CDS II 2010 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

For a triangle with sides a, b, and c to be acute-angled, the square of the longest side must be less than the sum of the squares of the other two sides. Given two sides a = 6 cm and b = 2 cm, let the third side be c. We must consider two scenarios based on which side is the longest.

In the first scenario, if c is the longest side, the condition for an acute triangle is c² < a² + b². Substituting the values, we get c² < 6² + 2², which simplifies to c² < 36 + 4 = 40. Therefore, c < √40, which can be written as c < 2√10.

In the second scenario, if the side of 6 cm is the longest side, the condition is a² < b² + c². Substituting the values, we get 6² < 2² + c², which simplifies to 36 < 4 + c². This implies c² > 32, so c > √32, which can be written as c > 4√2.

Combining these two conditions, the range for the third side c is 4√2 < c < 2√10. This matches option d.

Explanation

The last digit of powers of 3 follows a cycle of four: 3, 9, 7, 1. To find the last digit of 3^4798, divide the exponent 4798 by 4. 4798 divided by 4 leaves a remainder of 2. The second number in the cycle is 9.

Explanation

To solve this expression, use the rationalization formula for terms of the form 1/(√a - √b). Multiplying the numerator and denominator by the conjugate (√a + √b), we get 1/(√a - √b) = (√a + √b)/(a - b). In this specific problem, the difference between the radicands in every denominator is 1 (e.g., 9 - 8 = 1, 8 - 7 = 1$), so the formula simplifies to 1/(√a - √b) = √a + √b.

Applying this to each term in the expression: 1/(√9 - √8) = √9 + √8 1/(√8 - √7) = √8 + √7 1/(√7 - √6) = √7 + √6 1/(√6 - √5) = √6 + √5 1/(√5 - √4) = √5 + √4

Substituting these back into the original alternating series: (√9 + √8) - (√8 + √7) + (√7 + √6) - (√6 + √5) + (√5 + √4)

Expanding the brackets: √9 + √8 - √8 - √7 + √7 + √6 - √6 - √5 + √5 + √4

Observe that the terms √8, √7, √6, and √5 cancel out due to opposite signs. This is a telescoping series. The remaining terms are: √9 + √4 = 3 + 2 = 5

Thus, the value is 5, which corresponds to option c.

Explanation

To factorize the given expression x² + 4y² + 4y - 4xy - 2x - 8, first rearrange the terms to group related variables together. The expression can be written as (x² - 4xy + 4y²) - 2x + 4y - 8.

Observe that the first three terms x² - 4xy + 4y² form a perfect square trinomial, which follows the identity (a - b)² = a² - 2ab + b². Here, a = x and b = 2y, so x² - 4xy + 4y² = (x - 2y)².

Now, consider the next two terms -2x + 4y. Factoring out -2 gives -2(x - 2y). Substituting these back into the original expression, we get (x - 2y)² - 2(x - 2y) - 8.

To simplify further, let u = x - 2y. The expression becomes a quadratic in terms of u: u² - 2u - 8. To factorize this quadratic, we find two numbers that multiply to -8 and add to -2, which are -4 and 2. Thus, the quadratic factors as (u - 4)(u + 2).

Replacing u with the original expression x - 2y, we obtain (x - 2y - 4) and (x - 2y + 2), which matches option a.

Explanation

To find the Highest Common Factor (HCF) of the two polynomials, the fastest approach is to factorize the simpler expression first and then check if those factors divide the larger expression using the Factor Theorem.

First, consider the second polynomial Q(x, y) = x² + xy - 2y². This is a quadratic form that can be factorized by splitting the middle term: x² + 2xy - xy - 2y² = x(x + 2y) - y(x + 2y) = (x + 2y)(x - y). The factors of Q(x, y) are (x + 2y) and (x - y). The HCF must be one of these factors or their product.

Next, we test these factors in the first polynomial P(x, y) = 4x³ + 3x²y - 9xy² + 2y³ using the Factor Theorem. For (x - y) to be a factor, P(y, y) must be zero: P(y, y) = 4(y)³ + 3(y)²y - 9(y)y² + 2y³ = 4y³ + 3y³ - 9y³ + 2y³ = 0. Thus, (x - y) is a factor.

For (x + 2y) to be a factor, P(-2y, y) must be zero: P(-2y, y) = 4(-2y)³ + 3(-2y)²y - 9(-2y)y² + 2y³ = 4(-8y³) + 3(4y²)y + 18y³ + 2y³ = -32y³ + 12y³ + 18y³ + 2y³ = 0. Thus, (x + 2y) is also a factor.

Since both (x - y) and (x + 2y) are common factors to both polynomials, their product (x + 2y)(x - y) is the HCF, which corresponds to option c.

Explanation

The difference between each divisor and its corresponding remainder is constant: 52 - 33 = 19, 78 - 59 = 19, and 117 - 98 = 19. The least such number is LCM(52, 78, 117) - 19. The LCM of 52, 78, and 117 is 468. The number is 468 - 19 = 449. The sum of the digits is 4 + 4 + 9 = 17.

Explanation

According to the Factor Theorem, if a polynomial P(x) vanishes at x = a, then (x - a) is a factor of P(x). Given that both f(x) and g(x) vanish at x = 1/2, it implies that (x - 1/2) is a common factor of both polynomials. However, the problem specifies that f(x) and g(x) have integral coefficients. According to the Rational Root Theorem, if a polynomial with integer coefficients has a rational root p/q in simplest form, then (qx - p) is a factor of that polynomial. In this case, the root is 1/2, where p = 1 and q = 2. Therefore, the factor with integral coefficients is (2x - 1). Since (2x - 1) is a factor of both f(x) and g(x), it must also be a factor of their Highest Common Factor (HCF). Thus, option (c) 2x - 1 is the correct factor.

Explanation

According to the Factor Theorem, if a polynomial f(x) is divisible by (x - a), then f(a) = 0. Given the polynomial f(x) = px³ + x² - 2x - q, it is divisible by both (x - 1) and (x + 1). For the factor (x - 1), we substitute x = 1 into the expression: f(1) = p(1)³ + (1)² - 2(1) - q = 0, which simplifies to p + 1 - 2 - q = 0, giving the equation p - q = 1. For the factor (x + 1), we substitute x = -1 into the expression: f(-1) = p(-1)³ + (-1)² - 2(-1) - q = 0, which simplifies to -p + 1 + 2 - q = 0, giving the equation -p - q = -3 or p + q = 3. Adding these two simultaneous equations p - q = 1 and p + q = 3 yields 2p = 4, resulting in p = 2. Substituting p = 2 back into p + q = 3 gives 2 + q = 3, which means q = 1. Therefore, the values of p and q are 2 and 1 respectively, which corresponds to option d.

Explanation

Let the present ages of A and B be 2x and 5x respectively, based on the given ratio 2:5. After 8 years, their ages will be 2x + 8 and 5x + 8. According to the problem, the ratio of their ages at that time will be 1:2. This can be expressed as the equation (2x + 8) / (5x + 8) = 1 / 2. By cross-multiplying the terms, we get 2(2x + 8) = 1(5x + 8), which simplifies to 4x + 16 = 5x + 8. Rearranging the terms to solve for x, we find 5x - 4x = 16 - 8, which gives x = 8. The difference between their present ages is 5x - 2x = 3x. Substituting the value of x, the difference is 3 × 8 = 24 years. Therefore, option (c) is the correct answer.

Explanation

Let the initial quantity of milk be 5x and the initial quantity of water be 1x. According to the problem, 5 litres of water are added to the mixture, making the new quantity of water x + 5. The new ratio of milk to water is given as 5:2. Therefore, we can set up the equation 5x/(x + 5) = 5/2. Cross-multiplying gives 10x = 5(x + 5), which simplifies to 10x = 5x + 25. Solving for x, we get 5x = 25, so x = 5. The quantity of milk in the original mixture is 5x = 5 ⋅ 5 = 25 litres. Alternatively, using the ratio change method, the milk remains constant at 5 parts, while the water increases from 1 part to 2 parts. The increase of 1 part corresponds to the 5 litres of water added. Since 1 part equals 5 litres, the quantity of milk is 5 parts × 5 litres/part = 25 litres.

