CDS II 2021 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

The prime factors of 26381 can be found by testing the options. Multiplying 23, 31, and 37: 23 31 = 713, and 713 37 = 26381.

Explanation

According to the Factor Theorem, a linear polynomial (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0. To determine which of the given options is a factor of P(x) = (x - 1)(x - 2)(x - 4) - 90, we substitute the root of each option into the polynomial.

For option (d) x - 7, the root is x = 7. Substituting this value into the polynomial: P(7) = (7 - 1)(7 - 2)(7 - 4) - 90 P(7) = (6)(5)(3) - 90 P(7) = 90 - 90 = 0

Since the result is zero, (x - 7) is a factor of the given polynomial. Checking other options like (c) x - 6 gives P(6) = (5)(4)(2) - 90 = 40 - 90 = -50, which is not zero. Similarly, P(14) and P(-14) for options (b) and (a) will not yield zero. Thus, the fastest way to solve this in a competitive exam is to plug in the values of the options and check for zero.

Explanation

Rewrite the expression to match the identity (a - b)² = a² - 2ab + b²:

23 - 4√15 = 20 - 4√15 + 3 = (2√5)² - 2 ⋅ 2√5 ⋅ √3 + (√3)² = (2√5 - √3)²

√(23 - 4√15) = 2√5 - √3

Explanation

37 divided by 9 leaves a remainder of 1 (since 37 = 9*4 + 1). Therefore, 37^1000 mod 9 is equivalent to 1^1000 mod 9, which is 1.

Explanation

Let L be the LCM and H be the HCF. Given L + H = 536 and L - H = 296. Adding the two equations gives 2L = 832, so L = 416. Subtracting gives 2H = 240, so H = 120. Using the property that the product of two numbers equals the product of their LCM and HCF: 104 OtherNumber = 416 120. OtherNumber = (416 120) / 104 = 4 120 = 480.

Explanation

20 men in 5 days complete 1/4 of the work, which means 100 man-days = 1/4 work. The remaining 3/4 work requires 3 * 100 = 300 man-days. To finish this in the remaining 5 days, the number of men required is 300 / 5 = 60 men. Additional men needed = 60 - 20 = 40.

Explanation

Since x is a negative real number, we have x < 0. For statement 1, since k is a natural number, k > 0. The product of a positive number k and a negative number x is always negative, meaning kx < 0. Therefore, the claim that kx > 0 is incorrect. For statement 2, we examine the expression x² + x, which can be factored as x(x + 1). For x < 0, the sign of this product depends on (x + 1). If x lies in the interval (-1, 0), then x is negative and (x + 1) is positive, making x(x + 1) < 0. Thus, x² + x > 0 is not always true. For statement 3, because x < 0, multiplying by 2 makes it smaller (2x < x), and the additive inverse -x becomes a positive number. Thus, 2x < x < -x is always true. For statement 4, since x is any negative real number, it can be an irrational number such as -√π. In such a case, x² = π, which is an irrational number. Therefore, x² is not always a rational number. Since statements 1, 2, and 4 are not correct, the correct option is (b).

Explanation

To find the sum of the linear factors of the expression 2x² + xy - 3y², we first factorize the quadratic expression using the method of splitting the middle term. We look for two numbers whose product equals the product of the coefficients of x² and y², which is 2 × (-3) = -6, and whose sum equals the coefficient of the middle term xy, which is 1. These two numbers are 3 and -2.

Rewriting the expression by splitting the middle term, we get 2x² + 3xy - 2xy - 3y². Now, we group the terms and factor out the common elements: 2x(x + 1.5y) is not ideal, so we group as (2x² - 2xy) + (3xy - 3y²). Factoring these gives 2x(x - y) + 3y(x - y). Taking (x - y) as a common factor, the expression becomes (2x + 3y)(x - y). The two linear factors are therefore (2x + 3y) and (x - y).

To find the sum of these linear factors, we add them together: (2x + 3y) + (x - y). Combining like terms, 2x + x gives 3x and 3y - y gives 2y. Thus, the sum is 3x + 2y, which corresponds to option (c).

Explanation

For a quadratic equation of the form ax² + bx + c = 0, the nature of the roots is determined by the discriminant formula D = b² - 4ac. If D ≥ 0, the equation has real roots, and if D < 0, the equation does not have real roots.

Evaluating the options using this formula: For option a, 2x² + 16x + 3 = 0, D = 16² - 4(2)(3) = 256 - 24 = 232, which is greater than 0. For option b, 2x² + 10x - 1 = 0, D = 10² - 4(2)(-1) = 100 + 8 = 108, which is greater than 0. For option c, x² - 8x + 1 = 0, D = (-8)² - 4(1)(1) = 64 - 4 = 60, which is greater than 0. For option d, 4x² + 9x + 6 = 0, D = 9² - 4(4)(6) = 81 - 96 = -15.

Since the discriminant for option d is negative (D = -15), the quadratic equation 4x² + 9x + 6 = 0 does not have real roots.

Explanation

The standard form of a quadratic equation with roots α and β is given by the formula x² - (α + β)x + αβ = 0. Given that the sum of the roots is 7 and the product is 12, the original equation is x² - 7x + 12 = 0. Solving this by factoring, we have (x-3)(x-4) = 0, which gives the roots as 3 and 4.

Identifying the roots, the bigger root is 4 and the smaller root is 3. According to the conditions, the bigger root is halved and the smaller root is doubled. Therefore, the new roots are 4/2 = 2 and 3 × 2 = 6.

To find the resulting quadratic equation, we calculate the new sum and product of these roots. New sum = 2 + 6 = 8 and new product = 2 × 6 = 12. Substituting these values into the standard formula x² - (sum)x + (product) = 0, we get x² - 8x + 12 = 0. This corresponds to option (b).

Explanation

For the equation x^2 - kx + 2 = 0 to have real and distinct solutions, the discriminant D must be greater than 0. D = (-k)^2 - 4(1)(2) = k^2 - 8. k^2 - 8 > 0 implies k^2 > 8, so k > 2sqrt(2) or k < -2sqrt(2).

Explanation

To solve this problem, we use the general expansion formula for the product of three binomials: (x - a)(x - b)(x - c) = x³ - (a + b + c)x² + (ab + bc + ca)x - abc. By substituting x = 1, a = α, b = β, and c = γ, the expression becomes:

(1 - α)(1 - β)(1 - γ) = 1³ - (α + β + γ)1² + (αβ + βγ + γα)1 - αβγ

Simplifying this, we get: 1 - (α + β + γ) + (αβ + βγ + γα) - αβγ

According to the given condition, α + β + γ = αβ + βγ + γα. Substituting this value into the expanded expression, the terms -(α + β + γ) and +(αβ + βγ + γα) cancel each other out because they are equal in magnitude but opposite in sign.

The remaining terms are 1 - αβγ. Therefore, the expression (1 - α)(1 - β)(1 - γ) is equal to 1 - αβγ, which corresponds to option a.

Explanation

The equation is log10(x) + log10(x^2) = 2log10(x) + 1. This simplifies to log10(x) + 2log10(x) = 2*log10(x) + 1, which further simplifies to log10(x) = 1. Thus, x = 10^1 = 10.

Explanation

Since p and q are prime, their LCM is p * q = 2231. Testing prime factors, 2231 / 23 = 97. Both 23 and 97 are prime. Since p > q, p = 97 and q = 23. The value of p - q = 97 - 23 = 74.

Explanation

To find the time when the runners will next meet at the starting point, we calculate the Least Common Multiple (LCM) of their individual round completion times. The meeting time must be a multiple of each runner's lap time so that each has completed a whole number of rounds.

Given the lap times are 20, 30, and 35 minutes, we use prime factorization to find the LCM: 20 = 2² × 5 30 = 2 × 3 × 5 35 = 5 × 7

The LCM is determined by taking the highest power of each prime factor present in the numbers: LCM(20, 30, 35) = 2² × 3 × 5 × 7 LCM = 4 × 3 × 5 × 7 = 420 minutes

To convert the total minutes into hours, we divide by 60: Time in hours = 420/60 = 7 hours

Thus, the runners will meet at the starting point after 7 hours, which matches option d.

