SSC CGL Tier 1 2012 Quantitative Aptitude Questions with Solutions

Explanation

To determine the value of p, we utilize the fundamental trigonometric identity sin² Θ + cos² Θ = 1. Given the equations p sin Θ = √3 and p cos Θ = 1, we square both sides of each equation to obtain p² sin² Θ = 3 and p² cos² Θ = 1. Adding these two resulting equations gives p² sin² Θ + p² cos² Θ = 3 + 1.

By factoring out p² from the left side, the equation becomes p² (sin² Θ + cos² Θ) = 4. Substituting the identity sin² Θ + cos² Θ = 1 into this expression simplifies it to p² (1) = 4. Solving for p by taking the square root of both sides, we find that p = √4, which equals 2.

Explanation

The area of a trapezium is given by the formula Area = ½ × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them. Substituting the given values a = 5 cm, b = 9 cm, and Area = 35 sq.cm, we get 35 = ½ × (5 + 9) × h, which simplifies to 35 = 7h, yielding h = 5 cm.

In an isosceles trapezium where AB = CD, the projection of the non-parallel side CD on the longer base BC is calculated as x = (BC - AD)/(2). Substituting the lengths of the parallel sides, x = (9 - 5)/(2) = 2 cm. Using the Pythagorean theorem in the right-angled triangle formed by the height h, the projection x, and the slant side CD, we have CD² = h² + x². Substituting the values, CD² = 5² + 2² = 25 + 4 = 29. Therefore, CD = √(29) cm.

Explanation

From the given equation x sin Θ - y cos Θ = 0, we have the relation x sin Θ = y cos Θ. The second given equation is x sin³ Θ + y cos³ Θ = sin Θ cos Θ. We can rewrite this equation by splitting the terms as (x sin Θ) sin² Θ + (y cos Θ) cos² Θ = sin Θ cos Θ.

Substituting x sin Θ for y cos Θ in the expression, we get: (x sin Θ) sin² Θ + (x sin Θ) cos² Θ = sin Θ cos Θ

Factoring out x sin Θ from the left side: x sin Θ (sin² Θ + cos² Θ) = sin Θ cos Θ

Using the trigonometric identity sin² Θ + cos² Θ = 1, the equation simplifies to: x sin Θ (1) = sin Θ cos Θ

Dividing both sides by sin Θ (since sin Θ cos Θ ≠ 0), we find: x = cos Θ

Now, substituting x = cos Θ back into the relation x sin Θ = y cos Θ: cos Θ sin Θ = y cos Θ y = sin Θ

The value of x² + y² is calculated as: x² + y² = (cos Θ)² + (sin Θ)² x² + y² = cos² Θ + sin² Θ = 1

Explanation

Substituting x = 4 into the left-hand side of the equation √(4x - 9) + √(4x + 9) = 5 + √7 yields:

LHS = √(4(4) - 9) + √(4(4) + 9) LHS = √(16 - 9) + √(16 + 9) LHS = √7 + √(25) LHS = √7 + 5 LHS = 5 + √7

Since the resulting value matches the right-hand side of the equation exactly, x = 4 is the valid solution.

Alternatively, the value can be found by comparing the rational and irrational components of the equation. For the equation to hold, the terms can be equated as follows: √(4x + 9) = 5 ⟹ 4x + 9 = 25 ⟹ 4x = 16 ⟹ x = 4 √(4x - 9) = √7 ⟹ 4x - 9 = 7 ⟹ 4x = 16 ⟹ x = 4 Both conditions lead to the same consistent value for the variable.

Explanation

Given uₙ = cosⁿ α + sinⁿ α. We use the fundamental trigonometric identity sin² α + cos² α = 1 to express u₄ and u₆ in terms of sin² α cos² α.

Using the identity a² + b² = (a+b)² - 2ab for u₄: u₄ = cos⁴ α + sin⁴ α = (cos² α + sin² α)² - 2 sin² α cos² α u₄ = 1 - 2 sin² α cos² α

Using the identity a³ + b³ = (a+b)³ - 3ab(a+b) for u₆: u₆ = cos⁶ α + sin⁶ α = (cos² α + sin² α)³ - 3 sin² α cos² α (cos² α + sin² α) u₆ = 1³ - 3 sin² α cos² α (1) = 1 - 3 sin² α cos² α

Now, substitute these expressions into 2u₆ - 3u₄ + 1: 2(1 - 3 sin² α cos² α) - 3(1 - 2 sin² α cos² α) + 1 = 2 - 6 sin² α cos² α - 3 + 6 sin² α cos² α + 1 = 2 - 3 + 1 = 0

Alternatively, by substituting a specific value like α = 0, we find u₆ = 1⁶ + 0⁶ = 1 and u₄ = 1⁴ + 0⁴ = 1. The expression then becomes 2(1) - 3(1) + 1 = 0.

Explanation

To solve the expression √(-3) + √(3 + 8{7 + 4{3)}}, we use the algebraic identity (a + b)² = a² + b² + 2ab to simplify the nested radicals starting from the innermost layer.