Explanation

The difference between each divisor and its remainder is constant: 4 - 3 = 1, 5 - 4 = 1, 8 - 7 = 1, and 9 - 8 = 1. The smallest positive integer is LCM(4, 5, 8, 9) - 1. The LCM of 4, 5, 8, and 9 is 360. The number is 360 - 1 = 359.

Explanation

If x = 2, then the numbers are x = 2, x + 1 = 3, and x + 3 = 5. All three (2, 3, and 5) are prime numbers. For x = 0, 0 is not prime. For x = 1, 1 is not prime. For x = 101, x + 1 = 102 is not prime.

Explanation

To find the total monthly salary, first calculate the total percentage of the salary spent on all items. Total expenditure percentage = 30% (rent) + 25% (food) + 20% (education) + 12% (electricity) = 87%. Since the total salary is 100%, the remaining percentage is 100% - 87% = 13%.

The problem states that the remaining balance is Rs 1,040. Therefore, 13% of the total salary is equal to Rs 1,040. Using the formula for the total value when a percentage is known: Total Salary = Remaining Amount/Remaining Percentage × 100. Substituting the values: Total Salary = 1,040/13 × 100. Performing the division, 1,040 ÷ 13 = 80. Thus, Total Salary = 80 × 100 = 8,000. The monthly salary of the person is Rs 8,000, which corresponds to option (a).

Explanation

From the given equations, pˣ = rʸ = m, we can express p and r in terms of m: p = m¹/ˣ r = m¹/ʸ

Similarly, from rʷ = pᶻ = n, we can express r and p in terms of n: r = n¹/ʷ p = n¹/ᶻ

Now, we equate the two different expressions for p and r: For p: m¹/ˣ = n¹/ᶻ ⇒ m = nˣ/ᶻ For r: m¹/ʸ = n¹/ʷ ⇒ m = nʸ/ʷ

Equating the two expressions for m: nˣ/ᶻ = nʸ/ʷ

Since the bases are the same, we can equate the exponents: x/z = y/w

Cross-multiplying these terms gives: xw = yz

This matches option a, confirming the relationship between the variables.

Explanation

Let the distance between point A and point B be d km. The time taken by the father to cover this distance is t_f = d/12 hours, and the time taken by the son is tₛ = d/18 hours. According to the problem, the son starts 60 minutes (1 hour) after the father and arrives 60 minutes (1 hour) before the father. This means the son's total travel time is 2 hours less than the father's travel time.

We can express this relationship using the formula: d/12 - d/18 = 2

To solve for d, we find a common denominator for 12 and 18, which is 36: (3d - 2d)/36 = 2 d/36 = 2 d = 36 × 2 = 72 km

Alternatively, we can use the competitive exam shortcut formula for distance when the difference in time Δt is known: d = (v₁ × v₂)/(v₂ - v₁) × Δt Substituting the values v₁ = 12, v₂ = 18, and Δt = 2 hours: d = (12 × 18)/(18 - 12) × 2 = 216/6 × 2 = 36 × 2 = 72 km

Thus, the distance between A and B is 72 km, which corresponds to option (b).

Explanation

For the equation (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0 to have equal roots, the discriminant must be zero. D = [-2(ac + bd)]^2 - 4(a^2 + b^2)(c^2 + d^2) = 0 4(a^2c^2 + b^2d^2 + 2abcd) - 4(a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2) = 0 a^2c^2 + b^2d^2 + 2abcd - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2 = 0 2abcd - a^2d^2 - b^2c^2 = 0 -(ad - bc)^2 = 0 implies ad = bc. While Option D is ac = bd, there is a likely typo in the question's middle term. If the middle term were -2(ad + bc)x, the condition would be ac = bd. Given the answer key, D is the intended answer.

Explanation

To find the time taken by two trains to cross each other, we use the concept of relative speed. When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The total distance to be covered for the trains to completely cross each other is the sum of their lengths.

Total Distance D = L₁ + L₂ = 100 m + 150 m = 250 m. Relative Speed Vᵣₑₗ = V₁ + V₂ = 75 kmph + 50 kmph = 125 kmph.

To perform the calculation in seconds, we must convert the relative speed from kmph to m/s using the conversion factor 5/18: Vᵣₑₗ = 125 × 5/18 m/s.

Using the formula for time t = Distance/Relative Speed: t = 250 / (125 × 5/18) t = (250 × 18) / (125 × 5) t = (2 × 18) / 5 t = 36/5 = 7.2 s.

Therefore, the time taken by the trains to cross each other is 7.2 s, which corresponds to option b.

Explanation

Given the ratio (a - b)/(a + b) = 1/5, we can determine the relationship between a and b using cross-multiplication: 5(a - b) = 1(a + b), which simplifies to 5a - 5b = a + b. Rearranging the terms to group like variables, we get 5a - a = b + 5b, which results in 4a = 6b. This simplifies to a/b = 6/4 = 3/2. Let a = 3k and b = 2k for some constant k.

We need to find the value of the expression (a² - b²)/(a² + b²). Substituting the values of a and b into this expression, we get:

((3k)² - (2k)²)/((3k)² + (2k)²) = (9k² - 4k²)/(9k² + 4k²) = 5k²/13k² = 5/13

Thus, the required ratio is 5:13, which matches option c.

Explanation

Given the relationships described in the question, we can express the variations as mathematical equations. Since x varies as the mth power of y, we write x = k₁yᵐ, where k₁ is a constant. Similarly, since y varies as the nth power of z, we write y = k₂zⁿ, where k₂ is another constant. To determine the relationship between x and z, we substitute the expression for y into the equation for x: x = k₁(k₂zⁿ)ᵐ. According to the laws of exponents, specifically the power of a power rule (aᵇ)ᶜ = aᵇᶜ, the equation simplifies to x = k₁k₂ᵐzᵐⁿ. The problem also states that x varies as the pth power of z, which means x = k₃zᵖ. By comparing the exponents of z in the two resulting expressions for x, we find that p = mn. Therefore, option (c) is the correct relationship.

Explanation

The solution relies on the algebraic identity which states that if x + y + z = 0, then x³ + y³ + z³ = 3xyz. To determine if this condition holds, we expand the given expressions for x, y, and z:

x = (b - c)(a - d) = ab - bd - ac + cd y = (c - a)(b - d) = cb - cd - ab + ad z = (a - b)(c - d) = ac - ad - bc + bd

Now, we sum these three expressions: x + y + z = (ab - bd - ac + cd) + (bc - cd - ab + ad) + (ac - ad - bc + bd)

By grouping like terms: x + y + z = (ab - ab) + (-bd + bd) + (-ac + ac) + (cd - cd) + (bc - bc) + (ad - ad) = 0

Since the sum x + y + z is exactly zero, we can directly apply the formula x³ + y³ + z³ = 3xyz. This makes option (c) the correct result. This shortcut avoids the tedious calculation of cubing the binomial products and is the fastest way to solve such problems in competitive exams.

Explanation

The n-th term of an arithmetic progression is given by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. Given the 14th term a₁₄ = 6 and the 6th term a₆ = 14, we can find the common difference d using the relation aₘ - aₙ = (m-n)d. Substituting the given values, we get 6 - 14 = (14 - 6)d, which simplifies to -8 = 8d, giving d = -1. To find the 95th term a₉₅, we can use the 6th term as a reference: a₉₅ = a₆ + (95-6)d. Substituting the values, a₉₅ = 14 + 89(-1) = 14 - 89 = -75. Therefore, option (a) is correct.

