CDS II 2011 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The LCM of 5, 6, 7, and 8 is 840. To find the least number that leaves a remainder of 3 when divided by these numbers, we add the remainder to the LCM: 840 + 3 = 843.

Explanation

Let the numbers be 3x and 5x. According to the problem, (3x - 9) / (5x - 9) = 12 / 23. Cross-multiplying gives 23(3x - 9) = 12(5x - 9), which simplifies to 69x - 207 = 60x - 108. Solving for x: 9x = 99, so x = 11. The second number is 5x = 5 * 11 = 55.

Explanation

To find the Highest Common Factor (HCF) of polynomials, we identify the highest degree polynomial that divides each given expression. A fast way to check for common factors is using the Factor Theorem, which states that if P(a) = 0, then (x-a) is a factor of P(x).

We test x=1 for all three polynomials: For P₁(x) = x⁴ - 3x + 2, P₁(1) = 1⁴ - 3(1) + 2 = 1 - 3 + 2 = 0. For P₂(x) = x³ - 3x² + 3x - 1, P₂(1) = 1³ - 3(1)² + 3(1) - 1 = 1 - 3 + 3 - 1 = 0. For P₃(x) = x⁴ - 1, P₃(1) = 1⁴ - 1 = 0.

Since x=1 is a root for all three, (x-1) is a common factor. To check if a higher degree factor like (x² - 1) is the HCF, (x+1) must also be a common factor. Testing x=-1 in P₁(x): P₁(-1) = (-1)⁴ - 3(-1) + 2 = 1 + 3 + 2 = 6. Since P₁(-1) ≠ 0, (x+1) is not a factor of P₁(x), meaning (x² - 1) cannot be the HCF.

Additionally, P₂(x) is the binomial expansion of (x-1)³ and P₃(x) factors as (x-1)(x+1)(x²+1). The only common factor shared by all three polynomials is (x-1). Thus, the HCF is (x-1), which corresponds to option (a).

Explanation

Given the equalities (5x - 7y + 10)/1 = (3x + 2y + 1)/8 = (11x + 4y - 10)/9, we can solve for x and y by forming a system of linear equations. Equating the first two ratios gives 8(5x - 7y + 10) = 3x + 2y + 1, which simplifies to 40x - 56y + 80 = 3x + 2y + 1. Rearranging the terms, we get the first equation 37x - 58y = -79.

Next, we equate the second and third ratios: 9(3x + 2y + 1) = 8(11x + 4y - 10). This simplifies to 27x + 18y + 9 = 88x + 32y - 80. Rearranging these terms yields the second equation 61x + 14y = 89.

To solve for x, we can express y from the second equation as 14y = 89 - 61x or y = (89 - 61x)/14. Substituting this into the first equation: 37x - 58((89 - 61x)/14) = -79 37x - 29(89 - 61x)/7 = -79 Multiplying the entire equation by 7 to clear the fraction: 259x - 2581 + 1769x = -553 2028x = 2028 x = 1

Substituting x = 1 back into 61x + 14y = 89: 61(1) + 14y = 89 14y = 28 y = 2

The value of x + y is 1 + 2 = 3, which corresponds to option c.

Explanation

In simple interest, the interest earned remains constant every year. Let the principal be P and the interest for one year be I. The amount after n years is given by the formula A = P + (n × I).

Given that the amount after 8 years is Rs. 1620 and the amount after 5 years is Rs. 1350, the difference between these two amounts represents the simple interest for 3 years. Interest for 3 years = 1620 - 1350 = 270. Interest for 1 year I = 270/3 = 90.

To find the principal P, we subtract the simple interest for 5 years from the total amount after 5 years: Interest for 5 years = 90 × 5 = 450. Principal P = 1350 - 450 = 900.

Thus, the sum is Rs. 900, which corresponds to option c.

Explanation

In geometry, through any three non-collinear points, there exists one and only one unique circle, known as the circumcircle of the triangle formed by those three points.

Explanation

The total wetted surface area of a cylindrical tank filled with water consists of the area of the circular base and the curved surface area of the cylinder up to the depth of the water. The formula for the total wetted area is given by A = πr² + 2πrh, where r is the radius and h is the depth of the water.

Given the diameter d = 7m, the radius r = d/2 = 7/2 = 3.5m. The depth of the water h = 4m. Using π ≈ 22/7, we calculate the areas as follows:

Area of the base = πr² = 22/7 × (3.5)² = 22/7 × 12.25 = 38.5m². Curved surface area = 2πrh = 2 × 22/7 × 3.5 × 4 = 2 × 22 × 0.5 × 4 = 88m².

Adding these together, the total wetted surface area is 38.5 + 88 = 126.5m². This corresponds to option (b).

Explanation

To find the total cost of painting the wall, we first need to calculate the area of the trapezium-shaped wall using the standard area formula: Area = 1/2 × (sum of parallel sides) × height. Given the parallel sides as a = 3m and b = 5m, and the height as h = 4m, we substitute these values into the formula: Area = 1/2 × (3 + 5) × 4. This simplifies to Area = 1/2 × 8 × 4, which equals 4 × 4 = 16 sq m.

The total cost is then calculated by multiplying the area of the wall by the given rate of painting per square metre. Using the formula Total Cost = Area × Rate, we get Total Cost = 16 × 25. To solve this quickly, 16 × 25 can be viewed as 4 × (4 × 25) = 4 × 100 = 400. Therefore, the cost of painting the wall is Rs. 400, which corresponds to option (b).

Explanation

To find the area of the path surrounding the lawn, we use the formula for the area of a path surrounding a rectangular plot: Area of path = (Outer Length × Outer Width) - (Inner Length × Inner Width). The inner dimensions are given as length L = 30 m and width W = 16 m. Since the path of width w = 2 m surrounds the lawn on all sides, the outer length becomes L + 2w = 30 + 2(2) = 34 m and the outer width becomes W + 2w = 16 + 2(2) = 20 m. Substituting these values into the formula, we get Area of path = (34 × 20) - (30 × 16) = 680 - 480 = 200 m². Alternatively, using the shortcut formula Area = 2w(L + W + 2w), we calculate 2(2)(30 + 16 + 4) = 4(50) = 200 m². Thus, option (a) is correct.

Explanation

Height of incomplete pillar (h1) = 100 tan(45°) = 100 1 = 100m. Height of complete pillar (h2) = 100 tan(60°) = 100 √3. The increase required is h2 - h1 = 100√3 - 100 = 100(√3 - 1)m.

Explanation

For a system of linear equations in two variables a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the nature of the solution is determined by the ratios of their coefficients. The standard condition for infinitely many solutions is a₁/a₂ = b₁/b₂ = c₁/c₂, which indicates that the two lines are coincident (overlap perfectly).

Given the equations x + 2y = 3 and 3x + 6y = 9, we identify the coefficients as a₁ = 1, b₁ = 2, c₁ = 3 and a₂ = 3, b₂ = 6, c₂ = 9. Substituting these values into the ratio formula:

a₁/a₂ = 1/3 b₁/b₂ = 2/6 = 1/3 c₁/c₂ = 3/9 = 1/3

Since 1/3 = 1/3 = 1/3, the condition a₁/a₂ = b₁/b₂ = c₁/c₂ is satisfied. This implies that the second equation is simply a multiple of the first equation (multiplied by 3), meaning they represent the same straight line. Therefore, every point on the line is a solution, resulting in infinitely many solutions as stated in option c.

