CDS II 2016 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The equation is log10[1 - {1 - (1 - x²)⁻¹}⁻¹]^(1/2) = 1. Simplifying the inner expression: (1 - x²)⁻¹ = 1/(1 - x²). Then 1 - 1/(1 - x²) = (1 - x² - 1)/(1 - x²) = -x²/(1 - x²). The inverse of this is (1 - x²)/(-x²) = (x² - 1)/x². Then 1 - (x² - 1)/x² = 1/x². The equation becomes log10(1/x²)^(1/2) = log10(1/|x|) = 1. This implies 1/|x| = 10, so |x| = 0.1. None of the options match 0.1. However, if the exponent 1/2 were -1/2, the equation would be log10(x) = 1, giving x = 10. Given the answer key says B, there is a likely typo in the question's exponent, but B is the intended answer.

Explanation

Let f(x) = 4x^2 - 16x + lambda/4. For the roots alpha and beta to satisfy 1 < alpha < 2 and 2 < beta < 3, the conditions are f(1) > 0, f(2) < 0, and f(3) > 0. f(1) = 4 - 16 + lambda/4 = lambda/4 - 12 > 0 implies lambda > 48. f(2) = 16 - 32 + lambda/4 = lambda/4 - 16 < 0 implies lambda < 64. f(3) = 36 - 48 + lambda/4 = lambda/4 - 12 > 0 implies lambda > 48. Thus, 48 < lambda < 64. The integers lambda can take are {49, 50, ..., 63}, which is 63 - 49 + 1 = 15 values.

Explanation

Numerator: 6^2 + 7^2 + 8^2 + 9^2 + 10^2 = 36 + 49 + 64 + 81 + 100 = 330. Denominator: sqrt(7 + 4sqrt(3)) - sqrt(4 + 2sqrt(3)). Note that 7 + 4sqrt(3) = (2 + sqrt(3))^2 and 4 + 2sqrt(3) = (sqrt(3) + 1)^2. The denominator is (2 + sqrt(3)) - (sqrt(3) + 1) = 1. The result is 330 / 1 = 330.

Explanation

Given the equations x² = y + z, y² = z + x, and z² = x + y. By adding x, y, and z to the respective equations, we obtain: x² + x = x + y + z y² + y = x + y + z z² + z = x + y + z

Let S = x + y + z. This implies that x, y, and z are roots of the quadratic equation t² + t - S = 0. A quadratic equation can have at most two distinct roots, meaning at least two of the variables x, y, z must be equal.

For a fast competition-style approach, we test the symmetric case where x = y = z. Substituting this into the first equation: x² = x + x ⇒ x² = 2x ⇒ x(x - 2) = 0 Since the expression 1/(x + 1) + 1/(y + 1) + 1/(z + 1) must result in one of the given options, we test x = 2. If x = 2, then y = 2 and z = 2. The value of the expression is: 1/(2 + 1) + 1/(2 + 1) + 1/(2 + 1) = 1/3 + 1/3 + 1/3 = 3/3 = 1

This matches option B. In cases where x ≠ z, the roots of t² + t - S = 0 satisfy x + z = -1. It can be shown algebraically that 1/(x+1) + 1/(z+1) = 1/xz, and combining this with y = x and the relation x² = -1 also yields a final result of 1.

Explanation

The equation 5P9 + 3R7 + 2Q8 = 1114 can be written as (509 + 10P) + (307 + 10R) + (208 + 10Q) = 1114. This simplifies to 1024 + 10(P + R + Q) = 1114, so 10(P + R + Q) = 90, which means P + R + Q = 9. To maximize Q, let P = 0 and R = 0, then Q = 9.

Explanation

The given conditions state that the number n leaves remainders 1, 2, 3, 4, and 5 when divided by 2, 3, 4, 5, and 6 respectively. In each case, the difference between the divisor and the remainder is constant: 2-1 = 1, 3-2 = 1, 4-3 = 1, 5-4 = 1, and 6-5 = 1. This indicates that n+1 must be exactly divisible by 2, 3, 4, 5, and 6.

To find the smallest such positive integer, we use the formula for the Least Common Multiple (LCM). The LCM of 2, 3, 4, 5, and 6 is calculated as follows: the prime factors are 2=2¹, 3=3¹, 4=2², 5=5¹, and 6=2 × 3. The LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60.

The general form of the integer is n = 60k - 1, where k is a positive integer. For k=1, n = 60(1) - 1 = 59. For k=2, n = 60(2) - 1 = 119. Since we are looking for integers between 0 and 100, only n=59 satisfies the condition. Therefore, there is only one such integer, making option (b) the correct choice.

Explanation

Let the number be x. According to the problem, 8x - x/8 = 2016. This gives (63x)/8 = 2016. Solving for x: x = (2016 8) / 63 = 32 8 = 256.

Explanation

For real roots, the discriminant D = a^2 - 4b must be >= 0, so a^2 >= 4b. Given a, b are in {1, 2, 3, 4}: if a=1, 1 >= 4b (no b); if a=2, 4 >= 4b (b=1); if a=3, 9 >= 4b (b=1, 2); if a=4, 16 >= 4b (b=1, 2, 3, 4). Total pairs: 1 + 2 + 4 = 7.

Explanation

Let the integers be x and y (x > y). x^2 + y^2 = 208 and x^2 = 18y. Substituting x^2: 18y + y^2 = 208, so y^2 + 18y - 208 = 0. Solving the quadratic: y = (-18 + sqrt(324 + 832))/2 = (-18 + 34)/2 = 8. Then x^2 = 18*8 = 144, so x = 12. The difference x - y = 12 - 8 = 4.

Explanation

The set difference A - B consists of elements that are present in set A but not in set B. Given A = {7, 8, 9, 10, 11, 12} and B = {7, 10, 14, 15}, we first identify the intersection A ∩ B, which contains the elements common to both sets. Here, A ∩ B = {7, 10}. Using the formula for the cardinality of set difference, n(A - B) = n(A) - n(A ∩ B), we have 6 - 2 = 4. The specific elements are {8, 9, 11, 12}. Similarly, for the set difference B - A, we use the formula n(B - A) = n(B) - n(A ∩ B), which gives 4 - 2 = 2. The specific elements are {14, 15}. Thus, the number of elements in (A - B) and (B - A) are 4 and 2 respectively, which matches option (b).

Explanation

Daily savings = 4.65 = 465/100 = 93/20 rupees. For the total savings (93/20)*n to be an integer, n must be a multiple of 20. The least number of days is 20.

Explanation

To find how often the winner passes the slower runner in a circular race, we determine the number of laps the winner gains over the other. The total number of laps in the race is calculated using the formula Total Distance/Lap Length = 4 km/0.25 km = 16 laps. Let the speeds of A and B be 5v and 4v respectively. The time taken by the winner A to complete the race is T = 4/5v. In this duration, the distance covered by B is Distanceᵦ = 4v × 4/5v = 3.2 km. The relative distance gained by A over B throughout the race is 4 km - 3.2 km = 0.8 km. The number of times A passes B is the number of full laps A has gained over B, which is Relative Distance/Lap Length = 0.8 km/0.25 km = 3.2. Since a pass occurs only when a full lap is completed relative to the opponent, we take the integer part of this value, which is 3. Thus, the winner passes the other three times, matching option c).

Explanation

A rational number expressed in the form p/q in its simplest form has a terminating decimal expansion if and only if the prime factorization of the denominator q is of the form 2ⁿ 5ᵐ, where n and m are non-negative integers. If the prime factorization of the denominator contains any prime factor other than 2 or 5, the decimal expansion is non-terminating and repeating.

Evaluating the given options: In option (a), 15/1600 simplifies to 3/320. The denominator 320 = 2⁶ × 5, which consists only of 2 and 5, so it terminates. In option (b), 23/8 has a denominator 8 = 2³, so it terminates. In option (c), 35/50 simplifies to 7/10. The denominator 10 = 2 × 5, so it terminates. In option (d), 17/6 is already in simplest form. The prime factorization of the denominator is 6 = 2 × 3. Since it contains the prime factor 3, which is different from 2 or 5, the decimal expansion is non-terminating and repeating. Therefore, option (d) is the correct answer.

