SSC CGL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

Let h be the height of the building, where h = 10√3 m. Let the distances of points P and Q from the foot of the building be x and y respectively. For a building of height h and two points with complementary angles of elevation (or depression), the standard relation between the distances and the height is h² = xy.

Substituting the given height into this relation: xy = (10√3)² = 300

The distance between the two points is given as PQ = |x - y| = 20 m. To find the distance of P, we assume x > y and substitute y = x - 20 into the product equation: x(x - 20) = 300 x² - 20x - 300 = 0

Solving this quadratic equation by factoring: (x - 30)(x + 10) = 0

Since distance must be a positive value, we take x = 30 m.

Explanation

Since A and B are complementary angles, we have A + B = 90°, which implies B = 90° - A. Using the properties of complementary angles, we can determine that sin B = sin(90° - A) = cos A, cos B = cos(90° - A) = sin A, and tan B = tan(90° - A) = cot A. Furthermore, the relation cot B = cot(90° - A) = tan A holds.

The given expression is sin A cos B + cos A sin B tan A tan B + sec² A - cot² B. By substituting the equivalent values in terms of angle A, the expression becomes sin A (sin A) + cos A (cos A) (tan A cot A) + sec² A - tan² A.

Simplifying this, we obtain sin² A + cos² A(1) + (sec² A - tan² A). Using the fundamental trigonometric identities sin² A + cos² A = 1 and sec² A - tan² A = 1, the entire expression evaluates to 1 + 1 = 2.

Explanation

Since A, O, and B are points on a line segment, they form a straight angle where ∠ AOB = 180°. The angle ∠ BOC is the supplement to ∠ AOC, calculated as 180° - 40° = 140°. The external bisector of an angle is defined as the ray that bisects the supplementary angle formed by extending one of the sides of the original angle. In this geometry, the extension of the side AO through O is the ray OB. Therefore, the external angle of ∠ AOC is ∠ BOC. Given that OY is the external bisector of ∠ AOC, it bisects ∠ BOC. Thus, ∠ BOY = ½ ∠ BOC = ½ × 140° = 70°.

Explanation

The fundamental trigonometric identity states that sec² Θ - tan² Θ = 1. Given the expressions 4x = sec Θ and 4/x = tan Θ, we substitute these into the identity:

(4x)² - (4/x)² = 1

16x² - 16/x² = 1

By factoring out 16 from the left side of the equation, we obtain:

16(x² - 1/x²) = 1

Dividing both sides by 16 yields the required value:

x² - 1/x² = 1/16

Explanation

The given equation can be solved by converting all trigonometric terms into tan Θ. Dividing both sides of the equation 2sin² Θ + cos² Θ = 3sin Θ cos Θ by cos² Θ:

2 (sin² Θ)/(cos² Θ) + (cos² Θ)/(cos² Θ) = 3 (sin Θ cos Θ)/(cos² Θ)

2tan² Θ + 1 = 3tan Θ

Rearranging the terms into a standard quadratic form:

2tan² Θ - 3tan Θ + 1 = 0

Now, factor the quadratic equation:

(2tan Θ - 1)(tan Θ - 1) = 0

This yields two possible solutions: 2tan Θ - 1 = 0 ⟹ tan Θ = 1/2 tan Θ - 1 = 0 ⟹ tan Θ = 1

The condition sin Θ ≠ cos Θ implies that tan Θ ≠ 1. Consequently, the only valid solution is tan Θ = 1/2.

Explanation

The identity cos(90° - Θ) = sin Θ allows the original equation sin Θ + cos Θ = √2 cos(90° - Θ) to be simplified: sin Θ + cos Θ = √2 sin Θ

Dividing the entire equation by sin Θ yields: (sin Θ)/(sin Θ) + (cos Θ)/(sin Θ) = √2 sin Θ{sin Θ}

Applying the identity cot Θ = (cos Θ)/(sin Θ) simplifies this to: 1 + cot Θ = √2

Subtracting 1 from both sides gives the value for the cotangent: cot Θ = √2 - 1

Explanation

Given the equations x sin³ θ + y cos³ θ = sin θ cos θ and x sin θ = y cos θ, we can solve for x and y by using substitution. First, rewrite the first equation by splitting the powers of sine and cosine: (x sin θ) sin² θ + (y cos θ) cos² θ = sin θ cos θ. Using the second equation, we substitute x sin θ in place of y cos θ to get (x sin θ) sin² θ + (x sin θ) cos² θ = sin θ cos θ.

