CDS II 2009 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

Let the rates of work for P, Q, and R be P, Q, and R units per day respectively. The combined rate of P and Q is 1/2 units per day, the combined rate of Q and R is 1/4 units per day, and the combined rate of P and R is 1/(12/5) = 5/12 units per day. Summing these rates gives 2(P + Q + R) = 1/2 + 1/4 + 5/12. Finding the common denominator of 12, we have 2(P + Q + R) = (6 + 3 + 5)/12 = 14/12 = 7/6. Dividing by 2, the combined efficiency of P, Q, and R is P + Q + R = 7/12 units per day. To find P's individual efficiency, subtract the rate of Q and R from the total combined rate: P = (P + Q + R) - (Q + R) = 7/12 - 1/4. This calculation simplifies to P = 7/12 - 3/12 = 4/12 = 1/3 units per day. The total time required for P alone to complete the job is the reciprocal of the rate, which is 1 / (1/3) = 3 days. This corresponds to option b.

Explanation

To calculate the total amount to be paid at the end of two years, we use the compound interest formula A = P(1 + R/100)ⁿ. Substituting the given values, P = 7500, R = 16, and n = 2, we get A = 7500(1 + 16/100)². This simplifies to A = 7500(1.16)², which further becomes A = 7500 × 1.3456. Multiplying these values gives A = 10092.

Alternatively, for faster calculation in competitive exams, use the effective interest rate formula for two years: Rₑf_f = R + R + (R × R)/100. For R = 16, Rₑf_f = 16 + 16 + (16 × 16)/100 = 32 + 2.56 = 34.56%. The total compound interest is 7500 × 34.56% = 2592. Adding the principal, the total amount to be paid is 7500 + 2592 = 10092. Thus, the correct option is b.

Explanation

When two items are sold at the same selling price, one at a gain of x% and the other at a loss of x%, there is always an overall loss. The standard formula to calculate this net loss percentage is Loss % = x²/100. In this problem, the selling price of each cycle is Rs. 1,188, and the common percentage of gain and loss is x = 10%. Substituting this value into the formula, we get Loss % = 10²/100 = 100/100 = 1%. Therefore, the trader incurs a 1% loss on the whole transaction. This shortcut eliminates the need to calculate the cost price of each item individually. The result is independent of the actual selling price of Rs. 1,188, provided it is the same for both items. Thus, option (a) is the correct result.

Explanation

The least number is found by taking the LCM of the divisors and subtracting the common difference. The divisors are 42, 72, and 84. The differences between divisors and their respective remainders are 42-25=17, 72-55=17, and 84-67=17. The LCM of 42, 72, and 84 is 504. The required number is 504 - 17 = 487.

Explanation

Summing the three given equations (a+b) + (b+c) + (a+c) = 2m² + 6m + 2, we get 2(a+b+c) = 2(m² + 3m + 1), which simplifies to a+b+c = m² + 3m + 1. We can now express a, b, and c individually in terms of m: a = (a+b+c) - (b+c) = (m² + 3m + 1) - 6m = m² - 3m + 1 b = (a+b+c) - (a+c) = (m² + 3m + 1) - 2 = m² + 3m - 1 c = (a+b+c) - (a+b) = (m² + 3m + 1) - 2m² = -m² + 3m + 1

Applying the condition a ≤ b: m² - 3m + 1 ≤ m² + 3m - 1 2 ≤ 6m ⇒ m ≥ 1/3

Applying the condition b ≤ c: m² + 3m - 1 ≤ -m² + 3m + 1 2m² ≤ 2 ⇒ m² ≤ 1 ⇒ -1 ≤ m ≤ 1

Combining these two inequalities, the range of m is the intersection of m ≥ 1/3 and -1 ≤ m ≤ 1, which results in 1/3 ≤ m ≤ 1. This corresponds to option c.

Explanation

Let the daily charge per student be k. Total cost = number of students number of days k. 6300 = 35 24 k, which gives k = 6300 / 840 = 7.5. For the second case, 3375 = 25 days 7.5. Days = 3375 / (25 * 7.5) = 3375 / 187.5 = 18.

Explanation

The profit distribution in a partnership is based on the ratio of the product of capital and time period. The formula used is Profitₚ/Profit_Q = (Capitalₚ × Timeₚ)/(Capital_Q × Time_Q).

Given: Capital of P (Cₚ) = Rs. 2,525 Time of P (Tₚ) = 12 months Capital of Q (C_Q) = Rs. 1,200 Time of Q (T_Q) = x months Profit share of P (Pₚ) = Rs. 1,212 Profit share of Q (P_Q) = Total Profit - Pₚ = 1,644 - 1,212 = 432

Substituting these values into the formula: 1212/432 = (2525 × 12)/(1200 × x)

Simplifying the left side by dividing both numerator and denominator by 12: 101/36 = (2525 × 12)/(1200 × x)

Simplifying the right side: 101/36 = 2525/(100 × x) 101/36 = 101/4x

By comparing the numerators, we get: 36 = 4x x = 9 months

Since Q's investment was for 9 months in a year, Q joined the business after 12 - 9 = 3 months. Thus, option b is correct.

Explanation

To find the value of √0.0001, first convert the decimal into a fraction. We have 0.0001 = 1/10000. Now, apply the square root to both the numerator and the denominator: √0.0001 = √(1/10000) = √1/√10000 = 1/100. In decimal form, 1/100 is equal to 0.01. By definition, a rational number is any number that can be expressed in the form p/q, where p and q are integers and q is not zero. Since 0.01 can be written as the fraction 1/100, it is a rational number. Option (a) is incorrect because the value is exactly 0.01, not less than 0.01. Therefore, the most accurate description among the choices is that it is a rational number, making option (b) the correct choice.

Explanation

To find the value of k, we use the property that the product of a cycle of ratios equals 1. Given the ratios x/y = 1/3, y/z = 5/k, z/t = 2/5, and t/x = 3/4, we can set up the following equation: x/y ⋅ y/z ⋅ z/t ⋅ t/x = 1 Substituting the given values into the equation, we get: 1/3 ⋅ 5/k ⋅ 2/5 ⋅ 3/4 = 1 By canceling out the common terms in the numerator and denominator, specifically 3 and 5, the equation simplifies to: (1 ⋅ 5 ⋅ 2 ⋅ 3)/(3 ⋅ k ⋅ 5 ⋅ 4) = 1 30/60k = 1 1/2k = 1 Solving for k: 2k = 1 k = 1/2 Thus, the value of k is 1/2, which corresponds to option a.

Explanation

Let the distance be D and the initial speed be v. In the first scenario, the person rests for 9 hours a day, meaning the actual walking time per day is 24 - 9 = 15 hours. The total walking time over 114 days is 114 × 15 hours. Using the formula Distance = Speed × Time, we have D = v × (114 × 15).

In the second scenario, the distance to cover is 2D. The person now rests twice as long, which is 9 × 2 = 18 hours a day. Consequently, the walking time per day becomes 24 - 18 = 6 hours. The speed is also doubled, becoming 2v. Let T be the number of days required to cover the new distance. The total distance equation becomes 2D = 2v × (T × 6).

Substituting the value of D from the first equation into the second, we get 2 × (v × 114 × 15) = 12v × T. Simplifying the equation by canceling v from both sides, we have 2 × 1710 = 12T, which leads to 3420 = 12T. Solving for T, we find T = 3420/12 = 285 days. Thus, the person will take 285 days, which corresponds to option c.

Explanation

When two objects move in the same direction, their relative speed is the difference between their individual speeds. Here, the relative speed of the faster train with respect to the man in the slower train is given by the formula Vᵣₑₗ = V₁ - V₂. Substituting the given values, Vᵣₑₗ = 50 km/hr - 32 km/hr = 18 km/hr.

To perform calculations with time in seconds, we must convert the relative speed from km/hr to m/s by multiplying with 5/18. Thus, Vᵣₑₗ = 18 × 5/18 = 5 m/s.

Since the man is considered a point object, the distance to be covered by the faster train to pass him completely is equal to the length of the faster train itself. Using the distance formula Distance = Speed × Time, we get: Length of faster train = Vᵣₑₗ × t = 5 m/s × 15 s = 75 m.