First, we simplify √(7 + 43). We can rewrite the radicand as 7 + 4√3 = 4 + 3 + 2(2)(√3) = 2² + (√3)² + 2(2)(√3), which is (2 + √3)². Therefore, √(7 + 43) = 2 + √3.

Next, we substitute this value into the middle radical layer: √(3 + 8(2 + 3)) = √(3 + 16 + 83) = √(19 + 83). We simplify this by rewriting 19 + 8√3 as 16 + 3 + 2(4)(√3) = 4² + (√3)² + 2(4)(√3), which is (4 + √3)². Thus, √(19 + 83) = 4 + √3.

Finally, we substitute this result back into the outermost part of the expression: √(-3) + (4 + √3) = √(-3) + 4 + √3 = √4 = 2.

Explanation

In a circle, the radius to the midpoint of a chord is perpendicular to that chord. Therefore, OP ⊥ AB and OQ ⊥ AC, implying ∠ OPA = 90° and ∠ OQA = 90°. In quadrilateral APOQ, the sum of interior angles is 360°. With ∠ BAC = 32°, the angle at the center is ∠ POQ = 360° - (90° + 90° + 32°) = 148°. Since points R and S lie on the circle along the extensions of OP and OQ respectively, the central angle ∠ ROS is equal to ∠ POQ, which is 148°. According to the circle theorem, the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the circumference. Thus, the angle at point T on the major arc is ∠ RTS = ½ ∠ ROS = (148°)/(2) = 74°.

Explanation

The final quantity of milk remaining in a container after n successive operations of removing a volume x and replacing it with water from an initial volume V is calculated using the formula Q = V(1 - (x)/(V))ⁿ.

In this problem, the initial volume of milk V is 80 L, and the amount replaced in each step x is 8 L. The process was performed once and then repeated twice more, which means the total number of operations n is 3.

Substituting these values into the formula: Q = 80(1 - ⁸⁄₈₀)³ Q = 80(1 - ⅒)³ Q = 80(⁹⁄₁₀)³ Q = 80 × ⁷²⁹⁄₁₀₀₀ Q = (8 × 729)/(100) Q = ⁵⁸³²⁄₁₀₀ Q = 58.32 L.

Explanation

In a circle, a perpendicular drawn from the center to a chord bisects the chord. This construction creates a right-angled triangle where the radius (r) is the hypotenuse, the distance from the center to the chord (d) is one leg, and half the length of the chord (L/2) is the other leg. Applying the Pythagoras theorem:

r² = d² + ((L)/(2))²

Substituting the given values r = 13 cm and d = 12 cm:

13² = 12² + ((L)/(2))² 169 = 144 + ((L)/(2))² ((L)/(2))² = 169 - 144 ((L)/(2))² = 25 (L)/(2) = √(25) = 5 cm

The total length of the chord XY is 2 × 5 = 10 cm. This follows the standard Pythagorean triplet (5, 12, 13), where 13 is the hypotenuse.

Explanation

The sum of the three internal angles of a triangle is always π radians. Given the two angles as 1/2 radian and 1/3 radian, the third angle in radians is calculated by subtracting their sum from π: Third angle = π - (1/2 + 1/3) = π - 5/6 radians.

To convert this measure into degrees, we multiply the value in radians by 180/π: Third angle in degrees = (π - 5/6) × (180/π) = 180 - (5/6 × 180/π) = 180 - (150/π).

Using the value π = 22/7: Third angle in degrees = 180 - (150 × 7 / 22) = 180 - (75 × 7 / 11) = 180 - 525/11.

Dividing 525 by 11 gives: 525/11 = 47 ⁸⁄₁₁.

Subtracting this from 180: 180 - 47 ⁸⁄₁₁ = 132 ³⁄₁₁°.

Explanation

When two equal circles of radius r pass through each other's center, the distance between their centers is also r. Let the centers be O₁ and O₂, and the intersection points of the circles be A and B. In the triangle AO₁O₂, the side O₁O₂ = r, and the sides O₁A and O₂A are also equal to r as they are radii of the respective circles. This makes △ AO₁O₂ an equilateral triangle with side length r = 5 cm.

The common chord AB is perpendicular to the line joining the centers O₁O₂ and is bisected by it. The distance from point A to the line O₁O₂ represents the altitude of the equilateral triangle △ AO₁O₂. The formula for the altitude h of an equilateral triangle with side s is h = (√3/2)s. For s = 5 cm, the altitude is 5√3/2 cm. Since the common chord AB is twice this altitude, its length is 2 × 5√3/2 = 5√3 cm.

Explanation

When the grain dealer cheats by 10% while buying, it means he pays for 100 units but receives 110 units. Thus, his cost price for 110 units is equivalent to the price of 100 units. When he cheats by 10% while selling, it means he charges the customer for 100 units but only delivers 90 units. Thus, his selling price for 90 units is equivalent to the price of 100 units.