Explanation

For the given polynomial 25x² - 15x + 2, let the zeros be α and β. According to the relations between coefficients and zeros, the sum of zeros is α + β = -b/a = -(-15)/25 = 3/5 and the product of zeros is αβ = c/a = 2/25.

The new zeros are given as 1/(2α) and 1/(2β). Let S be the sum and P be the product of these new zeros. The sum S = 1/(2α) + 1/(2β) = (α + β)/(2αβ). Substituting the known values, S = (3/5)/(2(2/25)) = (3/5)/(4/25) = 3/5 × 25/4 = 15/4. The product P = 1/(2α) × 1/(2β) = 1/(4αβ). Substituting the known value, P = 1/(4(2/25)) = 1/(8/25) = 25/8.

A quadratic polynomial with sum S and product P is given by the formula k(x² - Sx + P). Setting k = 8 to eliminate the denominators, we get: 8(x² - 15/4x + 25/8) = 8x² - 30x + 25. This matches option b.

Explanation

By Fermat's Little Theorem, a^(p-1) mod p = 1 if p is prime. For p = 101 and a = 2, 2^(100) mod 101 = 1.

Explanation

Let the total number of workers in the office be x. Based on the problem, the number of women is 1/3x. Consequently, the number of men is x - 1/3x = 2/3x. To find the number of married women, we multiply the total women by the given fraction of married women: Married Women = 1/2 × 1/3x = 1/6x. Similarly, to find the number of married men, we multiply the total men by the given fraction of married men: Married Men = 3/4 × 2/3x = 1/2x. The ratio of married women to married men is calculated as 1/6x : 1/2x. Simplifying this ratio by dividing the two fractions gives 1/6 ÷ 1/2 = 1/6 × 2/1 = 2/6 = 1/3. Therefore, the ratio is 1:3, which matches option d. The data provided regarding the number of children is a distractor and is not required to solve for the ratio of married adults.

Explanation

Let the efficiency of B be Eᵦ. Since A can do 50% more work than B in the same time, the efficiency of A is Eₐ = 1.5 Eᵦ. The ratio of efficiencies Eₐ : Eᵦ = 1.5 : 1 = 3 : 2. To simplify calculations, let B's efficiency be 2 units per hour and A's efficiency be 3 units per hour.

The total work is calculated using the formula Work = Efficiency × Time. Given that B alone can do the work in 30 hours, the total work is W = 2 × 30 = 60 units.

B works alone for 12 hours, completing 2 × 12 = 24 units of work. The remaining work is 60 - 24 = 36 units. When A joins B, their combined efficiency becomes Eₐ₊ᵦ = 3 + 2 = 5 units per hour. The time required for them to complete the remaining work is calculated as Remaining Work / Combined Efficiency = 36 / 5 = 7.2 hours. Thus, option (d) represents 7.2 hours.

Explanation

Since the distance covered is the same, speed and time are inversely proportional. Let the original speed be S and the original time be T. The new speed is S + 20% of S = 1.2S. Using the formula Distance = Speed × Time, we can set up the equation S × T = 1.2S × (T - 20). Dividing both sides by S gives T = 1.2(T - 20), which simplifies to T = 1.2T - 24. Rearranging the terms to solve for T, we get 0.2T = 24, which leads to T = 24/0.2 = 120 minutes.

Alternatively, using the ratio method for faster calculation, the speed increases from 100% to 120%, which is a ratio of 5:6. Because time is inversely proportional to speed for a constant distance, the ratio of original time to new time is 6:5. The difference between the ratios is 6 - 5 = 1 unit. Given that this difference equals 20 minutes, one unit represents 20 minutes. Therefore, the original time, which is 6 units, is 6 × 20 = 120 minutes. This matches option (b).

Explanation

Let T represent the maximum marks for the examination. The passing marks can be expressed in terms of the scores and the differences provided for both candidates. For the first candidate, scoring x% marks leads to a failure by a marks, meaning the passing marks are x/100 T + a. For the second candidate, scoring y% marks results in b marks more than the passing threshold, meaning the passing marks are y/100 T - b. Since the passing marks are the same for both, we equate the two expressions: y/100 T - b = x/100 T + a. Rearranging the terms to solve for T, we get y/100 T - x/100 T = a + b, which simplifies to T/100(y - x) = a + b. Solving for T gives T = 100(a + b)/(y - x), which corresponds to option c.

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a. In the given equation x² + px + q = 0, we have a = 1, b = p, and c = q. Given that the roots are p and q, we can apply these formulas as follows:

Sum of roots: p + q = -p/1, which simplifies to p + q = -p, or 2p + q = 0. Product of roots: p · q = q/1, which simplifies to pq = q.

From the product equation pq = q, we have pq - q = 0, which factors to q(p - 1) = 0. This gives two possibilities: q = 0 or p = 1.

Case 1: If q = 0, substituting this into the sum equation 2p + q = 0 gives 2p + 0 = 0, which means p = 0. Case 2: If p = 1, substituting this into the sum equation 2p + q = 0 gives 2(1) + q = 0, which means q = -2.

Therefore, the possible values for p are 0 or 1, making option (a) the correct choice.

Explanation

To find the cost of 11 kg of nuts, we apply the chain rule of proportionality to determine the unit price of each item sequentially. First, the cost of 1 kg of rice is calculated as 125/2.5 = 50 Rs. Since the cost of 9 kg of rice is equal to that of 4 kg of pulses, the cost of 1 kg of pulses is (9 × 50)/4 = 450/4 = 112.5 Rs. Given that the cost of 14 kg of pulses is equal to that of 1.5 kg of tea, the cost of 1 kg of tea is (14 × 112.5)/1.5 = 1575/1.5 = 1050 Rs. Since the cost of 2 kg of tea is equal to that of 5 kg of nuts, the cost of 1 kg of nuts is (2 × 1050)/5 = 2100/5 = 420 Rs. Finally, the cost of 11 kg of nuts is 11 × 420 = 4620 Rs. This confirms that option d is the correct answer.

Explanation

1729 is famously known as the Hardy-Ramanujan number. This number possesses a unique mathematical property where it is the smallest positive integer that can be expressed as the sum of the cubes of two positive integers in two different ways. This fact was revealed by the Indian mathematician Srinivasa Ramanujan to G.H. Hardy during a visit, after Hardy remarked that the taxi number 1729 seemed rather dull.

The two distinct pairs of cubes are: First way: 1729 = 1³ + 12³ = 1 + 1728 = 1729 Second way: 1729 = 9³ + 10³ = 729 + 1000 = 1729

Since there are exactly two such pairs of positive integers whose cubes sum up to 1729, option (c) is the correct statement. This property makes 1729 the first taxicab number, denoted as Ta(2) in number theory.

Explanation

The fastest way to solve this problem is by using the successive percentage change formula: Net % Change = x + y + xy/100, where x is the first percentage change (increase in cost price) and y is the second percentage change (discount on marked price).

Substituting the given values: x = 20 y = -10 (since a discount represents a decrease)

Net % Gain = 20 + (-10) + (20 × (-10))/100 Net % Gain = 10 - 200/100 Net % Gain = 10 - 2 = 8%

Alternatively, using the base method, let the cost price CP = 100. The marked price MP is increased by 20%, so MP = 100 + 20% of 100 = 120. The selling price SP is calculated by applying a 10% discount on the MP: SP = 120 - (10% of 120) = 120 - 12 = 108. The gain percentage is (SP - CP)/CP × 100 = (108 - 100)/100 × 100 = 8%. Thus, option d is correct.

Explanation

Two numbers p and q are said to be co-prime or relatively prime if their Highest Common Factor HCF(p, q) = 1. This means they have no common positive integer factor other than 1. Statement 1 is correct because any two distinct prime numbers, such as p = 3 and q = 5, will always have an HCF of 1. Statement 2 is correct because a prime number and a composite number can be co-prime, for example, p = 7 (prime) and q = 10 (composite), where HCF(7, 10) = 1. Statement 3 is correct because two composite numbers can also be co-prime if they do not share any common prime factors, such as p = 8 (composite) and q = 9 (composite), where HCF(8, 9) = 1. Since all three scenarios are mathematically possible, option d is the correct choice.