Therefore, the length of the faster train is 75 m, which corresponds to option a).

Explanation

Total ration available = 100 soldiers 10 days = 1000 soldier-days. Ration consumed in 2 days = 100 soldiers 2 days = 200 soldier-days. Remaining ration = 1000 - 200 = 800 soldier-days. New number of soldiers = 100 + 60 = 160. Remaining days = 800 / 160 = 5 days.

Explanation

When two objects move in opposite directions from the same starting point, their relative speed is the sum of their individual speeds. Using the formula for relative speed in opposite directions, we get:

Relative Speed = S₁ + S₂

Substituting the given values: Relative Speed = 5 km/hr + 6.5 km/hr = 11.5 km/hr

To find the time taken to reach a specific distance apart, we use the standard time formula: Time = Distance / Relative Speed

Substituting the target distance of 92 km and the relative speed calculated above: Time = 92 km / 11.5 km/hr

To simplify the calculation, we can multiply both the numerator and denominator by 10: Time = 920 / 115 = 8 hours

Therefore, the time they will take to be 92 km apart is 8 hours, which corresponds to option (d).

Explanation

Let the initial number of workers be x. An increase of 15 workers corresponds to 20% of x. So, 0.20x = 15. x = 15 / 0.20 = 75.

Explanation

The amount for compound interest is calculated using the formula A = P(1 + r/n)ⁿᵗ, where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this problem, the principal P = 400, the annual rate r = 10% = 0.10, the compounding frequency n = 2 (half-yearly), and the time t = 3/2 years.

Substituting the values into the formula, the rate per half-year becomes r/n = 0.10/2 = 0.05 and the total number of compounding periods becomes nt = 2 × (3/2) = 3.

The calculation is as follows: A = 400(1 + 0.05)³ A = 400(1.05)³ A = 400 × 1.157625 A = 463.05

Therefore, the principal of Rs. 400 will amount to Rs. 463.05 in 3/2 years when compounded half-yearly. This matches option (b).

Explanation

The fastest way to solve problems involving successive percentage decreases is by using the formula for effective percentage change: Net Change % = x + y + xy/100. In this case, since the population is decreasing twice by 10%, we substitute x = -10 and y = -10 into the formula. This gives us -10 + (-10) + ((-10) × (-10))/100, which simplifies to -20 + 100/100, resulting in -20 + 1 = -19%. This means the total population decreased by 19%. Therefore, the remaining population percentage is 100% - 19% = 81%. Alternatively, using the multiplier method, the remaining population is calculated as 100 × (1 - 0.10) × (1 - 0.10) = 100 × 0.90 × 0.90 = 100 × 0.81 = 81%. This confirms that option d is the correct result.

Explanation

To determine the largest value, we expand the repeating decimals to several decimal places to allow for a direct digit-by-digit comparison. Option a is 0.725, which can be written as 0.725000... Option b is 0.725̅, where only the digit 5 repeats, resulting in 0.725555... Option c is 0.725̅, where the block 25 repeats, resulting in 0.725252... Option d is 0.725̅, where the block 725 repeats, resulting in 0.725725... Comparing these values from left to right, the first three decimal places are identical for all options as 7, 2, and 5. However, at the fourth decimal place, the values are 0 for a, 5 for b, 2 for c, and 7 for d. Since 7 is the largest digit at the tenth-thousandths position, 0.725̅ is the largest among the given options.

Explanation

Given m = n(n-1). The expression is m^2 - 2m = m(m-2) = n(n-1)(n^2 - n - 2). Factoring the quadratic gives n(n-1)(n-2)(n+1). This is the product of four consecutive integers. The product of k consecutive integers is always divisible by k!, and 4! = 24.

Explanation

In problems where a trader sells a mixture at the cost price of the pure ingredient, the profit percentage is directly proportional to the ratio of the added impurity (water) to the pure quantity (milk). The applicable formula is Profit % = (Quantity of water/Quantity of milk) × 100.

Given that the profit is 12.5%, we can express this as a fraction: 12.5% = 12.5/100 = 1/8. Substituting the known values into the formula: 1/8 = Quantity of water/28.

Multiplying both sides by 28 to isolate the quantity of water: Quantity of water = 28/8 = 3.5 litres.

This calculation shows that the vendor added 3.5 litres of water to the 28 litres of milk to achieve a gain of 12.5% while keeping the selling price equal to the cost price. This corresponds to option c.

Explanation

Let the ratio of x to y be denoted as r = x/y. According to the problem, the ratio of x to y is 25 times the ratio of y to x. This relationship can be expressed as a mathematical equation: x/y = 25 × y/x. Substituting r for x/y and 1/r for y/x, the equation becomes r = 25 × 1/r. Multiplying both sides of the equation by r gives r² = 25. Taking the square root of both sides, we find r = √25 = 5. Thus, the ratio x/y = 5/1, which is expressed as 5:1. This corresponds to option b.

Explanation

Let the total number of swans be x. The equation is x = (7/2)sqrt(x) + 2. Let u = sqrt(x), then u^2 - (7/2)u - 2 = 0, or 2u^2 - 7u - 4 = 0. Solving for u using the quadratic formula: u = (7 + sqrt(49 + 32))/4 = (7 + 9)/4 = 4. Since u = 4, x = 4^2 = 16.

Explanation

Let the full fare be F and the reservation charge be R. Based on the problem, the total cost of one reserved first class ticket is given by the formula: F + R = 362. The cost for one full and one half reserved tickets is the sum of one full ticket and one child's ticket, where the child's fare is 0.5F but the reservation charge R remains the same. Thus, (F + R) + (0.5F + R) = 554. Substituting the value of F + R = 362 into this equation, we get 362 + 0.5F + R = 554. Subtracting 362 from both sides yields 0.5F + R = 192. To find R, we can subtract this resulting equation from the first one: (F + R) - (0.5F + R) = 362 - 192. This simplifies to 0.5F = 170, meaning the full fare F = 340. Substituting F = 340 back into the first equation 340 + R = 362, we find R = 362 - 340 = 22. Therefore, the reservation charge is Rs. 22, which matches option b.

Explanation

Let r be the number of rows and s be the students per row. Total students N = rs. Condition 1: (s+1)(r-2) = rs => r - 2s = 2. Condition 2: (s-1)(r+3) = rs => 3s - r = 3. Adding the two equations: s = 5. Substituting s = 5 into the first equation: r - 10 = 2 => r = 12. Total students = 12 * 5 = 60.

Explanation

To find the Least Common Multiple (LCM) of two algebraic expressions, we must first factorize them completely into their irreducible components. The LCM is then the product of the highest power of every factor that appears in either expression.

First, we factorize the numerical coefficients: the LCM of 3 and 11 is 3 × 11 = 33.

Next, we factorize the algebraic parts. For the first expression, we use the difference of cubes formula: 3(a³ - b³) = 3(a - b)(a² + ab + b²)

For the second expression, we use the difference of squares formula twice: 11(a⁴ - b⁴) = 11(a² - b²)(a² + b²) = 11(a - b)(a + b)(a² + b²)

Now, we identify all unique factors present in both expressions: The numerical factor 33. The common factor (a - b). The factor (a² + ab + b²). The factor (a + b). The factor (a² + b²).

Multiplying these together, we get: LCM = 33(a - b)(a² + ab + b²)(a + b)(a² + b²)

Since (a - b)(a² + ab + b²) = a³ - b³, we can rewrite the expression as: LCM = 33(a³ - b³)(a + b)(a² + b²)

This matches option a.

Explanation

Let the number be 10x + y. x + y = 10. Also, (10x + y) - 18 = 10y + x, which simplifies to 9x - 9y = 18 or x - y = 2. Solving x + y = 10 and x - y = 2 gives x = 6 and y = 4. The product of digits is 6 * 4 = 24.

Explanation

The first polynomial x³ - 27 can be factorized using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²). Thus, x³ - 3³ = (x - 3)(x² + 3x + 9). The factors of this polynomial are a linear factor (x - 3) and a quadratic factor (x² + 3x + 9). Since the Highest Common Factor (HCF) of the two given polynomials is a quadratic polynomial, the HCF must be x² + 3x + 9.