To calculate the total profit percentage, we compare the effective cost and selling price per unit. Let the nominal price of 1 unit be 1. Cost Price (CP) of 1 unit = ¹⁰⁰⁄₁₁₀ = ¹⁰⁄₁₁ Selling Price (SP) of 1 unit = ¹⁰⁰⁄₉₀ = ¹⁰⁄₉

Profit Percentage = (SP - CP)/(CP) × 100 Substituting the values: Profit Percentage = (10)/(9) - ¹⁰⁄₁₁{¹⁰⁄₁₁} × 100 Profit Percentage = (1)/(9) - ¹⁄₁₁{¹⁄₁₁} × 100 Profit Percentage = (11 - 9)/(9) × 100 Profit Percentage = ²⁄₉ × 100 = 22.22%

Explanation

In △ ACD, it is given that CD = CA, which implies that △ ACD is an isosceles triangle. Therefore, the angles opposite to these sides are equal, so ∠ CAD = ∠ CDA. Since ∠ ADB = 20° is given, then ∠ CDA = 20°, which leads to ∠ CAD = 20°.

The angle ∠ ACB is an exterior angle to △ ACD at vertex C. By the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Thus, ∠ ACB = ∠ CAD + ∠ CDA = 20° + 20° = 40°.

In △ ABC, the problem states that AB = AC, meaning △ ABC is also an isosceles triangle. The angles opposite to these equal sides must be equal, so ∠ ABC = ∠ ACB. Substituting the calculated value of ∠ ACB, we find that ∠ ABC = 40°.

Explanation

Using the trigonometric identity sin θ = cos (90° - θ), or more generally, if sin A = cos B, then A + B = 90°. By applying this to the given equation sin (x + y) = cos [3(x + y)], we treat A = (x + y) and B = 3(x + y).

Summing these angles gives: (x + y) + 3(x + y) = 90° 4(x + y) = 90°

To find the value of tan [2(x + y)], we divide the above equation by 2: 2(x + y) = 45°

Substituting this value into the tangent function: tan [2(x + y)] = tan 45° = 1

Explanation

A garden roller is cylindrical in shape, and the area it covers in one complete revolution is equal to its curved surface area. The curved surface area of a cylinder is calculated using the formula π × d × h, where d represents the diameter and h represents the length (or height) of the roller.

Given: Diameter d = 1.4 m Length h = 2 m Number of revolutions n = 5

First, calculate the area covered in a single revolution: Area per revolution = ²²⁄₇ × 1.4 × 2 Area per revolution = 22 × 0.2 × 2 Area per revolution = 8.8 sq.m

To determine the total area covered in 5 revolutions, multiply the area of one revolution by the total number of revolutions: Total area = 5 × 8.8 Total area = 44 sq.m

Explanation

Let t be the time in hours spent traveling by train. Since the total time for the journey is 6 hours, the time spent traveling by bus is (6 - t) hours. Using the relation Distance = Speed × Time, the total distance is the sum of the distances covered by bus and train:

40(6 - t) + 55t = 285

Expanding the equation: 240 - 40t + 55t = 285 15t = 285 - 240 15t = 45 t = 3 hours

The distance traveled by train is calculated by multiplying the speed of the train by the time spent on the train: 55 km/hr × 3 hr = 165 km

Alternatively, applying the rule of allegation to the speeds, the average speed of the entire journey is 285 / 6 = 47.5 km/hr. The ratio of time spent on the bus to time spent on the train is (55 - 47.5) : (47.5 - 40), which is 7.5 : 7.5 or 1 : 1. Dividing the total time of 6 hours in this ratio gives 3 hours for each part. The train distance is therefore 55 × 3 = 165 km.

Explanation

The sum of a set of numbers is equal to the average multiplied by the count of numbers. For the five numbers, the total sum is 5 × 56.4 = 282.

The sum of the first and the second number is 2 × 53 = 106. The sum of the fourth and the fifth number is 2 × 56 = 112.

To find the third number, subtract the sum of the first, second, fourth, and fifth numbers from the total sum of all five numbers: Third number = 282 - (106 + 112) Third number = 282 - 218 Third number = 64

Explanation

Using the complementary angle relationships cot 70° = tan(90° - 70°) = tan 20° and tan 70° = cot(90° - 70°) = cot 20°, the expression is simplified to: (1 + sec 20° + tan 20°)(1 - csc 20° + cot 20°) Expressing these terms using sine and cosine functions: (1 + (1)/(cos 20°) + (sin 20°)/(cos 20°)) (1 - (1)/(sin 20°) + (cos 20°)/(sin 20°)) Taking common denominators in both parts: ((cos 20° + 1 + sin 20°)/(cos 20°)) ((sin 20° - 1 + cos 20°)/(sin 20°)) The product of the numerators follows the algebraic identity (a + b)(a - b) = a² - b² where a = (sin 20° + cos 20°) and b = 1: (sin 20° + cos 20°)² - 1²{sin 20° cos 20°} Expanding the square: sin² 20° + cos² 20° + 2 sin 20° cos 20° - 1{sin 20° cos 20°} Using the identity sin² θ + cos² θ = 1, the numerator simplifies to: (1 + 2 sin 20° cos 20° - 1)/(sin 20° cos 20°) = (2 sin 20° cos 20°)/(sin 20° cos 20°) = 2