Factoring out x sin θ yields x sin θ (sin² θ + cos² θ) = sin θ cos θ. According to the fundamental trigonometric identity, sin² θ + cos² θ = 1. Thus, the equation simplifies to x sin θ = sin θ cos θ. Since the problem states sin θ ≠ 0, we divide both sides by sin θ to find x = cos θ.

Substituting x = cos θ back into the relation x sin θ = y cos θ gives cos θ sin θ = y cos θ. Since cos θ ≠ 0, we find y = sin θ. To calculate the final expression x² + y², we substitute the values found: x² + y² = (cos θ)² + (sin θ)² = cos² θ + sin² θ. Applying the identity again, we obtain the result of 1.

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 opposite interior angles. In △ ABC, the exterior angle formed by producing side BC to D is ∠ ACD. The interior opposite angles are ∠ A and ∠ B.

The relationship is given by the formula: ∠ ACD = ∠ A + ∠ B

Substituting the given values ∠ ACD = 108° and ∠ B = (∠ A)/(2) into the formula: 108° = ∠ A + (∠ A)/(2)

To solve for ∠ A, simplify the right side of the equation: 108° = (2∠ A + ∠ A)/(2) 108° = (3∠ A)/(2)

Rearranging the equation to isolate ∠ A: 3∠ A = 108° × 2 3∠ A = 216° ∠ A = (216°)/(3) ∠ A = 72°

Explanation

The length of the direct common tangent PQ between two circles of radii r₁ and r₂ that touch each other externally is given by the formula PQ = 2√(r₁₎ᵣ___₂. Given the radii are 4 cm and 9 cm, we substitute these values into the formula:

PQ = 2√(4 × 9) PQ = 2√(36) PQ = 2 × 6 = 12 cm

The side of the square is equal to the length of the common tangent PQ. The area of a square is calculated as side². Therefore, the area of the square with side PQ is:

Area = PQ² = 12² = 144 sq. cm

Explanation

In a circle, the radius drawn to the point of contact of a tangent is perpendicular to that tangent. Consequently, in the right-angled triangle OAP, ∠ OAP = 90°. Given that ∠ AOP = 60°, the third angle of the triangle is calculated as ∠ APO = 180° - (90° + 60°) = 30°. A standard geometric property of circles states that the line segment OP connecting the external point P to the center O bisects the angle between the two tangents PA and PB. Therefore, the total angle ∠ APB = 2 × ∠ APO = 2 × 30° = 60°.

Explanation

In a regular polygon, the sum of an interior angle and its adjacent exterior angle is always 180° because they form a linear pair. Let the exterior angle be E and the interior angle be I. According to the problem, the interior angle is double the exterior angle, which gives the relation I = 2E.

Substituting this into the linear pair equation: I + E = 180° 2E + E = 180° 3E = 180° E = 60°

For any regular polygon with n sides, the measure of each exterior angle is given by the formula E = (360°)/(n). By substituting the calculated value of E: 60° = (360°)/(n) n = (360°)/(60°) n = 6

Explanation

In a rhombus, all sides are equal in length, so PQ = QR = RS = SP = 6 cm. Adjacent angles in a rhombus are supplementary, meaning their sum is 180°. Given that ∠ PQR = 120°, the adjacent angle ∠ SPQ is calculated as 180° - 120° = 60°. Now, consider △ PQS. Since PQ = PS = 6 cm, △ PQS is an isosceles triangle. In an isosceles triangle, if the angle between the two equal sides is 60°, the triangle must be equilateral because the remaining two angles must also be equal to (180° - 60°) / 2 = 60°. In an equilateral triangle, all sides are equal. Therefore, the length of the diagonal QS is equal to the side length PQ, which is 6 cm.

Explanation

The angle θ between the hour hand and the minute hand of a clock at a given time H:M can be calculated using the standard formula θ = |30H - ¹¹⁄₂M|. For the time 2 : 15 p.m., the value of H is 2 and the value of M is 15. Substituting these values into the formula:

θ = |30(2) - ¹¹⁄₂(15)| θ = |60 - ¹⁶⁵⁄₂| θ = |60 - 82.5| θ = |-22.5| θ = 22.5°

This can also be understood by calculating the individual movement of the hands from the 12 o'clock position. At 2:00, the hour hand is at 60°. In 15 minutes, the hour hand moves further by 15 × 0.5° = 7.5°, reaching a total of 67.5°. Meanwhile, the minute hand moves 15 × 6° = 90°. The angle between them is the absolute difference between their positions, which is |90° - 67.5°| = 22.5°.

