CDS II 2013 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The relationship between the Highest Common Factor (HCF), Least Common Multiple (LCM), and two numbers a and b is given by the formula: a × b = HCF × LCM. Let the two numbers be represented as a = 98x and b = 98y, where x and y are integers. The LCM of these two numbers is expressed as: LCM = HCF × x × y Substituting the given values: 2352 = 98 × x × y xy = 2352/98 = 24

The sum of the numbers is: S = a + b = 98x + 98y = 98(x + y) To find the possible sum, we examine the pairs of factors of 24: If x = 1, y = 24, then S = 98(1 + 24) = 98 × 25 = 2450 If x = 2, y = 12, then S = 98(2 + 12) = 98 × 14 = 1372 If x = 3, y = 8, then S = 98(3 + 8) = 98 × 11 = 1078 If x = 4, y = 6, then S = 98(4 + 6) = 98 × 10 = 980

Comparing these results with the given options, the value 1372 matches option (a). Thus, the sum of the numbers may be 1372.

Explanation

The fundamental property of composite numbers states that any composite number n must have at least one prime factor p such that p ≤ √n. For numbers less than 121, the maximum value we consider is 120. The square root of 120 is √120 ≈ 10.95. The prime numbers less than or equal to 10.95 are 2, 3, 5, and 7. Therefore, every composite number less than 121 must be divisible by at least one of these primes: 2, 3, 5, or 7. This directly confirms that statement 2 is correct. Since any composite number in this range is a multiple of 2, 3, 5, or 7, removing all multiples of these four primes from the set of integers up to 120 will leave only the prime numbers. This confirms that statement 1 is correct. Thus, both statements 1 and 2 are mathematically sound, making option (c) the right choice.

Explanation

To factorize the given expression x² + 1/x² + 8(x + 1/x) + 14, we use the method of substitution. Let y = x + 1/x. Squaring both sides of this substitution, we get y² = (x + 1/x)² = x² + 1/x² + 2. This allows us to express the first two terms of the original equation as x² + 1/x² = y² - 2. Substituting these into the expression, it becomes (y² - 2) + 8y + 14, which simplifies to the quadratic form y² + 8y + 12. To factorize this quadratic, we look for two numbers that multiply to 12 and add up to 8, which are 6 and 2. Thus, the expression factorizes as (y + 6)(y + 2). By substituting back y = x + 1/x, we obtain the factors (x + 1/x + 6)(x + 1/x + 2). Comparing these factors with the given options, option (c) x + 1/x + 6 is clearly one of the factors.

Explanation

The first expression is a^2b^4 + 2a^2b^2 = a^2b^2(b^2 + 2). The second expression is (ab)^7 - 4a^2b^9 = a^7b^7 - 4a^2b^9 = a^2b^7(a^5 - 4b^2). The highest common factor (HCF) is the product of the lowest powers of common factors, which is a^2b^2.

Explanation

Factorizing the expressions: x^2 + 2x - 8 = (x + 4)(x - 2); x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x - 2)^2; x^2 + 4x = x(x + 4). The LCM is the product of the highest powers of all factors involved: x^1 (x + 4)^1 (x - 2)^2.

Explanation

A's work rate is 1/4 per day and B's work rate is 1/12 per day. Together, their rate is 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 per day. The number of days required is the reciprocal of the combined rate, which is 3 days.

Explanation

According to the Factor Theorem, a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0. Let the given polynomial be P(x) = x³ + 2x² + 3x + 8.

To verify statement 1, we substitute x = -3 into the polynomial: P(-3) = (-3)³ + 2(-3)² + 3(-3) + 8 P(-3) = -27 + 2(9) - 9 + 8 P(-3) = -27 + 18 - 9 + 8 = -10 Since P(-3) ≠ 0, the expression (x + 3) is not a factor of the polynomial.

To verify statement 2, we substitute x = 2 into the polynomial: P(2) = (2)³ + 2(2)² + 3(2) + 8 P(2) = 8 + 2(4) + 6 + 8 P(2) = 8 + 8 + 6 + 8 = 30 Since P(2) ≠ 0, the expression (x - 2) is not a factor of the polynomial.

As neither statement 1 nor statement 2 is correct, option (d) is the correct choice.

Explanation

To find the profit percentage for the second scenario, we first determine the cost price of the article using the formula CP = (SP × 100)/(100 + Profit%). Substituting the given values SP = 3600 and Profit% = 20%, we get CP = (3600 × 100)/120 = 3000. Now, consider the second case where the selling price SPₙₑᵥᵥ = 3150. The profit amount is calculated as SPₙₑᵥᵥ - CP, which is 3150 - 3000 = 150. The new profit percentage is then found using the formula Profit% = (Profit/CP) × 100. Substituting the values, we have Profit% = (150/3000) × 100 = 15/3 = 5%. Therefore, the person would have earned a profit of 5%, which corresponds to option b).

Explanation

To solve this problem, we use the string method or bundling technique for permutations where certain items must remain together. We treat the two largest books as a single unit or bundle. Since there are 5 different books in total, replacing the two largest books with one bundle leaves us with 3 individual books plus the bundle, totaling 5 - 2 + 1 = 4 entities to arrange. The number of ways to arrange these 4 entities in a row is 4!. Additionally, the two largest books within the bundle can be arranged among themselves in 2! ways. According to the fundamental principle of counting, the total number of ways to arrange the books such that the two largest are always together is the product of these two permutations: 4! × 2!. This matches option (c).

Explanation

For x^2 - 3x + 2 to be a factor of P(x) = x^3 - 3x^2 + 3x - 2, both roots of x^2 - 3x + 2 = 0 (which are x = 1 and x = 2) must be roots of P(x). Checking x = 1: P(1) = 1 - 3 + 3 - 2 = -1. Since P(1) is not 0, (x - 1) is not a factor, and thus x^2 - 3x + 2 is not a factor. Both statements 1 and 2 are identical and incorrect.

Explanation

According to the Factor Theorem, if a linear polynomial (x + a) is a factor of a polynomial P(x), then P(-a) = 0. In this problem, the given factor is (x + 5), which implies that x = -5 must be a root of the expression P(x) = x³ + 3x² + 4x + k. To find the value of k, we substitute x = -5 into the polynomial and set the total sum to zero:

(-5)³ + 3(-5)² + 4(-5) + k = 0

Calculating the powers and products: -125 + 3(25) - 20 + k = 0 -125 + 75 - 20 + k = 0 -50 - 20 + k = 0 -70 + k = 0 k = 70

The result shows that k must be 70 for (x + 5) to be a factor of the given expression, which corresponds to option (b).

Explanation

Using the principle of inclusion-exclusion for sets: Total = Mathematics + Statistics - Both. 110 = (2x + 20) + (2x + 30) - x. Simplifying: 110 = 3x + 50. 3x = 60, so x = 20.

Explanation

Let y = sqrt(6 + sqrt(6 + ...)). Then y = sqrt(6 + y), which means y^2 - y - 6 = 0. Solving for y (where y > 0), we get (y - 3)(y + 2) = 0, so y = 3. The problem states x^2 = 6 + y, so x^2 = 6 + 3 = 9. Thus, x = 3 or x = -3. One of the values is 3.

Explanation

Let Q be the quantity that must be added. According to the problem, the mathematical relationship is given by the equation (1 - x)(1 + x²) + Q = x³. To find the value of Q, we rearrange the equation as Q = x³ - (1 - x)(1 + x²). First, expand the product (1 - x)(1 + x²) using the distributive property: (1 - x)(1 + x²) = 1(1 + x²) - x(1 + x²) = 1 + x² - x - x³. Now, substitute this expanded expression back into the equation for Q: Q = x³ - (-x³ + x² - x + 1). By distributing the negative sign across the terms inside the parentheses, we get Q = x³ + x³ - x² + x - 1. Combining the like terms x³ and x³, we obtain Q = 2x³ - x² + x - 1. This result corresponds to option (c).

Explanation

Statement 1: For 7710312401, the alternating sum of digits is (1+4+1+0+7) - (0+2+3+1+7) = 13 - 13 = 0. Since the result is divisible by 11, the number is divisible by 11. Statement 2: 173 is not divisible by 2, 3, 5, 7, 11, or 13 (the primes up to sqrt(173)), so 173 is a prime number. Both are correct.

Explanation

To solve this problem, we use the algebraic identity which states that for any positive odd integer n, the expression aⁿ + bⁿ is always divisible by (a + b).