Explanation

The equation 2 sin a + 15 cos² a = 7 can be solved by expressing all trigonometric terms in terms of sin a. Using the fundamental identity cos² a = 1 - sin² a, we substitute into the given equation:

2 sin a + 15(1 - sin² a) = 7 2 sin a + 15 - 15 sin² a = 7 15 sin² a - 2 sin a - 8 = 0

This is a quadratic equation in terms of sin a. We can solve it using the quadratic formula sin a = -b ± √(b² - 4ac){2a}:

sin a = 2 ± √((-2)² - 4(15)(-8)){2(15)} sin a = 2 ± √(4 + 480){30} sin a = (2 ± 22)/(30)

This yields two possible values: sin a = ²⁴⁄₃₀ = ⅘ or sin a = -²⁰⁄₃₀ = -⅔. Given the constraint 0 ≤ a ≤ π/2, a lies in the first quadrant where the sine value must be non-negative. Therefore, we have:

sin a = ⅘

In a right-angled triangle where the opposite side is 4 and the hypotenuse is 5, the adjacent side is found using the Pythagorean theorem: √(5² - 4²) = √(25 - 16) = 3. From this, the cosine value is:

cos a = ⅗

The value of cot a is defined as the ratio of the cosine to the sine:

cot a = (cos a)/(sin a) = (3/5)/(4/5) = ¾

Explanation

The net profit percentage can be calculated using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x is the markup percentage and y is the discount percentage. Given a markup of 30% (x = 30) and a discount of 10% (y = -10), the calculation is:

Gain % = 30 - 10 + (30 × (-10))/(100) Gain % = 20 - 3 = 17%

Alternatively, if the cost price is assumed to be 100, the marked price is 130 because it is 30% higher. Applying a 10% discount on the marked price requires subtracting 10% of 130, which is 13, resulting in a selling price of 130 - 13 = 117. The gain is the difference between the selling price and the cost price, which is 117 - 100 = 17, representing a 17% gain on the original cost price.

Explanation

From the given equation sin 2Θ = 1/2, we can equate 2Θ to 30° because sin 30° = 1/2. Solving for Θ, we get 2Θ = 30°, which simplifies to Θ = 15°. Substituting this value into the expression cos (75° - Θ), we obtain cos (75° - 15°) = cos 60°. Using the standard trigonometric table value, cos 60° = 1/2.

Explanation

The triangle is bounded by the x-axis (y = 0) and the lines 2x + y = 4 and x - y + 1 = 0. Since the base of the triangle lies along the x-axis, the altitude corresponding to this base is the perpendicular distance from the third vertex to the x-axis. This distance is equal to the absolute value of the y-coordinate of the point of intersection of the two lines 2x + y = 4 and x - y + 1 = 0.

To find the intersection point, solve the system of equations: 2x + y = 4 x - y = -1

Adding the two equations eliminates y: (2x + y) + (x - y) = 4 + (-1) 3x = 3 x = 1

Substituting x = 1 into the equation x - y + 1 = 0: 1 - y + 1 = 0 2 - y = 0 y = 2

The coordinates of the vertex not lying on the x-axis are (1, 2). The altitude of the triangle, measured as the vertical distance from this vertex to the x-axis, is equal to the y-coordinate of the vertex. Thus, the altitude is 2 units.

Explanation

First, rationalize the given values of x and y: x = (1)/(2 + √(3)) = 2 - √3{(2 + √3)(2 - √3)} = 2 - √3{4 - 3} = 2 - √3 y = (1)/(2 - √(3)) = 2 + √3{(2 - √3)(2 + √3)} = 2 + √3{4 - 3} = 2 + √3

Observe that xy = (2 - √3)(2 + √3) = 4 - 3 = 1, which means y = ¹⁄ₓ. The expression to be evaluated is (1)/(x + 1) + (1)/(y + 1). Substituting y = ¹⁄ₓ into the expression:

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

Alternatively, substituting the rationalized values directly: (1)/((2 - √(3)) + 1) + (1)/((2 + √(3)) + 1) = (1)/(3 - √(3)) + (1)/(3 + √(3)) Taking the common denominator: (3 + √3) + (3 - √3){(3 - √3)(3 + √3)} = (6)/(9 - 3) = ⁶⁄₆ = 1

Explanation

In a partnership business, the ratio of profits is directly proportional to the product of the capital invested and the time duration of the investment. The relationship is expressed by the formula:

Profit_A{Profit_B} = Capital_A × Time_A{Capital_B × Time_B}

From the question, A's capital (I_A) is Rs. 7,000 and the investment duration (T_A) is 12 months. B joins 5 months after A, which means B's investment duration (T_B) is 12 - 5 = 7 months. The profit sharing ratio is given as 2:3. Let B's capital be I_B. Substituting these values into the ratio formula:

⅔ = (7000 × 12)/(I_B₎_×₇

Simplifying the equation by dividing 7000 by 7:

⅔ = (1000 × 12)/(I_B₎

⅔ = (12000)/(I_B₎

Cross-multiplying to solve for I_B:

2 × I_B = 12000 × 3

2 × I_B = 36000

I_B = ³⁶⁰⁰⁰⁄₂ = 18000

The capital invested by B is Rs. 18,000.