Explanation

Let p, n, e be the costs of a pencil, notebook, and eraser. We have: (1) 5p + 6n + 7e = 250 and (2) 6p + 4n + 2e = 180. To find 2n + 4e, we eliminate p. Multiply (1) by 6 and (2) by 5: 30p + 36n + 42e = 1500 and 30p + 20n + 10e = 900. Subtracting the two gives 16n + 32e = 600. Dividing by 8, we get 2n + 4e = 75.

Explanation

The product 1^50 2^49 ... * 50^1 is the sequence under consideration (assuming 50! is a notation for the product sequence). The number of zeros is determined by the power of 5 in the prime factorization. The powers of 5 are found in 5^46, 10^41, 15^36, 20^31, 25^26 (which is 5^52), 30^21, 35^16, 40^11, 45^6, and 50^1. Summing these: 46+41+36+31+52+21+16+11+6+1 = 261.

Explanation

Given the quadratic equation x² - 60x + 899 = 0, we use the quadratic formula x = (-b ± √(b² - 4ac))/(2a) to find the roots. Here, a = 1, b = -60, and c = 899.

First, calculate the discriminant D: D = b² - 4ac = (-60)² - 4(1)(899) = 3600 - 3596 = 4.

Now, find the roots: x = (60 ± √4)/(2(1)) = (60 ± 2)/2. The two roots are x₁ = 62/2 = 31 and x₂ = 58/2 = 29.

Since it is given that p > q, we have p = 31 and q = 29.

Now we substitute these values into the given options to check for equality to zero: Option (a): p - q - 1 = 31 - 29 - 1 = 1 ≠ 0. Option (b): p - 2q + 27 = 31 - 2(29) + 27 = 31 - 58 + 27 = 58 - 58 = 0. Option (c): 2p - q - 30 = 2(31) - 29 - 30 = 62 - 59 = 3 ≠ 0. Option (d): 3p - 2q - 43 = 3(31) - 2(29) - 43 = 93 - 58 - 43 = 93 - 101 = -8 ≠ 0.

Thus, the equation p - 2q + 27 = 0 is satisfied, making option (b) the correct choice.

Explanation

For a quadratic equation ax² + bx + c = 0 to have real roots, the discriminant D must be greater than or equal to zero. The formula for the discriminant is D = b² - 4ac. In the given equation x² - 4x - log₁₀N = 0, the coefficients are a = 1, b = -4, and c = -log₁₀N.

Substituting these values into the discriminant formula: (-4)² - 4(1)(-log₁₀N) ≥ 0 16 + 4log₁₀N ≥ 0

To find the minimum value of N, we solve the inequality: 4log₁₀N ≥ -16 log₁₀N ≥ -4

Converting the logarithmic equation to exponential form: N ≥ 10⁻⁴ N ≥ 0.0001

The minimum value of N is therefore 0.0001, which corresponds to option (d) 0-0001.

Explanation

5^(x-1) = (2.5)^log10(5). Taking log10 of both sides: (x-1)log10(5) = log10(5) log10(2.5). Dividing by log10(5): x-1 = log10(2.5) = log10(5/2) = log10(5) - log10(2). Thus, x = 1 + log10(5) - log10(2). Since 1 = log10(10), x = log10(105/2) = log10(25) = 2*log10(5).

Explanation

The equation is 96 - 64a^3 + 8/a^6 - 48/a^3 - t^3 = 0. Note that (4a - 2/a^2)^3 = 64a^3 - 96 + 48/a^3 - 8/a^6. The equation can be rewritten as -(4a - 2/a^2)^3 - t^3 = 0, so t = -(4a - 2/a^2) = 2/a^2 - 4a. Then a^2*t + 4a^3 = a^2(2/a^2 - 4a) + 4a^3 = 2 - 4a^3 + 4a^3 = 2.

Explanation

To find the value of B, we use the standard algebraic relation B = ((A + B) - (A - B))/2. By substituting the given expressions into this formula, we get 2B = (x² - 8)/(x + 2) - (-x² + 2x + 4)/(x + 2). Since the denominators are common, we subtract the numerators directly: 2B = (x² - 8 - (-x² + 2x + 4))/(x + 2) = (x² - 8 + x² - 2x - 4)/(x + 2) = (2x² - 2x - 12)/(x + 2). Dividing both sides by 2, we find B = (x² - x - 6)/(x + 2). We can factor the numerator x² - x - 6 as (x - 3)(x + 2), so B = ((x - 3)(x + 2))/(x + 2) = x - 3. To match this with the given options, we check option (c), where (2x² - 7x + 3)/(2x - 1) can be simplified by factoring the numerator as (x - 3)(2x - 1). This yields ((x - 3)(2x - 1))/(2x - 1) = x - 3, which matches our result. Thus, option (c) is the correct representation.

Explanation

The expression is a specific case of the identity Sum[x^n / ((x-y)(x-z))] for x, y, z distinct. For n=2, the sum a^2/((a-b)(a-c)) + b^2/((b-a)(b-c)) + c^2/((c-a)(c-b)) is always equal to 1. Substituting a=x^2, b=y^2, c=z^2, the result is 1.

Explanation

Given the ratio (2ab - b²) : (6a² - ab) = 1 : 6, we can express this as the equation (2ab - b²)/(6a² - ab) = 1/6. Cross-multiplying the terms gives 6(2ab - b²) = 6a² - ab, which simplifies to 12ab - 6b² = 6a² - ab. Rearranging all terms to one side results in the quadratic equation 6a² - 13ab + 6b² = 0. To find the relationship between a and b, we factor the quadratic: 6a² - 9ab - 4ab + 6b² = 0, which leads to 3a(2a - 3b) - 2b(2a - 3b) = 0. This factors completely into (3a - 2b)(2a - 3b) = 0. This gives us two possible ratios for a/b: either a/b = 2/3 or a/b = 3/2. The target expression is (a + b) : (a - b), which can be simplified by dividing the numerator and denominator by b to get (a/b + 1)/(a/b - 1). Substituting the first value a/b = 3/2 gives (3/2 + 1)/(3/2 - 1) = (5/2)/(1/2) = 5. Substituting the second value a/b = 2/3 gives (2/3 + 1)/(2/3 - 1) = (5/3)/(-1/3) = -5. Thus, the value is either -5 or 5, making option (d) the correct choice.

Explanation

The given expression x² + y² + z² - 2xy - 2yz + 2zx is a perfect square of a trinomial. According to the standard algebraic identity (A + B + C)² = A² + B² + C² + 2AB + 2BC + 2AC, we can rewrite the target expression by letting A = x, B = -y, and C = z. This gives us: x² + y² + z² - 2xy - 2yz + 2zx = (x - y + z)²

From the given ratios, we express x, y, and z in terms of k: x = k(b + c - a) y = k(b - c - a) z = k(a + b - c)

Substitute these values into the trinomial (x - y + z): x - y + z = k(b + c - a) - k(b - c - a) + k(a + b - c) x - y + z = k(b + c - a - b + c + a + a + b - c)

Combining the like terms within the parentheses: Terms of a: -a + a + a = a Terms of b: b - b + b = b Terms of c: c + c - c = c x - y + z = k(a + b + c)

To find the value of the original expression, square the result: (x - y + z)² = [k(a + b + c)]² = k²(a + b + c)²

This result matches option (c).

Explanation

Let the numbers be 2k, 3k, and 5k. Sum of squares: (2k)^2 + (3k)^2 + (5k)^2 = 4k^2 + 9k^2 + 25k^2 = 38k^2. Given 38k^2 = 1368, k^2 = 36, so k = 6. Sum of numbers = 2k + 3k + 5k = 10k = 10 * 6 = 60.