Explanation

The volume of a hollow sphere is given by the formula Vₛ = 4/3π(R³ - r³), where R is the external radius and r is the internal radius. Given the external diameter is 8 cm and the internal diameter is 4 cm, the radii are R = 4 cm and r = 2 cm. Substituting these values, the volume of the hollow sphere is Vₛ = 4/3π(4³ - 2³) = 4/3π(64 - 8) = 4/3π(56) = 224π/3 cm³.

The volume of a cone is given by the formula V꜀ = 1/3πr꜀²h. Given the base diameter of the cone is 8 cm, its radius r꜀ = 4 cm. Thus, V꜀ = 1/3π(4²)h = 16πh/3 cm³. Since the sphere is melted to form the cone, their volumes are equal: 224π/3 = 16πh/3. Canceling π/3 from both sides, we get 224 = 16h, which simplifies to h = 224/16 = 14 cm. Therefore, the height of the cone is 14 cm, which corresponds to option b.

Explanation

Let the digits be x and y. The number is 10x + y and the sum of digits x + y = 8. The reversed number is 10y + x. The difference is (10x + y) - (10y + x) = 18, which simplifies to 9x - 9y = 18, or x - y = 2. The difference between the digits is 2.

Explanation

To find the gain percent, we use the formula: Profit Percentage = (SP - CP)/CP × 100. To calculate this efficiently, we should find the cost price (CP) and selling price (SP) for a common quantity of lemons. The quantities given are 2 lemons and 5 lemons, so we take the LCM of 2 and 5, which is 10.

Assume the fruit-seller buys and sells 10 lemons. CP of 10 lemons = 10/2 × 1 = 5 rupees. SP of 10 lemons = 10/5 × 3 = 2 × 3 = 6 rupees.

Now, calculate the profit: Profit = SP - CP = 6 - 5 = 1 rupee. Gain Percent = 1/5 × 100 = 20%.

Alternatively, using the unitary method, the CP of one lemon is 1/2 = 0.50 rupees and the SP of one lemon is 3/5 = 0.60 rupees. The profit per lemon is 0.60 - 0.50 = 0.10 rupees. The gain percentage is 0.10/0.50 × 100 = 20%. This matches option (c).

Explanation

For any transaction where two articles are sold at the same selling price, one at a profit of x% and the other at a loss of x%, there is always an overall loss. The standard shortcut formula to calculate this net loss percentage is x²/100%. In this specific problem, the value of x is 10. Substituting this value into the formula, we get 10²/100 = 100/100 = 1%. Therefore, the man suffers a net loss of 1% in the overall transaction, which corresponds to option b. This formula-based approach is the most efficient way to solve such problems in competitive exams as it eliminates the need to calculate the individual cost prices of the watches. The given selling price of Rs. 495 is a constant and does not affect the final percentage of loss.

Explanation

If 6 ≤ x ≤ 8, then (x - 6) is non-negative (≥ 0) and (x - 8) is non-positive (≤ 0). The product of a non-negative number and a non-positive number is always non-positive. Therefore, (x - 6)(x - 8) ≤ 0.

Explanation

The square root of 1780 is approximately 42.19. The next perfect square is 43^2 = 1849. The number that must be added is 1849 - 1780 = 69.

Explanation

Using the quadratic formula for a^2b^2x^2 - (a^2 + b^2)x + 1 = 0, the discriminant is (a^2 + b^2)^2 - 4(a^2b^2)(1) = (a^2 - b^2)^2. The roots are x = [(a^2 + b^2) ± (a^2 - b^2)] / (2a^2b^2). Root 1 = 2a^2 / 2a^2b^2 = 1/b^2. Root 2 = 2b^2 / 2a^2b^2 = 1/a^2.

Explanation

The volume of a cylindrical vessel is given by the formula V꜀ᵧₗᵢₙₐₑᵣ = πr²h, where r is the base radius and h is the height. Substituting the given values, V꜀ᵧₗᵢₙₐₑᵣ = π × (60)² × 10 = π × 3600 × 10 = 36000π cm³. The volume of a spherical vessel is given by the formula Vₛₚₕₑᵣₑ = 4/3 πR³, where R is the radius of the sphere. Since the volumes are equal, we set 4/3 πR³ = 36000π. Canceling π from both sides, we get 4/3 R³ = 36000. Solving for R³, we have R³ = 36000 × 3/4 = 9000 × 3 = 27000. Taking the cube root of both sides, R = ∛27000 = 30 cm. The diameter d of the spherical vessel is twice its radius, so d = 2 × 30 = 60 cm. Therefore, option b is correct.

Explanation

The surface area of a sphere is given by the formula S = 4πr². The initial surface area S₁ is 616 cm². According to the problem, the area is reduced by 75%, which means the new surface area S₂ is 25% of the original area. Therefore, S₂ = 616 × 25/100 = 616/4 = 154 cm². To find the new radius r₂, we use the formula 4πr₂² = 154. Substituting π = 22/7, we get 4 × 22/7 × r₂² = 154. Solving for r₂², we have r₂² = (154 × 7)/(4 × 22) = (7 × 7)/4 = 49/4. Taking the square root of both sides, r₂ = 7/2 = 3.5 cm, which corresponds to option (d).

Explanation

The distance of the first ship from the base is h / tan(30°) = h√3. The distance of the second ship from the base is h / tan(45°) = h. Since they are on opposite sides, the total distance between them is h√3 + h = (√3 + 1)h.

Explanation

Since (x, y) = (4, 1) is the solution to the pair of linear equations mx + y = 5 and 2x + ny = 5, these values must satisfy both equations simultaneously. We substitute x = 4 and y = 1 into the equations to find the values of the constants m and n.

For the first equation mx + y = 5: m(4) + 1 = 5 4m = 5 - 1 4m = 4 m = 1

For the second equation 2x + ny = 5: 2(4) + n(1) = 5 8 + n = 5 n = 5 - 8 n = -3

Now, we calculate the required sum m + n: m + n = 1 + (-3) m + n = 1 - 3 m + n = -2

Thus, the value of m + n is -2, which corresponds to option (a).

Explanation

The expression is in the form of a standard algebraic identity. Let a = 2.3 and b = 0.3. We observe that 0.027 = (0.3)³, and the denominator terms 0.69 and 0.09 are equal to 2.3 × 0.3 and (0.3)² respectively. Therefore, the expression can be rewritten as (a³ - b³)/(a² + ab + b²). According to the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), the expression simplifies directly to a - b. Substituting the identified values into this simplified form, we get 2.3 - 0.3 = 2. Thus, the value of the expression is 2, which corresponds to option (d).

Explanation

The earth completes one full rotation of 360° in 24 hours. To find the angle turned in a specific duration, we use the formula: Angle = Given Time / Total Time for Full Rotation × 360°.

First, convert the given time of 4 hours and 12 minutes into a single decimal unit of hours. Since 60 minutes make 1 hour, 12 minutes is equal to 12/60 = 0.2 hours. Thus, the total time elapsed is 4 + 0.2 = 4.2 hours.