Explanation

Let x = cot A. The given expression is x + 1/x = 2. Squaring both sides of the equation using the algebraic identity (a + b)² = a² + b² + 2ab, we obtain (x + 1/x)² = 2², which expands to x² + 1/x² + 2(x)(1/x) = 4. This simplifies to x² + 1/x² + 2 = 4. Subtracting 2 from both sides yields x² + 1/x² = 2. Substituting the original trigonometric term back into the equation, we find that cot² A + 1/cot² A = 2.

Alternatively, the equation x + 1/x = 2 can be rearranged as x² - 2x + 1 = 0, which is the expansion of (x - 1)² = 0. Solving for x gives x = 1, implying cot A = 1. Consequently, cot² A + 1/cot² A = 1² + 1/1² = 1 + 1 = 2.

Explanation

The expression is f(x) = sin 2x + csc 2x. We use the property that for any positive real number a, the sum a + ¹⁄ₐ ≥ 2. This is derived from the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that (a + b)/(2) ≥ √(ab). By setting a = sin 2x and b = csc 2x = (1)/(sin 2x), we get:

(sin 2x + 1)/(sin 2x){2} ≥ √(sin 2x · (1)/(sin 2x)) = 1

Multiplying both sides by 2 yields sin 2x + csc 2x ≥ 2. The equality holds when sin 2x = csc 2x, which occurs when sin² 2x = 1. Since the maximum value of sin 2x is 1, the minimum value of the function in its positive domain is 1 + ¹⁄₁ = 2.

Explanation

Let h be the height of the tower. According to the properties of right-angled triangles, the relationship between height, distance from the base, and the angle of elevation θ is given by h = distance × tan(θ). For the initial position at a distance of 500m: h = 500 × tan(30°) = 500 × (1)/(√(3)) = (500)/(√(3)) meters

After driving for 10 seconds, the car is at a new distance x from the tower where the angle of elevation is 60°. Using the height calculated above: x = (h)/(tan(60°)) = 500/√3{√3} = ⁵⁰⁰⁄₃ meters

The distance covered by the car during these 10 seconds is the difference between the initial distance and the final distance: Distance covered = 500 - ⁵⁰⁰⁄₃ = (1500 - 500)/(3) = ¹⁰⁰⁰⁄₃ meters

The speed v in meters per second is calculated as: v = (Distance)/(Time) = (1000/3)/(10) = ¹⁰⁰⁄₃ m/s

To convert the speed from m/s to km/hr, we multiply by the conversion factor ¹⁸⁄₅: Speed = ¹⁰⁰⁄₃ × ¹⁸⁄₅ = 20 × 6 = 120 km/hr

Explanation

The given equation is tan Θ + cot Θ = 2. Since cot Θ = (1)/(tan Θ), the equation can be written as tan Θ + (1)/(tan Θ) = 2. Multiplying through by tan Θ gives tan² Θ + 1 = 2 tan Θ, which simplifies to tan² Θ - 2 tan Θ + 1 = 0. This is the expansion of the perfect square (tan Θ - 1)² = 0, which implies tan Θ = 1. For a positive acute angle, tan Θ = 1 occurs when Θ = 45°. Substituting this value into the expression for sec Θ, we get sec 45° = √2.

Explanation

In a right-angled triangle formed by the tower and the ground, the angle of depression from the top of the tower to the point is equal to the angle of elevation from the point to the top of the tower because they are alternate interior angles. Let h be the height of the tower and d be the horizontal distance of the point from the base of the tower.

Using the trigonometric ratio for tangent: tan(θ) = (Height)/(Distance)

Given h = 200m and θ = 45°: tan(45°) = (200)/(d)

Since tan(45°) = 1, the equation becomes: 1 = (200)/(d) d = 200m

Explanation

Using the complementary angle identity sin (90° - Θ) = cos Θ, the given equation sin Θ + cos Θ = √2 sin (90° - Θ) can be rewritten as:

sin Θ + cos Θ = √2 cos Θ

To find cot Θ, we divide the entire equation by sin Θ:

1 + cot Θ = √2 cot Θ

Rearranging the terms to group cot Θ on one side gives:

√2 cot Θ - cot Θ = 1

cot Θ (√2 - 1) = 1

cot Θ = (1)/(√(2) - 1)

To simplify, rationalize the denominator by multiplying the numerator and the denominator by the conjugate (√2 + 1):

cot Θ = 1 × (√2 + 1){(√2 - 1)(√2 + 1)}

cot Θ = √2 + 1{(√2)² - (1)²} = √2 + 1{2 - 1} = √2 + 1

Explanation

Given the trigonometric equations sin (A - B) = 1/2 and cos (A + B) = 1/2, we identify the standard angles for these values. Since A and B are positive acute angles, A - B and A + B must also fall within the range where these basic inverse values apply.