Explanation

The radius r of the circle is the distance between the points P(2, 0) and Q(5, 4). Using the distance formula r = √((x₂₎₋ₓ___₁₎_² ₊ ₍_y_₂ ₋ _y_₁₎_²:

r = √((5 - 2)²⁾⁺⁽⁴⁻⁰⁾^^^² = √(3²⁾⁺⁴^^^² = √(9 + 16) = √(25) = 5

The area A of a circle is given by the formula A = π r². Substituting the calculated radius into the formula:

A = π × (5)² = 25π

Explanation

To find the value of (x - ¹⁄ₓ)², we utilize basic algebraic identities to relate the given power of four to the required expression. We are given the equation x⁴ + (1)/(x⁴) = 23. By adding 2 to both sides of the equation, we can form a perfect square on the left-hand side: x⁴ + (1)/(x⁴) + 2 = 23 + 2. This is equivalent to x⁴ + (1)/(x⁴) + 2(x²)((1)/(x²)) = 25. According to the identity (a + b)² = a² + b² + 2ab, this expression simplifies to (x² + (1)/(x²))² = 25. Taking the square root of both sides, we obtain x² + (1)/(x²) = 5.

We now expand the expression we need to evaluate: (x - ¹⁄ₓ)². Using the identity (a - b)² = a² + b² - 2ab, we have (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2(x)(¹⁄ₓ), which simplifies to x² + (1)/(x²) - 2. By substituting the previously derived value of x² + (1)/(x²) = 5 into this formula, we calculate the final result as 5 - 2 = 3.

Explanation

Let x = √(6 + 6 + {6 + …)}. Due to the infinite nature of the expression, the inner part of the radical is also equal to x, leading to the equation x = √(6 + x). Squaring both sides gives x² = 6 + x, which can be rearranged into the quadratic equation x² - x - 6 = 0. Factoring this equation yields (x - 3)(x + 2) = 0, giving possible values of x = 3 or x = -2. Since the result of a square root of positive terms must be positive, the value is 3.

A shortcut for solving infinite nested radicals of the form √(n + n + …) is to find two consecutive integers whose product is n. If n = k(k + 1), the value of the expression is the larger integer (k + 1). For n = 6, the consecutive integers are 2 and 3 (2 × 3 = 6). Therefore, the value is 3.

Explanation

The algebraic identity for the square of a binomial is (x + 1/x)² = x² + (1/x)² + 2(x)(1/x), which simplifies to (x + 1/x)² = x² + 1/x² + 2. By rearranging this formula to solve for the target expression, we obtain x² + 1/x² = (x + 1/x)² - 2. Substituting the given value x + 1/x = 12 into the equation yields x² + 1/x² = 12² - 2. Since 12² = 144, the result is 144 - 2 = 142.

Explanation

In Δ ABC, since AD is the median to the side BC, D is the midpoint of BC, which implies BD = DC = ½ BC. The problem states AD = ½ BC, so AD = BD = DC. This equality indicates that the point D is equidistant from vertices A, B, and C, making D the circumcenter of the triangle. Because the circumcenter lies on the side BC, BC must be the diameter of the circumcircle, and Δ ABC is a right-angled triangle with ∠ BAC = 90°.

Using the given value ∠ BAD = 30°, we calculate ∠ DAC = ∠ BAC - ∠ BAD = 90° - 30° = 60°. In Δ ADC, since AD = DC, the triangle is isosceles, meaning the angles opposite these sides are equal. Therefore, ∠ ACB = ∠ DAC = 60°.

Explanation

The product of square roots follows the property √a × √b = √(ab). Applying this to the given expression:

√6 × √(15) = √(6 × 15) = √(90)

To simplify √(90), factorize it into a perfect square and another factor:

√(90) = √(9 × 10) = 3√(10)

Given the equation √6 × √(15) = x√(10), substitute the simplified value into the equation:

3√(10) = x√(10)

By dividing both sides by √(10), we find x = 3. Since the principal square roots of positive numbers are positive, the product is positive, yielding the specific value for x.

Explanation

To solve the expression 3 - 3 + √5{4} - (1)/(3 + √(5)), first rationalize the third term by multiplying the numerator and denominator by the conjugate of the denominator, 3 - √5.