Explanation

The expression is a summation of terms in the form √(1 + 1/n² + 1/(n+1)²). To solve this efficiently, we use the algebraic identity √(1 + 1/n² + 1/(n+1)²) = 1 + 1/n - 1/(n+1). This can be verified by squaring the right side: (1 + 1/(n(n+1)))² = 1 + 2/(n(n+1)) + 1/(n²(n+1)²), which simplifies to the expression inside the square root after common denominators are taken.

The given series is the sum of these terms from n=1 to n=2007. The general term is Tₙ = 1 + 1/n - 1/(n+1). The total sum S is: S = Σⁿ⁼¹₂₀₀₇ (1 + 1/n - 1/(n+1)) S = Σⁿ⁼¹₂₀₀₇ 1 + Σⁿ⁼¹₂₀₀₇ (1/n - 1/(n+1))

The first part of the sum is 1 × 2007 = 2007. The second part is a telescoping series: (1/1 - 1/2) + (1/2 - 1/3) + … + (1/2007 - 1/2008) All intermediate terms cancel out, leaving only the first and last terms: 1 - 1/2008.

Adding the two parts together: S = 2007 + 1 - 1/2008 = 2008 - 1/2008

This matches option a.

Explanation

First, calculate the downstream speed S_d and upstream speed S_u using the formula Speed = Distance/Time. Downstream speed S_d = 20 km/2 hours = 10 km/hr. Upstream speed S_u = 4 km/2 hours = 2 km/hr.

To find the speed of the current, we use the standard relation: Speed of current = (S_d - S_u)/2 Substituting the calculated values into the formula: Speed of current = (10 - 2)/2 = 8/2 = 4 km/hr.

Therefore, the speed of the current is 4 km/hr, which corresponds to option (d). This method is the fastest way to solve boat and stream problems in competitive exams.

Explanation

Let x and y be any two positive odd integers. Any odd integer can be represented in the form 2n + 1 where n is a non-negative integer.

For statement 1, we know that the square of an odd number is always odd. Therefore, x² and y² are both odd integers. Since the sum of two odd integers is always an even integer, x² + y² must be even. Thus, statement 1 is correct.

For statement 2, consider the algebraic expansion using the general form of odd integers: x² + y² = (2n + 1)² + (2m + 1)² x² + y² = 4n² + 4n + 1 + 4m² + 4m + 1 x² + y² = 4(n² + n + m² + m) + 2

The expression 4(n² + n + m² + m) + 2 shows that x² + y² always leaves a remainder of 2 when divided by 4. Therefore, x² + y² is never divisible by 4, making statement 2 incorrect. For example, if x = 1 and y = 1, x² + y² = 2, which is even but not divisible by 4. If x = 1 and y = 3, x² + y² = 1 + 9 = 10, which is also even but not divisible by 4. Consequently, only statement 1 is correct, leading to option a.

Explanation

The equation 2^(x+2) 27^(x-1) = 9 simplifies to 2^(x+2) (3^3)^(x-1) = 3^2. Taking logs or testing values, the equation only has one real root. However, the provided answer key is B. As analyzed, x=2 is not a root of the given equation (2^4 * 27^1 = 432, not 9). The question contains significant errors in its expression or options, but based on the requested answer key, B is chosen.

Explanation

The profit share of each partner in a business is directly proportional to the product of their investment and the time duration for which the money was invested. The standard formula used is Profit = Investment × Time. Given the investment ratio as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the time ratio as 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, we calculate the profit shares as follows:

A: 1 × 12 = 12 B: 2 × 11 = 22 C: 3 × 10 = 30 D: 4 × 9 = 36 E: 5 × 8 = 40 F: 6 × 7 = 42 G: 7 × 6 = 42 H: 8 × 5 = 40 I: 9 × 4 = 36 J: 10 × 3 = 30 K: 11 × 2 = 22 L: 12 × 1 = 12

Comparing all the calculated profit values, the maximum value obtained is 42. This value is shared by both F and G. Therefore, both F and G will receive the maximum profit at the end of the year, making option c the correct choice.

Explanation

The sum is S = 2^122 + (2^2)^62 + (2^3)^42 + (2^2)^64 + 2^130 = 2^122 + 2^124 + 2^126 + 2^128 + 2^130. Factoring out 2^122: S = 2^122(1 + 2^2 + 2^4 + 2^6 + 2^8) = 2^122(1 + 4 + 16 + 64 + 256) = 2^122(341). Since 341 = 11 * 31, the sum is divisible by 11.

Explanation

The expression 4p² + 4pq - 3q² can be factorized by splitting the middle term. By rewriting 4pq as 6pq - 2pq, the expression becomes 4p² + 6pq - 2pq - 3q², which simplifies to 2p(2p + 3q) - q(2p + 3q) = (2p - q)(2p + 3q). Given that 2p + 3q = 12, we substitute this value into the factored equation: (2p - q) × 12 = 126. Dividing both sides by 12, we find 2p - q = 126 / 12 = 10.5 or 21/2.

We now have a system of two linear equations:

1. 2p + 3q = 12
2. 2p - q = 21/2

Subtracting the second equation from the first to eliminate p gives (2p + 3q) - (2p - q) = 12 - 21/2, which results in 4q = 3/2, so q = 3/8. Substituting q = 3/8 back into the second equation, we get 2p - 3/8 = 21/2, which means 2p = 84/8 + 3/8 = 87/8, leading to p = 87/16. The required value is p + 2q = 87/16 + 2(3/8) = 87/16 + 12/16 = 99/16, which matches option d.

Explanation

n = floor(log10(3^30)) + 1 = floor(30 log10 3) + 1 = floor(30 0.4771) + 1 = floor(14.313) + 1 = 14 + 1 = 15.

Explanation

The units digits of powers of 7 follow a cycle of 4: 7, 9, 3, 1. 139 mod 4 = 3. The 3rd digit in the cycle is 3.

Explanation

Using the algebraic identities for the difference and sum of cubes, we know that x³ - a³ = (x - a)(x² + ax + a²) and x³ + a³ = (x + a)(x² - ax + a²). Substituting these into the given expression allows us to cancel the quadratic terms in the numerators and denominators:

(x² + ax + a²)/((x - a)(x² + ax + a²)) - (x² - ax + a²)/((x + a)(x² - ax + a²)) = 1/(x - a) - 1/(x + a)

Combining these fractions using a common denominator gives:

((x + a) - (x - a))/((x - a)(x + a)) = 2a/(x² - a²)

From the given equation 4x + 3a = 0, we find that x = -3a/4. Substituting this value of x into the simplified expression:

2a/((-3a/4)² - a²) = 2a/(9a²/16 - a²) = 2a/((9a² - 16a²)/16) = 2a/(-7a²/16) = 2a × -16/7a² = -32a/7a² = -32/7a

Thus, the value of the expression is -32/(7a), which corresponds to option c.

Explanation

Let incomes be I_M = k, I_K = 2k and expenses be E_M = m, E_K = 3m. Savings S_M = k - m and S_K = 2k - 3m. S_K - S_M = k - 2m. Whether this is positive, negative, or zero depends on the values of k and m, which are not provided. Thus, it is impossible to determine.

Explanation

The profit in a partnership is divided in the ratio of the product of the capital invested and the time period for which it was invested. Total equivalent capital for X is calculated by summing the products of the amounts and their respective durations: (700 × 3) + (700 - 2/7 × 700) × 3 + (500 + 3/5 × 200) × 6. First, calculate X's withdrawals and additions: withdrawal = 2/7 × 700 = 200; amount put back = 3/5 × 200 = 120. X's equivalent capital = (700 × 3) + (500 × 3) + (620 × 6) = 2100 + 1500 + 3720 = 7320. Y's equivalent capital = 600 × 12 = 7200. The ratio of profit sharing between X and Y is 7320 : 7200 = 61 : 60. The total profit is Rs. 726. The sum of the ratio parts is 61 + 60 = 121. X's share of the profit = 61/121 × 726. Since 726 ÷ 121 = 6, X's share = 61 × 6 = 366. Therefore, X should receive Rs. 366, which corresponds to option B.