Now, substitute the values into the formula: Angle = 4.2 / 24 × 360°

To simplify the calculation, first divide 360 by 24: 360 / 24 = 15° per hour

Then, multiply this hourly rate by the total hours elapsed: Angle = 4.2 × 15° Angle = (4 × 15) + (0.2 × 15) Angle = 60° + 3° = 63°

Therefore, the earth turns through an angle of 63° in 4 hours and 12 minutes, which corresponds to option a.

Explanation

Let the monthly gross pay of the employee be X. According to the problem, the contribution to the General Provident Fund (GPF) is 10% of the gross pay, which is 0.10X. The net pay is the amount remaining after this deduction, expressed by the formula: Net Pay = Gross Pay - GPF Contribution. Substituting the given values, we get 13,500 = X - 0.10X, which simplifies to 13,500 = 0.90X. To find the gross pay, we calculate X = 13,500/0.90 = 15,000. The monthly GPF contribution is therefore 10% of 15,000, which equals 0.10 × 15,000 = 1,500. Alternatively, using the shortcut ratio method, if the deduction is 10%, the net pay represents 90% of the total. Since 90% corresponds to Rs. 13,500, then 10% corresponds to 13,500/9 = 1,500. Thus, option c is the correct choice.

Explanation

To solve this problem, first convert the fractional ratio into a whole number ratio by multiplying each term by the Least Common Multiple (LCM) of the denominators 5, 6, and 10. The LCM of 5, 6, and 10 is 30.

The simplified ratio A : B : C is calculated as: A = 1/5 × 30 = 6 B = 1/6 × 30 = 5 C = 1/10 × 30 = 3

Thus, the ratio is 6 : 5 : 3. The sum of the ratio parts is 6 + 5 + 3 = 14.

The share of A is determined by the formula: Share of A = (Ratio of A/Sum of ratios) × Total Amount Share of A = 6/14 × 8400

Simplifying the fraction 6/14 gives 3/7. Share of A = 3/7 × 8400 = 3 × 1200 = 3600.

Therefore, the share of A is Rs. 3600, which corresponds to option c.

Explanation

First, simplify the given fraction-based ratios into whole number ratios. For P:Q = 3/5 : 5/7, multiply both terms by the LCM of the denominators (35) to get P:Q = 21:25. For Q:R = 3/4 : 2/5, multiply both terms by the LCM of the denominators (20) to get Q:R = 15:8.

To find the combined ratio P:Q:R, we must equalize the value of Q in both ratios. The LCM of 25 and 15 is 75. Multiplying the ratio P:Q by 3 gives P:Q = (21 × 3) : (25 × 3) = 63:75. Multiplying the ratio Q:R by 5 gives Q:R = (15 × 5) : (8 × 5) = 75:40. Therefore, the combined ratio is P:Q:R = 63:75:40.

To verify if option (b) matches this result, simplify the ratio 9/20 : 15/28 : 2/7 by multiplying each term by the LCM of 20, 28, and 7, which is 140: P = 9/20 × 140 = 9 × 7 = 63 Q = 15/28 × 140 = 15 × 5 = 75 R = 2/7 × 140 = 2 × 20 = 40 This results in the ratio 63:75:40, which perfectly matches our derived value. Thus, option (b) is correct.

Explanation

Using the man-days formula (M1 D1) / W1 = (M2 D2) / W2: (10 8) / 20 = (M2 10) / 100. This simplifies to 4 = M2 / 10, so M2 = 40.

Explanation

To find the time taken by the train to cross the platform, we use the formula Time = Total Distance/Speed. When a train crosses a platform, the total distance to be covered is the sum of the length of the train and the length of the platform. Therefore, Total Distance = 280 m + 220 m = 500 m.

The speed of the train is given in kmph, which must be converted to m/s to maintain consistency with the distance in meters. The conversion factor is 5/18. Thus, Speed = 60 × 5/18 = (10 × 5)/3 = 50/3 m/s.

Substituting these values into the time formula: Time = 500/(50/3) = (500 × 3)/50 = 10 × 3 = 30 s. Comparing this result with the options, we find that option d is the correct answer.

Explanation

Given the condition x + y + z = 0, we can express the binomial terms in the denominator in terms of the single remaining variable. From the given equation x + y + z = 0, we derive the following substitutions: x + y = -z y + z = -x z + x = -y

Now, substitute these derived values into the denominator of the expression xyz/((x + y)(y + z)(z + x)). The denominator becomes: (x + y)(y + z)(z + x) = (-z)(-x)(-y)

Multiplying the three negative terms together, the result is negative because the product of an odd number of negative signs is negative: (-z)(-x)(-y) = -(xyz)

Substituting this result back into the full expression, we obtain: xyz/-(xyz)

By canceling xyz from the numerator and the denominator, we get: -1

Therefore, the value of the expression is -1, which matches option a.

Explanation

By the Remainder Theorem, when a polynomial P(x) is divided by (x + 1), the remainder is P(-1). P(-1) = (-1)^11 + 1 = -1 + 1 = 0.

Explanation

According to the Factor Theorem, if (x - a) is a factor of a polynomial f(x), then f(a) = 0. In this problem, the given polynomial is f(x) = x² + 4px - 11p and the factor is (x - 3). Therefore, substituting x = 3 into the polynomial must result in zero.

Applying the formula: f(3) = (3)² + 4p(3) - 11p = 0 9 + 12p - 11p = 0 9 + p = 0 p = -9

The value of p is calculated to be -9, which matches option (a). Using the Factor Theorem is the most efficient shortcut for such problems as it avoids the time-consuming process of polynomial long division or synthetic division.

Explanation

Factorizing the terms: (a^3b - ab^3) = ab(a^2 - b^2) = ab(a - b)(a + b); (a^3b^2 + a^2b^3) = a^2b^2(a + b); ab(a + b) = ab(a + b). The LCM is the product of the highest powers of all factors: a^2 b^2 (a + b) * (a - b) = a^2b^2(a^2 - b^2).

Explanation

When a circle circumscribes a rectangle, the diagonal of the rectangle is equal to the diameter of the circle. To find the length of the diagonal d of a rectangle with length l = 16 cm and width w = 12 cm, we use the Pythagorean theorem: d = √(l² + w²) = √(16² + 12²) = √(256 + 144) = √400 = 20 cm. Since the diameter of the circle is 20 cm, the radius r is 20/2 = 10 cm. The area of the circle is calculated using the standard formula A = πr². Substituting the value of the radius, we get A = π × (10)² = 100π square cm. This matches option c).

Explanation

The altitude of an equilateral triangle is (√3/2) side. So, 2√3 = (√3/2)s, which means s = 4. The area is (√3/4) s^2 = (√3/4) * 16 = 4√3 square cm.

Explanation

The point where the three altitudes of a triangle intersect is known as the orthocenter. An altitude of a triangle is the perpendicular line segment drawn from a vertex to the opposite side or the line containing the opposite side. Since the three altitudes are concurrent, they all meet at a single point, which is the orthocenter. The position of the orthocenter depends on the type of triangle: in an acute-angled triangle, it lies inside the triangle; in a right-angled triangle, it coincides with the vertex where the right angle is located; and in an obtuse-angled triangle, it lies outside the triangle. This point is distinct from the centroid, which is the concurrence of medians, the in-center, which is the concurrence of angle bisectors, and the circum-center, which is the concurrence of perpendicular bisectors of the sides.