Explanation

For inequality 1, we have (a²-b²)/(a²+b²) > (a-b)/(a+b). Given a > b > 0, the term a-b is positive. We can rewrite the left side as ((a-b)(a+b))/(a²+b²). Dividing both sides by the positive term (a-b), the inequality becomes (a+b)/(a²+b²) > 1/(a+b). Cross-multiplying the denominators, which are both positive, yields (a+b)² > a²+b². Expanding this gives a² + 2ab + b² > a² + b², which simplifies to 2ab > 0. Since a and b are both positive, this condition is always satisfied, proving that inequality 1 is correct.

For inequality 2, (a³+b³)/(a²+b²) > (a²+b²)/(a+b), we cross-multiply the positive denominators to obtain (a³+b³)(a+b) > (a²+b²)². Expanding both sides, we get a⁴ + a³b + ab³ + b⁴ > a⁴ + 2a²b² + b⁴. Subtracting a⁴ + b⁴ from both sides results in a³b + ab³ > 2a²b², which can be factored as ab(a² + b² - 2ab) > 0, or ab(a-b)² > 0. This inequality holds for all a, b > 0 as long as a ≠ b. The statement claims the inequality holds only when a > b > 0, but since it also holds for b > a > 0, the restrictive phrase only when makes inequality 2 incorrect. Therefore, only inequality 1 is correct.

Explanation

x = (3n-1)(2n+1) and y = (3n+1)(4n-3). Given x/y = 6/11, then 11(6n^2 + n - 1) = 6(12n^2 - 5n - 3). Simplifying: 66n^2 + 11n - 11 = 72n^2 - 30n - 18, which leads to 6n^2 - 41n - 7 = 0. Solving via quadratic formula: n = (41 + sqrt(1681 + 168))/12 = (41 + 43)/12 = 7.

Explanation

Let (ay-bx)/c = (cx-az)/b = (bz-cy)/a = k. This gives us a system of three linear equations:

1. ay - bx = ck
2. cx - az = bk
3. bz - cy = ak

To solve for k, we express x and z in terms of y and k. From equation 1, bx = ay - ck, which implies x = (ay-ck)/b. From equation 3, bz = cy + ak, which implies z = (cy+ak)/b. Substituting these expressions for x and z into equation 2: c((ay-ck)/b) - a((cy+ak)/b) = bk (cay - c²k - acy - a²k)/b = bk cay - acy - (a²+c²)k = b²k 0 = (a²+b²+c²)k

Since a, b, c are denominators in the original expression, they are non-zero, meaning a²+b²+c² ≠ 0. Thus, k must be 0. When k=0, the equations become: ay - bx = 0 → x/a = y/b cx - az = 0 → x/a = z/c bz - cy = 0 → y/b = z/c

This implies x/a = y/b = z/c = λ for some constant λ. Statement 1 is x/a = y/b, which is directly derived above and is therefore correct. For Statement 2, we substitute x = aλ, y = bλ, and z = cλ: (x+y+z)/(a+b+c) = (aλ + bλ + cλ)/(a+b+c) = (λ(a+b+c))/(a+b+c) = λ Since z/c = cλ/c = λ, the equality (x+y+z)/(a+b+c) = z/c holds true. Both statements 1 and 2 are correct, making option c the right choice.

Explanation

Let the length of the rectangular garden be l and the width be w. The area of the garden is given by the formula A = l × w = 375. Since only three sides are fenced with 65m of wire, the total length used is either 2l + w = 65 or l + 2w = 65. Substituting w = 65 - 2l into the area formula, we get l(65 - 2l) = 375, which simplifies to the quadratic equation 2l² - 65l + 375 = 0.

Using the quadratic formula l = (-b ± √(b² - 4ac))/(2a), we have l = (65 ± √(65² - 4(2)(375)))/(2(2)) = (65 ± √(4225 - 3000))/4 = (65 ± √1225)/4 = (65 ± 35)/4. This gives two possible values for l: l = 100/4 = 25 or l = 30/4 = 7.5.

If l = 25m, then w = 65 - 2(25) = 15m. The perimeter of the rectangle is calculated using the formula P = 2(l + w) = 2(25 + 15) = 2(40) = 80m. This matches option a. If we take l = 7.5m, then w = 50m, and the perimeter would be 2(7.5 + 50) = 115m, which is not in the options. Thus, the perimeter of the garden is 80m.

Explanation

Since x = 3/2 is a root of the equation ax²-4ax+15=0, it must satisfy the equation. Substituting x = 3/2 into the equation gives a(3/2)²-4a(3/2)+15=0. Simplifying this, we get 9a/4-6a+15=0. Multiplying the entire equation by 4 to clear the denominator, we obtain 9a-24a+60=0, which simplifies to -15a=-60, resulting in a=4.

Substituting a=4 back into the original equation, we get 4x²-16x+15=0. For a quadratic equation Ax²+Bx+C=0, let the roots be x₁ and x₂. The sum of the roots is x₁+x₂ = -B/A = -(-16)/4 = 4 and the product of the roots is x₁x₂ = C/A = 15/4.

To find the sum of the squares of the roots, we use the algebraic identity x₁²+x₂² = (x₁+x₂)²-2x₁x₂. Substituting the values, we get x₁²+x₂² = 4²-2(15/4) = 16-15/2. Converting 16 to a fraction with denominator 2, we have 32/2-15/2 = 17/2. This matches option b.

Explanation

Let the digits be x and y. xy = 8 and (10x + y) + 63 = 10y + x. The second equation simplifies to 9(y - x) = 63, so y - x = 7. Substituting y = x + 7 into xy = 8 gives x(x + 7) = 8, so x^2 + 7x - 8 = 0. The positive digit solution is x = 1, then y = 8. The sum of the digits is 1 + 8 = 9.

Explanation

Let the speed of the stream be v km/hr. The speed of the boat downstream is (30 + v) km/hr and the speed upstream is (30 - v) km/hr. The total time taken for the round trip is calculated using the formula T = d/(u + v) + d/(u - v). Substituting the given values d = 60 km, u = 30 km/hr, and T = 9/2 hours:

60/(30 + v) + 60/(30 - v) = 9/2

Taking 60 as a common factor:

60 ( ((30 - v) + (30 + v))/((30 + v)(30 - v)) ) = 9/2

60 ( 60/(900 - v²) ) = 9/2

3600/(900 - v²) = 9/2

Cross-multiplying gives:

7200 = 9(900 - v²)

7200 = 8100 - 9v²

9v² = 8100 - 7200

9v² = 900

v² = 100

v = 10 km/hr.

Hence, the speed of the stream is 10 km/hr, which corresponds to option c.

Explanation

Let the present age of the father be F and the sum of the present ages of his four children be S. According to the problem, the present age of the father is equal to the sum of the ages of his children, so F = S. After ten years, the father's age will be F + 10. Since there are four children, the sum of their ages will increase by 4 × 10 = 40 years, making the new sum S + 40. The problem states that after ten years, this sum will be 1.6 times the age of the father. This gives the equation S + 40 = 1.6(F + 10). Substituting S = F into the equation, we have F + 40 = 1.6F + 16. Rearranging the equation to solve for F: 40 - 16 = 1.6F - F, which simplifies to 24 = 0.6F. Solving for F, we get F = 24 / 0.6 = 240 / 6 = 40. Thus, the present age of the father is 40 years, which corresponds to option b.

Explanation

Let N be the numerator and D be the denominator. N + D = 10. Also (N + 3) / (D - 1) = 1, so N + 3 = D - 1, which implies D - N = 4.

Explanation

For a unique solution in a system of linear equations a1x + b1y = c1 and a2x + b2y = c2, the condition is a1/a2 != b1/b2. Here, 7/k != k/7, which means k^2 != 49, so k != 7 and k != -7. Option A (k != 7) is a correct necessary condition.