For x² + 3x + 9 to be the HCF, it must divide the second polynomial Q(x) = x³ + 4x² + 12x + k without leaving a remainder. We can use the method of comparing coefficients. Let Q(x) = (x² + 3x + 9)(x + a). Expanding this product, we get x³ + ax² + 3x² + 3ax + 9x + 9a, which simplifies to x³ + (a + 3)x² + (3a + 9)x + 9a.

Comparing the coefficients of x² with the given polynomial x³ + 4x² + 12x + k: a + 3 = 4 a = 1

To verify, compare the coefficients of x: 3a + 9 = 3(1) + 9 = 12 This matches the coefficient of x in Q(x). Now, compare the constant terms: k = 9a k = 9(1) = 9

Thus, the value of k is 9, which corresponds to option (b).

Explanation

Let x^4 = u. The problem becomes dividing u^10 + 2 by u + 1. According to the Remainder Theorem, the remainder is the value of the polynomial when u = -1. Remainder = (-1)^10 + 2 = 1 + 2 = 3.

Explanation

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - a), the remainder is equal to f(a). Mathematically, this is expressed as f(x) = (x - a)q(x) + f(a), where q(x) is the quotient. The Factor Theorem is a specific application of this principle. It states that (x - a) is a factor of f(x) if and only if f(a) = 0. Since the condition f(a) = 0 is simply the particular case where the remainder provided by the Remainder Theorem is zero, the Factor Theorem is a special case of the Remainder Theorem. Therefore, option B is the correct statement.

Explanation

x = 2 + sqrt(3), then 1/x = 2 - sqrt(3). x + 1/x = (2 + sqrt(3)) + (2 - sqrt(3)) = 4. Squaring both sides: (x + 1/x)^2 = 4^2 => x^2 + 2 + 1/x^2 = 16. Therefore, x^2 + x^-2 = 14.

Explanation

For 7x^2 + 12x + 18 = 0, the sum of roots (a+b) = -12/7 and the product (ab) = 18/7. The sum of squares a^2 + b^2 = (a+b)^2 - 2ab = (-12/7)^2 - 2(18/7) = 144/49 - 36/7 = (144 - 252)/49 = -108/49. The ratio of the sum of squares to the product is (-108/49) / (18/7) = (-108/49) * (7/18) = -6/7. While -6/7 is the mathematically correct result for the given equation, the answer key provided is C (-6:1), which would occur if the coefficient of x^2 was 1 and the constant was -36 or other specific values. Based on the provided answer key, we select C.

Explanation

According to the polynomial division algorithm, for any polynomial P(x) and a non-zero divisor D(x), there exist unique polynomials Q(x) and R(x) such that P(x) = D(x)Q(x) + R(x), where either R(x) = 0 or the degree of R(x) is strictly less than the degree of D(x). In this case, the divisor D(x) is a linear polynomial, which means its degree is exactly 1. Consequently, the degree of the remainder R(x) must be less than 1. A polynomial with a degree less than 1 can only be a polynomial of degree 0, which is a non-zero constant polynomial, or it can be the zero polynomial. Therefore, the remainder is either a constant polynomial or the zero polynomial, making option (c) the correct choice. If the linear polynomial is a factor of P(x), the remainder is the zero polynomial; otherwise, it is a constant.

Explanation

Let aˣ = bʸ = cᶻ = k. From this equality, we can express a, b, and c in terms of k as a = k¹/ˣ, b = k¹/ʸ, and c = k¹/ᶻ. Given the condition abc = 1, we substitute the values of a, b, and c into the equation: k¹/ˣ · k¹/ʸ · k¹/ᶻ = 1 Using the law of exponents kᵐ · kⁿ = kᵐ⁺ⁿ, we simplify the left side: k⁽¹/ˣ ⁺ ¹/ʸ ⁺ ¹/ᶻ⁾ = 1 Since any non-zero base raised to the power of 0 equals 1, we can write 1 as k⁰: k⁽¹/ˣ ⁺ ¹/ʸ ⁺ ¹/ᶻ⁾ = k⁰ By comparing the exponents on both sides, we get: 1/x + 1/y + 1/z = 0 To solve for the required expression, we take the common denominator xyz: (yz + xz + xy)/xyz = 0 Multiplying both sides by xyz, we find: xy + yz + zx = 0 This result matches option (c).

Explanation

Let (3.7)ˣ = 10000 and (0.037)ʸ = 10000. From these equations, we can express the bases in terms of the constant 10000 as 3.7 = 10000¹/ˣ and 0.037 = 10000¹/ʸ. We observe that the two bases are related as 0.037 = 3.7/100 = 3.7 × 10⁻². Substituting the expressions for the bases, we get 10000¹/ʸ = 10000¹/ˣ × 10⁻². Since 10000 = 10⁴, we can write 10 = 10000¹/⁴, and therefore 10⁻² = (10000¹/⁴)⁻² = 10000⁻²/⁴ = 10000⁻¹/². Substituting this value back into the equation gives 10000¹/ʸ = 10000¹/ˣ × 10000⁻¹/². Using the law of exponents aᵐ × aⁿ = aᵐ⁺ⁿ, we get 10000¹/ʸ = 10000¹/ˣ⁻¹/². By equating the exponents of the same base, we obtain 1/y = 1/x - 1/2. Rearranging the terms to find the required value, we get 1/x - 1/y = 1/2, which matches option c.

Explanation

Female coffee drinkers in Town A = 5% of 10,000 = 500. Female coffee drinkers in Town B = 15% of 20,000 = 3,000. Total female coffee drinkers = 500 + 3,000 = 3,500.

Explanation

First, simplify the given expression by finding a common denominator for the two fractions. The expression is E = 1/(a + 1) + (2a + 1)/(a² - 1). Since a² - 1 = (a + 1)(a - 1), we can rewrite the expression as E = (1(a - 1) + (2a + 1))/(a² - 1), which simplifies to E = (a - 1 + 2a + 1)/(a² - 1) = 3a/(a² - 1).

Next, substitute the value of a = (1 + x)/(2 - x) into the simplified expression. To calculate a² - 1, use the identity A² - B² = (A - B)(A + B) where A = (1 + x)/(2 - x) and B = 1: a² - 1 = ((1 + x)/(2 - x) - 1)((1 + x)/(2 - x) + 1) = ((1 + x - 2 + x)/(2 - x))((1 + x + 2 - x)/(2 - x)) = ((2x - 1)/(2 - x))(3/(2 - x)) = 3(2x - 1)/(2 - x)².

Now, substitute both a and a² - 1 back into E: E = (3((1 + x)/(2 - x)))/(3(2x - 1)/(2 - x)²) = 3(1 + x)/(2 - x) · (2 - x)²/(3(2x - 1)) = (1 + x)(2 - x)/(2x - 1).

This matches option C.

Explanation

Given the condition pq + qr + rp = 0, we can express the product of any two variables in terms of the third variable to simplify the denominators. For the first term, we have qr = -pq - rp. Substituting this into the denominator, we get p² - qr = p² - (-pq - rp) = p² + pq + rp = p(p + q + r). Therefore, the first term simplifies to p²/p(p + q + r) = p/(p + q + r).

Applying the same substitution logic to the second term, we know rp = -pq - qr, so the denominator becomes q² - rp = q² - (-pq - qr) = q² + pq + qr = q(q + p + r). Thus, the second term simplifies to q²/q(p + q + r) = q/(p + q + r).

For the third term, since pq = -qr - rp, the denominator is r² - pq = r² - (-qr - rp) = r² + qr + rp = r(r + q + p). Consequently, the third term simplifies to r²/r(p + q + r) = r/(p + q + r).

Adding these three simplified terms together, we obtain p/(p + q + r) + q/(p + q + r) + r/(p + q + r) = (p + q + r)/(p + q + r) = 1. This result corresponds to option B.

Explanation

Let the length of CB be x units. Since the total length of the line segment AB is 2 units and C is a point on this segment, the length of AC can be expressed as 2 - x units. According to the given condition AC² = AB × CB, we substitute the known values into the equation: (2 - x)² = 2 × x.