Explanation

Let the total capacity of the tank be the Least Common Multiple (LCM) of 2 and 6, which is 6 units. The efficiency of pipe X is 6/2 = 3 units per hour, and the efficiency of pipe Y is 6/6 = 1 unit per hour. Since pipe X was opened at 10 a.m. and pipe Y was opened at 11 a.m., pipe X worked alone for 1 hour. In this 1 hour, pipe X filled 3 × 1 = 3 units of the tank. The remaining work to be completed is 6 - 3 = 3 units. After 11 a.m., both pipes X and Y work together. Their combined efficiency is 3 + 1 = 4 units per hour. The time required to fill the remaining 3 units is given by the formula: Time = Remaining Work/Combined Efficiency = 3/4 hours. Converting 3/4 hours into minutes, we get (3/4) × 60 = 45 minutes. Therefore, the tank will be completely filled at 11 a.m. plus 45 minutes, which is 11:45 a.m. Option (c) is correct.

Explanation

Relative speed for two trains moving in opposite directions is given by vᵣₑₗ = v₁ + v₂. Here, vᵣₑₗ = 48 + 42 = 90 km/hr. To convert this to m/s, we multiply by 5/18, so vᵣₑₗ = 90 × 5/18 = 25 m/s. The total distance covered when two trains cross each other is the sum of their lengths. Let the length of the first train be L₁ and the second be L₂ = 1/2L₁. Using the formula Distance = Speed × Time, we get L₁ + 1/2L₁ = 25 × 12. This simplifies to 3/2L₁ = 300, giving L₁ = 200 m.

Now, consider the first train crossing a platform of length Lₚ. The speed of the first train in m/s is 48 × 5/18 = 40/3 m/s. The total distance to cross the platform is L₁ + Lₚ. Using the formula Distance = Speed × Time, we have 200 + Lₚ = 40/3 × 45. Solving this, 200 + Lₚ = 40 × 15, which means 200 + Lₚ = 600. Therefore, Lₚ = 600 - 200 = 400 m. This corresponds to option b.

Explanation

For a constant distance, speed is inversely proportional to the time taken to cover that distance. This relationship is defined by the formula t = d/s, where t is time, d is distance, and s is speed. Therefore, if the speeds of the three cars are in the ratio s₁ : s₂ : s₃ = 2 : 3 : 4, the ratio of the times taken will be the inverse of the speed ratio, which is t₁ : t₂ : t₃ = 1/2 : 1/3 : 1/4. To convert this into a whole number ratio, multiply each term by the Least Common Multiple (LCM) of the denominators 2, 3, and 4, which is 12. Multiplying each part gives 1/2 × 12 : 1/3 × 12 : 1/4 × 12, resulting in the ratio 6 : 4 : 3. Thus, option (d) is the correct answer.

Explanation

Let B be the number of boys and G be the number of girls. (20B + 16G) / (B + G) = 18. 20B + 16G = 18B + 18G, so 2B = 2G, meaning B = G. The percentage of boys is B / (B + G) = 1/2 = 50%.

Explanation

The area of the four walls of a room is given by the formula for lateral surface area: Area = 2h(l + b), where h is the height, l is the length, and b is the breadth. According to the question, the area of the four walls is 120 m², the height h = 4 m, and the length l = 2b. Substituting these values into the formula, we get 120 = 2 × 4 × (2b + b). This simplifies to 120 = 8 × 3b, which means 120 = 24b. Solving for b, we find b = 120 / 24 = 5 m. Since the length is twice the breadth, l = 2 × 5 = 10 m. The area of the floor is calculated as length multiplied by breadth, so Area of floor = l × b = 10 × 5 = 50 m². Thus, option (b) is correct.

Explanation

For a square of side x inscribed in a circle of radius R, the diagonal of the square is equal to the diameter of the circle. Using the Pythagorean theorem, the diagonal is √(x² + x²) = x√2. Therefore, x√2 = 2R, which simplifies to x = 2R/√2 = R√2.

For an equilateral triangle of side y inscribed in a circle of radius R, the relationship between the side and the radius is given by the formula y = R√3. This is derived from the fact that the distance from the center to a vertex is R, and the angle subtended by a side at the center is 120°.

To find the ratio of x to y, we divide the two expressions: x/y = R√2/R√3 = √2/√3 = √(2/3).

Comparing this result with the given options, the ratio is √(2/3), which corresponds to option (a).

Explanation

The Curved Surface Area (CSA) of a cylinder is 2πrh and the Total Surface Area (TSA) is 2πr(h+r). Given the ratio CSA/TSA = 1/2, we substitute the formulas: (2πrh)/(2πr(h+r)) = 1/2. Simplifying this gives h/(h+r) = 1/2, which implies 2h = h+r, therefore h=r.

Using the given TSA of 616 cm² and the fact that h=r, we have 2πr(r+r) = 616, which simplifies to 4πr² = 616. Solving for r²: πr² = 616/4 = 154. Substituting π = 22/7, we get (22/7)r² = 154 ⇒ r² = (154 × 7)/22 = 49. Thus, r = 7 cm and since h=r, h = 7 cm.

The volume of the cylinder is calculated using V = πr²h. Substituting the values: V = 22/7 × 7² × 7 = 22 × 49 = 1078 cm³. This matches option (c).

Explanation

The volume of the copper is given as 1 m³ and its weight is 9000 kg, which implies a density ρ = 9000 kg/m³. The copper is rolled into a square bar of length L = 9 m. Let the side of the square cross-section be s. Since the volume remains constant, we use the formula for the volume of a prism: V = Area of cross-section × Length. Substituting the values, we get 1 = s² × 9, which gives s² = 1/9 m², and thus s = 1/3 m.

An exact cube cut from this bar will have a side length equal to the side of the square cross-section, so the side of the cube is a = 1/3 m. The volume of this cube is calculated using the formula V꜀ᵤ₆ₑ = a³ = (1/3)³ = 1/27 m³. The weight of the cube is the product of its volume and the density of copper: Weight = V꜀ᵤ₆ₑ × ρ = 1/27 m³ × 9000 kg/m³ = 9000/27 kg = 1000/3 kg. Therefore, option B is correct.

Explanation

Volume of cone V = (1/3)pir^2h = (1/3)(22/7)(8.4)^2(3.5) = 258.72 m^3. Number of bags = 258.72 / 1.96 = 132.

Explanation

In triangle ABC, since it is a right angled triangle with legs BC = 5 cm and AC = 12 cm, the hypotenuse AB = √(5² + 12²) = √169 = 13 cm. Let the angle ∠ABC be β. From the given dimensions, sin β = AC/AB = 12/13 and cos β = BC/AB = 5/13.

In triangle BCD, we are given ∠BCD = 30° and we know ∠CBD = β. Using the sine rule in triangle BCD, we have CD/sin β = BC/sin ∠BDC. Since the sum of angles in a triangle is 180°, ∠BDC = 180° - (30° + β). Therefore, sin ∠BDC = sin(180° - (30° + β)) = sin(30° + β).

Substituting the values, we get CD = (BC sin β)/sin(30° + β). We use the identity sin(A + B) = sin A cos B + cos A sin B to find sin(30° + β) = sin 30° cos β + cos 30° sin β = 1/2 · 5/13 + √3/2 · 12/13 = (5 + 12√3)/26.

Now, substitute all values into the formula for CD: CD = (5 · (12/13))/((5 + 12√3)/26) = (60/13)/((5 + 12√3)/26) = 60/13 · 26/(5 + 12√3) = 120/(5 + 12√3) cm. This matches option d.

Explanation

Let the radii of the two circles be r₁ and r₂. Since the circles touch each other externally, the distance between their centres is equal to the sum of their radii. Therefore, r₁ + r₂ = 14 cm. The sum of their areas is given as 130π, which provides the equation πr₁² + πr₂² = 130π. Dividing by π, we get r₁² + r₂² = 130. To find the difference in the radii, |r₁ - r₂|, we use the algebraic identity (r₁ + r₂)² = r₁² + r₂² + 2r₁r₂. Substituting the known values, 14² = 130 + 2r₁r₂, which simplifies to 196 = 130 + 2r₁r₂. Solving for the product term, 2r₁r₂ = 196 - 130 = 66. Now, apply the identity for the square of the difference: (r₁ - r₂)² = r₁² + r₂² - 2r₁r₂. Substituting the values, (r₁ - r₂)² = 130 - 66 = 64. Taking the square root of both sides, r₁ - r₂ = √64 = 8 cm. Thus, option d is the correct answer.