Explanation

For compound interest, the amount formula is A = P(1 + r/100)ⁿ. According to the problem, the sum doubles itself in 5 years, so we can write 2P = P(1 + r/100)⁵, which simplifies to (1 + r/100)⁵ = 2. We need to find the number of years n for the sum to become four times itself, which means 4P = P(1 + r/100)ⁿ, simplifying to (1 + r/100)ⁿ = 4. Since 4 can be expressed as 2², we substitute the value of 2 from the first equation: (1 + r/100)ⁿ = [ (1 + r/100)⁵ ]². This simplifies to (1 + r/100)ⁿ = (1 + r/100)¹⁰. By comparing the exponents, we find that n = 10 years. A competitive exam shortcut for this pattern is that if a sum becomes k times in t years, it will become kᵐ times in m × t years. In this case, k = 2, t = 5, and the target is 4 times (2²), so m = 2. The total time required is 2 × 5 = 10 years, making option a the correct choice.

Explanation

The time at which the hands of a clock coincide can be calculated using the formula t = 60/11 × H, where t is the number of minutes past the hour H. In this problem, the hands coincide between 3 and 4 o'clock, so we substitute H = 3 into the formula. This gives t = (60 × 3)/11 = 180/11. Dividing 180 by 11, we get 16 4/11 minutes. This value represents the exact time past 3 o'clock when the hour hand and minute hand are in the same position. Since 16 4/11 is greater than 16 but less than 17, the hands coincide between 16-17 minutes. This matches option (b).

Explanation

Sheela stitches 1/2 suit per day; Meena stitches 1/(2 1/2) = 2/5 suit per day.\n\nCombined rate = 1/2 + 2/5 = 5/10 + 4/10 = 9/10 suits per day.\n\nTime for 30 suits = 30/(9/10) = 300/9 = 33 1/3 days ≈ 33 days.

Explanation

The expression 8x^3 - 36x^2y + 54xy^2 - 27y^3 is the expansion of (2x - 3y)^3. Given 2x - 3y - 7 = 0, we have 2x - 3y = 7. Thus, (2x - 3y)^3 - 340 = 7^3 - 340 = 343 - 340 = 3.

Explanation

p = kq/r^2. Let new values be q' = 1.2q and r' = 0.8r. Then p' = k(1.2q) / (0.8r)^2 = (1.2 / 0.64) (kq/r^2) = 1.875p. The percentage increase is (1.875 - 1) 100 = 87.5%.

Explanation

Let w be days worked and a be days absent. w + a = 30 and 500w - 100a = 11400. Substituting a = 30 - w: 500w - 100(30 - w) = 11400 => 600w - 3000 = 11400 => 600w = 14400 => w = 24.

Explanation

The fastest way to solve this problem is by using the rule of alligation, as it deals with the weighted average of two different gain percentages. Let the cost price of the chair be CP꜀ₕₐᵢᵣ and the cost price of the table be CPₜₐ₆ₗₑ. The gain on the chair is 5%, the gain on the table is 20%, and the overall gain is 16%.

Applying the alligation formula: Ratio of CP꜀ₕₐᵢᵣ to CPₜₐ₆ₗₑ = (20% - 16%) : (16% - 5%) Ratio = 4% : 11% Ratio = 4 : 11

The total cost price is given as ₹750. The total parts in the ratio are 4 + 11 = 15 units. Value of one unit = 750 / 15 = ₹50. Since the table represents 11 units in the ratio, its original cost is 11 × 50 = ₹550. Thus, option (c) is correct.

Explanation

Let the total distance of the journey be D km. Using the fundamental formula Time = Distance/Speed, the total time taken in the first scenario T₁ is the sum of times for each one-third of the journey: T₁ = (D/3)/60 + (D/3)/50 + (D/3)/40 = D/3 (1/60 + 1/50 + 1/40). Finding a common denominator of 600 for the terms in the bracket, we get T₁ = D/3 (10 + 12 + 15/600) = 37D/1800 hours.

In the second scenario, the total time T₂ is the sum of times for each half of the journey: T₂ = (D/2)/60 + (D/2)/40 = D/2 (1/60 + 1/40). Using a common denominator of 120, we get T₂ = D/2 (2 + 3/120) = 5D/240 = D/48 hours.

The problem states that T₂ is 4 minutes longer than T₁. Converting 4 minutes to hours, we have 4/60 = 1/15 hours. Thus, the equation is T₂ - T₁ = 1/15, which substitutes to D/48 - 37D/1800 = 1/15. To solve for D, we use the least common multiple of 48 and 1800, which is 3600. This gives (75D - 74D)/3600 = 1/15, simplifying to D/3600 = 1/15. Consequently, D = 3600/15 = 240 km.

Explanation

This problem follows an Arithmetic Progression (AP) where the savings increase by a constant amount every year. Let the first year savings be a = 2000 and the common difference be d = 1000. The total savings over n years is the sum of the AP, denoted by Sₙ = 170000.

Using the sum formula for an AP: Sₙ = n/2 [2a + (n - 1)d]

Substituting the given values into the formula: 170000 = n/2 [2(2000) + (n - 1)1000] 170000 = n/2 [4000 + 1000n - 1000] 170000 = n/2 [3000 + 1000n] 170000 = 1500n + 500n²

Dividing the entire equation by 500 to simplify: 340 = 3n + n² n² + 3n - 340 = 0

Solving this quadratic equation using the quadratic formula n = (-b ± √(b² - 4ac))/2a: n = (-3 ± √(3² - 4(1)(-340)))/2(1) n = (-3 ± √(9 + 1360))/2 n = (-3 ± √1369)/2 n = (-3 ± 37)/2

Since the number of years cannot be negative, we take the positive root: n = 34/2 = 17

Thus, the person will save ₹170000 in 17 years, which corresponds to option b.

Explanation

Let u = cos(theta). For 0 <= theta < 90, 0 < u <= 1. We seek the minimum of u^3 + 1/u^3. By AM-GM inequality, (u^3 + 1/u^3) / 2 >= sqrt(u^3 * 1/u^3) = 1. Thus, u^3 + 1/u^3 >= 2. The minimum value is 2.

Explanation

14sin^2(theta) + 10cos^2(theta) = 11 can be written as 10(sin^2(theta) + cos^2(theta)) + 4sin^2(theta) = 11. 10 + 4sin^2(theta) = 11 => sin^2(theta) = 1/4 => sin(theta) = 1/2. Thus theta = 30 degrees. tan(30) + cot(30) = 1/sqrt(3) + sqrt(3) = 4/sqrt(3).

Explanation

The expression is (sin^3 + cos^3)/(sin + cos) + (sin^3 - cos^3)/(sin - cos). Using the identities a^3+b^3=(a+b)(a^2-ab+b^2) and a^3-b^3=(a-b)(a^2+ab+b^2), the expression simplifies to (sin^2 - sincos + cos^2) + (sin^2 + sincos + cos^2) = 1 - sincos + 1 + sincos = 2.

Explanation

Let H be the height of the flagstaff. The ladder of length 10m reaches a point (H-10)m high. Let x be the distance from the foot of the ladder to the base of the staff. In the right triangle formed by the ladder, x^2 + (H-10)^2 = 10^2, which simplifies to x^2 + H^2 - 20H + 100 = 100, so x^2 = 20H - H^2. The angle of elevation to the top of the staff is 60 degrees, so tan(60) = H/x, meaning x = H/sqrt(3). Substituting x into the first equation: (H/sqrt(3))^2 = 20H - H^2, which gives H^2/3 = 20H - H^2. Adding H^2 to both sides: 4H^2/3 = 20H. Dividing by H (H is not 0): 4H/3 = 20, so H = 15m.

Explanation

The expression is 1 + 2sin^2(theta)cos^2(theta) - sin^4(theta) - cos^4(theta). We know that sin^4(theta) + cos^4(theta) = (sin^2(theta) + cos^2(theta))^2 - 2sin^2(theta)cos^2(theta) = 1 - 2sin^2(theta)cos^2(theta). Substituting this into the expression: 1 + 2sin^2(theta)cos^2(theta) - (1 - 2sin^2(theta)cos^2(theta)) = 4sin^2(theta)cos^2(theta). This can be written as (2sin(theta)cos(theta))^2 = sin^2(2theta). The maximum value of sin^2(2theta) is 1, which occurs when 2*theta = 90 degrees, or theta = 45 degrees.