(1)/(3 + √(5)) × 3 - √5{3 - √5} = 3 - √5{3² - (√5)²} = 3 - √5{9 - 5} = 3 - √5{4}

Now, substitute this value back into the original expression:

3 - 3 + √5{4} - 3 - √5{4}

Since the two fractions have a common denominator of 4, they can be combined:

3 - [ (3 + √5) + (3 - √5){4} ]

In the numerator, the positive and negative √5 terms cancel out:

3 - (3 + 3)/(4) = 3 - ⁶⁄₄

Simplifying the fraction ⁶⁄₄ to ³⁄₂ gives:

3 - ³⁄₂ = (6 - 3)/(2) = ³⁄₂

Explanation

The algebraic identity states that for any three variables x, y, and z, the expression x³ + y³ + z³ - 3xyz can be factored as (x + y + z)(x² + y² + z² - xy - yz - zx). A direct consequence of this identity is that if x + y + z = 0, then the entire expression x³ + y³ + z³ - 3xyz must equal zero, as it becomes a product of zero and another factor.

In the given problem, let x = a, y = b, and z = 1. The expression to be evaluated is a³ + b³ + 1³ - 3(a)(b)(1), which simplifies to a³ + b³ + 1 - 3ab. Since it is given that a + b + 1 = 0, substituting these values into the identity yields a³ + b³ + 1 - 3ab = (0)(a² + b² + 1² - ab - b - a) = 0.

Explanation

The line equation is x = 3y - 7. Converting this to the slope-intercept form y = mx + c gives 3y = x + 7, or y = ⅓x + ⁷⁄₃. Thus, the slope m of the line is ⅓.

For any two points (x₁, y₁) and (x₂, y₂) on a line, the slope is defined as m = (y₂ - y₁)/(x₂ - x₁). Substituting the given points (a, b) and (a + 3, b + k) into this formula: m = ((b + k) - b)/((a + 3) - a) = ᵏ⁄₃

Setting the calculated slope equal to the known slope of the line: ᵏ⁄₃ = ⅓ k = 1

This can also be verified by direct substitution. Since (a, b) is on the line, it satisfies the equation a = 3b - 7. Since (a + 3, b + k) is also on the line, it must satisfy (a + 3) = 3(b + k) - 7. Expanding the second equation gives a + 3 = 3b + 3k - 7. Substituting the first equation into the second: (3b - 7) + 3 = 3b + 3k - 7 3b - 4 = 3b + 3k - 7 Subtracting 3b from both sides and adding 7 results in 3 = 3k, which simplifies to k = 1.

Explanation

The sum of the ages of the four boys five years ago was 4 × 9 = 36 years. To find their current total age, we add 5 years for each of the four boys to the previous sum, resulting in 36 + (4 × 5) = 36 + 20 = 56 years. Alternatively, the present average age of the four boys is 9 + 5 = 14 years, making their current total age 14 × 4 = 56 years.

When the new boy is included, the group consists of five boys with an average age of 15 years. The total sum of their present ages is 5 × 15 = 75 years. The present age of the new boy is calculated by subtracting the current total age of the original four boys from the total age of the group of five, which gives 75 - 56 = 19 years.

Explanation

The average of a set of numbers is calculated as the sum of the numbers divided by the total count of numbers. In this case, there are 8 numbers: 39, 48, 51, 63, 75, 83, x, and 69. The problem states that their average is 60.

Using the basic formula for average: Sum of observations = Average × Total number of observations Sum = 60 × 8 = 480

Now, calculate the sum of the seven known values: 39 + 48 + 51 + 63 + 75 + 83 + 69 = 428

To find the value of x, subtract the sum of the known numbers from the total sum: x = 480 - 428 = 52

Alternatively, using the deviation method, the sum of deviations of all values from the average (60) must be zero. The deviations of the known numbers are: 39 - 60 = -21 48 - 60 = -12 51 - 60 = -9 63 - 60 = 3 75 - 60 = 15 83 - 60 = 23 69 - 60 = 9

Summing these deviations: -21 - 12 - 9 + 3 + 15 + 23 + 9 = 8

Since the total deviation must be zero, the deviation for x must be -8: x - 60 = -8 x = 52

Explanation

The cost C of the diamond is proportional to the square of its weight W, expressed by the formula C = kW². Let the weights of the three pieces be x, 2x, and 3x based on the given ratio 1:2:3. The total weight of the original diamond was W = x + 2x + 3x = 6x.

The initial value of the diamond is Cᵢₙᵢₜᵢₐₗ = k(6x)² = 36kx². We are given that Cᵢₙᵢₜᵢₐₗ = 5184, so 36kx² = 5184. Solving for the constant factor kx² gives kx² = 5184 / 36 = 144.