Explanation

Let r = 3 be the radius of the circle and L = 2 be the length of the chord. Since the diameter AB intersects the chord perpendicularly at point P, it bisects the chord. Therefore, the distance from P to either endpoint of the chord is L/2 = 2/2 = 1 unit.

Let O be the center of the circle. Using the Pythagorean theorem in the right-angled triangle formed by the center O, the intersection point P, and one end of the chord, we have: OP² + (L/2)² = r² OP² + 1² = 3² OP² = 9 - 1 = 8 OP = √8

The diameter AB is divided by point P into two segments, AP and BP. The center O is the midpoint of AB, so AO = OB = 3. Since we are given that AP > BP, point P must lie on the segment OB. Thus, the lengths of the segments are: AP = AO + OP = 3 + √8 BP = OB - OP = 3 - √8

The ratio of AP to BP is therefore (3 + √8) : (3 - √8), which matches option (b).

Explanation

Circumference C = pid = (22/7)(5/11) = 10/7 m. Distance D = 7000 m. Number of rounds = 7000 / (10/7) = 700 * 7 = 4900.

Explanation

Let the radius of the base of the cylinder be r. Since the hemispherical dome is surmounted on the diameter of the cylinder, its radius is also r. Let the total height of the building be H and the height of the cylindrical part be h. According to the problem, H = 3r. Since the total height is the sum of the cylinder height and the radius of the hemisphere, H = h + r, which implies h = 3r - r = 2r.

The total volume V of the building is the sum of the volume of the cylinder and the volume of the hemisphere: V = πr²h + 2/3 πr³ Substitute h = 2r: V = πr²(2r) + 2/3 πr³ = 2πr³ + 2/3 πr³ = 8/3 πr³

Given that V = 67 1/21 = 1408/21 m³. Substituting π = 22/7: 8/3 × 22/7 × r³ = 1408/21 176/21 r³ = 1408/21 176r³ = 1408 r³ = 1408/176 = 8 r = 2 m

The total height of the building is H = 3r = 3 × 2 = 6 m. Thus, option a is correct.

Explanation

Let the radius r and height h of the cylinder be 2x and 3x respectively, based on the given ratio 2:3. The formula for the volume of a right circular cylinder is V = πr²h. Substituting the given volume of 1617 cm³ and the ratio variables:

1617 = π(2x)²(3x) 1617 = π(4x²)(3x) 1617 = 12πx³

Using π = 22/7, we solve for x:

1617 = 12 × 22/7 × x³ x³ = (1617 × 7)/(12 × 22) = 11319/264 = 42.875 x = ∛42.875 = 3.5 cm

The formula for the total surface area (TSA) of a cylinder is TSA = 2πr(r + h). Substituting the values of r and h in terms of x:

TSA = 2π(2x)(2x + 3x) = 2π(2x)(5x) = 20πx²

Substitute x = 3.5 and π = 22/7:

TSA = 20 × 22/7 × (3.5)² TSA = 20 × 22/7 × 12.25 TSA = 20 × 22 × 1.75 TSA = 440 × 1.75 = 770 cm²

Thus, the total surface area of the cylinder is 770 cm², which corresponds to option C.

Explanation

To find the cost of painting per square centimetre, we first calculate the total surface area of the spherical vessel using the formula SA = 4πr² or SA = πd². Given the diameter d = 14 cm, the radius r = 7 cm. Substituting the values, SA = 4 × 22/7 × 7² = 4 × 22 × 7 = 616 sq cm.

The cost of painting per square centimetre is determined by dividing the total cost by the total surface area. Using the formula Rate = Total Cost/Surface Area, we get Rate = 8008/616. Simplifying this division, 8008/616 = 13. Thus, the cost of painting per square centimetre is Rs. 13, which corresponds to option (c).

Explanation

This property is based on the Pitot Theorem, which states that in a tangential quadrilateral (a quadrilateral that has an inscribed circle), the sum of the lengths of opposite sides is equal. To prove this, we use the property that tangents drawn from an external point to a circle are equal in length. Let the sides of the quadrilateral be AB, BC, CD, and DA, and let the points where the inscribed circle touches these sides be P, Q, R, and S respectively.

The lengths of the tangents from the vertices are equal: AP = AS, BP = BQ, CQ = CR, and DR = DS. The sum of one pair of opposite sides is AB + CD = (AP + PB) + (CR + RD). Substituting the equal tangent lengths, we get AB + CD = AS + BQ + CQ + DS. Rearranging these terms gives (AS + SD) + (BQ + QC), which equals AD + BC. Thus, the sum of one pair of opposite sides equals the sum of the other pair of opposite sides, making option b the correct choice.

Explanation

Let the radius of the circle be r and the side of the square be s. The perimeter of the circle is given by the formula 2πr and the perimeter of the square is given by 4s. Since both perimeters are equal, we have 2πr = 4s, which simplifies to s = πr/2.

Next, we calculate the areas of both shapes. The area of the circle is A꜀ᵢᵣ꜀ₗₑ = πr². The area of the square is Aₛᵩᵤₐᵣₑ = s² = (πr/2)² = π²r²/4.

To compare the two areas, we find their ratio: A꜀ᵢᵣ꜀ₗₑ/Aₛᵩᵤₐᵣₑ = (πr²)/(π²r²/4) = 4πr²/π²r² = 4/π.

Using the approximate value of π ≈ 3.14159, the ratio 4/3.14159 ≈ 1.27. Since this ratio is greater than 1, the area of the circle is larger than the area of the square. This confirms that option (b) is correct. This result is a specific instance of the isoperimetric inequality, which states that among all closed curves of a given perimeter, the circle encloses the maximum area.

Explanation

Volume of frustum V = (1/3)pih(r1^2 + r2^2 + r1r2). Here h=24, r1=2, r2=9. V = (1/3)pi24(4 + 81 + 18) = 8pi(103) = 824pi. Thus x = 824.

Explanation

When the mid-points of the sides of an equilateral triangle are joined, the resulting inner triangle has sides that are half the length of the original triangle. According to the area ratio theorem for similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, the area of each subsequent triangle is (1/2)² = 1/4 of the area of the previous triangle.

Let the area of the first triangle be A₁. The area of the n-th triangle in such a sequence is given by the formula Aₙ = A₁ ⋅ (1/4)ⁿ⁻¹. To find the ratio of the area of the fourth triangle A₄ to the seventh triangle A₇, we substitute the values into the formula:

A₄ = A₁ ⋅ (1/4)⁴⁻¹ = A₁ ⋅ (1/4)³ A₇ = A₁ ⋅ (1/4)⁷⁻¹ = A₁ ⋅ (1/4)⁶

The ratio is calculated as follows: A₄/A₇ = (A₁ ⋅ (1/4)³)/(A₁ ⋅ (1/4)⁶) = 1/((1/4)⁶⁻³) = 1/((1/4)³) = 4³ = 64

Thus, the ratio of the area of the fourth triangle to that of the seventh triangle is 64:1, which corresponds to option c).

Explanation

The volume of rain collected from a roof is calculated using the formula Volume = Area of roof × Rainfall depth, while the volume of a cylindrical vessel is given by Volume = πr²h. Since the vessel is just full, we equate the two volumes: πr²h = Area of roof × Rainfall depth.

Given the roof area is 22 m × 20 m = 440 m², and for the cylindrical vessel, the radius r = 2/2 = 1 m and height h = 3.5 m. Substituting these values into the equation: 22/7 × (1)² × 3.5 = 440 × Rainfall depth. Simplifying the left side, 22/7 × 3.5 = 22 × 0.5 = 11. Therefore, 11 = 440 × Rainfall depth, which leads to Rainfall depth = 11/440 = 1/40 m. To convert this value into centimeters, we multiply by 100: 1/40 × 100 = 2.5 cm. Thus, option (c) is correct.

Explanation

For two similar cones, the ratio of their volumes is equal to the cube of the ratio of their corresponding heights. Let V be the volume and H be the height of the original cone, and v be the volume and h be the height of the small cone cut from the top. The relationship is given by the formula v/V = (h/H)³.