Explanation

Let h = 1200m be the altitude of the aeroplane and W = 3000m be the width of the river. The angles of depression to the opposite banks are 30° and θ. The horizontal distances from the point directly below the aeroplane to the two banks, x₁ and x₂, are given by the formula x = h cot(α).

For the first bank, x₁ = 1200 cot(30°) = 1200 √3. Using √3 ≈ 1.732, we get x₁ ≈ 1200 × 1.732 = 2078.4m. Since the banks are on opposite sides, the total width is W = x₁ + x₂. Thus, x₂ = 3000 - 2078.4 = 921.6m.

Now, we find tan(θ) using the relation tan(θ) = h/x₂: tan(θ) = 1200/921.6 ≈ 1.302.

Comparing this value with standard trigonometric ratios: tan(45°) = 1 and tan(60°) = √3 ≈ 1.732. Since 1 < 1.302 < 1.732, it follows that tan(45°) < tan(θ) < tan(60°). Therefore, 45° < θ < 60°, which matches option c.

Explanation

Given the equation (cos²θ - 3cosθ + 2)/sin²θ = 1, we cross-multiply to obtain cos²θ - 3cosθ + 2 = sin²θ. Using the trigonometric identity sin²θ = 1 - cos²θ, the equation becomes cos²θ - 3cosθ + 2 = 1 - cos²θ. Rearranging all terms to one side gives the quadratic equation 2cos²θ - 3cosθ + 1 = 0.

Factorizing the quadratic expression, we get (2cosθ - 1)(cosθ - 1) = 0. This yields two possible values: cosθ = 1 or cosθ = 1/2. Given the constraint 0° < θ < 90°, we must have cosθ ≠ 1 because cosθ = 1 implies θ = 0° and sinθ = 0, which would make the original expression undefined. Therefore, we use cosθ = 1/2.

To find the value of sin²θ + cosθ, we substitute cosθ = 1/2 and sin²θ = 1 - cos²θ: sin²θ + cosθ = (1 - (1/2)²) + 1/2 = 1 - 1/4 + 1/2 = 3/4 + 2/4 = 5/4. This matches option (a).

Explanation

For statement 1, consider the left hand side (LHS) as sin²θ(sin²θ - 1). Using the fundamental identity sin²θ + cos²θ = 1, we know sin²θ - 1 = -cos²θ. Thus, LHS = sin²θ(-cos²θ) = -sin²θ cos²θ. Similarly, the right hand side (RHS) is cos²θ(cos²θ - 1). Since cos²θ - 1 = -sin²θ, the RHS = cos²θ(-sin²θ) = -sin²θ cos²θ. Since LHS = RHS, statement 1 is an identity.

For statement 2, use the algebraic identity (a+b)² = a² + b² + 2ab with a = sin²θ and b = cos²θ. Then (sin²θ + cos²θ)² = sin⁴θ + cos⁴θ + 2sin²θ cos²θ. Substituting sin²θ + cos²θ = 1, we get 1² = sin⁴θ + cos⁴θ + 2sin²θ cos²θ, which simplifies to sin⁴θ + cos⁴θ = 1 - 2sin²θ cos²θ. The given expression uses a plus sign instead of a minus sign, making statement 2 not an identity.

For statement 3, the LHS is tan²θ(tan²θ + 1). Using the identity 1 + tan²θ = sec²θ, the expression becomes tan²θ sec²θ. Now, substitute tan²θ = sec²θ - 1 into the expression to get (sec²θ - 1)sec²θ = sec⁴θ - sec²θ. This matches the RHS, so statement 3 is an identity. Therefore, only 1 and 3 are identities, making option c correct.

Explanation

Note that 66 degrees = 90 degrees - 24 degrees. Therefore, sin(66) = cos(24), cos(66) = sin(24), tan(66) = cot(24), and cot(66) = tan(24). The expression becomes: sin(24)cos(24) - cos(24)sin(24) + tan(24)cot(24) - cot(24)tan(24). The first two terms cancel out (0), and the last two terms are 1 - 1 = 0. The final result is 0.

Explanation

x^2 + y^2 + z^2 = (p sin A cos B)^2 + (p sin A sin B)^2 + (p cos A)^2 = p^2 sin^2 A cos^2 B + p^2 sin^2 A sin^2 B + p^2 cos^2 A. Factor out p^2 sin^2 A from the first two terms: p^2 sin^2 A (cos^2 B + sin^2 B) + p^2 cos^2 A. Since cos^2 B + sin^2 B = 1, this simplifies to p^2 sin^2 A + p^2 cos^2 A = p^2 (sin^2 A + cos^2 A). Since sin^2 A + cos^2 A = 1, the result is p^2.

Explanation

The expression x² - y² can be solved efficiently using the algebraic identity x² - y² = (x + y)(x - y).

First, calculate the sum x + y: x + y = (m sec A + n tan A) + (m tan A + n sec A) x + y = m(sec A + tan A) + n(sec A + tan A) x + y = (m + n)(sec A + tan A)

Next, calculate the difference x - y: x - y = (m sec A + n tan A) - (m tan A + n sec A) x - y = m(sec A - tan A) - n(sec A - tan A) x - y = (m - n)(sec A - tan A)

Now, multiply these two results together: x² - y² = [(m + n)(sec A + tan A)] · [(m - n)(sec A - tan A)] x² - y² = (m + n)(m - n) · (sec A + tan A)(sec A - tan A)

Using the identity (a + b)(a - b) = a² - b², we get: x² - y² = (m² - n²)(sec² A - tan² A)

Applying the fundamental trigonometric identity sec² A - tan² A = 1, the expression simplifies to: x² - y² = (m² - n²) · 1 = m² - n²

Thus, the result corresponds to option a.

Explanation

Given tan θ = 1 for 0° < θ < 90°, we can identify that θ = 45° since tan 45° = 1. For θ = 45°, the values of the sine and cosine functions are sin 45° = 1/√2 and cos 45° = 1/√2. Substituting these values into the expression sin² θ - 2 sin θ cos θ, we obtain (1/√2)² - 2(1/√2)(1/√2). Simplifying the terms, we have 1/2 - 2(1/2) = 1/2 - 1 = -1/2. Alternatively, using the double angle formula 2 sin θ cos θ = sin 2θ, the expression simplifies to sin² 45° - sin 90°, which equals 1/2 - 1 = -1/2. This matches the value in option (d).

Explanation

Numerator: 2sin^3(theta) - sin(theta) = sin(theta)(2sin^2(theta) - 1). Using sin^2(theta) = 1 - cos^2(theta), the numerator is sin(theta)(2(1 - cos^2(theta)) - 1) = sin(theta)(1 - 2cos^2(theta)). Denominator: cos(theta) - 2cos^3(theta) = cos(theta)(1 - 2cos^2(theta)). The expression is [sin(theta)(1 - 2cos^2(theta))] / [cos(theta)(1 - 2cos^2(theta))] = sin(theta)/cos(theta) = tan(theta).

Explanation

In a triangle ABC, the sum of the interior angles is always 180°, so A + B + C = 180°. From this, we can derive that B + C = 180° - A. Dividing both sides by 2 gives (B+C)/2 = (180° - A)/2 = 90° - A/2.

Using the trigonometric complementary angle identities, where tan(90° - θ) = cot(θ) and sin(90° - θ) = cos(θ), we can substitute θ = A/2. This gives tan((B+C)/2) = tan(90° - A/2) = cot(A/2) and sin((B+C)/2) = sin(90° - A/2) = cos(A/2).

Now, substituting these values into the given expression: tan((B+C)/2) + sin((B+C)/2) - cot(A/2) - cos(A/2) = cot(A/2) + cos(A/2) - cot(A/2) - cos(A/2).

The terms cot(A/2) and -cot(A/2) cancel each other out, and the terms cos(A/2) and -cos(A/2) also cancel each other out. The resulting value is 0, which matches option a.