Explanation

Let y = sqrt(x/(x + 3)). The equation becomes y - 1/y = -3/2. Multiplying by 2y gives 2y^2 + 3y - 2 = 0. Solving the quadratic equation, we get y = 1/2 or y = -2. Since y is a square root, y must be positive, so y = 1/2. Thus, sqrt(x/(x + 3)) = 1/2, which implies x/(x + 3) = 1/4. Solving for x, we get 4x = x + 3, so 3x = 3, which means x = 1.

Explanation

For any positive integer n, the unit digit of 3^(2n + 1) is either 3 (if n is even) or 7 (if n is odd). The unit digit of 2^(2n + 1) is either 2 (if n is even) or 8 (if n is odd). In either case, the sum of the unit digits is 3 + 2 = 5 or 7 + 8 = 15. Both result in a unit digit of 5.

Explanation

Since k is given as an even positive integer, we can represent it as k = 2n, where n is any positive integer. Substituting this into the given expression (k² + 2k), we get (2n)² + 2(2n) = 4n² + 4n. Factoring out the common term, the expression simplifies to 4n(n + 1). In any pair of consecutive integers n and (n + 1), one of them must be even, which means their product n(n + 1) is always divisible by 2. Consequently, the total expression 4n(n + 1) is always divisible by 4 × 2 = 8. To check for divisibility by 24, we can test values of k. For k = 2, the expression is 2² + 2(2) = 8, which is divisible by 8 but not by 24. For k = 4, the expression is 4² + 2(4) = 24, which is divisible by 24. Thus, the expression is always divisible by 8 but may not always be divisible by 24, making option (b) the correct choice.

Explanation

To solve this problem quickly in competitive exams, use the standard formula for percentage decrease when an increase is given: if A is r% more than B, then B is less than A by (r/(100 + r) × 100)%. In this case, the salary of X is r = 20% more than the salary of Y. Substituting the value into the formula, we get 20/(100 + 20) × 100. This simplifies to 20/120 × 100. Further simplifying the fraction 20/120 gives 1/6. Therefore, the calculation becomes 1/6 × 100, which equals 50/3%. This corresponds to option (c). Alternatively, if we assume the salary of Y is 100, then the salary of X is 120. The difference between the two salaries is 20. To find by how much percentage Y is less than X, we divide the difference by the salary of X and multiply by 100: (20/120) × 100, which also results in 50/3%.

Explanation

Let 2^x = y. The equation 4^x - 3 2^(x + 2) + 32 = 0 becomes y^2 - 3 4y + 32 = 0, which is y^2 - 12y + 32 = 0. Factoring gives (y - 8)(y - 4) = 0. So, y = 8 or y = 4. This means 2^x = 8 (x = 3) or 2^x = 4 (x = 2). The roots are 2 and 3.

Explanation

For x^2 - x - 1 = 0, the sum of roots alpha + beta = 1 and the product alpha beta = -1. Then alpha^2 + beta^2 = (alpha + beta)^2 - 2 alpha beta = 1^2 - 2(-1) = 3. Further, alpha^4 + beta^4 = (alpha^2 + beta^2)^2 - 2 (alpha * beta)^2 = 3^2 - 2(-1)^2 = 9 - 2 = 7.

Explanation

For a pair of linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the condition for a unique solution is that the lines must intersect at exactly one point, which occurs when the ratio of the coefficients of x is not equal to the ratio of the coefficients of y. This is expressed by the standard formula a₁/a₂ ≠ b₁/b₂. In the given equations kx - y = 2 and 6x - 2y = 3, the coefficients are a₁ = k, b₁ = -1, a₂ = 6, and b₂ = -2. Substituting these values into the unique solution formula, we get k/6 ≠ (-1)/(-2). Simplifying the fraction on the right side yields k/6 ≠ 1/2. To solve for k, multiply both sides of the inequality by 6, which gives k ≠ (1/2) × 6, resulting in k ≠ 3. Therefore, the equations have a unique solution provided that k is not equal to 3, which matches option (b).

Explanation

The standard formula to form a quadratic equation when the sum and product of its roots are known is x² - (sum of roots)x + (product of roots) = 0. In this problem, it is given that the sum of the roots is 9 and the product of the roots is also 9. Substituting these given values into the formula, we obtain x² - (9)x + 9 = 0, which simplifies directly to x² - 9x + 9 = 0. Comparing this result with the given options, we find that it matches option (d) perfectly. This approach is the fastest way to solve such problems in competitive exams as it avoids the need to find individual roots.

Explanation

A rational number expressed in the form p/q has a terminating decimal expansion if and only if the prime factorization of the denominator q consists only of the primes 2 and 5, i.e., q = 2ᵐ · 5ⁿ where m, n ≥ 0. If the denominator contains any prime factor other than 2 or 5, the decimal expansion will be non-terminating and repeating.

Applying this rule to the given options: In option (a), the denominator is 8, which can be written as 2³. Since it only contains the prime factor 2, it is a terminating decimal. In option (b), the denominator is 16, which can be written as 2⁴. Since it only contains the prime factor 2, it is a terminating decimal. In option (c), the denominator is 11. Since 11 is a prime number other than 2 or 5, the decimal expansion is non-terminating and repeating. Performing the division gives 3/11 = 0.272727... = 0.27. In option (d), the denominator is 25, which can be written as 5². Since it only contains the prime factor 5, it is a terminating decimal.

Therefore, option (c) is the correct choice.

Explanation

To find the value of the expression a²/bc + b²/ca + c²/ab, we first find a common denominator for the three fractions. The common denominator is abc. By adjusting the numerators accordingly, the expression becomes (a³ + b³ + c³)/abc.

According to the algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), we can determine the value of the numerator. Since it is given that a + b + c = 0, the entire right side of the identity becomes zero. This implies that a³ + b³ + c³ - 3abc = 0, which further simplifies to a³ + b³ + c³ = 3abc.

Substituting this value back into our simplified expression, we get 3abc/abc. By cancelling the abc term from both the numerator and the denominator, the final result is 3. Therefore, the value of the given expression is 3, which corresponds to option d.

Explanation

Let the required fraction be represented as x/y. According to the first condition, adding 1 to the denominator makes the fraction 1/2, which can be written as the equation x/(y+1) = 1/2. Cross-multiplying this gives 2x = y + 1, or 2x - y = 1. According to the second condition, adding 1 to the numerator makes the fraction 1, which can be written as (x+1)/y = 1. This simplifies to x + 1 = y, or y - x = 1. To find the values of x and y, we solve these two linear equations simultaneously. Adding the equations (2x - y = 1) and (y - x = 1) results in x = 2. Substituting x = 2 back into the equation y - x = 1 gives y - 2 = 1, which means y = 3. Therefore, the original fraction x/y is 2/3, which corresponds to option (b). Verification shows that 2/(3+1) = 2/4 = 1/2 and (2+1)/3 = 3/3 = 1.

Explanation

To find the speed of the boat in still water, we use the formulas for downstream and upstream speeds. The downstream speed s_d is calculated as distance divided by time: s_d = 9 km/2 hours = 4.5 kmph. The upstream speed sᵤ is similarly calculated as sᵤ = 9 km/6 hours = 1.5 kmph.

The speed of the boat in still water v_b is the arithmetic mean of the downstream and upstream speeds. The formula is v_b = (s_d + sᵤ)/2. Substituting the calculated values into the formula, we get v_b = (4.5 + 1.5)/2 = 6/2 = 3 kmph. Therefore, option (a) is the correct value.

Explanation

Let A's efficiency be 100%. B's efficiency is 160%. The ratio of efficiency A:B is 100:160 or 5:8. Since time is inversely proportional to efficiency, the ratio of time taken A:B is 8:5. Given A takes 24 days, 8 units = 24 days, so 1 unit = 3 days. B takes 5 units = 5 * 3 = 15 days.