The total value of the pieces after cutting is the sum of the values of each individual piece: C_fᵢₙₐₗ = k(x)² + k(2x)² + k(3x)² C_fᵢₙₐₗ = kx² + 4kx² + 9kx² = 14kx²

The loss involved in the cutting process is the difference between the initial value and the final value of the pieces: Loss = 36kx² - 14kx² = 22kx² Loss = 22 × 144 = 3168

Explanation

The reduction in the selling price is equivalent to the discount amount. Let the marked price of the toy be M. The problem provides that a 30% discount on M results in a reduction of Rs. 30. This relationship is expressed as: 0.30 × M = 30 M = (30)/(0.30) = 100 The marked price of the toy is Rs. 100. The selling price is calculated by subtracting the discount from the marked price: Selling Price = Marked Price - Discount Selling Price = 100 - 30 = 70 Alternatively, if the discount is 30%, the remaining selling price is 70% of the marked price. Since 30% equals Rs. 30, then 1% equals Rs. 1, and 70% equals Rs. 70.

Explanation

The volume V of a hemisphere is calculated using the formula V = ⅔ π r³. This formula shows that the volume is directly proportional to the cube of the radius, V ∝ r³. Consequently, the ratio of the radii of two hemispherical vessels is equal to the cube root of the ratio of their capacities.

Given the capacities V₁ = 6.4 litres and V₂ = 21.6 litres, the ratio of their volumes is:

V₁{V₂} = (6.4)/(21.6) = ⁶⁴⁄₂₁₆

To find the ratio of the radii r₁:r₂, we take the cube root of the volume ratio:

r₁{r₂} = ³√(⁶⁴⁄₂₁₆)

r₁{r₂} = ⁴⁄₆ = ⅔

Explanation

Let the capacity of the tank be the least common multiple of 8 and 6, which is 24 units. The efficiency of pipe A is 24 / 8 = 3 units per hour, and the efficiency of pipe B is 24 / 6 = 4 units per hour. When both pipes are opened together, their combined efficiency is 3 + 4 = 7 units per hour. In the first 2 hours, they fill 7 × 2 = 14 units of the tank. The remaining volume to be filled is 24 - 14 = 10 units. Since pipe A is closed, pipe B must fill these remaining 10 units at its individual rate of 4 units per hour. The time taken by pipe B to finish filling the tank is 10 / 4 = 2.5 hours.

Explanation

The algebraic identity for the numerator is a³ + b³ + c³ - 3abc = (a + b + c) · ½ [(a - b)² + (b - c)² + (c - a)²]. When we divide this expression by the denominator (a + b + c), the term (a + b + c) cancels out, simplifying the required value to ½ [(a - b)² + (b - c)² + (c - a)²].

By substituting the given values (a - b) = 3, (b - c) = 5, and (c - a) = 1 into this simplified formula, we get ½ [3² + 5² + 1²]. Calculating the squares results in ½ [9 + 25 + 1] = ³⁵⁄₂, which equals 17.5.

Explanation

The total increase in the population is 16300 - 15000 = 1300. Let M represent the number of males and F represent the number of females. Given the total population, we have the equation M + F = 15000. The increase in population based on the given percentages is 0.08M + 0.10F = 1300.

To find the number of females, we can multiply the first equation by 0.08 to get 0.08M + 0.08F = 1200. Subtracting this result from the increase equation gives (0.08M + 0.10F) - (0.08M + 0.08F) = 1300 - 1200, which simplifies to 0.02F = 100. Solving for F, we get F = 100 / 0.02 = 5000.

Alternatively, using the method of alligation, the average percentage increase is (1300 / 15000) × 100 = 26/3%. The ratio of males to females is calculated as (10 - 26/3) : (26/3 - 8), which equals 4/3 : 2/3, or 2 : 1. Dividing the total population of 15000 into three parts (2 + 1 = 3), the female population corresponds to one part: 15000 / 3 = 5000.

Explanation

The simple interest (SI) earned on Rs. 5,000 in one year is the difference between the amount and the principal: SI = 5700 - 5000 = 700. Using the simple interest formula SI = (P × R × T)/(100), we substitute the values to find the rate (R): 700 = (5000 × R × 1)/(100), which simplifies to 700 = 50R, giving R = 14% per annum.

To find what Rs. 7,000 becomes in 5 years at the same rate, we calculate the new simple interest: SI = (7000 × 14 × 5)/(100) = 70 × 70 = 4900. The total amount at the end of 5 years is the sum of the principal and the interest: Amount = 7000 + 4900 = 11900.