In the given expression 19⁵ + 21⁵, we have a = 19, b = 21, and n = 5. Since the exponent n = 5 is an odd number, the expression 19⁵ + 21⁵ must be divisible by (19 + 21).

Calculating the sum, we get 19 + 21 = 40. Therefore, 19⁵ + 21⁵ is divisible by 40. Any number that is divisible by 40 is also necessarily divisible by all the factors of 40. The factors of 40 include 1, 2, 4, 5, 8, 10, 20, and 40. Since both 10 and 20 are factors of 40, the expression 19⁵ + 21⁵ is divisible by both 10 and 20. This confirms that option c is the correct choice.

Explanation

Using the identity a^2 - b^2 = (a - b)(a + b), we have (49^2 - 25^2) = (49 - 25)(49 + 25) = 24 74. We are given 37x = 24 74. Since 74 = 2 37, the equation is 37x = 24 2 * 37. Dividing by 37, we get x = 48.

Explanation

According to the Factor Theorem, if (x - a) is a factor of a polynomial P(x), then P(a) = 0. In this problem, for (x - 1) to be a factor of the polynomial P(x) = 4x⁵ + 9x⁴ - 7x³ - 5x² - 4kx + 3k², we must have P(1) = 0.

Substituting x = 1 into the equation: P(1) = 4(1)⁵ + 9(1)⁴ - 7(1)³ - 5(1)² - 4k(1) + 3k² = 0 4 + 9 - 7 - 5 - 4k + 3k² = 0 13 - 12 - 4k + 3k² = 0 3k² - 4k + 1 = 0

To find the values of k, we solve this quadratic equation using the factorization method: 3k² - 3k - k + 1 = 0 3k(k - 1) - 1(k - 1) = 0 (3k - 1)(k - 1) = 0

Setting each factor to zero gives: k - 1 = 0 ⟹ k = 1 3k - 1 = 0 ⟹ k = 1/3

Thus, the values of k for which (x - 1) is a factor are 1 and 1/3, which corresponds to option d.

Explanation

The given expression is a cyclic polynomial f(x, y, z) = x(y² - z²) + y(z² - x²) + z(x² - y²). According to the Factor Theorem, if substituting a value for a variable makes the entire expression zero, then the corresponding binomial is a factor. To check if (y - z) is a factor, we substitute y = z into the expression: f(x, z, z) = x(z² - z²) + z(z² - x²) + z(x² - z²) = x(0) + z(z² - x²) - z(z² - x²) = 0. Since the result is zero, (y - z) is a factor. Similarly, to check if (z - x) is a factor, we substitute z = x: f(x, y, x) = x(y² - x²) + y(x² - x²) + x(x² - y²) = x(y² - x²) + y(0) - x(y² - x²) = 0. Since the result is zero, (z - x) is also a factor. Because the expression is cyclic, (x - y) will also be a factor. The product of these three linear factors (x - y)(y - z)(z - x) is a cubic polynomial, which matches the degree of the original expression. Thus, the expression is divisible by both (y - z) and (z - x), making option c the correct choice.

Explanation

From the first equation, 3x + 2y = 24. From the second equation, (2y + 3x)/xy = 1, which means 3x + 2y = xy. Substituting the first into the second, we get xy = 24. Using 3x + 2(24/x) = 24, we get 3x^2 - 24x + 48 = 0, or x^2 - 8x + 16 = 0, which means (x - 4)^2 = 0, so x = 4. Then y = 24/4 = 6. x + y = 4 + 6 = 10.

Explanation

The formula for calculating percentage increase is given by: Percentage Increase = (Final Value - Initial Value)/(Initial Value) × 100

Substituting the given values, the initial salary is ₹7,200 and the final salary is ₹8,100. First, calculate the absolute increase in salary, which is 8,100 - 7,200 = 900. Now, apply the formula to find the percentage increase: Percentage Increase = 900/7,200 × 100

By simplifying the fraction 900/7,200, we get 9/72, which further reduces to 1/8. To find the percentage, multiply this fraction by 100: 1/8 × 100 = 12.5%

Converting the decimal 0.5 to a fraction, 12.5% is equivalent to 12 1/2%. This matches option (d).

Explanation

Efficiency and time taken to complete a work are inversely proportional. The relationship is given by the formula: Efficiency × Time = Total Work. Let the efficiency of X be Eₓ and the time taken by X be Tₓ = 16 days. According to the question, the efficiency of Y is 60% more than that of X, which means Eᵧ = Eₓ + 0.60Eₓ = 1.6Eₓ. Since the total work remains constant for both, we can set up the equation: Eₓ × Tₓ = Eᵧ × Tᵧ. Substituting the given values, we get Eₓ × 16 = 1.6Eₓ × Tᵧ. Dividing both sides by Eₓ, we have 16 = 1.6 × Tᵧ. Solving for Tᵧ, we get Tᵧ = 16/1.6 = 160/16 = 10 days. Therefore, option (a) is the correct choice.

Explanation

Since the quantities of the two lots are equal, the average cost price per kg is the simple arithmetic mean of the individual cost prices. Using the formula for average cost price, Avg CP = (CP₁ + CP₂) / 2, we get Avg CP = (10 + 15) / 2 = 12.5 per kg. The selling price is given as SP = 15 per kg.

To find the profit percentage, use the formula Profit % = ((SP - Avg CP) / Avg CP) × 100. Substituting the values, we have Profit % = ((15 - 12.5) / 12.5) × 100 = (2.5 / 12.5) × 100. Simplifying the fraction, 2.5 / 12.5 = 1 / 5, which equals 0.2. Multiplying by 100 gives a profit of 20%. Thus, option c is the correct answer.

Explanation

This problem relies on a fundamental property of coprime numbers in number theory. Two integers are said to be coprime or relatively prime if their highest common factor (HCF) is 1. The given conditions are HCF(a, b) = 1 and HCF(a, c) = 1, which means a shares no prime factors with b and no prime factors with c.

According to the coprime product property, if an integer a is coprime to b and also coprime to c, then it must be coprime to their product bc. To understand this, consider the prime factorization of a. Since HCF(a, b) = 1, no prime factor of a divides b. Similarly, since HCF(a, c) = 1, no prime factor of a divides c. The prime factors of the product bc are simply the collection of all prime factors of b and all prime factors of c. Because a shares no prime factors with either b or c, it cannot share any prime factors with the product bc. Consequently, the only common positive divisor between a and bc is 1, which implies HCF(a, bc) = 1. This matches option a.

Explanation

Let the prime factorization of c be given by c = p₁ᵉ¹ p₂ᵉ² ... pₙᵉⁿ, where pᵢ are distinct primes and eᵢ are their respective exponents. The largest square divisor b of c is obtained by taking the maximum even power of each prime factor available in c. Thus, b = p₁²⌊ᵉ¹/²⌋ p₂²⌊ᵉ²/²⌋ ... pₙ²⌊ᵉⁿ/²⌋.

It is given that a² divides c. This means the prime factorization of a² can be written as a² = p₁²ᶠ¹ p₂²ᶠ² ... pₙ²ᶠⁿ, where 2fᵢ ≤ eᵢ for all i. From this inequality, we have fᵢ ≤ eᵢ/2, and since fᵢ is an integer, fᵢ ≤ ⌊e₁/2⌋.

Now, consider the prime factorization of a, which is a = p₁ᶠ¹ p₂ᶠ² ... pₙᶠⁿ. Comparing this with the expression for b, we see that each prime factor of a appears with an exponent fᵢ that is less than or equal to the exponent 2⌊eᵢ/2⌋ in b. Specifically, since fᵢ ≤ ⌊eᵢ/2⌋ ≤ 2⌊eᵢ/2⌋, it follows that a must be a divisor of b. Therefore, option c is correct.

Explanation

Any integer n can be categorized as either even or odd. If n is an even integer, it can be expressed as n = 2m for some integer m. Squaring this expression gives n² = (2m)² = 4m². By letting k = m², we see that the square of an even number is always of the form 4k. If n is an odd integer, it can be expressed as n = 2m + 1 for some integer m. Squaring this expression yields n² = (2m + 1)² = 4m² + 4m + 1. This can be rewritten as 4(m² + m) + 1. By letting k = m² + m, we see that the square of an odd number is always of the form 4k + 1. Consequently, every square integer must be of the form 4k or 4k + 1, which corresponds to option (d). This property is a fundamental result in modular arithmetic, stating that n² ≡ 0 (mod 4) or n² ≡ 1 (mod 4) for any integer n.