Explanation

For simple interest problems where a sum becomes a certain multiple of itself, the relationship between time T and the multiple n is given by the formula (T₁)/(n₁ - 1) = (T₂)/(n₂ - 1), where T is the time and n is the number of times the principal increases.

In the first case, the sum trebles, so n₁ = 3, and the time T₁ = 15 years and 6 months, which is 15.5 years or 31/2 years. In the second case, the sum doubles, so n₂ = 2, and we need to find the time T₂.

Substituting the values into the formula: 15.5/(3 - 1) = T₂/(2 - 1) 15.5/2 = T₂/1 T₂ = 7.75 years

To convert the decimal part into months, multiply 0.75 by 12: 0.75 × 12 = 9 months

Therefore, the sum will double in 7 years and 9 months, which corresponds to option b.

Explanation

To solve this problem quickly, use the direct relation between the two selling prices based on the cost price. The formula for the new selling price when transitioning from a loss to a gain is SP₂ = SP₁ × (100 + Gain%)/(100 - Loss%).

Given: SP₁ = 240 Loss% = 4% Gain% = 10%

Substituting the values into the formula: SP₂ = 240 × (100 + 10)/(100 - 4) SP₂ = 240 × 110/96

Simplifying the fraction by dividing 240 and 96 by their greatest common divisor, 24: 240 ÷ 24 = 10 96 ÷ 24 = 4

SP₂ = 10 × 110/4 SP₂ = 1100/4 SP₂ = 275

Thus, the trader must sell the article for Rs 275 (option a) to achieve a gain of 10%.

Explanation

Let the total strength of the class be T. Let the number of boys be B and the number of girls be G. According to the given condition, the difference between the number of boys and girls is 12% of the total strength, which can be represented as B - G = 0.12 T. We also know that the total strength is the sum of boys and girls, so B + G = T.

To solve for B and G, we can use the system of linear equations. Adding the two equations (B + G) + (B - G) = T + 0.12 T gives 2B = 1.12 T, which means B = 0.56 T. Subtracting the first equation from the second (B + G) - (B - G) = T - 0.12 T gives 2G = 0.88 T, which means G = 0.44 T.

The required ratio of the number of boys to the number of girls is B : G = 0.56 T : 0.44 T. Simplifying this ratio by dividing both sides by 0.04 T, we get 56 : 44 = 14 : 11. Thus, the correct ratio is 14:11, which corresponds to option (b).

Explanation

Let the numbers be 3k, 2k, and 5k. The sum of their squares is (3k)^2 + (2k)^2 + (5k)^2 = 1862. This simplifies to 9k^2 + 4k^2 + 25k^2 = 1862, so 38k^2 = 1862. k^2 = 1862 / 38 = 49, so k = 7. The numbers are 37 = 21, 27 = 14, and 5*7 = 35.

Explanation

Since x = -1 is a root of the quadratic equation ax² + x - 3 = 0, it must satisfy the equation. Substituting x = -1 into the equation, we get a(-1)² + (-1) - 3 = 0, which simplifies to a - 1 - 3 = 0, giving a = 4.

The quadratic equation is now 4x² + x - 3 = 0. To find the other root, we use the formula for the product of roots of a quadratic equation Ax² + Bx + C = 0, which is x₁ · x₂ = C/A. Here, A = 4, C = -3, and one root x₁ = -1. Substituting these values, we get -1 · x₂ = -3/4. Multiplying both sides by -1, we find that the other root x₂ = 3/4. Thus, option (c) is correct.

Explanation

1/2 log10 25 - 2 log10 3 + log10 18 = log10(25^1/2) - log10(3^2) + log10 18 = log10 5 - log10 9 + log10 18 = log10(5 / 9 * 18) = log10(10) = 1.

Explanation

Let r be a rational number and x be an irrational number. A rational number is defined as any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. An irrational number is one that cannot be expressed in this form. To determine the nature of their product, consider the property that the product of a non-zero rational number and an irrational number is always irrational. This can be proven by contradiction: if the product r · x = q, where q is a rational number, then x = q/r. Since the division of two rational numbers is also rational, x would have to be rational, which contradicts the given fact that x is irrational. Therefore, the result must be an irrational number, making option (b) the correct choice. While the product is rational if the rational number is zero, in standard competitive exam contexts, the general property for non-zero rationals is applied.

Explanation

Let the height of the building be h and the distance between the building and the tower be x. The height of the tower H is given as 75 m. According to the problem, the angle of elevation of the top of the tower from the bottom of the building is θ₁ = 60°, and the angle of elevation from the top of the building is θ₂ = 60°/2 = 30°.

Using the formula tan θ = Perpendicular/Base, we analyze the two right-angled triangles. For the triangle formed from the bottom of the building: tan 60° = H/x √3 = 75/x x = 75/√3 = 25√3 m.

Now, for the triangle formed from the top of the building, the perpendicular distance is the difference in heights (H - h): tan 30° = (H - h)/x 1/√3 = (75 - h)/(25√3)

Multiplying both sides by 25√3: (25√3)/√3 = 75 - h 25 = 75 - h h = 75 - 25 = 50 m.

Thus, the height of the building is 50 m, which corresponds to option c.

Explanation

In the first quadrant, where 0 ≤ θ ≤ π/2, the cosine function cos θ is a monotonically decreasing function. This means that as the angle θ increases, the value of cos θ decreases. We know from standard trigonometric values that cos(π/3) = 1/2. Since the function is decreasing, for the value of cos θ to be greater than or equal to 1/2, the angle θ must be less than or equal to the angle whose cosine is 1/2. Therefore, if cos θ ≥ cos(π/3), it directly implies that θ ≤ π/3. This range covers all values from 0 up to π/3 radians, making option (a) the correct choice.

Explanation

The given expression is a product of cosine values for angles starting from 1° up to 90°. The sequence of terms is cos 1° × cos 2° × cos 3° × ... × cos 89° × cos 90°. In trigonometry, the value of cos 90° is exactly 0. According to the zero product property in mathematics, if any single factor in a multiplication chain is zero, the entire product regardless of the values of other terms becomes zero. Therefore, since cos 90° = 0 is the final term of the sequence, the entire expression evaluates to 0 × (product of other terms) = 0. This makes option (b) the correct choice.

Explanation

To find the distance between the tops of two vertical poles, we use the Pythagorean theorem by considering a right-angled triangle. The base of this triangle is the distance between the feet of the poles, and the perpendicular is the difference between the heights of the two poles.

The formula for the distance between the tops D is: D = √(d² + (h₂ - h₁)²)

Substituting the given values: Distance between feet d = 12 m Height of pole 1 h₁ = 6 m Height of pole 2 h₂ = 11 m

First, calculate the difference in heights: h₂ - h₁ = 11 - 6 = 5 m

Now, substitute these into the formula: D = √(12² + 5²) D = √(144 + 25) D = √169 D = 13 m

The distance between the tops is 13 m, which corresponds to option (c). This is a standard application of the Pythagorean triplet (5, 12, 13), allowing for a quick mental calculation.

Explanation

To find the value of sin θ cos θ, use the algebraic identity (a + b)² = a² + b² + 2ab. Squaring both sides of the given equation sin θ + cos θ = 1 yields (sin θ + cos θ)² = 1². Expanding the left side, we get sin² θ + cos² θ + 2 sin θ cos θ = 1. We know from the fundamental trigonometric identity that sin² θ + cos² θ = 1. Substituting this value into the equation, we obtain 1 + 2 sin θ cos θ = 1. Subtracting 1 from both sides gives 2 sin θ cos θ = 0. Dividing by 2, we find that sin θ cos θ = 0. This directly leads to option b.