For the first equation, sin (A - B) = 1/2 implies A - B = 30° because sin 30° = 0.5.

For the second equation, cos (A + B) = 1/2 implies A + B = 60° because cos 60° = 0.5.

We now have a system of two linear equations:

1. A - B = 30°
2. A + B = 60°

Adding equations (1) and (2) gives 2A = 90°, which results in A = 45°. Subtracting equation (1) from equation (2) gives 2B = 30°, which results in B = 15°.

Both values, A = 45° and B = 15°, are positive acute angles that satisfy the initial conditions.

Explanation

The equation 7sin² θ + 3cos² θ = 4 can be simplified using the trigonometric identity cos² θ = 1 - sin² θ. Substituting this into the expression, we get: 7sin² θ + 3(1 - sin² θ) = 4 7sin² θ + 3 - 3sin² θ = 4 4sin² θ = 1 sin² θ = 1/4

Since θ is a positive acute angle, we take the positive square root: sin θ = √(1/4) = 1/2 In the first quadrant, sin θ = 1/2 when θ = 30°. Therefore, tan θ = tan 30° = 1 / √3.

Alternatively, dividing the entire original equation by cos² θ allows us to solve for tan θ directly: 7(sin² θ / cos² θ) + 3(cos² θ / cos² θ) = 4 / cos² θ 7tan² θ + 3 = 4sec² θ Applying the identity sec² θ = 1 + tan² θ: 7tan² θ + 3 = 4(1 + tan² θ) 7tan² θ + 3 = 4 + 4tan² θ 3tan² θ = 1 tan² θ = 1/3 tan θ = 1 / √3

Explanation

The number of revolutions per second is calculated by dividing the total revolutions per minute by 60, which gives ³⁶⁰⁄₆₀ = 6 revolutions per second. Because a single full revolution corresponds to an angle of 2π radians, the total angle in radians turned by the wheel in one second is 6 × 2π = 12π radians.

Explanation

The expression can be simplified using basic trigonometric identities and complementary angle relations. First, applying the identity cot(90° - θ) = tanθ, the numerator becomes sec²θ - tan²θ. According to the identity sec² A - tan² A = 1, the numerator equals 1.

Next, consider the denominator csc² 67° - tan² 23°. Since 23° and 67° are complementary angles, tan 23° = tan(90° - 67°) = cot 67°. Substituting this, the denominator becomes csc² 67° - cot² 67°, which equals 1 based on the identity csc² A - cot² A = 1. Thus, the value of the fraction is 1/1 = 1.

Finally, simplify the remaining terms sin² 40° + sin² 50°. Using the relation sin(90° - A) = cos A, we find sin 50° = cos 40°. Substituting this gives sin² 40° + cos² 40°, which equals 1 per the Pythagorean identity sin² A + cos² A = 1. Adding the simplified parts together results in 1 + 1 = 2.

Explanation

In any triangle ABC with sides a, b, c and perimeter P = a + b + c, let O be a point inside the triangle and S = OA + OB + OC. A fundamental property of triangles states that if a point O lies inside △ ABC, then OA + OB < AC + BC. This is derived by extending AO to meet BC at a point D and applying the triangle inequality to △ ABD and △ ODC. Applying this principle cyclically for all pairs of vertices: OA + OB < a + b OB + OC < b + c OC + OA < c + a

Summing these three inequalities gives 2(OA + OB + OC) < 2(a + b + c). Dividing both sides by 2 results in OA + OB + OC < a + b + c, which simplifies to S < P. Therefore, the sum of the distances from an interior point to the angular points is always less than the perimeter of the triangle.

Explanation

Let the common tangent at the point of contact P intersect the direct common tangent QR at point S. According to the property that tangents drawn from an external point to a circle are equal in length, for the first circle, SQ = SP. Similarly, for the second circle, SR = SP. Combining these gives SQ = SR = SP.

In △ SQP, since SQ = SP, the angles opposite these sides are equal, so ∠ SQP = ∠ SPQ. In △ SRP, since SR = SP, it follows that ∠ SRP = ∠ SPR. Consider △ QPR, where the sum of the interior angles must be 180°: ∠ PQR + ∠ PRQ + ∠ QPR = 180°

Since point S lies on the line segment QR, ∠ PQR is the same as ∠ SQP and ∠ PRQ is the same as ∠ SRP. Substituting the equal angle relations derived from the isosceles triangles: ∠ SPQ + ∠ SPR + ∠ QPR = 180°

Observing that ∠ SPQ + ∠ SPR = ∠ QPR, the equation becomes: ∠ QPR + ∠ QPR = 180° 2∠ QPR = 180° ∠ QPR = 90°

Explanation

In the right-angled triangle ABD, the angle ∠ ADB is 90° because AD is the perpendicular from A to BC. Given that ∠ ABD = 60° and the hypotenuse AB = 12 cm, the length of the base BD can be calculated using the cosine ratio: BD = AB cos 60°. Substituting the values, BD = 12 × ½ = 6 cm.