Explanation

sinθ/cosθ + cosθ/sinθ = tanθ + cotθ = 2. Let tanθ = t. Then t + 1/t = 2, which simplifies to t^2 - 2t + 1 = 0, or (t - 1)^2 = 0. Thus tanθ = 1, and θ = 45°.

Explanation

Classification of data refers to the process of arranging raw data into groups or classes based on specific characteristics. When data is organized according to geographical locations, such as countries, states, districts, or cities, it is termed as geographical or spatial classification. In this specific instance, the population figures are being grouped and presented according to the various States of India. Since the distinguishing factor used to categorize the data is the spatial entity (the state), the classification is geographical. While the population figures themselves are quantitative in nature, the method of classification depends on the basis of grouping, which in this case is the geography of India. Therefore, option d is the most appropriate classification for this dataset.

Explanation

Assume the starting position of the residence is at the origin (0, 0) on a Cartesian plane. The student first moves √2x km east, reaching the point (√2x, 0). Next, he moves x km north, bringing his position to (√2x, x). The third movement is x km north-east. Since north-east is at an angle of 45° to the east axis, the displacement components are x cos 45° east and x sin 45° north. Using cos 45° = sin 45° = 1/√2, the new position becomes (√2x + x/√2, x + x/√2). Finally, a 90° right turn from north-east points him south-east. Moving x km south-east adds x cos 45° to the x-coordinate and subtracts x sin 45° from the y-coordinate. The final coordinates of the school are (√2x + x/√2 + x/√2, x + x/√2 - x/√2), which simplifies to (2√2x, x). The shortest distance from the residence (0, 0) to the school is calculated using the distance formula √((x₂ - x₁)² + (y₂ - y₁)²), which gives √((2√2x)² + x²) = √(8x² + x²) = √(9x²) = 3x. Thus, option b is correct.

Explanation

The total area is the sum of the area of the right angled triangle and the areas of the three squares constructed on its sides. For a right angled triangle with sides a and b containing the right angle, the area of the triangle is given by the formula Areaₜᵣᵢₐₙ₉ₗₑ = 1/2 ab = 0.5 ab. The areas of the squares on sides a and b are a² and b² respectively. According to the Pythagorean theorem, the square of the hypotenuse c is c² = a² + b², which also represents the area of the square drawn on the hypotenuse. Summing these values, the total area is a² + b² + (a² + b²) + 0.5 ab, which simplifies to 2a² + 2b² + 0.5 ab. Factoring out 2 from the first two terms, we get 2(a² + b²) + 0.5 ab, which corresponds to option (c).

Explanation

The question as written contains typos, but based on the provided answer key and the structure of the options (which appear to be multiplication expressions like 2 * 3100191), Option A is the designated correct answer.

Explanation

Let the two numbers be x and y. According to the problem, the sum of the numbers is x + y = 10 and their product is xy = 20. We are required to find the sum of their reciprocals, which is represented by the expression 1/x + 1/y. By combining the fractions using a common denominator, we utilize the standard algebraic identity 1/x + 1/y = (x + y)/xy. By substituting the given sum and product into this formula, we obtain 10/20. Simplifying this fraction by dividing both the numerator and the denominator by 10 gives 1/2, which corresponds to option (b). This shortcut avoids the need to solve for the individual values of x and y, making it the most efficient method for competitive exams.

Explanation

To solve this division of rational expressions, we use the rule that dividing by a fraction is equivalent to multiplying by its reciprocal. The expression becomes (x² - 3x + 2)/(x² - 5x + 6) × (x² - 7x + 12)/(x² - 5x + 4). Next, we factorize each quadratic trinomial using the formula x² + (a+b)x + ab = (x+a)(x+b).

For the first term, x² - 3x + 2 factors into (x-1)(x-2) and x² - 5x + 6 factors into (x-2)(x-3). For the second term, x² - 7x + 12 factors into (x-3)(x-4) and x² - 5x + 4 factors into (x-1)(x-4). Substituting these factors back into the expression, we get:

((x-1)(x-2))/((x-2)(x-3)) × ((x-3)(x-4))/((x-1)(x-4))

By cancelling out the common terms in the numerator and the denominator, we see that (x-1) cancels with (x-1), (x-2) cancels with (x-2), (x-3) cancels with (x-3), and (x-4) cancels with (x-4). Since all terms cancel out, the final result is 1, which corresponds to option b.

Explanation

The total surface area of the remaining solid consists of three distinct parts: the curved surface area of the cylinder, the area of the remaining circular base, and the internal curved surface area of the conical cavity.

Using the given dimensions r = 3 cm and h = 4 cm, we apply the following formulas: Curved surface area of the cylinder = 2πrh = 2π(3)(4) = 24π sq cm. Area of the circular base = πr² = π(3)² = 9π sq cm. Curved surface area of the cone = πrl, where l is the slant height.

First, calculate the slant height l using the Pythagorean theorem: l = √(r² + h²) = √(3² + 4²) = √(9 + 16) = √25 = 5 cm. Now, the curved surface area of the conical cavity = π(3)(5) = 15π sq cm.

The total surface area is the sum of these three components: Total Surface Area = 24π + 9π + 15π = 48π sq cm.

Thus, option (d) is the correct result.

Explanation

Given that A = π/6 and B = π/3, we observe that A + B = π/6 + π/3 = π/2. This means A and B are complementary angles. Using the complementary angle identities, we know that sin A = cos B, cos A = sin B, tan A = cot B, and cot A = tan B.

For Statement 1, the left hand side is sin A + sin B. Substituting the identity sin A = cos B and sin B = cos A, we get cos B + cos A, which is exactly the right hand side cos A + cos B. Numerically, sin(π/6) + sin(π/3) = 1/2 + √3/2 and cos(π/6) + cos(π/3) = √3/2 + 1/2. Both sides equal (1 + √3)/2.

For Statement 2, the left hand side is tan A + tan B. Substituting the identities tan A = cot B and tan B = cot A, we get cot B + cot A, which is exactly the right hand side cot A + cot B. Numerically, tan(π/6) + tan(π/3) = 1/√3 + √3 = 4/√3 and cot(π/6) + cot(π/3) = √3 + 1/√3 = 4/√3.

Since both statements are mathematically identical due to complementary angle properties, option c is correct.

Explanation

Using the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca): 6^2 = 26 + 2(ab + bc + ca). 36 - 26 = 2(ab + bc + ca), so 10 = 2(ab + bc + ca), giving ab + bc + ca = 5.

Explanation

The largest four-digit number is 9999. Dividing 9999 by 88 gives 113 with a remainder of 55. The largest divisible number is 9999 - 55 = 9944.

Explanation

The most efficient way to solve time and work problems is by using the LCM method to determine total work units. Let the total work be the LCM of 8, 10, and 15, which is 120 units. The daily efficiencies of A, B, and C are calculated as follows: efficiency of A = 120/8 = 15 units/day, efficiency of B = 120/10 = 12 units/day, and efficiency of C = 120/15 = 8 units/day.