Explanation

To solve the expression √((1 + sin θ)/(1 - sin θ)), we use the method of rationalizing the denominator. Multiply both the numerator and the denominator inside the square root by the conjugate of the denominator, which is (1 + sin θ). This results in √(((1 + sin θ)(1 + sin θ))/((1 - sin θ)(1 + sin θ))). The numerator simplifies to (1 + sin θ)² and the denominator becomes (1 - sin² θ). Using the fundamental trigonometric identity sin² θ + cos² θ = 1, we can substitute 1 - sin² θ with cos² θ. The expression then becomes √((1 + sin θ)²/cos² θ). Removing the square root gives (1 + sin θ)/cos θ. Dividing each term in the numerator by the denominator, we get 1/cos θ + sin θ/cos θ. Since 1/cos θ = sec θ and sin θ/cos θ = tan θ, the final simplified form is sec θ + tan θ, which corresponds to option b.

Explanation

Let the height of the tower be h and the length of the shadow be s. The relationship between the height, shadow, and the altitude angle θ of the Sun is given by the trigonometric formula tan(θ) = h/s. Since the height of the tower h remains constant in both scenarios, we can use the relation s₁ tan(θ₁) = s₂ tan(θ₂). Substituting the given values s₁ = 15 m, θ₁ = 30°, and θ₂ = 60°, the equation becomes 15 tan(30°) = s₂ tan(60°). Using the standard values tan(30°) = 1/√3 and tan(60°) = √3, we get 15 × 1/√3 = s₂ × √3. To find s₂, we rearrange the formula: s₂ = 15/(√3 × √3) = 15/3 = 5 m. Therefore, the length of the shadow when the Sun's altitude is 60° is 5 m, which corresponds to option c.

Explanation

To find the correct arithmetic mean, we use the formula for mean correction based on the difference between the correct and incorrect values. The correct mean is calculated by adding the net change in the sum of observations to the incorrect mean and dividing by the total number of observations n.

The formula is: Correct Mean = Incorrect Mean + (Σ Correct Values - Σ Incorrect Values)/n

Substituting the given values: Incorrect Mean = 89.05 n = 100 Correct Values = 92 and 83 Incorrect Values = 192 and 33

Correct Mean = 89.05 + ((92 + 83) - (192 + 33))/100 Correct Mean = 89.05 + (175 - 225)/100 Correct Mean = 89.05 + -50/100 Correct Mean = 89.05 - 0.50 = 88.55

Thus, the correct arithmetic mean of the 100 numbers is 88.55, which corresponds to option a).

Explanation

Let the interior angles of the triangle be denoted as ∠A, ∠B, and ∠C. According to the angle sum property of a triangle, the sum of all three interior angles is always 180°, expressed by the formula ∠A + ∠B + ∠C = 180°. The given condition states that the sum of two angles is equal to the third angle, which can be written as ∠A + ∠B = ∠C. By substituting this value into the angle sum formula, we get ∠C + ∠C = 180°, which simplifies to 2∠C = 180°. Solving for ∠C gives ∠C = 90°. Since one of the interior angles is exactly 90°, the triangle is by definition a right-angled triangle, making option (a) the correct choice.

Explanation

According to the exterior angle theorem of a triangle, the measure of an exterior angle is equal to the sum of the measures of its two opposite interior angles. In triangle ABC, the exterior angle ∠ACD is formed by extending the side BC to D, making ∠BAC and ∠ABC the opposite interior angles. Therefore, we use the formula ∠ACD = ∠BAC + ∠ABC.

Given that ∠ACD = 120° and ∠ABC = 2/3 ∠CAB, let ∠BAC = x. Then, ∠ABC = 2/3x. Substituting these values into the formula, we get: 120° = x + 2/3x 120° = 5/3x To solve for x, multiply both sides by 3/5: x = 120° × 3/5 x = 24° × 3 x = 72°

Thus, ∠BAC = 72°, which corresponds to option d.

Explanation

According to the Tangent-Secant Theorem in geometry, when a tangent segment PC and a secant segment PAB are drawn to a circle from an external point P, the square of the length of the tangent segment is equal to the product of the lengths of the whole secant segment and its external part. The mathematical formula for this relationship is PC² = PA × PB.

In geometric terms, PC² represents the area of a square with side PC, and the product PA × PB represents the area of a rectangle with sides PA and PB. Therefore, the area of the rectangle with sides PA and PB is equal to the area of the square with side PC. This directly confirms that option a is the correct statement based on the properties of circles.

Explanation

Since AB and CD are straight lines intersecting at O, ∠BOC and ∠AOC form a linear pair. By the linear pair axiom, the sum of adjacent angles on a straight line is 180°. Therefore, ∠AOC = 180° - ∠BOC = 180° - 130° = 50°.

Because OF is the internal bisector of ∠BOC, we have ∠FOC = 1/2 ∠BOC = 130°/2 = 65°. Similarly, since OE is the internal bisector of ∠AOC, we have ∠EOC = 1/2 ∠AOC = 50°/2 = 25°. The measure of ∠FOE is the sum of these two angles: ∠FOE = ∠FOC + ∠EOC = 65° + 25° = 90°.

As a shortcut for competitive exams, remember the theorem that the angle between the internal bisectors of any two adjacent supplementary angles is always 90°. This is because x/2 + (180° - x)/2 = 180°/2 = 90°, regardless of the specific measures of the angles. Thus, option a is correct.

Explanation

To find the radius of the inscribed circle (inradius) of a triangle, we use the formula r = A/s, where A is the area of the triangle and s is the semi-perimeter. First, calculate the semi-perimeter s using the formula s = (a + b + c)/2. Substituting the given side lengths a = 35 cm, b = 44 cm, and c = 75 cm, we get s = (35 + 44 + 75)/2 = 154/2 = 77 cm. Next, we find the area A using Heron's formula A = √s(s-a)(s-b)(s-c). Substituting the values, A = √77(77-35)(77-44)(77-75) = √77 × 42 × 33 × 2. Breaking these into prime factors, we get A = √(7 × 11) × (7 × 3 × 2) × (3 × 11) × 2 = √2² × 3² × 7² × 11² = 2 × 3 × 7 × 11 = 462 sq cm. Finally, the inradius r = 462/77 = 6 cm. Therefore, the correct option is (d).

Explanation

The maximum length of a rod that can be placed inside a cuboid is equivalent to the length of its space diagonal, which connects two opposite vertices of the box. The formula to calculate the diagonal d of a cuboid with length l, breadth b, and height h is given by d = √(l² + b² + h²).

Substituting the given values into the formula: l = 30 cm, b = 24 cm, and h = 18 cm. d = √(30² + 24² + 18²) d = √(900 + 576 + 324) d = √1800

To simplify √1800, we factor out the largest perfect square: d = √(900 × 2) d = 30√2 cm.

Therefore, the maximum length of the rod that can fit inside the cuboid is 30√2 cm, which corresponds to option (b).

Explanation

Let the radii of the two spheres be r₁ and r₂. The formula for the volume of a sphere is V = 4/3 π r³. According to the problem, the ratio of their volumes is V₁/V₂ = 8/1. This gives (4/3 π r₁³)/(4/3 π r₂³) = 8/1, which simplifies to (r₁/r₂)³ = 8/1. Taking the cube root of both sides, we find that the ratio of their radii is r₁/r₂ = ∛8/∛1 = 2/1. The formula for the surface area of a sphere is A = 4 π r². Therefore, the ratio of their surface areas is A₁/A₂ = (4 π r₁²)/(4 π r₂²) = (r₁/r₂)²$. Substituting the calculated ratio of the radii, we get A₁/A₂ = (2/1)² = 4/1. Thus, the ratio of their surface areas is 4:1, which matches option (b).

Explanation

Let the width of the uniform border be x metres. The dimensions of the outer board are 4m and 3m. Since the border is applied to all sides, the dimensions of the inner painted rectangle are reduced by 2x in both length and width. Therefore, the inner length is (4 - 2x) and the inner width is (3 - 2x).