Explanation

The relative speed of the policeman with respect to the thief is the difference in their speeds because they are running in the same direction. Relative speed = 11 km/h - 10 km/h = 1 km/h. To convert this into meters per minute, multiply by 1000/60, which gives 50/3 m/min.

The reduction in the gap between the two over 6 minutes is calculated as Distance = Relative Speed × Time. Substituting the values, we get Distance = (50/3) × 6 = 100 meters. Since the initial distance between the thief and the policeman was 200 meters, the remaining distance after 6 minutes is 200 - 100 = 100 meters.

Explanation

The final price is calculated by applying successive profit multipliers to the original cost price. Let the original cost price be x.

A sells to B at 20% profit, making the price 1.20x. B sells to C at 25% profit, making the price 1.20x × 1.25.

Given that C pays Rs. 1,200: x × 1.20 × 1.25 = 1200 x × ⁶⁄₅ × ⁵⁄₄ = 1200 x × ⁶⁄₄ = 1200 x = (1200 × 4)/(6) x = 200 × 4 = 800

Explanation

When 75070 is divided by 65, the remainder is 60. To find the nearest multiple of 65, we compare subtracting the remainder (75070 - 60 = 75010) or adding the difference between the divisor and remainder (75070 + (65 - 60) = 75075). Since 75075 is only 5 units away whereas 75010 is 60 units away, 75075 is the nearest number.

Explanation

To calculate a value that is x% more than a given number y, the formula used is y × (1 + ˣ⁄₁₀₀). In this case, the base value y is 80 and the percentage increase x is 20%.

Substituting these values into the formula: Number = 80 × (1 + ²⁰⁄₁₀₀) Number = 80 × (1 + 0.2) Number = 80 × 1.2 Number = 96

Alternatively, the increment can be calculated by finding 20% of 80, which is ²⁰⁄₁₀₀ × 80 = 16. Adding this increment to the original value gives 80 + 16 = 96.

Explanation

The total height of the tree H is the sum of the vertical part still standing and the broken part that stretches to the ground as a hypotenuse. In the right-angled triangle formed, the distance from the root to where the top touches the ground is the base d = 30 m, and the angle with the ground is θ = 30°. The original height is determined by the formula H = d(tan θ + sec θ).

Substituting the given values into the formula: H = 30(tan 30° + sec 30°) H = 30((1)/(√(3)) + (2)/(√(3))) H = 30((3)/(√(3))) H = 30√3 m

The standing part 30 tan 30° = 10√3 m and the broken part 30 sec 30° = 20√3 m together make the full height of the tree.

Explanation

The given equation is cos A + cos² A = 1. By rearranging this expression, we get cos A = 1 - cos² A. According to the trigonometric identity sin² A + cos² A = 1, the term 1 - cos² A is equal to sin² A. Thus, we establish the relation cos A = sin² A. Squaring both sides of this relation yields cos² A = (sin² A)² = sin⁴ A. The expression to be evaluated is sin² A + sin⁴ A. Substituting the equivalent cosine terms into this expression gives sin² A + sin⁴ A = cos A + cos² A. Since the initial given condition states that cos A + cos² A = 1, the value of the required expression is 1.

Explanation

The total number of cows n is distributed among the four sons such that the sum of their individual shares equals n. Based on the given proportions and the fixed number for the fourth son, the equation is:

ⁿ⁄₂ + ⁿ⁄₄ + ⁿ⁄₅ + 7 = n

To find the value of n, combine the fractional parts by finding a common denominator for 2, 4, and 5, which is 20.

(10n + 5n + 4n)/(20) + 7 = n

¹⁹ⁿ⁄₂₀ + 7 = n

Rearrange the equation to isolate the constant:

n - ¹⁹ⁿ⁄₂₀ = 7

(20n - 19n)/(20) = 7

ⁿ⁄₂₀ = 7

n = 7 × 20 = 140

Explanation

To find the least number by which 675 must be multiplied to make it a perfect cube, we first determine its prime factorization. 675 = 5 × 135 135 = 5 × 27 27 = 3 × 3 × 3 So, the prime factorization is 675 = 3³ × 5².

For a number to be a perfect cube, every prime factor in its prime factorization must have an exponent that is a multiple of 3. In the factorization of 675, the factor 3 has an exponent of 3, which already satisfies the condition for a perfect cube. However, the factor 5 has an exponent of only 2. To make the exponent of 5 a multiple of 3, we must multiply the number by 5¹ to reach 5³.

Multiplying 675 by 5 yields 3³ × 5³ = (3 × 5)³ = 15³ = 3375. Therefore, the least number required is 5.