A and B work together for the first 2 days. The work completed in this period is 2 × (15 + 12) = 2 × 27 = 54 units. The remaining work is 120 - 54 = 66 units. After A quits, B and C work together to finish the remaining work. Their combined efficiency is 12 + 8 = 20 units/day. The time taken by B and C to complete the remaining 66 units is 66/20 = 33/10 days.

The total time taken to complete the work is the sum of the initial 2 days and the time spent by B and C: Total days = 2 + 33/10 = (20 + 33)/10 = 53/10 days. Thus, option d is correct.

Explanation

The bucket is in the shape of a frustum of a cone. The volume V of a frustum of a cone is given by the formula: V = 1/3 πh (R² + r² + Rr) Given the height h = 40 cm, the radius of the larger end R = 35 cm, and the radius of the smaller end r = 14 cm, we substitute these values into the formula using π = 22/7: V = 1/3 × 22/7 × 40 × (35² + 14² + 35 × 14) Calculating the terms inside the bracket: 35² = 1225 14² = 196 35 × 14 = 490 Sum = 1225 + 196 + 490 = 1911 Now, substitute the sum back into the volume equation: V = 1/3 × 22/7 × 40 × 1911 Since 1911 is divisible by 21 (where 21 = 3 × 7): 1911 ÷ 21 = 91 V = 22 × 40 × 91 V = 880 × 91 = 80080 cubic cm. Therefore, option (c) is the correct value.

Explanation

Let the side of the cube be denoted by a. The formula for the total surface area S of a cube is S = 6a² and the formula for its volume V is V = a³. To find the relationship between S and V, we can express a in terms of both variables. From the volume formula, we have a = V¹/³, which implies a⁶ = V². From the surface area formula, we have S = 6a². Cubing both sides of the surface area equation gives S³ = (6a²)³, which simplifies to S³ = 216a⁶. Substituting the value of a⁶ = V² into this equation, we get S³ = 216V². This directly matches option (b). Thus, the relationship is established by eliminating the side length a from the two basic geometric formulas for a cube.

Explanation

In a cyclic quadrilateral, opposite angles are supplementary (A + C = 180° and B + D = 180°). Therefore, cos C = cos(180° - A) = -cos A and cos D = cos(180° - B) = -cos B. The sum cos A + cos B + cos C + cos D = cos A + cos B - cos A - cos B = 0.

Explanation

The LCM of 2, 3, and 5 is 30. A number that leaves a remainder of 1 when divided by these must be of the form 30n + 1. For n=1, the number is 31.

Explanation

The median is the middle value of a data set when observations are arranged in increasing or decreasing order. For a set of n observations, if n is odd, the median is the value at the (n+1)/2 position. If n is even, the median is the average of the values at the n/2 and n/2 + 1 positions.

When the smallest and the largest observations are removed, the total number of observations becomes n-2. If n was odd, the new median is at the ((n-2)+1)/2 = (n-1)/2 position of the new set, which is exactly the (n+1)/2 position of the original set. If n was even, the new median is the average of the (n-2)/2 and (n-2)/2 + 1 positions of the new set, which correspond to the n/2 and n/2 + 1 positions of the original set. Thus, the median remains unchanged.

In contrast, the mean x̄ = Σxᵢ/n and standard deviation σ = √((Σ(xᵢ - x̄)²)/n) are sensitive to extreme values. Removing the minimum and maximum observations typically changes the sum and the spread of the data, thereby altering these measures. The mode may also change if the extreme values were the most frequent observations.

Explanation

For a quadratic equation of the form ax² - bx + c = 0, the sum and product of its roots are given by the formulas: Sum of roots = -(-b)/a = b/a Product of roots = c/a

Given that sin θ and cos θ are the roots of the equation, we can write: sin θ + cos θ = b/a sin θ cos θ = c/a

Using the trigonometric identity sin² θ + cos² θ = 1, we square the sum of the roots: (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ

Substituting the known values into this expression: (b/a)² = 1 + 2(c/a) b²/a² = 1 + 2c/a

Multiplying the entire equation by a² to clear the denominators: b² = a² + 2ac

Rearranging the terms to match the given options: a² - b² + 2ac = 0

This matches option b.

Explanation

The value of one unit radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. To convert radians to degrees, we use the standard formula: 1 radian = 180/π degrees.

Substituting the value of π ≈ 3.14159265, we get: 1 rad = 180/3.14159265 ≈ 57.2957795°.

To convert this decimal degree value into degrees, minutes, and seconds (DMS), we perform the following calculations: The integer part is 57°. Multiply the remaining fractional part by 60 to find the minutes: 0.2957795 × 60 = 17.74677'. The integer part is 17'. Multiply the remaining fractional part of the minutes by 60 to find the seconds: 0.74677 × 60 = 44.8062".

Rounding 44.8062" to the nearest whole number gives 45". Therefore, 1 radian is approximately equal to 57°17'45", which corresponds to option b.

Explanation

For any two positive numbers, the relationship between the arithmetic mean (AM), geometric mean (GM), and harmonic mean (HM) is given by the fundamental formula GM² = AM × HM. This relation shows that the geometric mean is the geometric mean of the arithmetic and harmonic means.

Given values are AM = 14 and GM = 12. Substituting these values into the formula:

12² = 14 × HM 144 = 14 × HM

To find the harmonic mean, divide both sides by 14:

HM = 144/14

Simplifying the fraction by dividing both the numerator and the denominator by 2:

HM = 72/7

Therefore, the harmonic mean of the two numbers is 72/7, which corresponds to option (d).

Explanation

By the property of circles, the angle subtended by a diameter at any point on the circumference is 90°. Therefore, the locus of point P such that angle APB = 90° is the circumference of the circle with AB as the diameter.

Explanation

Total work = 76 ladies * 33 days = 2508 lady-days. If the work was completed in 44 days, the number of ladies who worked is 2508 / 44 = 57. The number of ladies who did not report is 76 - 57 = 19.

Explanation

According to the Remainder Theorem, if a polynomial P(x) is divided by a linear divisor of the form (ax - b), the remainder is equal to P(b/a). In this problem, the polynomial is P(x) = 3x³ - 2x²y - 13xy² + 10y³ and the divisor is x - 2y. By treating y as a constant and setting the divisor to zero, we get x = 2y. Substituting x = 2y into the polynomial provides the remainder.

The calculation is as follows: Remainder = 3(2y)³ - 2(2y)²y - 13(2y)y² + 10y³ = 3(8y³) - 2(4y²)y - 26y³ + 10y³ = 24y³ - 8y³ - 26y³ + 10y³ = (24 - 8 - 26 + 10)y³ = (16 - 26 + 10)y³ = (-10 + 10)y³ = 0

Since the remainder is 0, it indicates that x - 2y is a factor of the given polynomial. Therefore, option a is the correct result.

Explanation

For problems where the angles of elevation from two points on the same side of a tower are complementary, the height of the tower can be calculated using the direct formula h = √(d₁ · d₂), where d₁ and d₂ are the distances of the two points from the base of the tower. This formula is derived from the fact that if one angle is θ, the other is 90° - θ, and since tan(θ) · tan(90° - θ) = tan(θ) · cot(θ) = 1, we have (h/d₁) · (h/d₂) = 1.

Substituting the given values d₁ = 36 m and d₂ = 64 m into the formula: h = √(36 · 64) h = √36 · √64 h = 6 · 8 h = 48 m.

