 = 14 cm. Since 14 cm is greater than the longest side of 12 cm (14 > 12), these three measurements satisfy the fundamental geometric requirement to form a triangle.

Explanation

Let the lengths of the two sides be 4x and 3x. In a right-angled triangle, the relationship between the sides and the hypotenuse is given by the Pythagorean theorem, a² + b² = c². Substituting the given values:

(4x)² + (3x)² = 20² 16x² + 9x² = 400 25x² = 400 x² = 16 x = 4

The lengths of the sides are calculated by multiplying the ratio parts by the common factor x. The first side is 4 × 4 = 16 cm and the second side is 3 × 4 = 12 cm. This set of dimensions corresponds to a scaled version of the basic 3-4-5 Pythagorean triplet, where each side is multiplied by 4 to result in 12, 16, and 20.

Explanation

From the given equation sin A + sin² A = 1, we can rearrange it to isolate sin A as sin A = 1 - sin² A. Using the fundamental trigonometric identity sin² θ + cos² θ = 1, we know that 1 - sin² A = cos² A. Therefore, we have the relationship sin A = cos² A.

By squaring both sides of this relationship, we obtain sin² A = (cos² A)² = cos⁴ A. The expression we need to evaluate is cos² A + cos⁴ A. Substituting the derived equivalents into this expression, we replace cos² A with sin A and cos⁴ A with sin² A. This transforms the expression into sin A + sin² A. Given that the value of sin A + sin² A is 1, it follows that the value of cos² A + cos⁴ A is also 1.