Explanation

A prime p of the form 3k + 1 must be odd (except for p=2, but 3k+1=2 implies k=1/3). If k is odd, k = 2m + 1, then p = 3(2m + 1) + 1 = 6m + 4, which is even and not prime. If k is even, k = 2m, then p = 3(2m) + 1 = 6m + 1. Thus, k must be even for p to be a prime of that form.

Explanation

Convert 10 meters to millimeters: 10 m = 10,000 mm. Divide the total length by the length of one piece: 10,000 mm / 200 mm = 50.

Explanation

26^2 + 97^2 = 676 + 9409 = 10085. Checking the options: a) 27^2 + 93^2 = 729 + 8649 = 9378; b) 34^2 + 93^2 = 1156 + 8649 = 9805; c) 82^2 + 41^2 = 6724 + 1681 = 8405; d) 79^2 + 62^2 = 6241 + 3844 = 10085. Option d is correct.

Explanation

Let M and W be the daily work rates of a man and woman. 14(2M + W) = 8(2M + 4W). 28M + 14W = 16M + 32W. 12M = 18W, so W = 2/3 M. Wages are proportional to the work rate. If M gets 90, W gets 2/3 * 90 = 60.

Explanation

Simplify the expressions: a) (5^-2)^-2 = 5^4, then (5^4)^-2 = 5^-8; b) (5^-2)^2 = 5^-4, then (5^-4)^-2 = 5^8; c) (2^-5)^-2 = 2^10, then (2^10)^-2 = 2^-20; d) (2^-5)^2 = 2^-10, then (2^-10)^-2 = 2^20. Comparing 5^-8 (1/390625) and 2^-20 (1/1048576), 2^-20 is the smallest.

Explanation

The last digits of 7^n follow a cycle: 7, 9, 3, 1. Since 402 mod 4 = 2, 7^402 ends in 9. The last digits of 3^n follow a cycle: 3, 9, 7, 1. Since 402 mod 4 = 2, 3^402 ends in 9. The sum of the last digits is 9 + 9 = 18, so the last digit is 8.

Explanation

To find the length of the train, use the basic formula: Distance = Speed × Time. When a train crosses a stationary point object like a pole, the distance covered is equal to the length of the train. First, convert the speed from kilometers per hour (kmph) to meters per second (m/s) using the standard conversion factor 5/18: Speed = 72 × 5/18 = 4 × 5 = 20 m/s. Now, substitute the speed and time into the distance formula: Length = 20 m/s × 15 s = 300 m. Thus, the length of the train is 300 m, which corresponds to option (c).

Explanation

This problem can be solved using the chain rule formula for work and wages, which is (M₁ × D₁)/W₁ = (M₂ × D₂)/W₂, where M represents the number of men, D represents the number of days, and W represents the total earnings or work done. Given values are M₁ = 18, D₁ = 5, and W₁ = 360. We need to find W₂ for M₂ = 15 and D₂ = 9. Substituting these values into the formula gives (18 × 5)/360 = (15 × 9)/W₂. Simplifying the left side, we get 90/360 = 1/4. Thus, 1/4 = 135/W₂, which leads to W₂ = 135 × 4 = 540. Therefore, 15 men will earn ₹ 540 in 9 days, which corresponds to option (b).

Explanation

To find the rational numbers lying between 1/4 and 3/4, first convert these boundary fractions into decimal form for easier comparison. The lower bound is 1/4 = 0.25 and the upper bound is 3/4 = 0.75. Any pair of numbers (x, y) must satisfy the condition 0.25 < x < 0.75 and 0.25 < y < 0.75. Evaluating the options by converting them to decimals, option (a) gives 0.262 and 0.752, where 0.752 exceeds the upper limit. Option (b) gives 0.24 and 0.74, where 0.24 is less than the lower limit. Option (c) gives 9 ÷ 40 = 0.225 and 31 ÷ 40 = 0.775, both of which fall outside the range. Option (d) gives 252/1000 = 0.252 and 748/1000 = 0.748. Since 0.25 < 0.252 < 0.75 and 0.25 < 0.748 < 0.75 are both true, option (d) is the correct pair.

Explanation

Let x be the initial number of passengers. After first stop: 4x/5 + 40. After second stop: 1/2(4x/5 + 40) + 30 = 70. 2x/5 + 20 + 30 = 70. 2x/5 = 20. x = 50.

Explanation

The share of wages when individuals work together is proportional to their efficiencies, which is the reciprocal of the time taken to complete the work. The efficiencies of A, B, and C are 1/15, 1/20, and 1/25 respectively. To simplify the ratio 1/15 : 1/20 : 1/25, we multiply each term by the least common multiple (LCM) of 15, 20, and 25, which is 300. This gives the ratio of their shares as (300/15) : (300/20) : (300/25) = 20 : 15 : 12. The sum of these ratio parts is 20 + 15 + 12 = 47. The share of C is calculated by taking his part over the total parts multiplied by the total earnings: (12 / 47) × 4700 = 12 × 100 = 1200. Thus, the share of C is ₹ 1,200.

Explanation

Given that x is a positive even integer and y is a negative odd integer, we can represent y as -n where n is a positive odd integer. Using the law of exponents a⁻ᵇ = 1/aᵇ, the expression xʸ can be rewritten as x⁻ⁿ = 1/xⁿ. Since x is a positive even integer and n is a positive integer, xⁿ will always result in a positive integer. A rational number is defined as any number that can be expressed in the form p/q where p and q are integers and q ≠ 0. In the resulting expression 1/xⁿ, both the numerator 1 and the denominator xⁿ are integers, and xⁿ cannot be zero because x is a positive integer. Therefore, xʸ is always a rational number, which corresponds to option c. For instance, if x = 2 and y = -3, then xʸ = 2⁻³ = 1/2³ = 1/8, which is a rational number.

Explanation

To solve this problem efficiently, use the direct formula for distance when two objects travel the same distance at different speeds with a given time difference:

D = (S₁ × S₂)/(S₂ - S₁) × Δt

In this formula, S₁ and S₂ represent the speeds of the two cars, and Δt represents the difference in their arrival times. Substituting the given values into the formula:

S₁ = 30 km/hr S₂ = 45 km/hr Δt = 2 hours

D = (30 × 45)/(45 - 30) × 2

Calculating the numerator and denominator: D = 1350/15 × 2

Simplifying the fraction: D = 90 × 2 = 180 km

The distance between the starting point and the destination is 180 km, which corresponds to option b. This approach is faster than setting up linear equations and is highly recommended for competitive exams.

Explanation

Since 4 goats or 6 sheep can graze the field in 50 days, we can establish an equivalence between their work rates. Given that 4 goats = 6 sheep, it follows that 1 goat = 6/4 = 1.5 sheep.

To find the time taken by 2 goats and 9 sheep, we first convert the entire group into a single unit of efficiency, specifically sheep. The total efficiency is 2 goats + 9 sheep = (2 × 1.5) + 9 = 3 + 9 = 12 sheep.

Now, we use the standard work formula M₁D₁ = M₂D₂, where M represents the number of animals and D represents the number of days. Substituting the known values: 6 sheep × 50 days = 12 sheep × D₂ 300 = 12 × D₂ D₂ = 300/12 = 25 days.

Thus, the field will be grazed in 25 days, which corresponds to option d.

Explanation

Let the distance to the office be D and the time of departure be T. The difference in arrival times is Δt = 1 p.m. - 11 a.m. = 2 hours. Using the standard distance formula for two different speeds s₁ and s₂ with a known time difference, we have D = (s₁s₂)/(s₂ - s₁) × Δt. Substituting the given values s₁ = 10 kmph and s₂ = 15 kmph, we get D = (10 × 15)/(15 - 10) × 2 = 150/5 × 2 = 30 × 2 = 60 km.

To find the departure time, calculate the time taken at 10 kmph: t = D/s₁ = 60/10 = 6 hours. Since the man reaches the office at 1 p.m., he must have started at 13 - 6 = 7 a.m. To reach the office at 12 noon, the total travel time required is 12 noon - 7 a.m. = 5 hours. The required speed is therefore s = D/t = 60/5 = 12 kmph, which corresponds to option (b).