Thus, the height of the tower is 48 m, which corresponds to option b).

Explanation

The total area of the equilateral triangle field is calculated using the formula A_field = (√3/4) a². Substituting the side length a = 6 m, we get A_field = (√3/4) × 36 = 9√3 ≈ 15.588 sq m. Since the horse is tied to one vertex, it can graze in a sector of a circle with radius r = 4.2 m and a central angle equal to the interior angle of the equilateral triangle, which is 60°. The area of this sector is given by A_grazing = (θ/360°) π r². Substituting θ = 60° and r = 4.2 m, we find A_grazing = (60/360) × π × (4.2)² = (1/6) × π × 17.64 = 2.94π ≈ 9.236 sq m. The percentage of the total area available for grazing is calculated as (A_grazing/A_field) × 100 = (9.236/15.588) × 100 ≈ 59.24%. This result is best approximated by option c) 59%.

Explanation

sqrt(0.0032 / 0.32) = sqrt(0.01) = 0.1.

Explanation

In set theory, a collection of objects is called a set only if it is well-defined. A collection is well-defined if there is an objective criterion to determine whether any specific element belongs to the collection or not. Statement 1 is incorrect because a geometric line consists of an infinite number of points, which means the set is infinite rather than finite. Statement 2 is incorrect because the term intelligent is subjective and varies from person to person; without a fixed mathematical or objective threshold, this collection is not well-defined. Similarly, statement 3 is incorrect because the word good is a relative term and lacks a standard objective measure, making the collection of good books not well-defined. Since none of the statements are correct, option d is the right choice.

Explanation

For x^2 - 6x + 6 = 0, alpha + beta = 6 and alphabeta = 6. alpha^2 + beta^2 = (alpha + beta)^2 - 2(alphabeta) = 36 - 12 = 24. alpha^3 + beta^3 = (alpha + beta)(alpha^2 + beta^2 - alphabeta) = 6(24 - 6) = 6 18 = 108. Total sum = 108 + 24 + 6 = 138.

Explanation

To solve the given equation 2ᵐ + 2¹⁺ᵐ = 24, we use the fundamental law of indices which states that aˣ⁺ʸ = aˣ · aʸ. Applying this rule to the second term of the equation, we can rewrite 2¹⁺ᵐ as 2¹ · 2ᵐ, which is equal to 2 · 2ᵐ. Substituting this back into the equation, we get 2ᵐ + 2 · 2ᵐ = 24. Combining the like terms on the left hand side, we have (1 + 2) · 2ᵐ = 24, which simplifies to 3 · 2ᵐ = 24. Dividing both sides of the equation by 3, we obtain 2ᵐ = 8. Since we know that 8 can be expressed as a power of 2, specifically 8 = 2³, we can write the equation as 2ᵐ = 2³. By comparing the exponents of the same base, we conclude that m = 3. This matches option (c).

Explanation

For any natural number n, the value of √n depends on whether n is a perfect square. If n is a perfect square, meaning n = k² for some natural number k, then √n = k, which is also a natural number. For example, if n = 4, then √4 = 2, and if n = 9, then √9 = 3. Conversely, if n is not a perfect square, then its square root √n is always an irrational number, meaning it cannot be expressed as a ratio of two integers p/q. Examples include √2, √3, √5, and √7. Since every natural number is either a perfect square or not, √n must be either a natural number or an irrational number, which aligns with option (d).

Explanation

Cumulative frequency for X = 5 is the sum of frequencies for all values of X up to 5: 2 + 7 + 14 + 22 = 45.

Explanation

For a set of numbers in an arithmetic progression, such as the set S = {1, 2, 3, …, n}, the median is always the average of the first and last terms. Applying the formula for the median of n consecutive integers: Median = (1 + n)/2. This holds true whether n is even or odd. For instance, if n=5, the median is (1+5)/2 = 3. If n=6, the median is (1+6)/2 = 3.5. Thus, statement 1 is mathematically correct.

The mode is defined as the value that appears most frequently in a data set. In the given set S, each integer from 1 to n appears exactly once. When every element in a set has the same frequency, the set is considered to have no mode. Since n does not occur more frequently than any other number in the set, it cannot be the mode. Therefore, statement 2 is incorrect. Consequently, only statement 1 is correct, making option a the right choice.

Explanation

Let the radii of the two cylinders be r₁ and r₂, and their heights be h₁ and h₂. Given the ratio of radii r₁ : r₂ = 2 : 3 and the ratio of curved surface areas CSA₁ : CSA₂ = 5 : 3.

The formula for the curved surface area of a cylinder is CSA = 2πrh. Taking the ratio: CSA₁/CSA₂ = 2πr₁h₁/2πr₂h₂ = r₁/r₂ × h₁/h₂ Substituting the known values: 5/3 = 2/3 × h₁/h₂ h₁/h₂ = 5/3 × 3/2 = 5/2

The formula for the volume of a cylinder is V = πr²h. Taking the ratio of the volumes: V₁/V₂ = πr₁²h₁/πr₂²h₂ = (r₁/r₂)² × h₁/h₂ Substituting the ratios of radii and heights: V₁/V₂ = (2/3)² × 5/2 = 4/9 × 5/2 = 20/18 = 10/9

The ratio of their volumes is 10:9, which corresponds to option b.

Explanation

Let the areas of the two circles be A_C1 and A_C2. Given that A_C1 : A_C2 = 1 : 2. Since the area of a circle is πr², the ratio of their radii r₁ : r₂ is √1 : √2, which is 1 : √2. The circumference of a circle is given by 2πr, so the ratio of their circumferences is also r₁ : r₂ = 1 : √2. When these circles are bent into squares, their circumferences become the perimeters of the squares. Let the perimeters be Pₛ₁ and Pₛ₂. Thus, Pₛ₁ : Pₛ₂ = 1 : √2. For any square, the area is Aₛ = (P/4)². Therefore, the ratio of the areas of the squares is Pₛ₁² : Pₛ₂². Substituting the values, we get 1² : (√2)², which simplifies to 1 : 2. Option a is correct.

Explanation

Let the width of the rectangle be denoted by w. According to the problem, the length l is 5 units more than twice the width, which can be written as l = 2w + 5. The formula for the area of a rectangle is Area = length × width. Substituting the given values, we get the equation 75 = (2w + 5)w, which simplifies to the quadratic equation 2w² + 5w - 75 = 0. To solve for w, we can use the factorization method: 2w² + 15w - 10w - 75 = 0, which further simplifies to w(2w + 15) - 5(2w + 15) = 0, giving (w - 5)(2w + 15) = 0. This yields w = 5 or w = -7.5. Since the width of a geometric figure cannot be negative, the width must be 5 units. Alternatively, using the option substitution method, if we test option (b) where w = 5, the length becomes 2(5) + 5 = 15. The area is then 5 × 15 = 75 square units, which matches the given condition perfectly. Thus, the width is 5 units.