Explanation

In problems where the difference between the divisors and their respective remainders is constant, the least number is found using the formula L = LCM(d₁, d₂, d₃, ...) - k, where k is the common difference.

First, calculate the common difference k: 35 - 18 = 17 45 - 28 = 17 55 - 38 = 17 Thus, k = 17.

Next, find the Least Common Multiple (LCM) of the divisors 35, 45, and 55: The prime factorizations are: 35 = 5 × 7 45 = 3² × 5 55 = 5 × 11 LCM(35, 45, 55) = 3² × 5 × 7 × 11 = 9 × 5 × 7 × 11 = 3465

The required least number is: L = LCM - k = 3465 - 17 = 3448

Explanation

Using the trigonometric identity csc² A - cot² A = 1, we can factor this as (csc A + cot A)(csc A - cot A) = 1. Given that csc A + cot A = 3, we substitute this value into the identity to get 3(csc A - cot A) = 1, which simplifies to csc A - cot A = ⅓.

We now have a system of two linear equations: csc A + cot A = 3 csc A - cot A = ⅓

Adding these two equations gives 2 csc A = 3 + ⅓ = ¹⁰⁄₃, which results in csc A = ⁵⁄₃. Since sin A = (1)/(csc A), we have sin A = ⅗.

Subtracting the second equation from the first gives 2 cot A = 3 - ⅓ = ⁸⁄₃, which results in cot A = ⁴⁄₃. Using the relation cos A = cot A × sin A, we find cos A = ⁴⁄₃ × ⅗ = ⅘. Alternatively, since A is an acute angle and sin A = ⅗, the adjacent side is √(5² - 3²) = 4, making cos A = ⅘.

Explanation

The number 13b7 is divisible by 11. According to the divisibility rule for 11, the difference between the sum of digits in the odd positions and the sum of digits in the even positions must be zero or a multiple of 11. For the four-digit number 13b7, we calculate (1 + b) - (3 + 7) = b - 9. For this value to be zero, b must be 9.

Using the given equation 4a3 + 984 = 13b7 and substituting b = 9, we have 4a3 + 984 = 1397. To find the value of a, we perform the subtraction 1397 - 984 = 413. Comparing the terms, we see that the three-digit number 4a3 corresponds to 413, which means a = 1. The value of a + b is calculated as 1 + 9 = 10.

Explanation

For two similar triangles Δ ABC and Δ DEF, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This geometric property is expressed by the formula:

(Area(Δ ABC))/(Area(Δ DEF)) = ((BC)/(EF))²

Substituting the given values into the formula:

⁶⁴⁄₁₂₁ = ((BC)/(15.4))²

Taking the square root of both sides to solve for the ratio of the sides:

√(64){√(121)} = (BC)/(15.4)

⁸⁄₁₁ = (BC)/(15.4)

To find BC, rearrange the equation:

BC = (8 × 15.4)/(11)

BC = 8 × 1.4 = 11.2 cm

Using the theorem of areas of similar triangles, the length of side BC is calculated to be 11.2 cm.

Explanation

Let AD be the median from vertex A to the side BC, where D is the midpoint of BC. The centroid G of a triangle divides each median in the ratio 2:1, measured from the vertex. Therefore, AG = 2GD. The problem provides the condition AG = BC, so substituting this into the ratio gives BC = 2GD, which simplifies to GD = (BC)/(2). In △ BGC, GD is the median to the side BC because D is the midpoint. Since D is the midpoint, the lengths BD and DC are both equal to (BC)/(2). Consequently, we have the equality GD = BD = DC. This geometry implies that D is equidistant from the three vertices of △ BGC, making D the circumcenter of the triangle. Since the circumcenter D lies on the side BC, BC must be the diameter of the circumcircle. According to the property that an angle subtended by a diameter at the circumference is a right angle, it follows that ∠ BGC = 90°.

Explanation

For any two angles A and B, the condition tan A tan B = 1 implies that tan A = 1 / tan B = cot B. Since cot B = tan(90° - B), it follows that A + B = 90°. Applying this property to the given equation tan(x + y) tan(x - y) = 1, we substitute A = x + y and B = x - y into the relation:

(x + y) + (x - y) = 90°

Simplifying the equation by combining like terms, we get:

2x = 90° x = 45°

The value of tan x is therefore tan 45° = 1. This can also be verified by expanding the tangent sum and difference identities, which results in the relation tan² x - tan² y = 1 - tan² x tan² y. Factoring this expression gives tan² x (1 + tan² y) = 1 + tan² y. Dividing both sides by (1 + tan² y) yields tan² x = 1, confirming tan x = 1.