Explanation

This problem is based on the chain rule of proportionality where the total work remains constant. The formula used to solve such problems is M₁ D₁ H₁ = M₂ D₂ H₂, where M represents the number of workers, D represents the number of days, and H represents the hours worked per day. Given values are M₁ = 20 workers, D₁ = 10 days, and H₁ = 5 hours per day. For the second scenario, M₂ = 25 workers and H₂ = 10 hours per day, while the number of days D₂ is to be calculated. Substituting these values into the formula, we get 20 × 10 × 5 = 25 × D₂ × 10. Simplifying the left side gives 1000 = 250 × D₂. Dividing both sides by 250, we find D₂ = 1000 / 250 = 4. Thus, the time required to complete the work is 4 days, which corresponds to option a.

Explanation

The expression m√n in the context of the provided options refers to the m-th root of n, denoted as ᵐ√n. For any natural numbers m and n, the m-th root ᵐ√n is rational if and only if n is a perfect m-th power of some integer k, such that n = kᵐ. In such a case, ᵐ√n = ᵐ√kᵐ = k, which is a rational number. However, if n is not a perfect m-th power, the value of ᵐ√n cannot be expressed as a fraction of two integers, which by definition makes it an irrational number. For example, if m = 3 and n = 8, then ∛8 = 2, which is rational because 8 is the 3rd power of 2. Conversely, if m = 3 and n = 9, then ∛9 is irrational because 9 is not a perfect cube of any integer. Therefore, the value is irrational unless n is the mth power of an integer, which makes option (b) the correct choice.

Explanation

For the quadratic equation ax² + bx + b = 0 with a ≠ 0, we use the standard properties of roots. The sum of the roots is given by the formula -B/A and the product of the roots is given by C/A. Here, A = a, B = b, and C = b. Therefore, the sum of the roots is -b/a and the product of the roots is b/a. For the sum to equal the product, -b/a = b/a must hold, which implies 2b/a = 0, or b = 0. Since the equation does not specify that b must be zero, Statement 1 is not generally correct.

Regarding Statement 2, the nature of the roots is determined by the discriminant D = B² - 4AC. Substituting the values, we get D = b² - 4ab, which can be factored as b(b - 4a). For the roots to be always real and unequal, D must be strictly greater than zero for all values of a and b. However, if we take a = 1 and b = 1, then D = 1(1 - 4) = -3, which results in imaginary roots. If a = 1 and b = 4, then D = 16 - 16 = 0, which results in equal roots. Thus, Statement 2 is also incorrect. Consequently, option d is the correct choice.

Explanation

For the quadratic equation x² - x - 1 = 0, the sum of the roots is α + β = -(-1)/1 = 1 and the product of the roots is αβ = -1/1 = -1 according to Vieta formulas. The numerator of the expression is α² + β², which can be expressed as (α + β)² - 2αβ. Substituting the known values, we get 1² - 2(-1) = 1 + 2 = 3. The denominator of the expression is (α² - β²)(α - β), which simplifies to (α + β)(α - β)(α - β) = (α + β)(α - β)². To find (α - β)², we use the identity (α - β)² = (α + β)² - 4αβ, which yields 1² - 4(-1) = 1 + 4 = 5. Substituting these into the denominator, we get 1 × 5 = 5. Therefore, the value of the expression is 3/5, which corresponds to option b.

Explanation

Let the sides of the rectangle be 4x and 9x based on the given ratio 4:9. Using the formula for the area of a rectangle, Area = length × breadth, we have 4x × 9x = 144. This simplifies to 36x² = 144. Dividing both sides by 36, we get x² = 4, which means x = 2 cm. The actual lengths of the sides are 4 × 2 = 8 cm and 9 × 2 = 18 cm. Now, apply the formula for the perimeter of a rectangle, Perimeter = 2(length + breadth). Substituting the dimensions, we get Perimeter = 2(18 + 8) = 2(26) = 52 cm. Therefore, option (a) is the correct answer.

Explanation

For a pyramid on a square base (Object A), the number of edges a is 8 (4 on the base and 4 meeting at the apex), the number of faces b is 5 (1 square base and 4 triangular faces), and the number of corners c is 5 (4 at the base and 1 apex). Checking the given condition a = b + c gives 8 = 5 + 5, which simplifies to 8 = 10. This is false.

For a cube (Object B), the number of edges a is 12, the number of faces b is 6, and the number of corners c is 8. Checking the condition a = b + c gives 12 = 6 + 8, which simplifies to 12 = 14. This is also false.

According to Euler's formula for convex polyhedra, the relationship between vertices V, edges E, and faces F is given by V - E + F = 2. Substituting the variables from the question, we have c - a + b = 2, which rearranges to a = b + c - 2. Since the given condition a = b + c differs from this fundamental law by a constant of 2, it cannot be true for any convex polyhedron, including a square pyramid or a cube. Therefore, the result is true for neither A nor B, making option (d) the correct choice.

Explanation

The area of the square is calculated using the formula Areaₛ₀ᵤₐᵣₑ = s², where s is the side length. For a side of 10 cm, Areaₛ₀ᵤₐᵣₑ = 10² = 100 cm². The problem describes two semicircles each with a radius r = 5 cm. Since two semicircles of the same radius combine to form one full circle, their total area is determined by the formula Area꜀ᵢᵣ꜀ₗₑ = πr². Substituting the given radius, we get Area꜀ᵢᵣ꜀ₗₑ = π × 5² = 25π. Using the standard approximation of π ≈ 3.14159, the combined area of the two semicircular cross-sections is 25 × 3.14159 ≈ 78.54 cm². The area between the square and these two semicircles is the difference between the area of the square and the area of the full circle: Areaᵣₑₘₐᵢₙᵢₙ_g = 100 - 78.54 = 21.46 cm². Rounding this value to one decimal place yields 21.5 cm², which corresponds to option d.

Explanation

The area of a parallelogram is calculated using the standard formula Area = base × height. In this problem, the side of the parallelogram is treated as the base b = 8.06 cm, and the perpendicular distance from the opposite side is the height h = 2.08 cm. Substituting these values into the formula, we get Area = 8.06 × 2.08. Performing the multiplication, 8.06 × 2 = 16.12 and 8.06 × 0.08 = 0.6448. Adding these components together, 16.12 + 0.6448 = 16.7648 cm squared. When rounded to two decimal places to find the approximate area as requested in the question, the result is 16.76 cm squared, which matches option c.

Explanation

To find the volume of a cone, use the formula V = 1/3 π r² h, where r represents the base radius and h represents the height. Substituting the given values r = 10 cm and h = 21 cm, and using π = 22/7 for ease of calculation, the expression becomes:

V = 1/3 × 22/7 × (10)² × 21

A quick mental shortcut for competitive exams is to notice that the product of the denominators 3 × 7 = 21, which perfectly cancels the height 21 in the numerator. This simplifies the calculation significantly:

V = 22 × 100 = 2200 cm³

Thus, the volume of the cone is 2200 cm³, which corresponds to option (a).

Explanation

First, identify the type of triangle using the Pythagorean theorem. For sides 7 cm, 24 cm, and 25 cm, we check if a² + b² = c²: 7² + 24² = 49 + 576 = 625 25² = 625 Since the sum of the squares of the two shorter sides equals the square of the longest side, the triangle is a right-angled triangle.

The area of a right-angled triangle is given by the formula: Area = 1/2 × base × height Substituting the values: Area = 1/2 × 7 × 24 = 7 × 12 = 84 cm²

According to the problem, the area of the circle is equal to the area of this triangle. Therefore, the area of the circle is 84 cm², which corresponds to option (b).

Explanation

Let the side of the equilateral triangle be a. The formula for the perimeter of an equilateral triangle is y = 3a, which gives a = y/3. The formula for the area of an equilateral triangle is x = (√3/4)a².

Substituting the value of a in terms of y into the area formula: x = (√3/4) (y/3)² x = (√3/4) · y²/9 x = (√3y²)/36

To find the relationship between x and y, rearrange the equation to isolate y²: y² = 36x/√3 Rationalizing the denominator gives: y² = 36√3x/3 = 12√3x

Now, square both sides of the equation to remove the square root: (y²)² = (12√3x)² y⁴ = 12² · (√3)² · x² y⁴ = 144 · 3 · x² y⁴ = 432x²

This matches option (a).