Expanding the left side of the equation gives 4 - 4x + x² = 2x. Rearranging all terms to one side to form a standard quadratic equation, we get x² - 6x + 4 = 0. To solve for x, we use the quadratic formula x = (-b ± √(b² - 4ac))/2a where a = 1, b = -6, and c = 4. Substituting these values, we have x = (6 ± √((-6)² - 4(1)(4)))/2(1) = (6 ± √(36 - 16))/2 = (6 ± √20)/2.

Simplifying the square root √20 as 2√5 gives x = (6 ± 2√5)/2 = 3 ± √5. Since the point C lies within the segment AB of length 2, the length CB must be less than 2 units. The value 3 + √5 is approximately 5.236, which exceeds the total length of the segment. Therefore, the only physically possible length is x = 3 - √5, which matches option b.

Explanation

Given x + y + z = 0, we can use the property of squares to simplify the denominators. Since x + y = -z, squaring both sides gives (x + y)² = (-z)², which expands to x² + 2xy + y² = z². Rearranging this equation, we find that x² + y² - z² = -2xy.

Similarly, by taking y + z = -x and squaring both sides, we get (y + z)² = (-x)², which simplifies to y² + 2yz + z² = x². Rearranging this gives y² + z² - x² = -2yz. Following the same logic for z + x = -y, we square both sides to get (z + x)² = (-y)², which results in z² + 2zx + x² = y², so z² + x² - y² = -2zx.

Substituting these three derived values into the original expression: 1/-2xy + 1/-2yz + 1/-2zx

Factoring out -1/2, the expression becomes: -1/2 (1/xy + 1/yz + 1/zx)

To add these fractions, we find a common denominator, which is xyz: -1/2 ( (z + x + y)/xyz )

Since we are given that x + y + z = 0, the numerator is 0: -1/2 ( 0/xyz ) = 0

Thus, the value of the expression is 0, which corresponds to option (d).

Explanation

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the condition for the system to have no solution (which occurs when the lines are parallel and distinct) is given by the formula a₁/a₂ = b₁/b₂ ≠ c₁/c₂. Comparing the given equations x + 2y - 3 = 0 and 5x + ky + 7 = 0 with the standard form, we identify the coefficients as a₁ = 1, b₁ = 2, c₁ = -3 and a₂ = 5, b₂ = k, c₂ = 7. Substituting these values into the formula, we obtain the relation 1/5 = 2/k ≠ -3/7. Solving the equation 1/5 = 2/k through cross multiplication yields k = 2 × 5 = 10. Since 1/5 is not equal to -3/7, the condition for no solution is fully satisfied. Thus, the value of k is 10, making option (d) the correct choice.

Explanation

To verify the given statements, we can use the method of counterexamples, which is the fastest way to disprove general algebraic claims in competition exams.

For statement 1, the claim is gcd(m, nk) = gcd(mn, k). Let us take m=6, n=2, and k=3. The left hand side gives gcd(6, 2 × 3) = gcd(6, 6) = 6. The right hand side gives gcd(6 × 2, 3) = gcd(12, 3) = 3. Since 6 ≠ 3, statement 1 is not generally true for all natural numbers.

For statement 2, the claim is gcd(m, n) × gcd(n, k) = gcd(m, k). Let us take m=2, n=2, and k=2. The left hand side gives gcd(2, 2) × gcd(2, 2) = 2 × 2 = 4. The right hand side gives gcd(2, 2) = 2. Since 4 ≠ 2, statement 2 is not generally true. Another simple case is m=2, n=3, k=2, where gcd(2, 3) × gcd(3, 2) = 1 × 1 = 1, while gcd(2, 2) = 2.

Thus, neither statement 1 nor statement 2 is correct, which makes option d the right choice.

Explanation

Use the algebraic identity which states that if a + b + c = 0, then a³ + b³ + c³ = 3abc. In the given expression, let a = x - y, b = y - z, and c = z - x. By calculating their sum, we get a + b + c = (x - y) + (y - z) + (z - x) = x - x + y - y + z - z = 0. Since the condition a + b + c = 0 is satisfied, the numerator (x - y)³ + (y - z)³ + (z - x)³ simplifies directly to 3(x - y)(y - z)(z - x). Substituting this value back into the full expression, we have 3(x - y)(y - z)(z - x) / (4(x - y)(y - z)(z - x)). The common terms (x - y)(y - z)(z - x) in the numerator and denominator cancel each other out, leaving the final result as 3/4, which corresponds to option c.

Explanation

To find the value of m, we first simplify the given equation (x(x - 1) - (m + 1))/((x - 1)(m - 1)) = x/m by cross-multiplication. m(x² - x - m - 1) = x(x - 1)(m - 1) Expanding both sides: mx² - mx - m² - m = (x² - x)(m - 1) mx² - mx - m² - m = mx² - x² - mx + x By canceling the common terms mx² and -mx from both sides, the equation reduces to a standard quadratic form: x² - x - (m² + m) = 0 For a quadratic equation ax² + bx + c = 0 to have equal roots, the discriminant must be zero, using the formula D = b² - 4ac = 0. Substituting the values a = 1, b = -1, and c = -(m² + m) into the formula: (-1)² - 4(1)(-(m² + m)) = 0 1 + 4m² + 4m = 0 The resulting expression is a perfect square: (2m + 1)² = 0. Solving for m, we get 2m + 1 = 0, which gives m = -1/2. This corresponds to option d.

Explanation

Let u = (a + √(a² + b³))¹/³ and v = (a - √(a² + b³))¹/³. The given expression is x = u + v. To find the value of the expression x³ + 3bx - 2a, we cube both sides using the identity (u + v)³ = u³ + v³ + 3uv(u + v).

First, calculate u³ + v³: u³ + v³ = (a + √(a² + b³)) + (a - √(a² + b³)) = 2a

Next, calculate the product uv: uv = [(a + √(a² + b³))(a - √(a² + b³))]¹/³ Using the difference of squares (A+B)(A-B) = A² - B²: uv = [a² - (√(a² + b³))²]¹/³ = [a² - (a² + b³)]¹/³ = (-b³)¹/³ = -b

Now, substitute these values into the cubing identity: x³ = (u + v)³ = (u³ + v³) + 3uv(u + v) x³ = 2a + 3(-b)(x) x³ = 2a - 3bx

Rearranging the terms to match the required expression: x³ + 3bx - 2a = 0

Thus, the value is 0, which corresponds to option d.

Explanation

A number is divisible by 9 if the sum of its digits is divisible by 9. Sum = 8 + 4 + 7 + 0 + 5 + ⊗ + 2 = 26 + ⊗. The smallest digit ⊗ that makes 26 + ⊗ divisible by 9 is 1 (26 + 1 = 27).

Explanation

Any integer n can be classified as either even or odd. If n is an even integer, it can be represented as n = 2k for some integer k. Squaring this gives n² = (2k)² = 4k². Since k² is also an integer, let p = k², which means the square is of the form 4p. If n is an odd integer, it can be represented as n = 2k + 1 for some integer k. Squaring this gives n² = (2k + 1)² = 4k² + 4k + 1 = 4(k² + k) + 1. Since k² + k is an integer, let p = k² + k, which means the square is of the form 4p + 1. Therefore, every square integer must be of the form 4p or 4p + 1, which corresponds to option d.

Explanation

Surface area of one brick = 2 (12.5 10 + 10 7.5 + 12.5 7.5) = 2 (125 + 75 + 93.75) = 2 293.75 = 587.5 sq cm. Converting to sq meters: 587.5 / 10,000 = 0.05875 m^2. Number of bricks = 5.875 / 0.05875 = 100.

Explanation

The volume of a cone is given by the formula V = 1/3 π r² h, where r is the radius of the base and h is the vertical height. For two cones with volumes V₁ and V₂, radii r₁ and r₂, and heights h₁ and h₂, the ratio of their volumes is V₁/V₂ = (1/3 π r₁² h₁)/(1/3 π r₂² h₂) = (r₁/r₂)² × h₁/h₂.