Explanation

In a right-angled triangle ABC with ∠B = 90°, the side AC is the hypotenuse. According to Pythagoras Theorem, AB² + BC² = AC². Given AC = 20 cm, we have AB² + BC² = 20² = 400.

We are given the sum of the other two sides as AB + BC = 10(1 + √3). Squaring both sides of this equation gives (AB + BC)² = (10(1 + √3))², which expands to AB² + BC² + 2(AB · BC) = 100(1 + 3 + 2√3). Substituting the value of AB² + BC² = 400, we get 400 + 2(AB · BC) = 100(4 + 2√3) = 400 + 200√3. Subtracting 400 from both sides yields 2(AB · BC) = 200√3, so AB · BC = 100√3.

The expression to evaluate is tan A + tan C. In a right triangle, tan A = BC/AB and tan C = AB/BC. Therefore, tan A + tan C = BC/AB + AB/BC = (BC² + AB²)/(AB · BC). Substituting the known values, tan A + tan C = 400/(100√3) = 4/√3. This matches option (a).

Explanation

In a right-angled triangle, the circum-radius R is always equal to half of the hypotenuse. This is because the hypotenuse of a right-angled triangle serves as the diameter of its circum-circle. Using the formula c = 2R, where c is the hypotenuse and R is the circum-radius, we substitute the given value of R = 10 cm: c = 2 × 10 = 20 cm.

The area of a triangle can be calculated using the formula Area = 1/2 × base × height. In this case, we take the hypotenuse as the base and the altitude drawn to the hypotenuse as the height. Given that the altitude is 8 cm and the hypotenuse is 20 cm, the calculation is as follows: Area = 1/2 × 20 cm × 8 cm Area = 10 cm × 8 cm = 80 cm².

Thus, the area of the triangle is 80 cm², which corresponds to option (b).

Explanation

Let radii be r1 and r2. Since they touch externally, the distance between centers is r1 + r2 = 9. The sum of areas is pi(r1^2 + r2^2) = 41pi, so r1^2 + r2^2 = 41. Using (r1 + r2)^2 = r1^2 + r2^2 + 2r1r2, we get 9^2 = 41 + 2r1r2, which means 81 - 41 = 2r1r2, so r1r2 = 20. The values that satisfy r1 + r2 = 9 and r1r2 = 20 are 4 and 5. The diameters are 8cm and 10cm. The difference is 10 - 8 = 2cm.

Explanation

Let the side of square X be s. The area of square X is given by the formula p = s². According to the properties of a square, the length of the diagonal d is related to the side s by the formula d = s√2. A new square is formed using this diagonal as its side, so the side length of the second square is d. The area of this square is q = d² = (s√2)² = 2s². To find the required value, we calculate the ratio p/q = s²/(2s²) = 1/2. Thus, the value of p/q is 1/2, which matches option (d).

Explanation

The area of a rhombus is given by the formula Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. Substituting the given values, 336 = 1/2 × 48 × d₂, which simplifies to 336 = 24 × d₂. Solving for the second diagonal, d₂ = 336/24 = 14 cm.

In a rhombus, the diagonals bisect each other at right angles, forming four right-angled triangles. The side of the rhombus s acts as the hypotenuse of these triangles. Using the Pythagorean theorem, the side length is calculated as s = √((d₁/2)² + (d₂/2)²). Substituting the values, s = √((48/2)² + (14/2)²) = √(24² + 7²) = √(576 + 49) = √625 = 25 cm.

The perimeter of a rhombus is 4 × s. Therefore, the perimeter is 4 × 25 = 100 cm, which corresponds to option c.

Explanation

The area described by the minute hand of a clock forms a sector of a circle. The area of a sector can be calculated using the formula: Area = θ/360 × πr² Alternatively, since the minute hand completes a full circle in 60 minutes, the area swept in t minutes is: Area = t/60 × πr²

Given the time interval is between 10:10 a.m. and 10:30 a.m., the time elapsed t is 30 - 10 = 20 minutes. The length of the minute hand r is 21 cm, and π = 22/7.

Substituting the values into the formula: Area = 20/60 × 22/7 × (21)² Area = 1/3 × 22/7 × 441 Area = 1/3 × 22 × 441/7 Area = 1/3 × 22 × 63 Area = 22 × 21 Area = 462 cm²

Thus, the area described by the minute hand is 462 cm², which corresponds to option C.

Explanation

The longest rod in a room is the space diagonal D. D^2 = L^2 + B^2 + H^2. Given D = 29, L = 21, and B = 16: 29^2 = 21^2 + 16^2 + H^2. 841 = 441 + 256 + H^2. 841 = 697 + H^2. H^2 = 841 - 697 = 144. H = sqrt(144) = 12m.

Explanation

Volume of hemispherical bowl = (2/3) pi R^3 = (2/3) pi 18^3 = (2/3) pi 5832 = 3888 pi cm^3. Volume of one cylindrical bottle = pi r^2 h = pi 3^2 4 = 36 pi cm^3. Number of bottles = (3888 pi) / (36 pi) = 3888 / 36 = 108.

Explanation

The mass of a hollow spherical shell is calculated by multiplying its volume by the density of the material. The volume of a hollow sphere is the difference between the volume of the outer sphere and the inner sphere, given by the formula V = 4/3 π (R³ - r³), where R is the external radius and r is the internal radius.

Given values: External radius R = 6 cm Internal radius r = 3 cm Density ρ = 7 g/cm³ π = 22/7

First, calculate the volume: V = 4/3 × 22/7 × (6³ - 3³) V = 4/3 × 22/7 × (216 - 27) V = 4/3 × 22/7 × 189 Simplifying the expression: 189/7 = 27 and 27/3 = 9. V = 4 × 22 × 9 V = 792 cm³

Now, calculate the mass using the formula Mass = Density × Volume: Mass = 7 g/cm³ × 792 cm³ Mass = 5544 g

Thus, the mass of the shell is 5544 g, which corresponds to option d.

Explanation

The total volume of the solid is the sum of the volume of the cone and the volume of the hemisphere. The radius r for both shapes is half of the diameter, so r = 14/2 = 7 cm. The height of the cone is h = 16 cm.

Using the volume formulas for a cone and a hemisphere, the total volume V is given by: V = 1/3 π r² h + 2/3 π r³

To simplify the calculation, we can factor out the common terms: V = 1/3 π r² (h + 2r)

Substituting the given values into the formula: V = 1/3 × 22/7 × 7² × (16 + 2 × 7) V = 1/3 × 22/7 × 49 × (16 + 14) V = 1/3 × 22 × 7 × 30 V = 22 × 7 × 10 V = 154 × 10 = 1540 cm³

Thus, the total volume of the combined solid is 1540 cm³, which corresponds to option a.

Explanation

The volume of each cube is given by V = a³ = 343 cm³, which allows us to find the side length a = ∛343 = 7 cm. When three such cubes are joined end to end, they form a cuboid with length L = 3a = 3 × 7 = 21 cm, breadth B = a = 7 cm, and height H = a = 7 cm. The total surface area (TSA) of a cuboid is determined by the formula TSA = 2(LB + BH + HL). Substituting the dimensions, we get TSA = 2(21 × 7 + 7 × 7 + 7 × 21) = 2(147 + 49 + 147) = 2(343) = 686 cm².

A quicker competition-style approach is to consider the number of exposed faces. Three separate cubes have 3 × 6 = 18 faces. Joining them end to end creates two internal interfaces, and each interface hides two faces from the total surface. Therefore, 4 faces are removed, leaving 18 - 4 = 14 exposed faces. The total surface area is thus 14 × a² = 14 × 7² = 14 × 49 = 686 cm², which corresponds to option c.

Explanation

The total surface area of the solid formed by surmounting a hemisphere on a cubical block is the sum of the total surface area of the cube and the curved surface area of the hemisphere, minus the area of the circular base of the hemisphere which is now covered.