Explanation

The volume of mud extracted from two hemispherical pitholes is given by the formula V = 2 × 2/3 πr³. Substituting π = 22/7 and r = 2 m, we get V = 4/3 × 22/7 × 2³ = (4 × 22 × 8)/21 = 704/21 m³. The total area of the rectangular field is L × W = 22 × 10 = 220 m². The area occupied by the two hemispherical holes is 2 × πr² = 2 × 22/7 × 2² = 176/7 m². Therefore, the remaining area of the field where the mud is spread is 220 - 176/7 = (1540 - 176)/7 = 1364/7 m². Let the rise in the level of the field be h. The volume of mud spread is equal to the remaining area multiplied by the rise in height: 704/21 = 1364/7 × h. Solving for h, we get h = 704/21 × 7/1364 = 704/(3 × 1364). Simplifying the fraction 704/1364 by dividing both numerator and denominator by 44, we get 16/31. Thus, h = 1/3 × 16/31 = 16/93 m, which corresponds to option c.

Explanation

Given the diameter of the circle is 50 cm, the radius r is 25 cm. Since CP is a radial segment and its extension intersects the circle at D, the total length CD = 25 cm. We are given DP = 18 cm, which means the segment CP = CD - DP = 25 - 18 = 7 cm.

To find the length of chord AB, we use the Intersecting Chords Theorem. Let the diameter passing through P and D be ED, where E is the point on the circle opposite to D. The length of the diameter ED is 50 cm. The point P divides this diameter into two segments: PD = 18 cm and EP = ED - PD = 50 - 18 = 32 cm.

Since chord AB is perpendicular to the diameter ED at point P, the diameter bisects the chord, meaning AP = PB. According to the Intersecting Chords Theorem, the product of the segments of the chords are equal: AP × PB = EP × PD AP² = 32 × 18 AP² = 576 AP = √576 = 24 cm

The total length of the chord AB is 2 × AP = 2 × 24 = 48 cm. Thus, the correct option is d).

Explanation

To find the area x inside the circle but outside the regular hexagon, we must subtract the area of the hexagon from the area of the circle.

The area of the circle is given by the formula A꜀ᵢᵣ꜀ₗₑ = πr². For a radius r = 5 cm, A꜀ᵢᵣ꜀ₗₑ = π × 5² = 25π. Using the approximation π ≈ 3.14159, the area is 25 × 3.14159 ≈ 78.54 cm².

A regular hexagon inscribed in a circle consists of six equilateral triangles, each with a side length equal to the radius of the circle. The area of a regular hexagon is given by the formula Aₕₑₓₐ₉ₒₙ = (3√3)/2 r². Substituting r = 5 cm, we get Aₕₑₓₐ₉ₒₙ = (3√3)/2 × 25 = 37.5√3. Using the approximation √3 ≈ 1.732, the area is 37.5 × 1.732 ≈ 64.95 cm².

The area x is the difference between these two values: x = A꜀ᵢᵣ꜀ₗₑ - Aₕₑₓₐ₉ₒₙ x ≈ 78.54 - 64.95 = 13.59 cm².

Since 13.59 cm² falls between 13 cm² and 15 cm², option a is correct.

Explanation

The curved surface area of a right circular cylinder is given by the formula x = 2πrh and the volume is given by y = πr²h, where r is the radius and h is the height. To find the ratio of height to radius h/r in terms of x and y, we first divide the volume by the curved surface area: y/x = (πr²h) / (2πrh) = r/2. This gives the radius as r = 2y/x. Next, we substitute this value of r into the surface area formula to solve for h: x = 2π(2y/x)h, which simplifies to x = (4πyh) / x, giving h = x² / (4πy). Now, we calculate the ratio h/r = (x² / 4πy) / (2y/x) = (x² / 4πy) × (x / 2y) = x³ / (8πy²). Since the ratio h/r is a function of both x and y, it is not independent of either variable. Thus, neither statement (a) nor (b) is correct, making option d the right choice.

Explanation

According to the midpoint theorem in geometry, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of its length. Consequently, the perimeter of a triangle formed by joining the midpoints of the sides of another triangle is exactly half the perimeter of the original triangle.

Let the perimeter of triangle PQR be Pₚ₀ᵣ, the perimeter of triangle DEF be P₀ₑ𝄴, and the perimeter of triangle ABC be Pₐᵦ꜀. The given sides of triangle PQR are 1, 2 and 3 units. Therefore, its perimeter is: Pₚ₀ᵣ = 1 + 2 + 3 = 6 units.

Since triangle PQR is formed by joining the midpoints of triangle DEF, the relationship between their perimeters is: P₀ₑ𝄴 = 2 × Pₚ₀ᵣ = 2 × 6 = 12 units.

Similarly, since triangle DEF is formed by joining the midpoints of triangle ABC, the relationship between their perimeters is: Pₐᵦ꜀ = 2 × P₀ₑ𝄴 = 2 × 12 = 24 units.

Thus, the perimeter of triangle ABC is 24 units, which matches option b.

Explanation

The curved surface area of a cone is given by the formula CSA = πrl, where r is the radius and l is the slant height. Given the diameter of the cylinder is 24 m, the radius r is 24 / 2 = 12 m. The height of the cylindrical portion is 11 m and the total height of the tent is 16 m, so the height of the conical portion h is 16 - 11 = 5 m.

To find the slant height l, we use the formula l = √(r² + h²). Substituting the values, we get l = √(12² + 5²) = √(144 + 25) = √169 = 13 m. Now, substituting r = 12 m and l = 13 m into the curved surface area formula with π = 22 / 7, we have CSA = (22 / 7) × 12 × 13. Calculating the product, 12 × 13 = 156, and 22 × 156 = 3432. Therefore, the curved surface area is 3432 / 7 square metre, which corresponds to option c.

Explanation

The whole surface area of a cone consists of the curved surface area plus the area of the circular base. The standard formula used to calculate this is TSA = πr(l + r), where r denotes the base radius and l denotes the slant height. Given the radius r = 7 cm and height h = 24 cm, we first find the slant height using the relation l = √(r² + h²). Substituting the values, we get l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. Now, substituting r = 7 and l = 25 into the surface area formula, we have TSA = 22/7 × 7 × (25 + 7). The 7 in the numerator and denominator cancel out, leaving 22 × 32, which equals 704 square cm. This matches option b.

Explanation

Radius r = 12 cm, height h = 16 cm. Slant height l = sqrt(12^2 + 16^2) = 20 cm. Curved surface area = pi r l = 3.14159 12 20 approx 753.98 sq cm. Cost = 753.98 * 0.70 = 527.786. Rounding gives 528.

Explanation

The area of a triangle is given by the formula Area = 1/2 × base × height. Substituting the given values, we have 12 = 1/2 × BC × 3. Solving for the base BC, we get BC = (12 × 2)/3 = 8 cm. In an isosceles triangle ABC where AB = AC, the altitude AD dropped from the vertex A to the base BC bisects the base. Therefore, BD = BC/2 = 8/2 = 4 cm. To find the length of the equal sides AB and AC, we apply the Pythagorean theorem in the right-angled triangle ABD: AB² = AD² + BD². Substituting the values, AB² = 3² + 4² = 9 + 16 = 25, which gives AB = √25 = 5 cm. Since AB = AC, then AC = 5 cm. The perimeter of the triangle is the sum of all sides: Perimeter = AB + AC + BC = 5 + 5 + 8 = 18 cm. This matches option (a).

Explanation

Total surface area of a cube = 6s^2. Given 6s^2 = 6, we have s^2 = 1, so s = 1. Volume = s^3 = 1^3 = 1 cubic unit.

Explanation

The volume of a sphere is calculated using the formula V = 4/3πr³, where r is the radius of the sphere. Since the diameter is twice the radius, the ratio of the diameters is equal to the ratio of the radii. Let the diameter of the earth be Dₑ and the diameter of the moon be Dₘ. According to the question, Dₘ = 1/4 Dₑ, which means the ratio of their diameters is Dₘ/Dₑ = 1/4.

To find the ratio of their volumes Vₘ and Vₑ, we use the relationship that the volume of a sphere is proportional to the cube of its diameter: Vₘ/Vₑ = (Dₘ/Dₑ)³

Substituting the given value: Vₘ/Vₑ = (1/4)³ = 1³/4³ = 1/64

Thus, the approximate ratio of the volume of the moon to the volume of the earth is 1:64, which corresponds to option b.

Explanation

Total Floor Area = Total patients × Floor area per patient = 56 × 2.2 = 123.2 m². Since Floor Area = Length × Breadth, we have: 14 × B = 123.2 B = 123.2/14 = 8.8 m.

Total Volume = Total patients × Space per patient = 56 × 8.8 = 492.8 m³. Since Volume = Floor Area × Height, we have: 123.2 × H = 492.8 H = 492.8/123.2 = 4 m.

Thus, the breadth of the room is 8.8 m and the height is 4 m, which corresponds to option a.