Given that V = 60 cm and v/V = 1/64, we substitute these values into the formula: 1/64 = (h/60)³

Taking the cube root of both sides: 1/4 = h/60 h = 60/4 = 15 cm

This h = 15 cm represents the height of the small cone from the apex. The question asks for the height from the base at which the section is made. Therefore, we subtract the height of the small cone from the total height of the original cone: Height from base = H - h = 60 - 15 = 45 cm.

Thus, the section is made at a height of 45 cm from the base, which corresponds to option d).

Explanation

In a circle, the length of the tangent from an external point is given by the formula L = r/tan(θ/2), where r is the radius of the circle and θ is the angle between the two tangents. In this problem, the radius r = 3 cm and the angle between the tangents θ = 60°. Substituting these values into the formula, we get L = 3/tan(60°/2) = 3/tan(30°). Since tan(30°) = 1/√3, the expression becomes L = 3/(1/√3) = 3√3 cm. This matches option a. Geometrically, the line connecting the center of the circle to the external point bisects the angle between the tangents, creating a right-angled triangle where the radius is the opposite side and the tangent is the adjacent side relative to the bisected angle. Using the trigonometric ratio tan(30°) = opposite/adjacent = 3/L, we find L = 3/(1/√3) = 3√3 cm.

Explanation

The volume of a sphere is calculated using the standard geometric formula V = 4/3πr³, where r represents the radius of the sphere. Given that the radius r = 3 cm, we substitute this value directly into the formula to find the volume.

Substituting the values, we get V = 4/3 × π × (3)³. First, calculate the cube of the radius: 3³ = 3 × 3 × 3 = 27. Now, multiply this by the constant factor: V = 4/3 × π × 27. By simplifying the fraction, 27 divided by 3 equals 9. Therefore, V = 4 × π × 9, which simplifies to 36π cm³. This matches option a.

Explanation

Consider a triangle with vertices (0, a), (-b, 0), and (c, 0). The sides are sqrt(a^2 + b^2), sqrt(a^2 + c^2), and (b + c). This matches the given sides. The area is (1/2) base height = (1/2) (b + c) a = a(b + c)/2.

Explanation

To find the area of the largest triangle inscribed in a semi-circle of radius r, we use the standard formula for the area of a triangle: Area = 1/2 × base × height.

For a triangle inscribed in a semi-circle, the maximum area is achieved when the base of the triangle is the diameter of the semi-circle. Therefore, the base b = 2r. The height h is the perpendicular distance from the base to the third vertex on the arc. The maximum possible height occurs when the vertex is at the highest point of the semi-circle, which is equal to the radius of the circle. Thus, the maximum height h = r.

Substituting these values into the area formula: Area = 1/2 × (2r) × (r) Area = 1/2 × 2r² Area = r²

This result indicates that the largest triangle inscribed in a semi-circle is an isosceles right triangle with the diameter as its hypotenuse. Consequently, option a is correct.

Explanation

The distance covered by a wheel in one complete revolution is equal to its circumference. The formula for the circumference of a circle is C = πd, where d is the diameter. For the front wheel, the diameter is 2x cm, so its circumference is 2πx. Given that the front wheel revolves n times, the total distance covered is n × 2πx.

The rear wheel must cover the same distance. Let the number of revolutions the rear wheel makes be m. Since the diameter of the rear wheel is 2y cm, its circumference is 2πy. The total distance covered by the rear wheel is m × 2πy. Equating the distances of both wheels, we get n × 2πx = m × 2πy. Simplifying the equation by canceling 2π from both sides, we have nx = my. Solving for m, we find m = nx/y. Therefore, the rear wheel revolves nx/y times, which corresponds to option c.

Explanation

To find the slant height of the conical part of the tent, we first identify the radius and the vertical height of the cone. The radius r of the base of the cone is equal to the radius of the cylindrical part, which is half of the given diameter: r = 126/2 = 63 m. The height h of the conical part is the difference between the total height of the tent and the height of the cylindrical part: h = 21 m - 5 m = 16 m.

Using the formula for the slant height l of a right circular cone, l = √(r² + h²), we substitute the known values: l = √(63² + 16²). Calculating the squares gives l = √(3969 + 256) = √4225. Taking the square root of 4225, we find l = 65 m. Thus, the slant height of the cone is 65 m, which matches option (b).

Explanation

The curved surface area of a cylinder is calculated using the formula CSA = 2πrh, which can also be simplified to CSA = πdh, where d is the diameter and h is the height of the cylinder. Given that the diameter d = 126 m and the height of the cylindrical part h = 5 m, we substitute these values into the formula using π = 22/7. The calculation is CSA = 22/7 × 126 × 5. Dividing 126 by 7 gives 18, so the expression becomes 22 × 18 × 5. Multiplying 18 by 5 equals 90, and 22 × 90 results in 1980 m². Thus, the curved surface area of the cylinder is 1980 m², which corresponds to option a. The total height of the tent is provided as extra information and is not required to find the surface area of the cylindrical section.

Explanation

CSA of cylinder = 2pirh = 2(22/7)635 = 1980. For the cone, l = sqrt(63^2 + 16^2) = sqrt(3969 + 256) = 65. CSA of cone = pirl = (22/7)6365 = 12870. Total canvas = 1980 + 12870 = 14850.

Explanation

Circumference C = pid = (22/7)0.8 = 17.6/7 m. Distance in 10 min = (66 1000 / 60) 10 = 11000 m. Revolutions = 11000 / (17.6/7) = 77000 / 17.6 = 4375.

Explanation

The given side lengths 5 cm, 12 cm, and 13 cm form a standard Pythagorean triplet since they satisfy the relation a² + b² = c², specifically 5² + 12² = 25 + 144 = 169 = 13². Therefore, the triangle is a right-angled triangle where the side of 13 cm is the hypotenuse and the sides of 5 cm and 12 cm are the legs. For any right-angled triangle, the area can be calculated using the formula Area = 1/2 × base × height. By substituting the values of the two legs as the base and height, we get Area = 1/2 × 5 × 12. This simplifies to Area = 5 × 6 = 30 cm², which corresponds to option a.

Explanation

Let the radius of the circle be r and the side of the square be a. According to the problem, the perimeter (circumference) of the circle is equal to the perimeter of the square. Using the formulas for perimeter, we have 2πr = 4a. This simplifies to a = πr/2.

The area of the circle is A꜀ᵢᵣ꜀ₗₑ = πr² and the area of the square is Aₛᵩᵤₐᵣₑ = a². To find the ratio of the area of the circle to that of the square, we divide the two formulas: Ratio = πr²/a²

Substituting the value of a from the perimeter equation: Ratio = πr²/(πr/2)² = πr²/(π²r²/4) = 4πr²/π²r² = 4/π

Using the value π = 22/7, the ratio becomes: Ratio = 4/(22/7) = (4 × 7)/22 = 28/22 = 14/11

Therefore, the ratio of the area of the circle to the area of the square is 14:11, which corresponds to option b.

Explanation

Since AB is the diameter of the circle with radius r, the length of AB is 2r. Given that CD is parallel to AB and is half the length of AB, the length of chord CD is r. Let M be the midpoint of the chord CD. In any circle, the line joining the center O to the midpoint of a chord is perpendicular to the chord. Therefore, triangle OMC is a right-angled triangle where OC is the radius of the circle, OM is the distance from the center to the chord, and MC is half the length of the chord CD.

Using the Pythagoras theorem in triangle OMC: OM² + MC² = OC² Substituting the known values, where OC = r and MC = CD/2 = r/2: OM² + (r/2)² = r² OM² = r² - r²/4 OM² = 3r²/4 OM = (√3r)/2

As the circle completes one rotation about its center O, the point M moves in a circular path centered at O. The radius of this locus is equal to the constant distance OM. Thus, the locus of the midpoint M is a circle with radius (√3r)/2, which corresponds to option d.

Explanation

(sin theta + cos theta)^2 = 7/4 => 1 + 2 sin theta cos theta = 7/4 => 2 sin theta cos theta = 3/4. Then (sin theta - cos theta)^2 = 1 - 2 sin theta cos theta = 1 - 3/4 = 1/4. So sin theta - cos theta = +/- 1/2.