The segment BE is the bisector of ∠ ABC, and since D lies on the line segment BC, BE is also the internal angle bisector of ∠ ABD in triangle ABD. According to the Angle Bisector Theorem, the internal bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. Therefore, in triangle ABD, the point E on AD satisfies the ratio AE / ED = AB / BD. By substituting the lengths of the sides, we get AE / ED = 12 / 6 = 2 / 1.

Explanation

To find the ratio a:d when given a series of intermediate ratios, we use the property of compounding ratios where the product of the individual ratios yields the ratio of the first term to the last term:

(a)/(d) = (a)/(b) × (b)/(c) × (c)/(d)

Substituting the given values into the equation:

(a)/(d) = ⅔ × ⅘ × ⁶⁄₇

Multiplying the numerators together and the denominators together:

(a)/(d) = (2 × 4 × 6)/(3 × 5 × 7) = ⁴⁸⁄₁₀₅

To simplify the resulting ratio, we divide both the numerator and the denominator by their common factor of 3:

(48 ÷ 3)/(105 ÷ 3) = ¹⁶⁄₃₅

Therefore, the ratio a:d is 16:35.

Explanation

The expression is solved by evaluating the nested roots and applying the given multiplier. The inner term is interpreted as the cube root of 0.000729, which is then multiplied by 3 before taking the final square root.

First, convert the decimal to scientific notation to simplify the root calculation. The value 0.000729 can be written as 729 × 10⁻⁶. Since 729 = 9³, the cube root is calculated as follows: ³√(0.000729) = (729 × 10⁻⁶)¹^/³ = 9 × 10⁻² = 0.09

Next, multiply this result by the coefficient 3 found within the outer square root: 3 × 0.09 = 0.27

Finally, calculate the square root of 0.27: √(0.27) = √(²⁷⁄₁₀₀) = √(9 × 3){10} = 3√3{10}

Using the standard approximation √3 ≈ 1.732: (3 × 1.732)/(10) = (5.196)/(10) = 0.5196

Rounding to two decimal places, the final value is approximately 0.52.

Explanation

The average of x and 1/x is given by the formula (x + 1/x)/(2) = 1. Multiplying both sides by 2, we obtain the equation x + ¹⁄ₓ = 2.

To solve for x, we multiply the entire equation by x to get x² + 1 = 2x. Rearranging the terms leads to the quadratic equation x² - 2x + 1 = 0, which is the expansion of the perfect square (x - 1)² = 0. This implies that x = 1.

Now, substitute x = 1 into the required expression x¹⁰ + (1)/(x¹⁰⁾. We get 1¹⁰ + (1)/(1¹⁰⁾, which simplifies to 1 + 1 = 2. In algebra, if x + ¹⁄ₓ = 2, then xⁿ + (1)/(xⁿ⁾ = 2 for all positive integer values of n because x is always equal to 1.

Explanation

The given operation is defined as a Θ b = a²b{3}. To find the value of 2 Θ {3 Θ (-1)}, first evaluate the expression within the braces. For 3 Θ (-1), substitute a = 3 and b = -1 into the formula:

3 Θ (-1) = 3² × (-1){3} = (9 × (-1))/(3) = -3

Now substitute this result back into the main expression to evaluate 2 Θ (-3). Using a = 2 and b = -3:

2 Θ (-3) = 2² × (-3){3} = (4 × (-3))/(3) = -4

Explanation

In a circle, the perpendicular from the center to a chord bisects that chord. Since OC is perpendicular to AB, C is the midpoint of AB, implying AB = 2CB. In the right-angled triangle OCB, OB represents the radius of the circle, given as 7 cm. Applying the Pythagorean theorem, OC² + CB² = OB². Substituting the given condition OC = CB, we have CB² + CB² = 7², leading to 2CB² = 49. This gives CB = (7)/(√(2)) cm. Therefore, the length of the chord AB is 2 × (7)/(√(2)), which simplifies to 7√2 cm.

Explanation

In the right-angled triangle ABC with ∠ B = 90°, the area is calculated as Area = ½ × AB × BC. Substituting the given values, Area of Δ ABC = ½ × 6 × 8 = 24 sq. cm. According to the midpoint theorem and properties of triangles, the four triangles formed by joining the midpoints D, E, F of the sides of a triangle are congruent and divide the original triangle into four equal areas. Therefore, the area of the triangle formed by the midpoints Δ DEF is exactly one-fourth of the area of the original triangle ABC. Calculating this gives Area (Δ DEF) = ¼ × Area(Δ ABC) = ¼ × 24 = 6 sq. cm.