Explanation

The surface area of a sphere is given by the formula S = 4πr². Since the radius r is half of the diameter d, the formula can be expressed in terms of the diameter as S = πd². According to the problem, the diameter is doubled, so the new diameter dₙₑᵥᵥ = 2d. Substituting this value into the surface area formula, the new surface area Sₙₑᵥᵥ becomes π(2d)² = 4πd². By comparing the new surface area to the original surface area, we see that Sₙₑᵥᵥ = 4 × (πd²) = 4S. Thus, the surface area increases four times, making option c the correct answer. This follows the square-cube law, where the area of a scaled object increases by the square of the scaling factor. Since the scaling factor is 2, the area increases by 2² = 4.

Explanation

To find the principal sum, we use the standard formula for the total amount under simple interest: A = P(1 + (RT)/100), where A is the amount, P is the principal, R is the rate of interest per annum, and T is the time in years. Substituting the given values, we have 364.80 = P(1 + (3.5 × 8)/100). First, calculate the product of the rate and time: 3.5 × 8 = 28. This simplifies the equation to 364.80 = P(1 + 28/100), which is 364.80 = P(1 + 0.28) or 364.80 = 1.28P. To isolate P, we divide the total amount by 1.28: P = 364.80/1.28. Performing the division, 36480 ÷ 128 = 285. Therefore, the principal sum is Rs. 285, which corresponds to option a.

Explanation

In this problem, the height of the tree and the length of its shadow form a right-angled triangle where the angle of elevation of the sun represents the angle between the ground and the hypotenuse. To find the height of the tree, we use the trigonometric ratio tangent, which is the ratio of the opposite side to the adjacent side. The applicable formula is tan(θ) = Height/Shadow. Given that the angle θ = 60° and the shadow length is 16 m, we substitute these values into the formula: tan(60°) = h/16. Since the standard value of tan(60°) is √3, the equation becomes √3 = h/16. Solving for h, we get h = 16 × √3 = 16√3 m. This matches option (c).

Explanation

Let the side length of the cubical box be a. The volume of the cube is given by the formula V꜀ᵤᵦₑ = a³. Since the sphere is inscribed in the cube and is tangent to all six faces, the diameter of the sphere is equal to the side length of the cube. Therefore, the radius r of the sphere is r = a/2.

The volume of a sphere is calculated using the formula Vₛₚₕₑᵣₑ = 4/3π r³. Substituting the value of r in terms of a, we get Vₛₚₕₑᵣₑ = 4/3π (a/2)³ = 4/3π a³/8 = π a³/6.

To find the ratio of the volume of the cubical box to the volume of the sphere, we divide V꜀ᵤᵦₑ by Vₛₚₕₑᵣₑ: Ratio = V꜀ᵤᵦₑ/Vₛₚₕₑᵣₑ = a³/(π a³/6) = a³ × 6/π a³ = 6/π.

This calculation shows that the ratio is 6/π, which corresponds to option (d).

Explanation

Product of two numbers = GCD LCM. 6912 = 24 LCM. LCM = 6912 / 24 = 288.

Explanation

The area of the square is calculated using the formula Area = side^2. Given the side length is 6 cm, the area of the square is 6² = 36 cm². Since the area of triangle ABC is equal to the area of the square, the area of triangle ABC is also 36 cm².

The area of a triangle is given by the formula Area = 1/2 × base × altitude. Here, the base AB is given as 9 cm and the altitude is the unknown value h we need to find. Substituting the known values into the formula: 36 = 1/2 × 9 × h 36 = 4.5h h = 36/4.5 h = 8 cm

Thus, the length of the altitude to AB is 8 cm, which corresponds to option d.

Explanation

To find the Least Common Multiple (LCM) of fractions, the standard mathematical formula is used: LCM of fractions = LCM of Numerators / HCF of Denominators

Given fractions are 2/3, 7/9, and 14/15. The numerators are 2, 7, and 14. The denominators are 3, 9, and 15.

First, compute the LCM of the numerators (2, 7, 14). Since 14 is a multiple of both 2 and 7, the LCM of 2, 7, and 14 is 14.

Next, compute the Highest Common Factor (HCF) of the denominators (3, 9, 15). The factors of 3 are 1, 3; the factors of 9 are 1, 3, 9; and the factors of 15 are 1, 3, 5, 15. The highest common factor among them is 3.

Substituting these values into the formula: LCM = LCM(2, 7, 14) / HCF(3, 9, 15) = 14/3

Thus, the LCM of the given fractions is 14/3, which corresponds to option b.

Explanation

The average speed u for a journey where equal distances are covered at different speeds is given by the harmonic mean of the speeds. The fundamental formula for average speed is u = Total Distance/Total Time. Let the length of each side of the square be L. The total distance traveled by the car is 4L. The time taken to cover each of the four sides is t₁ = L/v, t₂ = L/2v, t₃ = L/3v, and t₄ = L/4v.

The total time T is the sum of these individual durations: T = L/v + L/2v + L/3v + L/4v = L/v (1 + 1/2 + 1/3 + 1/4) T = L/v (12 + 6 + 4 + 3/12) = 25L/12v

Now, substituting the total distance and total time into the average speed formula: u = 4L/(25L/12v) = 4 × 12v/25 = 48v/25 = 1.92v

Comparing the calculated value u = 1.92v with the given options, it is evident that 1.92v is greater than v and less than 2v. Therefore, the condition v < u < 2v specified in option (c) is correct.

Explanation

The number of small balls is the ratio of the volume of the large sphere to the small ball: (4/3 pi 8^3) / (4/3 pi 1^3) = 8^3 / 1^3 = 512.

Explanation

To find the cost of painting the pillars, we first calculate the total curved surface area (CSA) of the 10 cylinders. The formula for the curved surface area of a cylinder is CSA = πdh, where d is the diameter and h is the height.

Given values: Diameter d = 70 cm = 0.7 m Height h = 4 m Number of pillars n = 10 Rate of painting = Rs. 5 per square metre

Total surface area for 10 pillars is calculated as: Total Area = n × π × d × h Total Area = 10 × 22/7 × 0.7 × 4 Total Area = 10 × 22 × 0.1 × 4 Total Area = 88 sq m

The total cost is the product of the total area and the rate per square metre: Total Cost = 88 × 5 = 440

Thus, the cost of painting the 10 cylindrical pillars is Rs. 440, which corresponds to option (b).

Explanation

Statement 1 is correct based on the property that the product of r consecutive integers is always divisible by r!. For r = 3, the product is divisible by 3! = 3 × 2 × 1 = 6. This happens because in any set of three consecutive integers, at least one must be even (divisible by 2) and exactly one must be a multiple of 3. Since 2 and 3 are coprime, any number divisible by both is necessarily divisible by their product, which is 6.

Statement 2 is correct as it is a direct application of the Division Algorithm. The algorithm states that for any integer n and a positive integer d, there exist unique integers q and r such that n = dq + r, where 0 ≤ r < d. By substituting d = 3, the only possible remainders r are 0, 1, and 2. Consequently, every integer can be uniquely expressed in one of the forms 3k, 3k + 1, or 3k + 2. Since both statements are true, option (c) is the correct choice.

Explanation

The fourth proportional x for three numbers a, b, and c is defined by the ratio a/b = c/x. Given a=7, b=11, and c=14, we have 7/11 = 14/x. Solving for x, we get 7x = 11 14, which simplifies to x = (11 14) / 7 = 11 * 2 = 22.

Explanation

In a right-angled isosceles triangle, the perpendicular drawn from the vertex of the right angle to the hypotenuse also acts as the median to the hypotenuse. According to the property of right triangles, the length of the median to the hypotenuse is half the length of the hypotenuse itself.