Explanation

In a partnership, the profit is divided in the ratio of the product of capital and time. Let the total capital be 1 unit and the total time be 1 unit.

The capital invested by C is calculated by subtracting the investments of A and B from the total: C = 1 - (⅙ + ¼) = 1 - ⁵⁄₁₂ = ⁷⁄₁₂

The profit sharing ratio is determined as: A : B : C = (Capital_A × Time_A) : (Capital_B × Time_B) : (Capital_C × Time_C) A : B : C = (⅙ × ⅙) : (¼ × ¼) : (⁷⁄₁₂ × 1) A : B : C = ¹⁄₃₆ : ¹⁄₁₆ : ⁷⁄₁₂

To simplify this ratio, multiply all terms by the least common multiple of the denominators (36, 16, 12), which is 144: A : B : C = (¹⁄₃₆ × 144) : (¹⁄₁₆ × 144) : (⁷⁄₁₂ × 144) = 4 : 9 : 84

The sum of the ratio parts is 4 + 9 + 84 = 97. Given the total profit is 19,400, B's share is calculated as: B's share = ⁹⁄₉₇ × 19400 B's share = 9 × 200 = 1800

Explanation

Let the initial quantities of A and B be 4x and x. After removing 10 liters of mixture (8 liters of A and 2 liters of B) and adding 10 liters of B, the new ratio is (4x - 8) / (x + 8) = 2/3, which gives x = 4. Thus, the initial volume of liquid A was 4x = 16 liters.

Explanation

The given equation is (5x² - 3y²)/(xy) = ¹¹⁄₂. To find the ratio k = (x)/(y), divide the terms in the numerator by xy to get 5((x)/(y)) - 3(ʸ⁄ₓ) = ¹¹⁄₂. Substituting k = (x)/(y) yields the equation 5k - ³⁄ₖ = ¹¹⁄₂.

Multiplying the entire equation by 2k to clear the fractions results in 10k² - 6 = 11k, which rearranges to the quadratic form 10k² - 11k - 6 = 0. Factoring the quadratic by splitting the middle term gives 10k² - 15k + 4k - 6 = 0, which simplifies to 5k(2k - 3) + 2(2k - 3) = 0, or (5k + 2)(2k - 3) = 0. The roots are k = ³⁄₂ and k = -⅖. Since x and y are specified as positive, the ratio (x)/(y) must be positive, hence (x)/(y) = ³⁄₂.

Explanation

The sum of the internal angles of a triangle is 180°. When each of the three angles is decreased by 15°, the total reduction in the sum of the angles is 3 × 15° = 45°. Therefore, the sum of the reduced angles is 180° - 45° = 135°.

The reduced angles are in the ratio 2:3:5. Let the reduced angles be 2x, 3x, and 5x. Their sum is 2x + 3x + 5x = 10x. Setting this equal to the sum of the reduced angles, we get 10x = 135°, which gives x = 13.5°.

The greatest reduced angle is 5x = 5 × 13.5° = 67.5°. To find the greatest original angle, we add back the 15° reduction: 67.5° + 15° = 82.5°.

To convert the degree measure to radians, we multiply by (π)/(180°): Radian measure = 82.5 × (π)/(180) = ¹⁶⁵⁄₂ × (π)/(180) = (33 × 5)/(2) × (π)/(36 × 5) = (33π)/(72) Simplifying by dividing the numerator and denominator by 3, we get (11π)/(24).

Explanation

For any two positive real numbers x and y, the Arithmetic Mean (AM) is always greater than or equal to the Geometric Mean (GM). Applying this inequality to the terms 4csc² a and 9sin² a, we have:

4csc² a + 9sin² a{2} ≥ √((4csc²⁾ᵃ⁾^·⁽⁹ˢⁱⁿ^^^² ᵃ⁾

Since csc² a = (1)/(sin²⁾ᵃ, the product of the two terms inside the square root simplifies to a constant:

4csc² a + 9sin² a ≥ 2√(4 · 9 · csc²⁾ᵃ^·ˢⁱⁿ^^^² ᵃ

4csc² a + 9sin² a ≥ 2√(36)

4csc² a + 9sin² a ≥ 2 · 6

4csc² a + 9sin² a ≥ 12

The minimum value of the expression is reached when 4csc² a = 9sin² a, which gives sin⁴ a = ⁴⁄₉, or sin² a = ⅔. Since this value of sin² a is within the valid range [0, 1], the least value is exactly 12.