Using the area formula Area = Length × Width, we set up the following equation based on the given painted area of 6m²: (4 - 2x)(3 - 2x) = 6

Expanding the product: 12 - 8x - 6x + 4x² = 6 4x² - 14x + 6 = 0

Dividing the equation by 2 to simplify: 2x² - 7x + 3 = 0

Solving the quadratic equation by factorisation: 2x² - 6x - x + 3 = 0 2x(x - 3) - 1(x - 3) = 0 (2x - 1)(x - 3) = 0

This yields two possible values for x: x = 0.5 or x = 3. Since the border width cannot be 3m because the total width of the board is only 3m, we reject x = 3. Thus, the width of the border is 0.5m, which corresponds to option (b).

Explanation

Let r be the radius of the base and h be the height of the solid cylinder. Given that h = 9 m. The formula for the Curved Surface Area (CSA) of a cylinder is 2πrh and the formula for the Total Surface Area (TSA) is 2πrh + 2πr². According to the problem, the CSA is equal to one-third of the TSA: 2πrh = 1/3(2πrh + 2πr²) Dividing both sides of the equation by 2πr simplifies the relation to: h = 1/3(h + r) Multiplying by 3 to clear the fraction gives: 3h = h + r Rearranging the terms to solve for r: r = 3h - h r = 2h Substituting the given value of h = 9 m: r = 2 × 9 = 18 m Thus, the radius of the base is 18 m, which corresponds to option b.

Explanation

When a rectangular sheet is revolved about one of its sides, the resulting three dimensional figure is a cylinder. In this problem, the sheet is revolved about its length, which means the length of the rectangle becomes the height h of the cylinder and the width of the rectangle becomes the radius r of the cylinder's base. Given the dimensions 7cm × 4cm, we have height h = 7cm and radius r = 4cm.

The volume V of a cylinder is given by the formula V = πr²h. Substituting the given values into the formula, we get V = 22/7 × (4)² × 7. The 7 in the denominator cancels out with the height 7 in the numerator, simplifying the expression to V = 22 × 16. Calculating the product, 22 × 16 = 352. Thus, the volume of the figure formed is 352cm³, which matches option a.

Explanation

Let the dimensions of the cuboid be a, b, c. The diagonals of the three faces are given by the Pythagorean theorem as x² = a² + b², y² = b² + c², and z² = c² + a². To find the volume V = abc, we first determine the values of a², b², c² in terms of x, y, z.

Summing the three equations gives 2(a² + b² + c²) = x² + y² + z², so a² + b² + c² = (x² + y² + z²)/2.

Subtracting the original equations from this sum, we get: a² = (a² + b² + c²) - (b² + c²) = (x² + y² + z²)/2 - y² = (x² + z² - y²)/2 b² = (a² + b² + c²) - (c² + a²) = (x² + y² + z²)/2 - z² = (x² + y² - z²)/2 c² = (a² + b² + c²) - (a² + b²) = (x² + y² + z²)/2 - x² = (y² + z² - x²)/2

The square of the volume is V² = a² b² c². Substituting the values: V² = ((x² + z² - y²)/2) · ((x² + y² - z²)/2) · ((y² + z² - x²)/2) V² = ((y² + z² - x²)(z² + x² - y²)(x² + y² - z²))/8

Taking the square root, we obtain V = √( (y² + z² - x²)(z² + x² - y²)(x² + y² - z²) ) / (2√2), which corresponds to option c.

Explanation

Let the ball radius be r. Then tank radius R = 4r and height H = 4r. Total tank volume = pi (4r)^2 4r = 64 pi r^3. Since the tank is half full, the available volume is 32 pi r^3. Volume of one ball = 4/3 pi r^3. Number of balls = (32 pi r^3) / (4/3 pi r^3) = 32 * 3 / 4 = 24.

Explanation

Radii are R1 = 18/2 = 9 cm and R2 = 8/2 = 4 cm. The distance between centers d = 13 cm. Since R1 + R2 = 9 + 4 = 13 cm, the distance between centers equals the sum of the radii. This means the two circles touch each other externally. For two externally touching circles, there are exactly 3 common tangents.

Explanation

The volume of a cone is given by the formula V = 1/3 πr²h, where r is the radius of the base and h is the height of the cone. In this problem, only the height is being increased, while the radius remains constant. Therefore, the volume of the cone is directly proportional to its height, represented as V ∝ h.

When the height is increased by 50%, the new height hₙₑᵥᵥ becomes h + 0.5h = 1.5h. Since the relationship is linear, the new volume Vₙₑᵥᵥ will be 1.5 times the original volume V. The percentage increase in volume is calculated using the formula: Percentage Increase = (Vₙₑᵥᵥ - Vₒₗ₀)/Vₒₗ₀ × 100. Substituting the values, we get (1.5V - V)/V × 100 = 0.5V/V × 100 = 50%. Thus, option c is correct.

Explanation

When three cubic objects are melted and formed into a single cube, the total volume remains conserved. The volume of a cube is given by the formula V = s³, where s is the side length.

The total volume of the three cubic biscuits is: Vₜₒₜₐₗ = 0.3³ + 0.4³ + 0.5³ Vₜₒₜₐₗ = 0.027 + 0.064 + 0.125 = 0.216 m³

Let the edge of the resulting single cubic biscuit be S. Then: S³ = 0.216 S = ∛0.216 = 0.6 m

The total surface area (TSA) of a cube is calculated using the formula TSA = 6S². Substituting the value of the edge S: TSA = 6 × (0.6)² TSA = 6 × 0.36 = 2.16 m²

Thus, the total surface area of the new cubic biscuit is 2.16 m², which corresponds to option d.

Explanation

To find the speed of the bicycle, we first determine the effective outer radius of the wheel, as the distance covered depends on the outermost circumference. The inner radius is half of the inner diameter, which is 50 / 2 = 25 cm. Adding the thickness of 10 cm to this gives the outer radius R = 25 + 10 = 35 cm = 0.35 m.

The formula for the distance covered in one revolution is the circumference C = 2πR. For n revolutions, the total distance D is given by D = n × 2πR. The speed v is the total distance divided by the total time t, expressed by the formula v = (n × 2πR)/t.

Substituting the given values: v = (10 × 2 × π × 0.35)/5 v = (20 × 3.14159 × 0.35)/5 v = 4 × 3.14159 × 0.35 v = 1.4 × 3.14159 v ≈ 4.398 m/s

Rounding to one decimal place, we get 4.4 m/s, which corresponds to option b.

Explanation

The length of a wire bent into the form of a semicircle represents the total perimeter of that semicircle. The perimeter P of a semicircle consists of the curved arc length plus the straight diameter. The formula for the perimeter is P = πr + 2r, where r is the radius.

Given that the length of the wire P = 36 cm, we substitute the values into the formula using π = 22/7:

36 = r(22/7 + 2)

To solve for r, first find a common denominator for the terms inside the parenthesis:

36 = r((22 + 14)/7)

36 = r(36/7)

Now, isolate r by multiplying both sides by 7/36:

r = (36 × 7)/36

r = 7 cm

Thus, the radius of the semicircle is 7 cm, which matches option (c).

Explanation

The total volume of earth removed is calculated using the formula: Total Volume = Number of strokes × Volume of earth removed per stroke. Substituting the given values, Total Volume = 250 × 500 cm³ = 125,000 cm³. The pit is in the shape of a cuboid, so its volume is also determined by the formula: Volume = Length × Width × Depth. Given the base dimensions are 50 cm × 50 cm, we set up the equation 50 cm × 50 cm × depth = 125,000 cm³. This simplifies to 2500 × depth = 125,000. Solving for depth, we find depth = 125,000 / 2500 = 50 cm. To express the final answer in meters as per the options, we use the conversion 1 m = 100 cm, which gives depth = 50 / 100 = 0.5 m. This matches option d.

Explanation

Let B be the number of boys and G be the number of girls. B + G = 150. The total weight is 150 * 60 = 9000 kg. The sum of weights is 70B + 55G = 9000. Substituting B = 150 - G, we get 70(150 - G) + 55G = 9000, which is 10500 - 70G + 55G = 9000. Then 15G = 1500, so G = 100.