Explanation

In a rhombus, the diagonals bisect each other at right angles, meaning the side of the rhombus acts as the hypotenuse of a right triangle formed by the half-lengths of the diagonals. The formula to find the side a from diagonals d₁ and d₂ is a = √((d₁/2)² + (d₂/2)²). Substituting the given values d₁ = 4.8 cm and d₂ = 1.4 cm, we get a = √(2.4² + 0.7²) = √5.76 + 0.49 = √6.25 = 2.5 cm. The perimeter of a rhombus is given by the formula P = 4a. Therefore, P = 4 × 2.5 = 10 cm, which matches option (b).

Explanation

According to the Intersecting Chords Theorem, if two chords of a circle intersect at a point P, the product of the segments of one chord is equal to the product of the segments of the other chord. For chords XY and ZT intersecting at P, the segments are PX, PY and PZ, PT. Thus, the relationship is given by the formula PX ⋅ PY = PZ ⋅ PT, which confirms that statement 1 is correct.

For statement 2, consider triangles PXZ and PTY. The angle ∠XPZ and angle ∠YPT are vertically opposite angles, so ∠XPZ = ∠YPT. Additionally, ∠PXZ and ∠PTY are inscribed angles that subtend the same arc ZY, making them equal. Similarly, ∠PZX and ∠PYT subtend the same arc XT, meaning ∠PZX = ∠PYT. Since all three corresponding angles are equal, ΔPXZ is similar to ΔPTY by the AA similarity criterion. Therefore, statement 2 is also correct. Both statements 1 and 2 are true, making option c the correct choice.

Explanation

To solve this, construct a diagonal joining vertices A and C, which divides the quadrilateral ABCD into two triangles, triangle ABC and triangle ADC. In triangle ABC, it is given that BA = BC, making it an isosceles triangle. According to the properties of isosceles triangles, angles opposite to equal sides are equal, so angle BAC = angle BCA. Now consider triangle ADC, where it is given that CD > AD. In any triangle, the angle opposite to the longer side is greater than the angle opposite to the shorter side. Therefore, angle DAC > angle DCA. The total angle BAD is the sum of angle BAC and angle DAC, while the total angle BCD is the sum of angle BCA and angle DCA. Since angle BAC = angle BCA and angle DAC > angle DCA, it mathematically follows that angle BAC + angle DAC > angle BCA + angle DCA. Consequently, angle BAD > angle BCD, which corresponds to option c.

Explanation

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This relationship is expressed by the formula: Area(ΔABC)/Area(ΔXYZ) = (AB/XY)²

Substituting the given values into the formula: 32/60.5 = (AB/7.7)²

To find AB, we first isolate AB²: AB² = (32 × 7.7²)/60.5 AB² = (32 × 59.29)/60.5 AB² = 1897.28/60.5 AB² = 31.36

Taking the square root of both sides: AB = √31.36 = 5.6 cm

Comparing this result with the given options, we find that it matches option a) 5.6 cm.

Explanation

A quadrilateral ABCD is cyclic if it is inscribed in a circle. Given that AB is parallel to CD, the figure satisfies the definition of a trapezium, as it possesses at least one pair of parallel opposite sides. In a circle, the arcs intercepted between two parallel chords are equal, meaning arc AD = arc BC. Since equal arcs subtend equal chords, it follows that AD = BC, which identifies the figure as an isosceles trapezium. The additional given condition that the diagonals are equal, AC = BD, is a standard property of isosceles trapeziums. While a rectangle is a special case of an isosceles trapezium that is also a parallelogram, the quadrilateral is not required to have the second pair of sides parallel. Thus, the most specific property that must hold for any such quadrilateral is that it is a trapezium.

Explanation

The area of a triangle is calculated using the formula Area = 1/2 · base · height. In triangle ABC, since it is right angled at A, we can consider AB as the base and AC as the height. Therefore, the area of the triangle is 1/2 · AB · AC.

Alternatively, if we consider the hypotenuse BC as the base, the perpendicular AD drawn from vertex A to BC serves as the height of the triangle. In this case, the area of the triangle is expressed as 1/2 · BC · AD.

Since both expressions represent the area of the same triangle ABC, we can equate them as follows: 1/2 · BC · AD = 1/2 · AB · AC

By multiplying both sides of the equation by 2, we eliminate the fraction: BC · AD = AB · AC

This matches the expression given in option (a).

Explanation

In any triangle, the sum of internal angles is 180°. Given ∠B = 90° and ∠C = 2∠A, we can write the equation as ∠A + 90° + 2∠A = 180°. Solving this gives 3∠A = 90°, which means ∠A = 30° and consequently ∠C = 60°. This identifies the triangle as a standard 30°-60°-90° right triangle.

In a 30°-60°-90° triangle, the sides opposite to the angles follow the ratio 1 : √3 : 2. Specifically, the side opposite to 30° is BC and the side opposite to 60° is AB. Using the tangent formula for ∠C, we have tan(60°) = AB/BC. Since tan(60°) = √3, we get √3 = AB/BC, which simplifies to AB = √3BC. Squaring both sides of this equation yields AB² = (√3BC)² = 3BC². Thus, option b is the correct result.

Explanation

The tops of the trees are at (0, x) and (s, y). The distance between them is z, so z^2 = (s - 0)^2 + (y - x)^2 = s^2 + y^2 + x^2 - 2xy. Rearranging for s^2, we get s^2 = z^2 - x^2 - y^2 + 2xy.

Explanation

Since the side AC is produced to D, the points A, C, and D lie on a straight line. This means that Angle ACB and Angle BCD form a linear pair. Given that ∠ACB = 70°, we can find ∠BCD using the linear pair property: ∠BCD = 180° - ∠ACB = 180° - 70° = 110°.

In triangle BCD, it is given that BC = CD, which makes it an isosceles triangle. In an isosceles triangle, the angles opposite to the equal sides are also equal. Therefore, ∠CBD = ∠CDB. Applying the angle sum property of a triangle to triangle BCD, we have ∠BCD + ∠CBD + ∠CDB = 180°. Substituting the known values, 110° + 2 × ∠CDB = 180°, which simplifies to 2 × ∠CDB = 70°. Thus, ∠CDB = 35°. Since ∠ADB is the same as ∠CDB, the value of ∠ADB is 35°, which corresponds to option a.

Explanation

Statement 1 is correct because a rhombus is defined as a parallelogram whose diagonals are perpendicular to each other. In any parallelogram, the diagonals already bisect each other; if they are also perpendicular, the figure must be a rhombus. Thus, saying it may be a rhombus is logically consistent with the property that it must be a rhombus.

Statement 2 is incorrect because the conditions of equal and perpendicular diagonals are insufficient to guarantee that a quadrilateral is a square. For a quadrilateral to be a square, its diagonals must be equal, perpendicular, and also bisect each other. If the diagonals are equal and perpendicular but do not bisect each other, the quadrilateral could be a kite or another irregular shape. For example, if diagonal d₁ and d₂ are both of length L and d₁ ⊥ d₂, but d₁ only bisects d₂ and not vice versa, the resulting figure is a kite, not a square. Therefore, Statement 2 is false.

Explanation

Statement 1 is incorrect because a fundamental property of circles is that the perpendicular bisector of any chord always passes through the center of the circle. Since the center of the circle is equidistant from the two endpoints of the chord (as both distances are equal to the radius r), it must lie on the perpendicular bisector of that chord.

Statement 2 is correct as it refers to Thales Theorem, which states that any angle subtended by a diameter at the circumference of a circle is a right angle. In other words, an angle inscribed in a semicircle is always 90°. Since statement 1 is false and statement 2 is true, the correct option is b) 2 only.

Explanation

To find the length of CF, we use the Midpoint Theorem. Draw a line through D parallel to BF, meeting the side AC at point G.

In ΔADG, E is given as the midpoint of the median AD, and EF is parallel to DG by construction. According to the Midpoint Theorem, a line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side. Therefore, F is the midpoint of AG, which implies AF = FG.

Now consider ΔCBF. Since D is the midpoint of BC (as AD is the median) and DG is parallel to BF, the Midpoint Theorem again implies that G is the midpoint of CF. Thus, CG = GF.

Combining these results, we have AF = FG = GC. Since these three segments are equal and their sum is AC, each segment is equal to 1/3 AC. The length of CF is the sum of CG and GF, which gives CF = 1/3 AC + 1/3 AC = 2/3 AC. This corresponds to option b.