Explanation

sin x + sin^2 x = 1 implies sin x = 1 - sin^2 x = cos^2 x. The expression cos^8 x + 2cos^6 x + cos^4 x = (cos^4 x + cos^2 x)^2. Substituting cos^2 x = sin x, we get (sin^2 x + sin x)^2 = 1^2 = 1.

Explanation

Let the height of the tower be h = 100 m. The distance of the car from the foot of the tower can be calculated using the formula x = h cot θ, where θ is the angle of depression. The initial distance x₁ when the angle of depression is 30° is x₁ = 100 cot 30° = 100√3 m. The final distance x₂ when the angle of depression becomes 60° is x₂ = 100 cot 60° = 100/√3 m. The distance traveled by the car is the difference between these two positions: d = x₁ - x₂ = 100√3 - 100/√3. Factoring out 100, we get d = 100(√3 - 1/√3) = 100((3-1)/√3) = 200/√3 m. To rationalize the denominator, multiply the numerator and denominator by √3, resulting in d = 200√3/3 m. Thus, option (b) is correct.

Explanation

Distance 1 = 75 / tan 30 = 75 sqrt(3). Distance 2 = 75 / tan 60 = 75 / sqrt(3) = 25 sqrt(3). Total distance = 75sqrt(3) + 25sqrt(3) = 100*sqrt(3).

Explanation

The expression is simplified using complementary angle identities and fundamental trigonometric identities. First, we identify complementary pairs: 68° and 22° since 68° + 22° = 90°, and 56° and 34° since 56° + 34° = 90°. According to the complementary angle formula tan θ = cot(90° - θ), we have tan 22° = cot 68°. Substituting this into the expression, the terms csc² 68° - tan² 22° become csc² 68° - cot² 68°, which equals 1 based on the identity csc² θ - cot² θ = 1. Similarly, using cot θ = tan(90° - θ), we find cot 34° = tan 56°. Thus, the terms sec² 56° - cot² 34° become sec² 56° - tan² 56°, which equals 1 based on the identity sec² θ - tan² θ = 1. Combining these results, the final value is 1 + 1 = 2, which corresponds to option d.

Explanation

From 2y cos theta = x sin theta, tan theta = 2y/x. From 2x sec theta - y csc theta = 3, (2x sin theta - y cos theta) / (sin theta cos theta) = 3. Substituting x sin theta = 2y cos theta gives (4y cos theta - y cos theta) / (sin theta cos theta) = 3, so 3y / sin theta = 3, which means y = sin theta. Then x = 2y cot theta = 2 cos theta. Thus x^2 + 4y^2 = 4 cos^2 theta + 4 sin^2 theta = 4.

Explanation

To find the value of tan θ + cot θ, we first express it in terms of sin θ and cos θ using the identities tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. Thus, tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin² θ + cos² θ)/(sin θ cos θ). Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the expression simplifies to 1/(sin θ cos θ).

Now, we use the given equation sin θ + cos θ = (1 + √3)/2 and square both sides to find the value of sin θ cos θ. Squaring gives (sin θ + cos θ)² = ((1 + √3)/2)², which expands to sin² θ + cos² θ + 2 sin θ cos θ = (1 + 3 + 2√3)/4. Substituting sin² θ + cos² θ = 1, we get 1 + 2 sin θ cos θ = (4 + 2√3)/4 = 1 + √3/2. Subtracting 1 from both sides, we find 2 sin θ cos θ = √3/2, which implies sin θ cos θ = √3/4. Substituting this value back into our simplified target expression, we get tan θ + cot θ = 1/(√3/4) = 4/√3, which corresponds to option d.

Explanation

Let A = sin^2 theta + cos^4 theta. Using the identity sin^2 theta = 1 - cos^2 theta, we have A = 1 - cos^2 theta + cos^4 theta. Let x = cos^2 theta. Since 0 <= theta < pi/2, 0 < x <= 1. The expression becomes A = x^2 - x + 1. This is a quadratic in x. The minimum value occurs at x = -(-1)/(2*1) = 1/2, where A = (1/2)^2 - 1/2 + 1 = 1/4 - 1/2 + 1 = 3/4. The maximum value occurs at the boundaries of the interval (0, 1]. At x = 1, A = 1^2 - 1 + 1 = 1. As x approaches 0, A approaches 1. Thus, 3/4 <= A <= 1.

Explanation

Using the identity csc^2 A - cot^2 A = 1, we can rewrite the numerator: cot A + csc A - 1 = (cot A + csc A) - (csc^2 A - cot^2 A) = (csc A + cot A) - (csc A - cot A)(csc A + cot A) = (csc A + cot A)(1 - (csc A - cot A)) = (csc A + cot A)(1 - csc A + cot A). The expression is then [(csc A + cot A)(1 - csc A + cot A)] / (cot A - csc A + 1). Since (1 - csc A + cot A) is the same as (cot A - csc A + 1), the expression simplifies to csc A + cot A = (1/sin A) + (cos A/sin A) = (1 + cos A)/sin A.

Explanation

The problem tests the fundamental property of equality in mathematics. According to the reflexive property of equality, any real number or trigonometric value is equal to itself. This means for any value x, the statement x = x is always true, while the strict inequality statements x > x or x < x are always logically false.

In statement 1, sin 1° > sin 1° suggests that a value is strictly greater than itself, which is impossible. In statement 2, cos 1° < cos 1° suggests that a value is strictly less than itself, which is also impossible. In statement 3, tan 1° > tan 1° similarly asserts a strict inequality between a value and itself. Since sin 1°, cos 1°, and tan 1° are specific constant real numbers, they must be equal to themselves. Therefore, all three statements 1, 2, and 3 are mathematically incorrect. This makes option d the correct choice as it identifies all three incorrect statements.

Explanation

Let tan^2 x = u. The equation is u + 1/u = 2. Multiplying by u gives u^2 + 1 = 2u, or u^2 - 2u + 1 = 0, which is (u - 1)^2 = 0. Thus, u = 1. Since tan^2 x = 1 and 0 < x < 90 degrees, tan x = 1, which means x = 45 degrees.

Explanation

The Sun's altitude is the angle of elevation of the Sun above the horizon. In trigonometry, the relationship between the height of an object, the length of its shadow, and the angle of elevation is given by the formula tan(θ) = height of the tower/length of the shadow. Let the height of the tower be h and the length of the shadow be s. According to the problem, the length of the shadow is equal to the height, so s = h. Substituting these values into the formula, we get tan(θ) = h/h = 1. We know from trigonometric tables that tan(45°) = 1. Therefore, θ = 45°. This indicates that when the height and shadow are equal, the Sun is at an altitude of 45° above the horizon, forming an isosceles right-angled triangle.

Explanation

Statement 1: cos 75 = sin 15, sin 12 = cos 78, cos 18 = sin 72. The expression is 1 + 1 - 1 = 1. Correct. Statement 2: cos 35 = sin 55, sin 11 = cos 79. The expression is 1 - 1 + (cos 28 / cos 62) = tan 62, which is not 1. Incorrect. Statement 3: sin 80 = cos 10, sin 59 = cos 31. The expression is 1 - (cos 31 / cos 31) = 1 - 1 = 0. Correct. Only 1 and 3 are correct.

Explanation

Let the pole height be h and the foot of the pole be P. Let the angle of elevation from corners A, B, and C be alpha. Then tan alpha = h/PA = h/PB = h/PC. This implies PA = PB = PC, meaning the point P is equidistant from the vertices of the triangle. This point is the circumcentre. Option B mentions "Circumference", which is a typo for "Circumcentre", but it is the only logically applicable option among the choices.

Explanation

To solve this product, we use the trigonometric identity tan(90° - θ) = cot θ and the property tan θ · cot θ = 1. The expression is the product of tan n° for n = 1 to 89. We can pair the terms from the start and the end of the series as follows: (tan 1° · tan 89°) · (tan 2° · tan 88°) · (tan 3° · tan 87°) … (tan 44° · tan 46°) · tan 45°. Using the identity, tan 89° = cot 1°, tan 88° = cot 2°, and so on, up to tan 46° = cot 44°. Substituting these into the pairs, we get (tan 1° · cot 1°) · (tan 2° · cot 2°) … (tan 44° · cot 44°) · tan 45°. Since tan θ · cot θ = 1 for all θ, each of the 44 pairs evaluates to 1. Additionally, the value of tan 45° is exactly 1. Thus, the product becomes 1 · 1 · 1 … · 1 · 1 = 1, which corresponds to option b.