Explanation

The sum of the internal angles of a triangle is always 180°. Let the measure of angle A be x. Based on the given conditions, angle B is 1.5x and angle C is 2.5x. Summing these values gives:

A + B + C = 180° x + 1.5x + 2.5x = 180° 5x = 180°

Solving for x: x = (180°)/(5) = 36°

To find the angle at B, we substitute the value of x into the expression for B: B = 1.5 × 36° = 54°

Explanation

The line L₁ bisects the 1st quadrant, which occupies the region from 0° to 90° relative to the positive x-axis. Therefore, the angle of inclination ₁ of line L₁ is 45°.

The line L₂ trisects the 2nd quadrant. The 2nd quadrant occupies the region from 90° to 180°. Trisecting this 90° range means dividing it into three equal parts of 30° each. Since L₂ is specified to be closer to the y-axis, it makes an angle of 30° with the y-axis (90° position). Thus, the angle of inclination ₂ of line L₂ with the positive x-axis is 90° + 30° = 120°.

The angle θ between two lines in the xy-plane is calculated as the absolute difference between their inclinations: θ = |₂ - ₁|. Substituting the values, we have θ = |120° - 45°| = 75°. Since 75° is less than 90°, it is the acute angle between L₁ and L₂.

Explanation

In a regular hexagon ABCDEF with side length s, all interior angles are equal to 120°. To determine the nature of Δ ACE, we calculate the lengths of its sides AC, CE, and EA. In Δ ABC, sides AB = BC = s and ∠ ABC = 120°. Applying the Law of Cosines, AC² = AB² + BC² - 2(AB)(BC) cos(120°). Substituting the values, AC² = s² + s² - 2s²(-1/2) = 3s², which gives AC = s√3. Due to the rotational symmetry of a regular hexagon, the triangles CDE and EFA are congruent to Δ ABC. Consequently, the lengths of the other two sides are also CE = s√3 and EA = s√3. Since all three sides AC, CE, and EA are of equal length, Δ ACE is an equilateral triangle.

Explanation

In a circle, the angle subtended by an arc at the center is double the angle subtended by it at any point on the circumference. For the arc CD, the angle at the center is ∠ COD = 120°. Therefore, the angle subtended by the same arc at the vertex A is ∠ CAD = ½ ∠ COD = ½ × 120° = 60°.

The total angle at vertex A of the quadrilateral is the sum of ∠ BAC and ∠ CAD. Given ∠ BAC = 30°, we have ∠ BAD = ∠ BAC + ∠ CAD = 30° + 60° = 90°.

Since ABCD is a cyclic quadrilateral, the sum of its opposite angles is 180°. Thus, ∠ BAD + ∠ BCD = 180°. Substituting the value of ∠ BAD, we get 90° + ∠ BCD = 180°, which simplifies to ∠ BCD = 90°.

Explanation

Let the total investment in the company be X. P owns 2/3 of the shares, so the remaining 1/3 is divided equally between Q and R. Therefore, R's share of the total investment is 1/2 × 1/3 = 1/6.

The increase in profit rate is 7% - 5% = 2%. It is given that P earns an extra Rs. 800 due to this increase. The extra earnings are calculated on P's portion of the investment.

P's extra earnings = Investment of P × increase in profit rate

800 = ⅔X × ²⁄₁₀₀

800 = (4X)/(300)

X = (800 × 300)/(4) = 60,000

The total investment of the company is Rs. 60,000. To find the investment of R, we multiply the total investment by R's share:

R's investment = ⅙ × 60,000 = 10,000

Explanation

When interest is compounded annually, the effective rate for 2 years at 20% is calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀. Thus, 20 + 20 + (20 × 20)/(100) = 44%.

When interest is compounded half-yearly, the rate becomes 10% per period and there are 4 periods in 2 years. The amount multiplier is (1 + ¹⁰⁄₁₀₀)⁴ = 1.1⁴ = 1.4641. The effective interest rate is (1.4641 - 1) × 100 = 46.41%.

The difference between the two effective rates is 46.41% - 44% = 2.41%. According to the question, this difference is equal to Rs. 482.

2.41% of P = 482 (2.41)/(100) × P = 482 P = (482 × 100)/(2.41) P = (48200)/(2.41) = 20,000

Explanation

The fraction of the journey covered by rail and car is ²⁄₁₅ + ⁹⁄₂₀ = ⁸⁺²⁷⁄₆₀ = ³⁵⁄₆₀ = ⁷⁄₁₂. This implies that out of 12 total units of distance, 7 units are covered by rail and car, leaving a remainder of 12 - 7 = 5 units. The remaining distance covered on foot is given as 10 km. Therefore, 5 units correspond to 10 km, which means 1 unit equals 2 km. The total journey is represented by 12 units, which equals 12 × 2 = 24 km.