Explanation

To find the average marks of all candidates, we use the weighted average formula: Overall Average = ((n₁ × A₁) + (n₂ × A₂))/(n₁ + n₂) In this case, we can use the percentages as weights. Let the total number of candidates be 100. Number of candidates who wrote in Hindi (n₁) = 40% of 100 = 40 Number of candidates who wrote in English (n₂) = (100% - 40%) of 100 = 60% of 100 = 60 Average marks for Hindi candidates (A₁) = 74 Average marks for English candidates (A₂) = 77

Substituting these values into the formula: Overall Average = ((40 × 74) + (60 × 77))/(40 + 60) Overall Average = (2960 + 4620)/100 Overall Average = 7580/100 = 75.8

Alternatively, using the shortcut method of deviation: The difference between the two averages is 77 - 74 = 3. The ratio of candidates is Hindi : English = 40 : 60 = 2 : 3. The overall average will be 74 + (3/(2+3) × 3) = 74 + (3/5 × 3) = 74 + 1.8 = 75.8. Thus, option b is the correct result.

Explanation

The given frequency distribution presents class intervals where the lower limit remains constant at 0, while the upper limit increases linearly in each step. These intervals, such as 0-10, 0-20, 0-30, 0-40, and 0-50, essentially categorize the data into groups that are less than 10, less than 20, less than 30, and so on. In this type of distribution, the frequency of each successive class is the sum of the frequencies of all preceding classes, which is evident as the frequency values increase monotonically from 3 to 25. This specific structure is known as a cumulative frequency distribution of the less than type. In a more than type distribution, the lower limit would change (e.g., more than 0, more than 10) and the frequencies would typically decrease as the limit increases. Therefore, option b is the correct identification of this statistical representation.

Explanation

A histogram is a powerful tool in statistics used to represent the frequency distribution of a continuous variable. Statement 1 is correct because a histogram serves as a visual equivalent to a frequency distribution table, where the x axis represents the class intervals and the y axis represents the frequency of observations within those intervals. The total area of the bars represents the total frequency of the data set, providing a clear picture of the data distribution, skewness, and modality.

Statement 2 is incorrect because it mischaracterizes the nature of continuous random variables. A histogram is suitable for any continuous random variable regardless of how the data is distributed. While the histogram represents the frequency as being evenly distributed across a specific bin (interval) to create a rectangular bar, the actual data points of a continuous random variable are rarely evenly distributed within those intervals. The uniform distribution within a bin is a simplifying assumption used for the graphical representation and for calculating the median or mode from the graph, not a prerequisite property of the variable itself.

Explanation

To find the median, first arrange the values in ascending order: 6, 7, 9, 11, 12, 15, 19. Since there are 7 values (an odd number), the median is the middle value (the 4th value), which is 11.

Explanation

M(x_i + y_i) = (1/n) sum(x_i + y_i) = (1/n) sum(x_i) + (1/n) * sum(y_i) = M(x_i) + M(y_i). Correct. G(x_i y_i) = (product(x_i y_i))^(1/n) = (product(x_i))^(1/n) (product(y_i))^(1/n) = G(x_i) G(y_i). Correct.

Explanation

Let the common perimeter of the circle and the square be P. For a circle, the perimeter (circumference) is given by 2πr = P, which implies the radius r = P/2π. The area of the circle is A꜀ = πr² = π(P/2π)² = P²/4π. For a square, the perimeter is 4s = P, where s is the side length, so s = P/4. The area of the square is Aₛ = s² = (P/4)² = P²/16. To compare the two areas, we compare the denominators 4π and 16. Since π is approximately 3.14159, 4π ≈ 12.566. Since 12.566 < 16, the value of P²/4π is greater than P²/16. Therefore, the area of the circle is larger than the area of the square, making option b correct. This is a general property in geometry where for a given perimeter, the circle encloses the maximum possible area among all closed plane figures.

Explanation

Assuming a typo in the question where 32 1/2 is actually 322: If x - 1/x = 4, then (x - 1/x)^2 = 16, which means x^2 + 1/x^2 - 2 = 16, so x^2 + 1/x^2 = 18. Then (x^2 + 1/x^2)^2 = 18^2 = 324, which means x^4 + 1/x^4 + 2 = 324, so x^4 + 1/x^4 = 322. This matches option D.

Explanation

Statement 1 is correct based on the AAA similarity criterion in geometry. Two triangles are said to be similar if their corresponding angles are equal and their corresponding sides are in the same ratio. If two triangles are equi-angular, it means all three corresponding angles are equal. According to the AAA theorem, this condition is sufficient to prove that the triangles are similar, although they may differ in size (scale).

Statement 2 is incorrect because equality of area does not imply similarity. The area of a triangle is given by the formula Area = 1/2 × base × height. Multiple triangles can have the same area but entirely different dimensions and angles. For example, a triangle with a base of 4 units and a height of 3 units has an area of 6 square units. Similarly, a triangle with a base of 6 units and a height of 2 units also has an area of 6 square units. These two triangles are not necessarily similar as their side ratios and internal angles can be completely different. Thus, only statement 1 is correct, making option a the right choice.

Explanation

For any two similar triangles, there is a fundamental geometric property stating that the ratio of their areas is equal to the square of the ratio of their corresponding sides. This relationship is defined by the formula: Ratio of Areas = (Ratio of Corresponding Sides)².

In this problem, the ratio of the corresponding sides is given as 9 : 4. By applying the formula directly, we substitute the given values: Ratio of Areas = (9/4)² Ratio of Areas = 9² / 4² Ratio of Areas = 81 / 16

Thus, the ratio of the areas of the two similar triangles is 81 : 16, which matches option (c). This is a standard result in geometry where the area of a figure scales with the square of its linear dimensions.

Explanation

A quadrilateral whose sides are tangent to a circle is known as a tangential quadrilateral. According to the property of tangents drawn from an external point to a circle, the lengths of the two tangent segments from a single vertex to the points of tangency are equal. Let the points of tangency on sides AB, BC, CD, and DA be P, Q, R, and S respectively. Therefore, AP = AS, BP = BQ, CQ = CR, and DR = DS.

To find the relationship between the sides, we calculate the sum of the opposite sides. The sum of sides AB and CD is given by AB + CD = (AP + PB) + (CR + RD). Substituting the equal tangent segments, we get AB + CD = AS + BQ + CQ + DS. Rearranging these terms, we see that (AS + DS) + (BQ + CQ) = AD + BC. This result is known as Pitot's Theorem, which states that for a tangential quadrilateral, the sums of opposite sides are equal. Thus, AB + CD = AD + BC, which corresponds to option c.

Explanation

Let the selling price of 1 dozen pencils be SP. The shopkeeper sells 8 dozen pencils, so the total selling price is 8 × SP. The problem states that the gain is equal to the selling price of 1 dozen pencils, which means Profit = 1 × SP.

Using the fundamental formula Cost Price (CP) = Selling Price (SP) - Profit, we can calculate the cost price for 8 dozen pencils as follows: Total CP = (8 × SP) - (1 × SP) = 7 × SP.

The gain percentage is calculated using the formula: Gain % = (Profit/CP) × 100

Substituting the values: Gain % = (SP / (7 × SP)) × 100 = 100/7%

Converting the improper fraction 100/7 into a mixed fraction: 100 ÷ 7 = 14 with a remainder of 2. Thus, the gain percentage is 14 2/7%. This corresponds to option c.

Explanation

To find the ratio x:z, we use the property of compound ratios where the ratio of A to C is the product of the ratio of A to B and the ratio of B to C. The formula is A/C = A/B × B/C. Substituting the given values into this formula, we get 2 = x/8 × 12/z. Simplifying the expression on the right side, we have 2 = 12x/8z. Further reducing the fraction 12/8 by dividing both numerator and denominator by 4, we get 2 = 3x/2z. To isolate the ratio x/z, multiply both sides by 2 to get 4 = 3x/z. Now, dividing both sides by 3 yields x/z = 4/3. Therefore, the ratio x:z is 4:3, which corresponds to option c.