Given the ratio of volumes V₁/V₂ = 4/5 and the ratio of radii r₁/r₂ = 2/3, we substitute these values into the formula to find the ratio of the heights: 4/5 = (2/3)² × h₁/h₂ 4/5 = 4/9 × h₁/h₂

To solve for h₁/h₂, multiply both sides by the reciprocal of 4/9: h₁/h₂ = 4/5 × 9/4 h₁/h₂ = 9/5

The ratio of their vertical heights is 9:5, which corresponds to option c.

Explanation

When a right triangle is revolved about one of its perpendicular sides, the resulting solid is a right circular cone. In this problem, the triangle is revolved about the side measuring 12 cm, which means the height of the cone is h = 12 cm and the other perpendicular side, measuring 5 cm, becomes the radius of the base r = 5 cm. The hypotenuse of 13 cm serves as the slant height of the cone. To find the volume V of the resulting cone, we use the standard formula V = 1/3 π r² h. Substituting the values into the formula, we get V = 1/3 π (5)² (12). Performing the calculation, V = 1/3 π × 25 × 12. By simplifying, V = π × 25 × 4 = 100π cm³. Therefore, the volume of the solid obtained is 100π cm³, which corresponds to option b.

Explanation

The volume of water flowing out of a pipe can be calculated using the formula for the volume of a cylinder, where the length of the cylinder is the distance the water travels in the given time. The total volume V is given by the product of the cross-sectional area A, the speed of the water v, and the time t: V = A × v × t.

Substituting the given values into the formula: Area A = 5 cm² Speed v = 30 cm/sec Time t = 1 minute = 60 seconds

V = 5 cm² × 30 cm/sec × 60 sec V = 150 × 60 cm³ V = 9000 cm³

To convert the volume from cubic centimeters to litres, we use the standard conversion 1000 cm³ = 1 litre. Total volume in litres = 9000/1000 = 9 litres.

Thus, option c is correct.

Explanation

The length of the wire remains constant regardless of its shape. Therefore, the circumference of the circle is equal to the perimeter of the square. The formula for the circumference of a circle is C = 2πr and the formula for the perimeter of a square is P = 4s, where s represents the side of the square. Given the radius r = 42 cm and using the value of π = 22/7, we compute the circumference as C = 2 × 22/7 × 42 = 2 × 22 × 6 = 264 cm. Equating the perimeter of the square to this length, we have 4s = 264. Dividing both sides by 4, we get s = 264/4 = 66 cm. Thus, the side of the square is 66cm, which matches option (a).

Explanation

Volume V = (1/3)pi r^2 h. If r and h increase by 20%, the new volume V' = (1/3)pi (1.2r)^2 (1.2h) = 1.2^3 V = 1.728 V. The percentage increase is (1.728 - 1) * 100 = 72.8%, which is approximately 73%.

Explanation

The volume of a hemispherical container is calculated using the formula V = 2/3 πr³. Given the radius r = 21 cm, we substitute the values into the formula using π = 22/7. The calculation is as follows: V = 2/3 × 22/7 × 21 × 21 × 21. By simplifying the terms, the 3 × 7 in the denominator cancels out one of the 21s in the numerator, leaving V = 2 × 22 × 21 × 21. This results in V = 44 × 441 = 19404 cm³. To convert the volume from cubic centimeters to litres, we use the conversion factor 1000 cm³ = 1 litre. Therefore, the volume in litres is 19404 / 1000 = 19.404 litres. Approximately, this is 19.4 litres, which corresponds to option (a).

Explanation

The volume of a bucket in the shape of a frustum of a cone is calculated using the standard formula V = 1/3 πh(R² + r² + Rr), where h is the vertical height, R is the radius of the larger circular base, and r is the radius of the smaller circular base. Given the parameters R = 5 units, r = 3 units, and h = 6 units, we substitute these values into the formula as follows: V = 1/3 π(6)(5² + 3² + 5 × 3). Simplifying the expression, we first solve the terms inside the parentheses: 25 + 9 + 15 = 49. Then, the coefficient becomes 1/3 × 6 × π = 2π. Multiplying these results gives V = 2π × 49 = 98π cubic units. Since 98π does not match any of the values provided in options (a) 33π, (b) 45π, or (c) 48π, the correct choice is (d) None of the above.

Explanation

In a triangle ABC right-angled at C, let the sides opposite to vertices A, B, and C be denoted by a, b, and c respectively. Here, a = BC and b = AC are the legs of the triangle, and c = AB is the hypotenuse. The area of a triangle is given by the formula Area = 1/2 × base × height.

Calculating the area in two different ways: Using the legs of the right triangle, the area is 1/2 × a × b. Using the hypotenuse c as the base and the perpendicular p as the height, the area is 1/2 × c × p.

Since both expressions represent the area of the same triangle, we set them equal to each other: 1/2 × c × p = 1/2 × a × b

Multiplying both sides by 2, we get: cp = ab or pc = ab

Comparing this result with the given options, we find that option (c) pc = ab is the correct relation.

Explanation

Circumference of the circular path = 2 pi 50 = 100pi meters. Circumference of the wheel = 2 pi 0.5 = pi meters. Revolutions for one full lap = 100pi / pi = 100 revolutions. One lap takes 60 minutes. In 15 minutes, the wheel makes (15/60) * 100 = 25 revolutions.

Explanation

When a solid object is completely immersed in a liquid, the volume of the liquid displaced is equal to the volume of the immersed object. In this case, the volume of the water that rises in the cylindrical can is equal to the volume of the solid sphere.

Let the internal radius of the cylindrical can be r and the radius of the solid sphere be R. Internal diameter of the cylinder = 24 cm, so r = 24/2 = 12 cm. Radius of the sphere R = 6 cm. Let the increase in water level be h.

The volume of the displaced water is the volume of a cylinder with radius r and height h: Vᵩₐₜₑᵣ = πr²h

The volume of the solid sphere is: Vₛₚₕₑᵣₑ = 4/3 πR³

Equating the two volumes: πr²h = 4/3 πR³ r²h = 4/3 R³ (12)² h = 4/3 (6)³ 144 h = 4/3 × 216 144 h = 4 × 72 144 h = 288 h = 288/144 = 2 cm

The water level increases by 2 cm, which corresponds to option c.

Explanation

When a plane intersects a sphere, the resulting curve of intersection is always a circle. To find the radius of this intersection circle, we use the Pythagorean theorem based on the right-angled triangle formed by the radius of the sphere, the distance of the plane from the center, and the radius of the intersection circle. The applicable formula is r = √(R² - d²), where R is the radius of the sphere, d is the perpendicular distance from the center of the sphere to the plane, and r is the radius of the circle of intersection.

Substituting the given values R = 10 cm and d = 8 cm into the formula, we get: r = √(10² - 8²) r = √(100 - 64) r = √36 r = 6 cm

Thus, the radius of the curve of intersection is 6 cm, which corresponds to option (b).

Explanation

Cylinder height h = 2r. Cylinder volume = pi r^2 2r = 2pir^3. Sphere volume = (4/3)pir^3. Volume cut away = 2pir^3 - (4/3)pir^3 = (2/3)pir^3. Fraction cut away = ((2/3)pir^3) / (2pir^3) = 1/3.

Explanation

To find the area of the square constructed on the side BD, we need to calculate the length of BD, which is the median from vertex B to the side AC. According to Apollonius's Theorem, for any triangle ABC where D is the midpoint of AC, the relationship between the sides and the median is given by the formula: AB² + BC² = 2(AD² + BD²).

Given the dimensions AB = 2 cm, BC = 3 cm, and AC = 4 cm, we first find AD. Since D is the midpoint of AC, AD = AC/2 = 4/2 = 2 cm. Now, substituting these values into the formula: 2² + 3² = 2(2² + BD²) 4 + 9 = 2(4 + BD²) 13 = 8 + 2 BD² 2 BD² = 13 - 8 2 BD² = 5 BD² = 2.5

The area of a square with side BD is defined as BD². Therefore, the area of the square is 2.5 cm², which corresponds to option b.