The formula is: Total Surface Area = Total Surface Area of Cube - Base Area of Hemisphere + Curved Surface Area of Hemisphere. Since the base area is πr² and the curved surface area is 2πr², the combined formula simplifies to: Total Surface Area = 6a² + πr²

Substituting the given values a = 14 cm and r = 7 cm: Total Surface Area = 6(14)² + 22/7(7)² Total Surface Area = 6(196) + 22(7) Total Surface Area = 1176 + 154 Total Surface Area = 1330 cm²

Thus, the total surface area of the resulting solid is 1330 cm², which corresponds to option a.

Explanation

Volume of one coin = pi r^2 h. Radius r = 3.5 / 2 = 1.75 cm. Height h = 4 mm = 0.4 cm. Volume = (22/7) (1.75)^2 0.4 = (22/7) 3.0625 0.4 = 3.85 cm^3. Volume of cuboid = 21 11 7 = 1617 cm^3. Number of coins = 1617 / 3.85 = 420.

Explanation

The slant height l of a frustum of a cone can be calculated using the formula l = √(h² + (R - r)²), where h is the vertical height, R is the radius of the larger circular end, and r is the radius of the smaller circular end. Given the values R = 105 cm, r = 42 cm, and h = 16 cm, we first find the difference between the radii: R - r = 105 - 42 = 63 cm. Substituting these values into the formula, we get l = √(16² + 63²). Calculating the squares, we have 16² = 256 and 63² = 3969. Adding these results together gives l = √(256 + 3969) = √4225. The square root of 4225 is 65, as 65 × 65 = 4225. Therefore, the slant height of the tub is 65 cm, which corresponds to option b.

Explanation

Revolving triangle ABC about BC creates a cone with radius r = AB = 8cm and height h = BC = 6cm. Slant height l = sqrt(8^2 + 6^2) = sqrt(64 + 36) = 10cm. Total Surface Area = pi r (l + r) = (22/7) 8 (10 + 8) = (22/7) 8 18 = 3168 / 7 approx 452.57 cm^2. The closest option is 452 cm^2.

Explanation

The total volume of the solid rod is the sum of the volumes of the two cylinders. Using the formula for the volume of a cylinder V = πr²h, the volume of the first cylinder is V₁ = 22/7 × 7² × 20 = 22 × 7 × 20 = 3080 cm³. The volume of the second cylinder is V₂ = 22/7 × 3.5² × 10 = 22/7 × 12.25 × 10 = 385 cm³. The total volume of the rod is Vₜₒₜₐₗ = 3080 + 385 = 3465 cm³.

To find the mass, we use the given density. Since 1 m³ = 100 × 100 × 100 = 1,000,000 cm³, the weight of 1 cm³ of the rod is 10000 kg/1,000,000 cm³ = 0.01 kg/cm³. The total mass of the rod is therefore 3465 cm³ × 0.01 kg/cm³ = 34.65 kg, which matches option a.

Explanation

The total volume of the solid is the sum of the volume of the right circular cone and the volume of the hemisphere. Using the formula for the volume of a cone, V꜀ₒₙₑ = 1/3 π r² h, and substituting the given values r = x and h = 2x, we get V꜀ₒₙₑ = 1/3 π (x)² (2x) = 2/3 π x³. Next, using the formula for the volume of a hemisphere, Vₕₑₘᵢ = 2/3 π r³, and substituting r = x, we get Vₕₑₘᵢ = 2/3 π x³. The total volume of the solid is Vₜₒₜₐₗ = V꜀ₒₙₑ + Vₕₑₘᵢ = 2/3 π x³ + 2/3 π x³ = 4/3 π x³. Since the formula for the volume of a sphere of radius x is Vₛₚₕₑᵣₑ = 4/3 π x³, the volume of the solid is equal to that of a sphere of radius x, which is option (a).

Explanation

The total surface area of the combined solid is the sum of the curved surface area of the right circular cone and the curved surface area of the hemisphere, as the base of the cone and the flat face of the hemisphere are joined internally.

The formula for the curved surface area of a cone is πrl, where l is the slant height. First, we find the slant height l using the formula l = √(r² + h²). Substituting the given values r = x and h = 2x, we get l = √(x² + (2x)²) = √(x² + 4x²) = √(5x²) = x√5. Thus, the curved surface area of the cone is π · x · x√5 = πx²√5.

The formula for the curved surface area of a hemisphere is 2πr². Substituting r = x, we get 2πx².

The total surface area (TSA) of the solid is: TSA = πx²√5 + 2πx² = πx²(√5 + 2)

Using the approximate values π = 22/7 ≈ 3.1428 and √5 ≈ 2.236: TSA ≈ 3.1428 × (2.236 + 2) × x² TSA ≈ 3.1428 × 4.236 × x² ≈ 13.313x²

Rounding to one decimal place, we get 13.3x², which corresponds to option (c).

Explanation

To find the volume of water left in the cylinder, we must subtract the volume of the solid from the total volume of the cylinder. The total volume of the cylinder is given by the formula V꜀ᵧₗᵢₙꜣₑᵣ = πr²h. Substituting the given values, V꜀ᵧₗᵢₙꜣₑᵣ = π(x)²(3x) = 3πx³.

The solid consists of a cone and a hemisphere. The volume of the cone is V꜀ₒₙₑ = 1/3 πr²h = 1/3 π(x)²(2x) = 2/3 πx³. The volume of the hemisphere is Vₕₑₘᵢₛₚₕₑᵣₑ = 2/3 πr³ = 2/3 π(x)³ = 2/3 πx³. Therefore, the total volume of the solid is Vₛₒₗᵢꜣ = 2/3 πx³ + 2/3 πx³ = 4/3 πx³.

The volume of water left in the cylinder is the difference between the cylinder volume and the solid volume: Vₗₑfₜ = 3πx³ - 4/3 πx³ = (9πx³ - 4πx³)/3 = 5/3 πx³

Using the value π = 22/7, we calculate the numerical value: Vₗₑfₜ = 5/3 × 22/7 × x³ = 110/21 x³ Dividing 110 by 21 gives approximately 5.238095 x³, which rounds to 5.24 x³. Thus, option d is the correct answer.

Explanation

The area of a minor segment of a circle is calculated by subtracting the area of the triangle formed by the chord and the radii from the area of the circular sector. The required formula is: Area of Minor Segment = θ/360 × πr² - 1/2 r² sin θ

Given the radius r = 2.1 cm, the central angle θ = 120°, π = 22/7, and √3 = 1.732, we substitute these values into the formula.

First, compute the area of the sector: Area of sector = 120/360 × 22/7 × (2.1)² = 1/3 × 22/7 × 4.41 Since 4.41 ÷ 7 = 0.63, the calculation becomes 1/3 × 22 × 0.63 = 22 × 0.21 = 4.62 cm².

Next, compute the area of the triangle: Area of triangle = 1/2 × (2.1)² × sin(120°) Using sin(120°) = sin(60°) = √3/2, the area is 1/2 × 4.41 × 1.732/2 = (4.41 × 1.732)/4 ≈ 7.63812/4 ≈ 1.9095 cm².

Subtracting the triangle area from the sector area: Area of minor segment = 4.62 - 1.9095 = 2.7105 cm².

The approximate area is 2.71 cm², which corresponds to option (a).

Explanation

The area of the major segment of a circle is calculated as the sum of the area of the major sector and the area of the triangle formed by the chord and the radii. The formula for the area of the major sector is (360 - θ)/360 × πr² and the area of the triangle is 1/2 r² sinθ. Given r = 2.1 cm and θ = 120°, we substitute the values as follows:

Area of major sector = (360 - 120)/360 × 22/7 × (2.1)² = 240/360 × 22/7 × 4.41 = 2/3 × 13.86 = 9.24 cm².

Area of triangle = 1/2 × (2.1)² × sin(120°) = 1/2 × 4.41 × √3/2 = (4.41 × 1.732)/4 = 7.63812/4 = 1.90953 cm².

Total area of major segment = 9.24 + 1.90953 = 11.14953 cm².

Rounding to two decimal places, we get 11.15 cm², which matches option d.