Explanation

To find the equation whose roots are -α⁻¹ and -β⁻¹, we use the transformation method which is the fastest approach for such problems. If the roots of the original equation x² + px + q = 0 are α and β, and we want to find an equation with roots y = -1/x, we substitute x = -1/y into the original equation.

Substituting x = -1/y into x² + px + q = 0: (-1/y)² + p(-1/y) + q = 0 1/y² - p/y + q = 0

To eliminate the denominators, multiply the entire equation by y²: y²(1/y²) - y²(p/y) + y²(q) = 0 1 - py + qy² = 0

Rearranging the terms in standard quadratic form gives: qy² - py + 1 = 0

Replacing y with x to match the given options, we get qx² - px + 1 = 0, which corresponds to option a).

Alternatively, using the sum and product of roots: Sum of original roots α + β = -p and product αβ = q. Sum of new roots S = -1/α - 1/β = -(α + β)/αβ = -(-p)/q = p/q. Product of new roots P = (-1/α)(-1/β) = 1/αβ = 1/q. The required equation is x² - Sx + P = 0: x² - (p/q)x + 1/q = 0 Multiplying by q yields qx² - px + 1 = 0.

Explanation

The geometric mean (GM) of a set of n positive numbers x₁, x₂, ..., xₙ is defined by the formula GM = (x₁ · x₂ · ... · xₙ)¹/ⁿ. For the geometric mean to exist as a positive real number, the product of the terms must be positive. If each number in the set is reduced by 2, some numbers may become zero or negative. For instance, if the set contains the number 1, subtracting 2 results in -1. If the product of the resulting numbers is negative and n is an even integer, the n-th root is not a real number, meaning the geometric mean does not exist in the real number system. Therefore, statement 1 is correct.

Statement 2 is incorrect because the geometric mean is not a linear operator. Unlike the arithmetic mean, where adding a constant k to every term increases the mean by k, the geometric mean does not follow this property. For example, consider the set {1, 4}. The original GM is √(1 · 4) = 2. If each number is increased by 2, the set becomes {3, 6}, and the new GM is √(3 · 6) = √18 ≈ 4.24. This value is not equal to 2 + 2 = 4. Thus, increasing each term by 2 does not simply increase the GM by 2. Option a is the correct choice.

Explanation

The slant height l of a frustum of a right-circular cone is calculated using the formula l = √(h² + (R - r)²), where h represents the vertical height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base. Given the values from the problem, the height h = 9 cm, the larger radius R = 6 cm, and the smaller radius r = 3 cm.

Substituting these values into the formula, we first find the difference between the radii: R - r = 6 cm - 3 cm = 3 cm. Now, substituting h and (R - r) into the slant height equation: l = √(9² + 3²) l = √(81 + 9) l = √90 l = √(9 × 10) l = 3√10 cm.

Thus, the slant height of the given frustum is 3√10 cm, which corresponds to option a.

Explanation

The height of the original cone can be determined using the property of similar triangles, as the smaller cone removed from the top and the larger original cone are similar. Let H be the height of the full cone, h be the height of the frustum, R be the radius of the larger base, and r be the radius of the smaller base. The relationship between these variables is given by the formula H = (R · h)/(R - r).

Substituting the given values R = 6 cm, r = 3 cm, and h = 9 cm into the formula, we get: H = (6 · 9)/(6 - 3) = 54/3 = 18 cm.

Alternatively, using the similarity ratio, the ratio of the radii is equal to the ratio of the heights: r/R = (H - h)/H 3/6 = (H - 9)/H 1/2 = (H - 9)/H H = 2(H - 9) H = 2H - 18 H = 18 cm.

Thus, the height of the cone is 18cm, which corresponds to option (d).

Explanation

To find the total surface area (TSA) of the frustum, we need the sum of its lateral surface area and the areas of its two circular bases. The given parameters are the larger radius R = 6 cm, the smaller radius r = 3 cm, and the height h = 9 cm.

First, calculate the slant height l using the formula l = √(h² + (R - r)²). Substituting the values: l = √(9² + (6 - 3)²) = √(81 + 3²) = √(81 + 9) = √90. Simplifying the radical gives l = 3√10 cm.

Next, we apply the total surface area formula for a frustum: TSA = π(R + r)l + πR² + πr².

Substitute the known values into the equation: TSA = π(6 + 3)(3√10) + π(6²) + π(3²) TSA = π(9)(3√10) + 36π + 9π TSA = 27π√10 + 45π.

To match the options, factor out 9π from the expression: TSA = 9π(3√10 + 5) cm².

This result matches option (b).

Explanation

Given that AO:OC = BO:OD and ∠AOB = ∠COD (vertically opposite angles), ΔAOB is similar to ΔCOD by SAS similarity. This implies that ∠OAB = ∠OCD, which proves that AB is parallel to CD since these are alternate interior angles. Consequently, the quadrilateral ABCD is a trapezoid.

The area of a trapezoid is given by the formula: Area = 1/2(AB + CD) × h

Substituting the given values: 1050 = 1/2(30 + 40) × h 1050 = 35h h = 1050/35 = 30 cm

The distance AE is the height h of the trapezoid, so AE = 30 cm. Let F be the projection of B onto the line CD. Since AB is parallel to CD and AE is perpendicular to CD, the figure ABFE is a rectangle. Thus, EF = AB = 30 cm and BF = AE = 30 cm.

In the right-angled triangle BFE, applying the Pythagoras theorem: BE² = BF² + FE² BE² = 30² + 30² BE² = 900 + 900 = 1800 BE = √1800 = 30√2 cm

This corresponds to option b.

Explanation

Given the condition AO/OC = BO/OD and the fact that ∠AOB = ∠COD (vertically opposite angles), ΔAOB is similar to ΔCOD by the SAS similarity criterion. The ratio of their corresponding sides is AB/CD = 30/40 = 3/4. Consequently, the ratio of the segments of the diagonals is BO/OD = 3/4.

The triangles ABC and ADC share a common base AC. The ratio of the areas of two triangles with the same base is equal to the ratio of their corresponding altitudes. Since the altitude from B to AC and the altitude from D to AC are proportional to the segments BO and OD respectively, we have Area(ΔABC)/Area(ΔADC) = BO/OD = 3/4.

Let the area of ΔADC be x. Then the area of ΔABC is 3/4x. The total area of the quadrilateral ABCD is the sum of these two triangles: 3/4x + x = 1050 7/4x = 1050 x = (1050 × 4)/7 x = 150 × 4 = 600cm²

Thus, the area of ΔADC is 600cm², which corresponds to option c.

Explanation

Given that AO:OC = BO:OD, the triangles ΔAOB and ΔCOD are similar by SAS similarity criterion (since ∠AOB = ∠COD are vertically opposite angles). This implies that AB is parallel to CD, making ABCD a trapezoid.

The area of a trapezoid is given by the formula Area = 1/2 × (AB + CD) × h, where h is the perpendicular distance between the parallel sides. Substituting the given values, 1050 = 1/2 × (30 + 40) × h, which simplifies to 1050 = 35h. Solving for h, we get h = 1050/35 = 30 cm. Since AE is the perpendicular drawn from A to CD, AE represents the height h of the trapezoid, so AE = 30 cm.

Because AB || CD and AE ⊥ CD, it follows that AE ⊥ AB, meaning ∠EAB = 90°. In ΔAEB, we have AE = 30 cm and AB = 30 cm. Since two sides are equal and the included angle is 90°, ΔAEB is a right-angled isosceles triangle. In such a triangle, the acute angles are equal, thus ∠AEB = ∠ABE = (180° - 90°)/2 = 45°. Therefore, option B is correct.