Explanation

Volume of large cone = (1/3)pi 6^2 8 = 96pi. Volume of one small cone = (1/3)pi 1^2 2 = (2/3)pi. Number of cones = 96pi / ((2/3)pi) = 96 * 3 / 2 = 144.

Explanation

The locus of a point P that maintains an equal distance from two fixed points A and B is the perpendicular bisector of the line segment joining them. Using the distance formula, if A is (x₁, y₁), B is (x₂, y₂), and P is (x, y), the condition PA = PB implies PA² = PB². This gives the equation (x - x₁)² + (y - y₁)² = (x - x₂)² + (y - y₂)²; Expanding and simplifying this equation leads to 2x(x₂ - x₁) + 2y(y₂ - y₁) + x₁² + y₁² - x₂² - y₂² = 0, which is a linear equation of the form ax + by + c = 0. Geometrically, this straight line is perpendicular to the segment AB and passes through its midpoint, making it the right bisector of AB. Therefore, option (a) is correct.

Explanation

According to the Angle Bisector Theorem in triangle ABC, since AD is the internal bisector of ∠BAC, the ratio of the sides is BD/CD = AB/AC. Given that CD = CX, we can substitute CX for CD to get BD/CX = AB/AC.

Now, consider triangle ABD and triangle ACX. We are given that ∠BAD = ∠CAX. To determine if these triangles are similar, we use the Law of Sines. In triangle ABD, AB/sin(∠ADB) = BD/sin(∠BAD). In triangle ACX, AC/sin(∠AXC) = CX/sin(∠CAX).

Dividing these two expressions gives AB/AC · sin(∠AXC)/sin(∠ADB) = BD/CX · sin(∠CAX)/sin(∠BAD). Since ∠BAD = ∠CAX and AB/AC = BD/CX, the terms cancel out, leaving sin(∠AXC) = sin(∠ADB). This implies that ∠AXC = ∠ADB or ∠AXC = 180° - ∠ADB. Given that X is outside the triangle and A, D, X are collinear, the geometry dictates ∠AXC = ∠ADB. With two corresponding angles being equal (∠BAD = ∠CAX and ∠ADB = ∠AXC), triangle ABD is similar to triangle ACX by the AA similarity criterion. Thus, option a is correct.

Explanation

Congruent triangles are identical in both shape and size, meaning their corresponding angles are equal and their corresponding sides are equal. Since similarity only requires the corresponding angles to be equal and the sides to be proportional, congruent triangles satisfy these conditions with a side ratio of 1:1. Therefore, statement 1 is correct.

Similar triangles, on the other hand, only share the same shape and may differ in size. For instance, two squares of different side lengths are similar but not congruent. Thus, statement 2 is incorrect.

Statement 3 describes the RHS (Right angle-Hypotenuse-Side) congruence criterion. In any right triangle, the relationship between the sides is governed by the Pythagorean theorem a² + b² = c², where c is the hypotenuse. If two right triangles have equal hypotenuses (c₁ = c₂) and one equal side (a₁ = a₂), then the third side must be b = √(c² - a²), which will be identical for both triangles. This leads to SSS congruence. Therefore, statement 3 is correct. Since statements 1 and 3 are correct, the correct option is d.

Explanation

In ΔABD and ΔACD, since AD is perpendicular to BC, both are right-angled triangles at D. We use the trigonometric ratio for tangent: tan B = AD/BD and tan C = AD/DC.

Multiplying these two equations, we get: tan B · tan C = AD/BD · AD/DC = AD²/(BD · DC)

Given the condition AD² = BD · DC, the expression simplifies to: tan B · tan C = 1

This implies tan B = 1/tan C = cot C. Using the complementary angle identity cot C = tan(90° - C), we have: tan B = tan(90° - C) B = 90° - C B + C = 90°

Since the sum of angles in ΔABC is 180°, ∠A = 180° - (B + C) = 180° - 90° = 90°. In any triangle where two angles sum to 90°, it is mathematically impossible for both angles to be less than 45°. If B < 45°, then C must be greater than 45° to satisfy B + C = 90°. If B = 45°, then C = 45°. Thus, at least one of the angles must be greater than or equal to 45°, which makes option (c) correct.

Explanation

In triangle ABC, since AB = AC, the triangle is isosceles, meaning the angles opposite to these sides are equal. Therefore, ∠ACB = ∠ABC = 65°. The sum of angles in a triangle is 180°, so ∠BAC = 180° - (65° + 65°) = 50°.

Since EF is parallel to AB, we can use the properties of parallel lines. Draw an auxiliary line L passing through point D such that L is parallel to AB. Because L ∥ AB and EF ∥ AB, it follows that L ∥ EF.

The line containing B, C, D acts as a transversal for the parallel lines AB and L. Thus, the corresponding angles are equal: ∠LDC = ∠ABC = 65°.

Next, consider the transversal FD crossing the parallel lines EF and L. The angles ∠EFD and ∠FDL are co-interior angles, which are supplementary: ∠EFD + ∠FDL = 180° 80° + ∠FDL = 180° ⇒ ∠FDL = 100°.

The angle ∠FDC is the difference between ∠FDL and ∠LDC: ∠FDC = ∠FDL - ∠LDC ∠FDC = 100° - 65° = 35°.

Explanation

The angle formed by the intersection of internal angle bisectors at the incenter O is given by the formula ∠BOC = 90° + ∠A/2. To derive this, we use the angle sum property of a triangle. In ΔABC, ∠A + ∠B + ∠C = 180°, which implies ∠B + ∠C = 180° - ∠A. In ΔBOC, the angles are ∠BOC, ∠B/2, and ∠C/2 because BO and CO are angle bisectors. Therefore, ∠BOC = 180° - (∠B/2 + ∠C/2). Substituting the expression for ∠B + ∠C, we get ∠BOC = 180° - (180° - ∠A)/2. Simplifying this gives ∠BOC = 180° - 90° + ∠A/2 = 90° + ∠A/2. This matches option b.

Explanation

The point equidistant from the vertices of a triangle is called the circumcenter. For any triangle, there is exactly one unique circumcenter.

Explanation

This geometric property is known as Varignon's Theorem. The solution relies on the Midpoint Theorem, which states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Let the midpoints of sides AB, BC, CD, and DA be P, Q, R, and S respectively. By drawing the diagonal AC, we create two triangles, ΔABC and ΔADC. In ΔABC, since P and Q are midpoints of AB and BC, the segment PQ ∥ AC and PQ = 1/2 AC. Similarly, in ΔADC, since S and R are midpoints of DA and CD, the segment SR ∥ AC and SR = 1/2 AC.

Since both PQ and SR are parallel to the same diagonal AC, they are parallel to each other (PQ ∥ SR). Furthermore, since both are equal to 1/2 AC, they are equal in length (PQ = SR). A quadrilateral with one pair of opposite sides that are both equal and parallel is always a parallelogram. Therefore, the resulting figure is a parallelogram.

Explanation

Given that cot θ = 8 / 15 and θ is a positive acute angle, we can determine the sides of a right-angled triangle. Since cot θ = adjacent / opposite, let the adjacent side be 8 and the opposite side be 15. Using the Pythagorean theorem, the hypotenuse is calculated as √(8² + 15²) = √(64 + 225) = √289 = 17. Consequently, the value of cos θ = adjacent / hypotenuse = 8 / 17.

The expression to evaluate is √((1 - cos θ)/(1 + cos θ)), which is the half-angle formula for tan(θ/2). Substituting the value of cos θ:

√((1 - 8/17)/(1 + 8/17)) = √(((17 - 8)/17)/((17 + 8)/17)) = √((9/17)/(25/17)) = √(9/25) = 3/5.

Thus, the value is 3 / 5, which corresponds to option c.

Explanation

The angle subtended by an arc at the centre of a circle is determined by the fundamental relationship between the arc length, the radius of the circle, and the angle in radians. The standard formula for arc length is s = rθ, where s represents the length of the arc traversed along the circumference, r is the radius of the circle, and θ is the angle subtended at the centre measured in radians.

In this specific problem, the given values are: Arc length s = 4π metres Radius r = 8 metres

Substituting these values into the formula to solve for θ: 4π = 8 × θ θ = 4π/8 θ = π/2 radians

Therefore, the angle subtended at the centre is π/2 radians, which matches option (b). In degrees, this angle is equivalent to 90°.