Explanation

In an equilateral triangle, the point of intersection of the altitudes is also the centroid, orthocenter, and circumcenter. The distance from a vertex to the centroid is the circumradius R of the triangle. The formula relating the side a of an equilateral triangle to its circumradius R is R = a/√3.

Given OP = 8 cm, we substitute this value into the formula to find the side a: 8 = a/√3 a = 8√3 cm.

The perimeter of an equilateral triangle is given by the formula Perimeter = 3a. Substituting the value of a: Perimeter = 3 × 8√3 = 24√3 cm.

Thus, the perimeter of triangle PQR is 24√3 cm, which corresponds to option d.

Explanation

In a circle, the angles subtended by the same arc at the circumference are equal. Here, we are given an equilateral triangle ABC inscribed in a circle. In an equilateral triangle, all interior angles are equal to 60°, so ∠BAC = ∠ABC = ∠ACB = 60°.

The angle ∠ADB subtends the arc AB. Similarly, the angle ∠ACB also subtends the same arc AB. According to the circle theorem which states that angles in the same segment of a circle are equal, we have ∠ADB = ∠ACB. Since ∠ACB = 60° because the triangle is equilateral, it follows that ∠ADB = 60°. Thus, option b is correct.

Explanation

The centroid G of a triangle is the point of concurrency of its medians. According to the properties of the centroid, it divides each median in the ratio 2:1, measured from the vertex to the midpoint of the opposite side. Thus, the distance from the vertex to the centroid is given by GA = 2/3 mₐ, GB = 2/3 mᵦ, and GC = 2/3 m꜀, where mₐ, mᵦ, and m꜀ are the lengths of the medians from vertices A, B, and C respectively. These distances are equal only if the triangle is equilateral. For any other triangle, the medians differ in length, making statement 1 incorrect.

Similarly, the orthocentre H is the intersection of the altitudes. The distance from a vertex to the orthocentre is given by the formula HA = 2R cos A, HB = 2R cos B, and HC = 2R cos C, where R is the circumradius of the triangle. These distances are equal if and only if cos A = cos B = cos C, which implies that the triangle must be equilateral. Since statement 2 does not specify that the triangle is equilateral, it is generally false. Therefore, option d is correct as neither statement holds true for a general triangle.

Explanation

In any triangle ABC, the sum of the interior angles is 180°, so ∠A + ∠B + ∠C = 180°. Since BI and CI are the angle bisectors of ∠B and ∠C respectively, the angles in triangle BIC at vertices B and C are ∠B/2 and ∠C/2.

Using the angle sum property for triangle BIC, we have ∠BIC + ∠B/2 + ∠C/2 = 180°. This can be rewritten as ∠BIC = 180° - 1/2(∠B + ∠C). From the first equation, we know that ∠B + ∠C = 180° - ∠A. Substituting this value into the expression for ∠BIC, we get ∠BIC = 180° - 1/2(180° - ∠A). Simplifying this gives ∠BIC = 180° - 90° + ∠A/2, which results in ∠BIC = 90° + ∠A/2. Therefore, option (b) is the correct relation.

Explanation

To find the value of tan θ + cot θ, we first express it in terms of sin θ and cos θ using the formula tan θ + cot θ = sin θ/cos θ + cos θ/sin θ = (sin² θ + cos² θ)/(sin θ cos θ). Since sin² θ + cos² θ = 1, the expression simplifies to 1/(sin θ cos θ). Now, we use the given equation sin θ + cos θ = √3 and square both sides to find the value of the product sin θ cos θ: (sin θ + cos θ)² = (√3)². Expanding the left side gives sin² θ + cos² θ + 2 sin θ cos θ = 3. Substituting the identity sin² θ + cos² θ = 1, we have 1 + 2 sin θ cos θ = 3. This simplifies to 2 sin θ cos θ = 2, which further implies sin θ cos θ = 1. Substituting this result back into our simplified expression for tan θ + cot θ, we get 1/1 = 1. Therefore, option a is correct.

Explanation

The problem involves a right-angled triangle where the pole represents the perpendicular (opposite side) and the shadow represents the base (adjacent side). To find the angle of elevation θ, we use the trigonometric ratio for tangent:

tan θ = Opposite/Adjacent

Substituting the given values into the formula:

tan θ = x/(x/√3)

Simplifying the fraction by multiplying the numerator by the reciprocal of the denominator:

tan θ = x · √3/x

tan θ = √3

From the standard trigonometric values, we know that tan 60° = √3. Therefore, the angle of elevation of the sun is 60°, which corresponds to option (c).

Explanation

To solve the given expression, we simplify the numerator and the denominator separately using trigonometric identities.

For the numerator, we use the compound angle formula cos(A ± B) = cos A cos B ∓ sin A sin B. Since cos 45° = sin 45° = 1/√2, we have: cos(45° + θ) = 1/√2(cos θ - sin θ) and cos(45° - θ) = 1/√2(cos θ + sin θ). Squaring these terms, the numerator becomes: [1/√2(cos θ - sin θ)]² + [1/√2(cos θ + sin θ)]² = 1/2(cos² θ + sin² θ - 2sin θ cos θ) + 1/2(cos² θ + sin² θ + 2sin θ cos θ) = 1/2(1 - sin 2θ) + 1/2(1 + sin 2θ) = 1/2 + 1/2 = 1.

For the denominator, we use the complementary angle identity tan(90° - x) = cot x. Note that 30° - θ can be written as 90° - (60° + θ). Thus, tan(30° - θ) = tan(90° - (60° + θ)) = cot(60° + θ). The denominator then becomes: tan(60° + θ) × cot(60° + θ) = tan(60° + θ) × 1/tan(60° + θ) = 1.

Substituting the results back into the original expression, we get 1/1 = 1, which corresponds to option c.

Explanation

We use the fundamental trigonometric identity sec² θ - tan² θ = 1. This expression is a difference of squares and can be factored as (sec θ + tan θ)(sec θ - tan θ) = 1. Given that sec θ + tan θ = m, we can substitute this into the identity:

m(sec θ - tan θ) = 1

sec θ - tan θ = 1/m

Now we have a system of two linear equations:

1. sec θ + tan θ = m
2. sec θ - tan θ = 1/m

Adding these two equations eliminates tan θ: (sec θ + tan θ) + (sec θ - tan θ) = m + 1/m 2 sec θ = (m² + 1)/m

Dividing by 2, we get: sec θ = (m² + 1)/2m

This corresponds to option b.

Explanation

To solve this problem efficiently, we use the trigonometric identity (a sin θ + b cos θ)² + (b sin θ - a cos θ)² = a² + b². This identity is derived from the fundamental property sin² θ + cos² θ = 1. Given the expression 5 sin θ + 12 cos θ = 13 (assuming the coefficient 5 for sin θ based on the 5, 12, 13 Pythagorean triplet), we identify a = 5 and b = 12. Substituting these values and the given sum into the identity, we get (13)² + (12 sin θ - 5 cos θ)² = 5² + 12². This simplifies to 169 + (12 sin θ - 5 cos θ)² = 25 + 144, which further reduces to 169 + (12 sin θ - 5 cos θ)² = 169. Subtracting 169 from both sides yields (12 sin θ - 5 cos θ)² = 0, which implies that 12 sin θ - 5 cos θ = 0. Multiplying this result by -1 gives 5 cos θ - 12 sin θ = 0, which corresponds to option C.

Explanation

Given the condition 4 tan θ = 3, we can find the value of tan θ = 3/4. To evaluate the expression (4 sin θ - cos θ)/(4 sin θ + 9 cos θ) efficiently, the best approach is to divide both the numerator and the denominator by cos θ. This converts the expression into terms of tan θ because tan θ = sin θ/cos θ.

Dividing the numerator and denominator by cos θ: ((4 sin θ)/(cos θ) - (cos θ)/(cos θ))/((4 sin θ)/(cos θ) + (9 cos θ)/(cos θ)) = (4 tan θ - 1)/(4 tan θ + 9)

Now, substitute the given value 4 tan θ = 3 directly into the expression: (3 - 1)/(3 + 9) = 2/12 = 1/6

Thus, the value of the expression is 1/6, which corresponds to option d.

Explanation

sin theta - cos theta = 0 implies sin theta = cos theta, which means theta = 45 degrees. Then sin^4(45) + cos^4(45) = (1/sqrt(2))^4 + (1/sqrt(2))^4 = 1/4 + 1/4 = 1/2.