Explanation

Using the weighted average formula or the rule of alligation, the ratio of the number of students in two groups can be found by comparing their individual averages with their combined average. For classes X and Y, the ratio is nₓ(83 - 79) = nᵧ(79 - 76), which simplifies to 4nₓ = 3nᵧ, giving nₓ : nᵧ = 3 : 4. Similarly, for classes Y and Z, the ratio is nᵧ(81 - 76) = n_z(85 - 81), which simplifies to 5nᵧ = 4n_z, giving nᵧ : n_z = 4 : 5.

Combining these two ratios, we get nₓ : nᵧ : n_z = 3 : 4 : 5. To find the overall average of all three classes, we use the weighted average formula: Averageₓᵧ_z = (nₓAₓ + nᵧAᵧ + n_zA_z)/(nₓ + nᵧ + n_z) Substituting the ratios and values: Averageₓᵧ_z = ((3 × 83) + (4 × 76) + (5 × 85))/(3 + 4 + 5) Averageₓᵧ_z = (249 + 304 + 425)/12 Averageₓᵧ_z = 978/12 Averageₓᵧ_z = 81.5

Thus, the average score of classes X, Y, and Z is 81.5, which corresponds to option a).

Explanation

To find the median of a set of discrete values, the data must first be arranged in ascending order. The given values are x + 4, x - 3.5, x - 2.5, x - 3, x - 2, x + 0.5, x - 0.5, and x + 5. Arranging these in ascending order gives the sequence: x - 3.5, x - 3, x - 2.5, x - 2, x - 0.5, x + 0.5, x + 4, x + 5.

The total number of observations is n = 8. Since n is an even number, the median is calculated as the arithmetic mean of the two middlemost terms, specifically the (n/2)th and (n/2 + 1)th terms. In this sorted sequence, the 4th term is x - 2 and the 5th term is x - 0.5.

Applying the median formula for an even number of observations: Median = (Xₙ/₂ + X₍ₙ/₂₎₊₁)/2 Median = ((x - 2) + (x - 0.5))/2 Median = (2x - 2.5)/2 Median = x - 1.25

This result corresponds to option (a), where the notation x - 1 · 25 represents x - 1.25.

Explanation

For a set of n observations arranged in ascending order, the median is the value of the (n+1)/2 th observation when n is odd. In this case, n = 9, so the median is the (9+1)/2 = 5th observation. Let the ordered observations be x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, x₉. The given median is x₅ = 20.5. The operation involves increasing the largest 4 observations, which are x₆, x₇, x₈, and x₉. Since only values greater than the median are being increased, the relative order of the observations remains unchanged, and the 5th observation remains x₅. Therefore, the value of the median does not change and remains 20.5, which corresponds to option d).

Explanation

Let the average of 10 innings be x. Total runs = 10x. After the 11th inning, new total = 10x + 108. New average = (10x + 108) / 11. We are told (10x + 108) / 11 = x + 6. Solving for x: 10x + 108 = 11x + 66, so x = 42. The new average is x + 6 = 42 + 6 = 48.

Explanation

Total sum of 20 observations = 20 * 17 = 340. To correct the sum, subtract the wrong values (3 and 6) and add the correct values (8 and 9): Correct Sum = 340 - 3 - 6 + 8 + 9 = 348. Correct mean = 348 / 20 = 17.4.

Explanation

The number of credit cards held by an individual is a quantitative variable because it represents a numerical count. Within quantitative data, it is specifically classified as discrete data because the values are countable and can only take specific integer values, such as 0, 1, 2, or 3. Discrete data is characterized by the fact that there are distinct gaps between the possible values; for instance, an individual cannot possess 1.5 or 2.75 credit cards. This differs from continuous data, which can take any value within a range, and qualitative or categorical data, which describe non-numerical attributes or labels. Since the number of cards is obtained through counting and results in whole numbers, it is treated as discrete data.

Explanation

Total marks for 100 students = 100 72 = 7200. Total marks for 70 boys = 70 75 = 5250. Total marks for 30 girls = 7200 - 5250 = 1950. Average marks for girls = 1950 / 30 = 65.

Explanation

To find the percentage of a particular sector in a pie chart, we use the standard formula: Percentage = (Central Angle of Sector / Total Central Angle of Circle) × 100

Given that the central angle for corporate tax is 108° and the total angle of a complete pie chart is 360°, we substitute these values into the formula as follows: Percentage = (108° / 360°) × 100

To simplify the calculation, we can first reduce the fraction 108/360. Both the numerator and the denominator are divisible by 36: 108 ÷ 36 = 3 360 ÷ 36 = 10 The simplified fraction is 3/10.

Now, multiplying this result by 100 to convert it into a percentage: 3/10 × 100 = 30%

Thus, the income from corporate tax represents 30% of the total funds, making option (c) the correct answer.

Explanation

Statement 1 describes inclusive classes (where there is a gap between the upper limit of one class and the lower limit of the next, e.g., 19 and 20), but calls them exclusive. Statement 2 describes exclusive classes (where the upper limit of one is the lower limit of the next, e.g., 20 and 20), but calls them inclusive. Both statements are incorrect.

Explanation

Frequency density = Frequency / Class Width. Here, Frequency = 30 and Class Width = 15 - 10 = 5. Frequency density = 30 / 5 = 6.

Explanation

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. We need pairs (a, b) where a != b and ab = 12k + 4 for k >= 1. Possible values for ab: 16, 28, 40, 52, 64, 76, 88, 100. Pairs from S: ab=16: (2,8), (8,2); ab=28: (4,7), (7,4); ab=40: (4,10), (10,4), (5,8), (8,5). Total = 8 pairs.

Explanation

Modulo 29: 13^5 + 14^5 + 15^5 + 16^5. Note that 15 = -14 (mod 29) and 16 = -13 (mod 29). Therefore, 15^5 = (-14)^5 = -14^5 and 16^5 = (-13)^5 = -13^5. The sum is 13^5 + 14^5 - 14^5 - 13^5 = 0.

Explanation

Sum of cubes of first 10 natural numbers = (10 11 / 2)^2 = 55^2 = 3025. Sum of squares of first 10 natural numbers = 10 11 * 21 / 6 = 385. Difference = 3025 - 385 = 2640.

Explanation

When a person travels equal distances at two different speeds, the average speed for the entire journey is calculated using the harmonic mean of the speeds. The formula used is Average Speed = 2v₁v₂/(v₁ + v₂), where v₁ is the speed of the first leg and v₂ is the speed of the return leg.

Substituting the given values, v₁ = 50 km/hour and v₂ = 30 km/hour: Average Speed = (2 × 50 × 30)/(50 + 30) Average Speed = 3000/80 Average Speed = 300/8 = 37.5 km/hour.

Alternatively, using the fundamental formula Average Speed = Total Distance / Total Time, the total distance is 150 + 150 = 300 km. The time for the onward journey is 150/50 = 3 hours and the time for the return journey is 150/30 = 5 hours. Total time is 3 + 5 = 8 hours. Therefore, Average Speed = 300/8 = 37.5 km/hour. This corresponds to option c).

Explanation

To solve the given equation x + 1/(1 + 1/(2 + 1/3)) = 2, we evaluate the continued fraction from the bottom up. First, calculate the innermost denominator: 2 + 1/3 = (6+1)/3 = 7/3. Substituting this back into the expression, we get 1 + 1/(7/3), which simplifies to 1 + 3/7 = (7+3)/7 = 10/7. Now, the entire fraction term becomes 1/(10/7), which equals 7/10. The equation now simplifies to x + 7/10 = 2. To find x, subtract 7/10 from both sides: x = 2 - 7/10. Converting 2 to a fraction with a denominator of 10, we have x = 20/10 - 7/10 = 13/10. Therefore, the value of x is 13/10, which corresponds to option (b).