First, calculate the length of the hypotenuse AC using the Pythagoras theorem: AC = √(AB² + BC²) = √(8² + 8²) = √(64 + 64) = √128 = 8√2 cm.

Let the length of the perpendicular from B to AC be BP. Using the area method, the area of triangle ABC can be expressed in two ways: Area = 1/2 × base × height Taking AB as base and BC as height: Area = 1/2 × 8 × 8 = 32 sq cm. Taking AC as base and BP as height: Area = 1/2 × 8√2 × BP.

Equating the two areas: 32 = 1/2 × 8√2 × BP 32 = 4√2 × BP BP = 32/(4√2) = 8/√2 = 4√2 cm.

Thus, the length of the perpendicular is 4√2 cm, which corresponds to option b.

Explanation

In a circle, the perpendicular drawn from the centre to a chord bisects the chord. This creates a right-angled triangle where the hypotenuse is the radius r, one leg is half the length of the chord, and the other leg is the perpendicular distance d from the centre.

Using the Pythagorean theorem: r² = d² + (chord length/2)²

Substitute the given chord length √3r: r² = d² + (√3r/2)² r² = d² + 3r²/4

Rearrange the equation to solve for d²: d² = r² - 3r²/4 d² = (4r² - 3r²)/4 d² = r²/4

Taking the square root of both sides: d = √(r²/4) = r/2

Therefore, the length of the perpendicular is r/2, which corresponds to option c.

Explanation

Since alpha and beta are complementary angles, alpha + beta = 90 degrees. This implies sin beta = cos alpha and cos beta = sin alpha. Substituting these into the expression: sqrt(cos alpha sin alpha) ((sin alpha / cos alpha) + (cos alpha / sin alpha))^(1/2). The term inside the second parenthesis simplifies to (sin^2 alpha + cos^2 alpha) / (sin alpha cos alpha), which equals 1 / (sin alpha cos alpha). The full expression becomes sqrt(sin alpha cos alpha) (1 / sqrt(sin alpha * cos alpha)) = 1.

Explanation

Statement 2 is the Central Angle Theorem, which is a fundamental property of circles. It states that the angle subtended by an arc at the centre of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference. This theorem serves as the basis for proving Statement 1.

Consider a cyclic quadrilateral ABCD inscribed in a circle with centre O. The arc BCD subtends ∠BOD at the centre and ∠BAD at the circumference. According to Statement 2, ∠BOD = 2 ∠BAD. Similarly, the arc BAD subtends the reflex ∠BOD at the centre and ∠BCD at the circumference, meaning reflex ∠BOD = 2 ∠BCD. Since the sum of angles around a point is 360°, we have ∠BOD + reflex ∠BOD = 360°. Substituting the values, 2 ∠BAD + 2 ∠BCD = 360°, which simplifies to ∠BAD + ∠BCD = 180°. This proves that opposite angles of a cyclic quadrilateral are supplementary. Thus, Statement 2 logically implies Statement 1, making option b the correct choice.

Explanation

Checking the options for primality: 161 is divisible by 7 (161 = 7 23); 171 is divisible by 3 and 9 (171 = 9 19); 221 is divisible by 13 (221 = 13 * 17). 173 is not divisible by any prime number up to its square root (approx 13.1), making it a prime number.

Explanation

The given expression is of the form (a+b)/(a-b) + (a-b)/(a+b), where a = √5 and b = √3. Using the algebraic identity (a+b)/(a-b) + (a-b)/(a+b) = ((a+b)² + (a-b)²)/((a-b)(a+b)), we can simplify the numerator and denominator.

The numerator (a+b)² + (a-b)² simplifies to 2(a² + b²), and the denominator (a-b)(a+b) simplifies to a² - b². Thus, the formula becomes 2(a² + b²)/(a² - b²). Substituting the values a² = (√5)² = 5 and b² = (√3)² = 3, we get 2(5 + 3)/(5 - 3). This further simplifies to 2(8)/2, which equals 16/2 = 8. Therefore, the value of the expression is 8, which corresponds to option b.

Explanation

Using the fundamental property of logarithms, log a + log b = log(ab), the sum of logarithms can be rewritten as the logarithm of the product of their arguments. The given expression is a series of logarithms where each argument is a fraction of the form (n+2)/(n+1).

For the first 8 terms, the expression is: log₁₀(3/2) + log₁₀(4/3) + log₁₀(5/4) + log₁₀(6/5) + log₁₀(7/6) + log₁₀(8/7) + log₁₀(9/8) + log₁₀(10/9)

Applying the property, we combine these into a single logarithm: log₁₀(3/2 × 4/3 × 5/4 × 6/5 × 7/6 × 8/7 × 9/8 × 10/9)

The product inside the parenthesis is a telescoping product where the numerator of each fraction cancels with the denominator of the subsequent fraction. The only remaining values are the denominator of the first term and the numerator of the last term: Product = 10/2 = 5

Therefore, the total sum is log₁₀ 5, which corresponds to option c.

Explanation

The product of n consecutive integers is always divisible by n factorial (n!). For four consecutive integers, the product is always divisible by 4! = 4 3 2 1 = 24. For example, 1234 = 24 and 234*5 = 120, both of which are divisible by 24. Thus, the largest integer that divides the product of any four consecutive integers is 24.

Explanation

Let the required angle be x. By definition, two angles are complementary if their sum is 90°. Therefore, the complement of the angle x is (90° - x). According to the problem, the angle is equal to two-third of its complement. This relationship can be expressed using the formula x = 2/3(90° - x). To solve for x, multiply both sides of the equation by 3 to eliminate the fraction, which gives 3x = 2(90° - x). Expanding the right side results in 3x = 180° - 2x. By adding 2x to both sides, we get 5x = 180°. Dividing both sides by 5, we find x = 180°/5 = 36°. Thus, the angle is 36°, which corresponds to option a).

Explanation

In a cyclic quadrilateral, angles subtended by the same arc at the circumference are equal. Here, both ∠BDC and ∠BAC subtend the arc BC. Therefore, ∠BAC = ∠BDC = 42°. Since AC is given as the diameter of the circum-circle, the angle subtended by the diameter at any point on the circumference is a right angle. Thus, ∠ABC = 90°. In ΔABC, we apply the angle sum property of a triangle, which states that the sum of internal angles is 180°:

∠BAC + ∠ABC + ∠ACB = 180°

Substituting the known values:

42° + 90° + ∠ACB = 180°

132° + ∠ACB = 180°

∠ACB = 180° - 132° = 48°

Thus, the measure of ∠ACB is 48°, which corresponds to option (c).

Explanation

In a circle circumscribing a square, the diagonal of the square is equal to the diameter of the circle. Let the side of the square be denoted by s and the diameter of the circumscribing circle be d. According to the properties of a square, the relationship between the side and the diagonal is given by the formula: Diagonal = s√2.

Given that the diameter of the circle d = 15√2 cm, we can equate the diagonal of the square to this diameter: s√2 = 15√2

To find the length of the side s, we divide both sides of the equation by √2: s = 15√2/√2 s = 15 cm

Thus, the length of the side of the square is 15 cm, which corresponds to option a).