Explanation

For the first statement, the left hand side is (cos²θ - sin²θ)/(cos²θ + sin²θ). Using the fundamental trigonometric identity cos²θ + sin²θ = 1, the expression simplifies to cos²θ - sin²θ. Now consider the right hand side: cos²θ(1 + tanθ)(1 - tanθ). Applying the algebraic identity (a+b)(a-b) = a² - b², this becomes cos²θ(1 - tan²θ). Substituting tan²θ = sin²θ/cos²θ, we get cos²θ(1 - sin²θ/cos²θ) = cos²θ - sin²θ. Since both sides are equal, statement 1 is correct.

For the second statement, the right hand side is (tanθ + secθ)². Expressing these in terms of sine and cosine, we have (sinθ/cosθ + 1/cosθ)² = ((1 + sinθ)/cosθ)² = (1 + sinθ)²/cos²θ. Using the identity cos²θ = 1 - sin²θ, the expression becomes (1 + sinθ)²/(1 - sin²θ). Factoring the denominator as a difference of squares, we get (1 + sinθ)²/((1 - sinθ)(1 + sinθ)). Simplifying the common term (1 + sinθ) results in (1 + sinθ)/(1 - sinθ), which matches the left hand side. Thus, statement 2 is also correct. Both statements are correct, making option c the right choice.

Explanation

Simplify the given equation by taking cosθ as a common factor from both terms on the left hand side: cosθ ( 1/(1 - sinθ) - 1/(1 + sinθ) ) = 2 Combine the fractions inside the bracket by taking the common denominator (1 - sinθ)(1 + sinθ): cosθ ( ((1 + sinθ) - (1 - sinθ))/((1 - sinθ)(1 + sinθ)) ) = 2 cosθ ( 2sinθ/(1 - sin²θ) ) = 2 Using the fundamental trigonometric identity cos²θ + sin²θ = 1, we can substitute 1 - sin²θ = cos²θ in the denominator: cosθ ( 2sinθ/cos²θ ) = 2 2sinθ/cosθ = 2 2tanθ = 2 tanθ = 1 The value of θ that satisfies tanθ = 1 within the provided options is θ = π/4, which corresponds to option c.

Explanation

Let the two angles in circular measure be x and y. According to the problem, their sum in circular measure is 1, which gives the equation x + y = 1. The difference of the angles in degree measure is also 1. To convert circular measure (radians) to degree measure, we use the standard relation Degree = Radian × 180/π. Therefore, the difference in degrees is expressed as x(180/π) - y(180/π) = 1. Simplifying this equation, we get x - y = π/180. We now have a system of two linear equations: x + y = 1 and x - y = π/180. Adding these two equations yields 2x = 1 + π/180, which simplifies to x = 1/2 + π/360. Subtracting the second equation from the first yields 2y = 1 - π/180, which simplifies to y = 1/2 - π/360. Thus, the two angles in circular measure are (1/2 - π/360) and (1/2 + π/360), which corresponds to option a.

Explanation

The sine function is strictly increasing in the interval [0, 90 degrees]. Since 0 < x < 45 and 45 < y < 90, it follows that x < y, and therefore sin x < sin y.

Explanation

Substitute the standard trigonometric values into the given expression. We have sin 60° = √3/2, cot 30° = √3, sec 45° = √2, cos 60° = 1/2, tan 45° = 1, and tan 60° = √3.

Plugging these values into the expression: sin³ 60° cot 30° - 2sec² 45° + 3cos 60° tan 45° - tan² 60° = (√3/2)³(√3) - 2(√2)² + 3(1/2)(1) - (√3)² = (3√3/8)(√3) - 2(2) + 3/2 - 3 = (3 × 3)/8 - 4 + 3/2 - 3 = 9/8 + 3/2 - 7

To solve, take a common denominator of 8: = (9 + 12 - 56)/8 = (21 - 56)/8 = -35/8

The result is -35/8, which corresponds to option b.

Explanation

Given the expression (p sec θ - q csc θ)/(p sec θ + q csc θ), we use the fundamental trigonometric identities sec θ = 1/cos θ and csc θ = 1/sin θ to rewrite the expression as (p/cos θ - q/sin θ)/(p/cos θ + q/sin θ).

By taking a common denominator of sin θ cos θ in both the numerator and the denominator, the expression simplifies to (p sin θ - q cos θ)/(p sin θ + q cos θ). To utilize the given value of tan θ, we divide both the numerator and the denominator by cos θ, which gives (p tan θ - q)/(p tan θ + q).

Now, substituting the given value tan θ = p/q into this simplified form: (p(p/q) - q)/(p(p/q) + q) = (p²/q - q)/(p²/q + q) = ((p² - q²)/q)/((p² + q²)/q)

Canceling the common denominator q from the numerator and denominator, we arrive at the final result: (p² - q²)/(p² + q²)

This matches option c.

Explanation

To solve this, we use the trigonometric identity sin² θ = 1 - cos² θ. Substituting this into the expression gives cos θ - 2 + (1 - cos² θ), which simplifies to -cos² θ + cos θ - 1. Let x = cos θ, where x must lie in the range [-1, 1]. The expression becomes a quadratic function f(x) = -x² + x - 1.

To determine the sign of this quadratic, we can check its discriminant D = b² - 4ac. Here, a = -1, b = 1, and c = -1, so D = 1² - 4(-1)(-1) = 1 - 4 = -3. Since the discriminant is negative and the coefficient of x² is negative, the quadratic expression is always negative for all real values of x.

Alternatively, by completing the square, we get -(x² - x + 1) = -[(x - 1/2)² + 3/4] = -(x - 1/2)² - 3/4. Since (x - 1/2)² is always non-negative, the maximum value of the expression is -3/4 when x = 1/2. Therefore, the value of the expression is always less than zero, making option a the correct choice.

Explanation

Given cot θ = 2xy/(x² - y²), we use the definition of the cotangent function in a right angled triangle, where cot θ = Adjacent side/Opposite side. Thus, we can let the adjacent side be 2xy and the opposite side be x² - y². To find cos θ, we first need the hypotenuse h, which is calculated using the Pythagorean theorem:

h = √((Adjacent)² + (Opposite)²) h = √((2xy)² + (x² - y²)²) h = √(4x²y² + (x⁴ - 2x²y² + y⁴)) h = √(x⁴ + 2x²y² + y⁴) h = √((x² + y²)²) h = x² + y²

Now, using the formula for cosine, cos θ = Adjacent side/Hypotenuse, we substitute the values obtained:

cos θ = 2xy/(x² + y²)

This matches option c.

Explanation

Using the fundamental trigonometric relation for a right-angled triangle formed by a person and their shadow, the tangent of the angle of elevation is the ratio of the height of the person to the length of the shadow. Therefore, tan θ = s/L, where s is the height and L is the shadow length.

For the first case with angle α and shadow length ρ, we have: tan α = s/ρ ⇒ ρ = s/tan α

For the second case with angle β = 3α and shadow length q, we have: tan 3α = s/q ⇒ q = s/tan 3α

To find the difference ρ - q, we substitute the expressions derived above: ρ - q = s/tan α - s/tan 3α

Factoring out s and taking the common denominator tan α tan 3α, we get: ρ - q = s (1/tan α - 1/tan 3α) ρ - q = s ((tan 3α - tan α)/(tan α tan 3α))

This expression matches option c perfectly.

Explanation

Let the angles of the triangle be a - d, a, and a + d. Since the sum of angles in a triangle is always 180°, we have (a - d) + a + (a + d) = 180°. Simplifying this gives 3a = 180°, which means a = 60°. According to the problem, the greatest angle is double the least angle, so a + d = 2(a - d). Substituting a = 60° into this equation, we get 60 + d = 2(60 - d), which simplifies to 60 + d = 120 - 2d. Solving for d, we find 3d = 60, so d = 20°. The three angles are therefore 60 - 20 = 40°, 60°, and 60 + 20 = 80°. The ratio of these angles is 40 : 60 : 80, which simplifies to 2 : 3 : 4. Since the conversion from degrees to radians involves multiplying by a constant factor π/180, the ratio remains unchanged in radian measure. Thus, the ratio is 2 : 3 : 4, matching option a.