Explanation

Assuming the typo "cc" refers to cosec, csc(75 + theta) = 1/sin(75 + theta). Since sin(75 + theta) = cos(90 - (75 + theta)) = cos(15 - theta), we have csc(75 + theta) = 1/cos(15 - theta) = sec(15 - theta). Therefore, sec(15 - theta) - sec(15 - theta) = 0.

Explanation

In any triangle ABC, the sum of all interior angles is always 180°. This can be expressed by the formula A + B + C = 180°. Since the triangle is given as right angled at C, we have C = 90°. Substituting this value into the angle sum formula, we get A + B + 90° = 180°, which simplifies to A + B = 90°. Now, substitute this result into the given expression cos(A + B) + sin(A + B). This becomes cos(90°) + sin(90°). Using the standard trigonometric values where cos(90°) = 0 and sin(90°) = 1, the calculation is 0 + 1 = 1. Thus, the value of the expression is 1, which corresponds to option (c).

Explanation

To find the value of α + β + γ, we determine the individual values of each acute angle using standard trigonometric ratios. For sin α = √3/2, the corresponding acute angle α is 60°. For cos β = √3/2, the corresponding acute angle β is 30°. For tan γ = 1, the corresponding acute angle γ is 45°.

Using these values, the sum is calculated as: α + β + γ = 60° + 30° + 45° = 135°

Comparing this result with the given options, the value matches option c).

Explanation

Using the identity (a + b)^3 = a^3 + b^3 + 3ab(a + b) with a = sin^2(theta) and b = cos^2(theta): (sin^2(theta) + cos^2(theta))^3 = sin^6(theta) + cos^6(theta) + 3sin^2(theta)cos^2(theta)(sin^2(theta) + cos^2(theta)). Since sin^2(theta) + cos^2(theta) = 1, the expression becomes 1^3 = sin^6(theta) + cos^6(theta) + 3sin^2(theta)cos^2(theta)(1). The value is 1.

Explanation

Statement 1 is correct. To compare the rates of increase of sin θ and tan θ, we analyze their derivatives with respect to θ. The rate of change of sin θ is given by its derivative cos θ, and the rate of change of tan θ is given by its derivative sec² θ. For θ in the first quadrant (0 < θ < π/2), cos θ starts at 1 and decreases toward 0, while sec² θ starts at 1 and increases toward infinity. Since sec² θ ≥ cos θ throughout this interval, tan θ increases faster than sin θ.

Statement 2 is incorrect. The value of sin θ + cos θ is not always greater than 1. For instance, if θ = 0, then sin 0 + cos 0 = 0 + 1 = 1, which is not strictly greater than 1. If θ = π, then sin π + cos π = 0 + (-1) = -1. Mathematically, the expression can be rewritten using the identity sin θ + cos θ = √2 sin(θ + π/4). The range of this function is [-√2, √2], which includes many values less than or equal to 1. Therefore, only statement 1 is correct, which corresponds to option (a).

Explanation

To solve the given expression, we use fundamental trigonometric identities to simplify the numerator and denominator. First, we simplify the term (tanθ + cotθ) by expressing it in terms of sinθ and cosθ: tanθ + cotθ = sinθ/cosθ + cosθ/sinθ = (sin²θ + cos²θ)/(sinθcosθ) Using the identity sin²θ + cos²θ = 1, this becomes 1/(sinθcosθ).

Next, we simplify the denominator secθ + cscθ using the reciprocal identities: secθ + cscθ = 1/cosθ + 1/sinθ = (sinθ + cosθ)/(sinθcosθ)

Now, substituting these simplified terms back into the original expression: ((sinθ + cosθ) × 1/(sinθcosθ)) / ((sinθ + cosθ)/(sinθcosθ)) ((sinθ + cosθ)/(sinθcosθ)) / ((sinθ + cosθ)/(sinθcosθ)) = 1

Since the numerator and denominator are identical, the expression simplifies to 1, which matches option a.

Explanation

The fastest way to solve this trigonometric expression is by substituting a convenient value for θ. Let θ = 0. Since sin 0 = 0, cos 0 = 1, tan 0 = 0, and sec 0 = 1, substituting these values into the expression gives ((1 + 1 - 0) · 1)/((1 + 1 + 0) · (1 - 0)) = (2 · 1)/(2 · 1) = 1. This immediately matches option a.

To verify this algebraically, simplify the numerator as (1 + sec θ - tan θ)cos θ = cos θ + sec θ cos θ - tan θ cos θ = cos θ + 1 - sin θ. Now, consider the denominator (1 + sec θ + tan θ)(1 - sin θ). By expressing sec θ and tan θ in terms of sin θ and cos θ, we get (1 + (1 + sin θ)/(cos θ))(1 - sin θ) = (cos θ + 1 + sin θ)/(cos θ) · (1 - sin θ). Expanding the numerator of this term results in cos θ - cos θ sin θ + 1 - sin θ + sin θ - sin² θ = cos θ - cos θ sin θ + (1 - sin² θ) = cos θ - cos θ sin θ + cos² θ = cos θ(1 - sin θ + cos θ). Dividing this by cos θ, the denominator simplifies to 1 - sin θ + cos θ. Since the simplified numerator and denominator are identical, the expression equals 1.

Explanation

The angle subtended by the balloon at the eye is 60 degrees, so the angle from the center to the tangent is 30 degrees. sin(30) = r / OC, so OC = 2r. The angle of elevation of the center is 60 degrees, so sin(60) = h / OC. h = 2r * (sqrt(3)/2) = sqrt(3)r.

Explanation

Data can be presented in various forms to make it understandable, including tables (tabular), graphs (graphical), and diagrams (diagrammatic). All three are standard methods of presentation.

Explanation

The average of u, v, w, x, y, z is 10, so their sum is 10 * 6 = 60. The sum of the new set is (u + 10) + (v + 20) + (w + 30) + (x + 40) + (y + 50) + (z + 60), which equals (u + v + w + x + y + z) + (10 + 20 + 30 + 40 + 50 + 60). This is 60 + 210 = 270. The average of the new set is 270 / 6 = 45.

Explanation

The formula for the arithmetic mean m of a set of n observations x₁, x₂, ..., xₙ is given by m = (Σ xᵢ)/n. In this problem, there are 7 observations: p, q, r, s, t, u, v. Therefore, the mean is expressed as m = (p + q + r + s + t + u + v)/7, which implies that 7m = p + q + r + s + t + u + v.

The expression given is the sum of the deviations of each observation from the mean: Σ (xᵢ - m) = (p - m) + (q - m) + (r - m) + (s - m) + (t - m) + (u - m) + (v - m). By grouping the terms, we can rewrite the expression as (p + q + r + s + t + u + v) - (m + m + m + m + m + m + m), which simplifies to (p + q + r + s + t + u + v) - 7m. Substituting the value 7m = p + q + r + s + t + u + v into the equation, we get (p + q + r + s + t + u + v) - (p + q + r + s + t + u + v) = 0. This confirms that the sum of deviations of all observations from their mean is always zero, making option a the correct choice.

Explanation

Given total frequency N = 20, so N/2 = 10. The median is 14.4, which falls in the class interval 12-18. Using the median formula: Median = L + ((N/2 - cf)/f) h, where L = 12, cf = 4 + x, f = 5, and h = 6. Substituting the values: 14.4 = 12 + ((10 - (4 + x)) / 5) 6. Simplifying: 2.4 = ((6 - x) / 5) * 6. Divide by 6: 0.4 = (6 - x) / 5. Multiply by 5: 2 = 6 - x. Therefore, x = 4.

Explanation

The total frequency N is given as 20. Summing the given frequencies, we get 4 + x + 5 + y + 1 = 20, which simplifies to x + y = 10. The median is given as 14.4, which falls within the class interval 12-18. Therefore, the median class is 12-18.

Using the median formula for grouped data: Median = L + ((N/2 - cf)/f) × h

Where: L = 12 (lower limit of the median class) N = 20 (total frequency) cf = 4 + x (cumulative frequency of the class preceding the median class) f = 5 (frequency of the median class) h = 6 (class width, 18 - 12)

Substituting the values: 14.4 = 12 + ((20/2 - (4 + x))/5) × 6 14.4 - 12 = ((10 - 4 - x)/5) × 6 2.4 = 6(6 - x)/5 12 = 36 - 6x 6x = 24 x = 4

Using the relation x + y = 10: 4 + y = 10 ⇒ y = 6

To find the relation between x and y: 3x = 3(4) = 12 2y = 2(6) = 12 Thus, 3x = 2y, which corresponds to option (b).