Explanation

The given equation is interpreted as sin θ + √3 cos θ = 2 based on the provided answer key and standard trigonometric patterns found in competitive exams. To solve this, we use the auxiliary angle formula a sin θ + b cos θ = √(a² + b²) cos(θ - α), where tan α = a/b.

Here, a = 1 and b = √3. The amplitude is √(1² + (√3)²) = √(1 + 3) = 2. The phase angle α is tan⁻¹(1/√3) = 30°.

Substituting these into the formula, we get: 2 cos(θ - 30°) = 2 cos(θ - 30°) = 1

Since we are looking for θ in the range 0 < θ ≤ 90°, the only solution for cos x = 1 is x = 0°. θ - 30° = 0° θ = 30°

Alternatively, by substituting the options: For θ = 30°, sin 30° = 1/2 and cos 30° = √3/2. LHS = 1/2 + √3(√3/2) = 1/2 + 3/2 = 4/2 = 2. Since LHS equals RHS, θ = 30° is the correct value. This corresponds to option a.

Explanation

Given the condition 0 < θ < φ < 90°, we analyze the range of the trigonometric functions involved. For any angle α such that 0° < α < 90°, the value of sin α lies strictly between 0 and 1, meaning 0 < sin α < 1. Similarly, the value of cos α also lies strictly between 0 and 1, meaning 0 < cos α < 1.

Squaring these values, we have 0 < sin² θ < 1 and 0 < cos² φ < 1. Adding these two strict inequalities together, we get 0 + 0 < sin² θ + cos² φ < 1 + 1, which simplifies to 0 < sin² θ + cos² φ < 2. This directly validates option c.

To check option a, we use the identity (sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ = 1 + sin 2θ. Since the maximum value of sin 2θ is 1, the maximum value of this expression is 2, which contradicts the claim that it is greater than 2. Since sin² θ and cos² φ are strictly less than 1, their sum can never equal 2, making the strict inequality in option c the most accurate description.

Explanation

The statement in option d is incorrect because the condition A ∩ B = ϕ defines disjoint sets, which are sets that have no elements in common. This does not imply that one of the sets must be empty. For example, consider two non-empty sets A = {1, 2} and B = {3, 4}. The intersection A ∩ B is ϕ because there are no shared elements, yet A ≠ ϕ and B ≠ ϕ. Thus, the claim that either A or B must be the empty set is logically false. The other options are standard properties of subsets: if A ⊆ B, then the union B ∪ A is B, the intersection B ∩ A is A, and the difference A - B is ϕ, making A ∩ (A - B) = ϕ.

Explanation

Average speed is calculated using the formula: Average Speed = Total Distance / Total Time. First, calculate the total distance covered by the boy: Total Distance = 8 km + 6 km + 4 km = 18 km. Next, find the time taken for each segment of the journey using the relation Time = Distance / Speed. For the first part, t₁ = 8/4 = 2 hours. For the second part, t₂ = 6/3 = 2 hours. For the third part, t₃ = 4/2 = 2 hours. The total time taken for the entire journey is t = 2 + 2 + 2 = 6 hours. Substituting these total values into the average speed formula, we get Average Speed = 18 km / 6 hours = 3 kmph, which corresponds to option b.

Explanation

To solve this problem, we use the change of base property of logarithms, specifically the rule concerning the power of the base: logₐⁿ b = 1/n logₐ b. In the given expression log₁₃(10)/log₁₆₉(10), we observe that the base in the denominator is 169, which can be expressed as 13².

Applying the base power rule to the denominator, we get: log₁₆₉(10) = log₁₃²(10) = 1/2 log₁₃(10).

Now, substitute this value back into the original fraction: log₁₃(10)/(1/2 log₁₃(10)).

The term log₁₃(10) appears in both the numerator and the denominator, so they cancel each other out: 1/(1/2) = 1 × 2 = 2.

Thus, the value of the expression is 2, which corresponds to option (b).

Explanation

Let the four numbers in geometric progression (GP) be a, ar, ar², ar³, where a is the first term and r is the common ratio. According to the problem, the third term is greater than the first by 9, which gives the equation ar² = a + 9, or a(r² - 1) = 9. The second term is greater than the fourth by 18, which gives the equation ar = ar³ + 18, or ar(1 - r²) = 18. This can be rewritten as -ar(r² - 1) = 18. To find r, we divide the second equation by the first equation: -ar(r² - 1) / a(r² - 1) = 18/9. This simplifies to -r = 2, hence r = -2. Substituting the value of r into the first equation a(r² - 1) = 9, we get a((-2)² - 1) = 9, which simplifies to a(4 - 1) = 9, or 3a = 9. Solving for a, we find a = 3. Thus, the first term of the GP is 3, which corresponds to option b.

Explanation

27 (81)^(2n+3) = 3^m => 3^3 (3^4)^(2n+3) = 3^m => 3^3 * 3^(8n+12) = 3^m => 3^(8n+15) = 3^m. Thus, m = 8n + 15.

Explanation

The harmonic mean (HM) of two numbers a and b is calculated using the formula HM = 2ab/(a + b). We can apply this formula directly to the three given pairs to determine their values.

For pair 1 (8, 12): HM = 2 × 8 × 12/(8 + 12) = 192/20 = 9.6

For pair 2 (9, 11): HM = 2 × 9 × 11/(9 + 11) = 198/20 = 9.9

For pair 3 (6, 24): HM = 2 × 6 × 24/(6 + 24) = 288/30 = 96/10 = 9.6

By comparing the calculated values, we see that the harmonic mean for pair 1 and pair 3 is the same, both equal to 9.6. Pair 2 has a harmonic mean of 9.9, which is different. Thus, only pairs 1 and 3 have the same harmonic mean, making option (c) the correct choice.

Explanation

Let the three distinct positive integers be a, b, and c such that a < b < c. The median of three numbers is the middle value when arranged in ascending order, so we have b = 5. To minimize the arithmetic mean, we must minimize the sum of the three integers, S = a + b + c. Since the integers must be positive and distinct, the smallest possible value for a is a = 1. Similarly, since c must be greater than b, the smallest possible value for c is c = 6. Substituting these values into the sum formula, we get S = 1 + 5 + 6 = 12. The formula for the arithmetic mean is Mean = S/n, where n is the number of observations. Substituting S = 12 and n = 3, we obtain Mean = 12/3 = 4. Thus, the least possible value of the arithmetic mean is 4, which corresponds to option c.

Explanation

A histogram is a graphical representation of a frequency distribution where the area of each rectangle is proportional to the frequency of the corresponding class interval. The mode of a dataset is the value that appears most frequently, which corresponds to the highest rectangle (the modal class) in a histogram.

To determine the exact mode graphically, lines are drawn from the top corners of the tallest rectangle to the top corners of the immediate rectangles to its left and right. The point where these two lines intersect is then projected vertically down to the x-axis, and the value at this point is the mode. This graphical method is based on the grouped data formula Mode = l + ((f₁ - f₀)/(2f₁ - f₀ - f₂)) × h, where l is the lower limit of the modal class, f₁ is the frequency of the modal class, f₀ is the frequency of the preceding class, f₂ is the frequency of the succeeding class, and h is the class width. While the mean and median typically require the original data or a frequency table for precise calculation, the mode can be identified directly from the visual peak of the histogram.

Explanation

Primary data refers to information collected for the first time by an investigator directly from the original source for a specific purpose. In the given examples, door-to-door visits (Example 1) and telephonic interviews (Example 3) involve the investigator interacting directly with the respondents to gather fresh information, which characterizes them as primary data. On the other hand, data collected from the records of the Census of India (Example 2) is secondary data because the information has already been gathered, processed, and published by a third party (the government). Therefore, only statements 1 and 3 are examples of primary data, making option b the correct choice.

Explanation

Geometric Mean = (125 729 1331)^(1/3). Since 125 = 5^3, 729 = 9^3, and 1331 = 11^3, the GM is 5 9 11 = 495.