Explanation

Let the radii of the circles centered at vertices A, B, and C be denoted as r₁, r₂, and r₃ respectively. Since the circles touch each other externally, the distance between the centers of any two touching circles is equal to the sum of their radii. This gives us the following system of linear equations based on the side lengths of triangle ABC: r₁ + r₂ = AB = 6 cm r₂ + r₃ = BC = 10 cm r₃ + r₁ = CA = 8 cm

To find the sum of the radii r₁ + r₂ + r₃, we add these three equations together: (r₁ + r₂) + (r₂ + r₃) + (r₃ + r₁) = 6 + 10 + 8 2(r₁ + r₂ + r₃) = 24

Dividing both sides by 2, we get: r₁ + r₂ + r₃ = 12 cm

Thus, the sum of the radii of the three circles is 12 cm, which corresponds to option (c). This approach is the fastest way to solve the problem as it avoids calculating the individual radii separately.

Explanation

First, we identify the nature of triangle ABC using its side lengths: AB = 6 cm, BC = 10 cm, and CA = 8 cm. We check if it is a right-angled triangle using the Pythagorean theorem: AB² + AC² = 6² + 8² = 36 + 64 = 100. Since BC² = 10² = 100, the equation AB² + AC² = BC² holds true. Thus, triangle ABC is a right-angled triangle with the right angle at vertex A and BC as the hypotenuse.

The area of a right-angled triangle can be calculated using its legs as the base and height. Using the formula Area = 1/2 × base × height, we have Area = 1/2 × 6 × 8 = 24 square cm.

The area can also be expressed using the hypotenuse BC as the base and the altitude from vertex A to BC, denoted as h, as the height. Therefore, Area = 1/2 × BC × h. Substituting the known values, we get 24 = 1/2 × 10 × h, which simplifies to 24 = 5h. Solving for h gives h = 24/5 = 4.8 cm. This corresponds to option d.

Explanation

Since the circles touch each other externally, the sum of the radii of two circles equals the length of the side of the triangle connecting their centers. Let the radii of the circles at A, B, and C be rₐ, rᵦ, and r꜀ respectively. rₐ + rᵦ = 6 rᵦ + r꜀ = 10 r꜀ + rₐ = 8 Adding these equations gives 2(rₐ + rᵦ + r꜀) = 24, so rₐ + rᵦ + r꜀ = 12. Subtracting the equations, we find r꜀ = 12 - 6 = 6 cm, rₐ = 12 - 10 = 2 cm, and rᵦ = 12 - 8 = 4 cm. Since 6² + 8² = 10², the triangle is right-angled at A, meaning ∠A = 90°. The area of sector P at vertex A is P = ∠A/360 · πrₐ² = 90/360 · π(2²) = 1/4 · 4π = π cm². Thus, statement 1 is correct. For the other sectors, Q = ∠B/360 · πrᵦ² = ∠B/360 · 16π and R = ∠C/360 · πr꜀² = ∠C/360 · 36π. Evaluating the expression 9Q + 4R: 9Q + 4R = 9(∠B/360 · 16π) + 4(∠C/360 · 36π) = 144π∠B/360 + 144π∠C/360 = 144π/360 (∠B + ∠C). Since ∠A = 90°, ∠B + ∠C = 90°. 9Q + 4R = 144π/360 · 90 = 144π/4 = 36π cm². Thus, statement 2 is also correct. Both statements 1 and 2 are correct, making option c the correct choice.

Explanation

This problem can be solved directly using Stewart's Theorem, which relates the lengths of the sides of a triangle to the length of a cevian. For any triangle ABC with a cevian AD that divides the opposite side BC into segments BD and CD, the formula is AB² · CD + AC² · BD = BC(AD² + BD · CD).

Given that AB = AC, let the length of these sides be s. Let BD = m and CD = n. The total length of the base BC is then m + n. Substituting these values into the Stewart's Theorem formula, we get s² · n + s² · m = (m + n)(AD² + mn). Factoring out s² on the left side gives s²(m + n) = (m + n)(AD² + mn). Since m + n is the length of side BC and cannot be zero, we divide both sides by (m + n) to obtain s² = AD² + mn. Rearranging this equation yields s² - AD² = mn. Substituting the original side lengths back into the equation, we get AC² - AD² = BD × CD. This matches option b.

Explanation

To find the length of the side AB in triangle ABC, we use Stewart's Theorem, which relates the lengths of the sides of a triangle to the length of a cevian (a line segment from a vertex to the opposite side). The formula is given by AB² ⋅ CD + AC² ⋅ BD = BC(AD² + BD ⋅ CD).

Given the parameters of the triangle: AB = AC = x (since it is an isosceles triangle) BD = 4 cm CD = 6 cm BC = BD + CD = 4 + 6 = 10 cm AD = 5 cm

Substituting these values into the Stewart's Theorem formula: x² ⋅ 6 + x² ⋅ 4 = 10(5² + 4 ⋅ 6) 6x² + 4x² = 10(25 + 24) 10x² = 10(49) Dividing both sides by 10: x² = 49 x = √49 = 7 cm

Thus, the length of AB is 7 cm, which corresponds to option (a).

Explanation

Given the side lengths of triangle ABC as AB = 1.6 cm, BC = 6.3 cm, and CA = 6.5 cm, we are asked to find the value of AB² + 4BQ². Since Q is the mid-point of side BC, the length of BQ is exactly half of BC, which is BQ = BC/2.

Substituting this relation into the required expression, we have 4BQ² = 4(BC/2)² = 4(BC²/4) = BC². Therefore, the expression AB² + 4BQ² simplifies directly to AB² + BC².

Using the given values: AB = 1.6 BC = 6.3

We calculate the squares as follows: AB² = (1.6)² = 2.56 BC² = (6.3)² = 39.69

Adding these two values together: AB² + BC² = 2.56 + 39.69 = 42.25

Thus, the value of AB² + 4BQ² is 42.25 cm², which corresponds to option (b).

Explanation

To find the value of AQ² + CP², we use Apollonius theorem, which relates the length of a median of a triangle to the lengths of its sides. For a triangle ABC with sides a, b, c, the length of the median mₐ to side a is given by the formula:

mₐ² = (2b² + 2c² - a²)/4

In this problem, AQ is the median to side BC and CP is the median to side AB. Applying the formula:

AQ² = (2AB² + 2AC² - BC²)/4 CP² = (2BC² + 2AC² - AB²)/4

Summing these two equations gives:

AQ² + CP² = (2AB² + 2AC² - BC² + 2BC² + 2AC² - AB²)/4 AQ² + CP² = (AB² + BC² + 4AC²)/4

Given the options, the triangle ABC is implied to be a right-angled triangle with the right angle at B. According to the Pythagorean theorem, AB² + BC² = AC². Substituting this into the sum:

AQ² + CP² = (AC² + 4AC²)/4 = 5AC²/4 = 1.25 AC²

The option C, written as 1 - 25AC², is a typographical representation of 1.25 AC². Thus, the result is 1.25 AC².

Explanation

Using the median formula m^2 = (2a^2 + 2b^2 - c^2) / 4. For CP (median to side AB=1.6): CP^2 = (26.3^2 + 26.5^2 - 1.6^2) / 4 = (79.38 + 84.5 - 2.56) / 4 = 161.32 / 4 = 40.33. For AQ (median to side BC=6.3): AQ^2 = (21.6^2 + 26.5^2 - 6.3^2) / 4 = (5.12 + 84.5 - 39.69) / 4 = 49.93 / 4 = 12.4825. Then 4(CP^2 - AQ^2) = 4(40.33 - 12.4825) = 4(27.8475) = 111.39.

Explanation

In triangle OPA, the segments OA and OP are both radii of the circle, which means OA = OP. This makes triangle OPA an isosceles triangle. It is given that the radius OP is perpendicular to the diameter AB, which implies that angle POA = 90°. In any isosceles triangle, the angles opposite the equal sides are equal, so angle OAP = angle OPA. Using the angle sum property of a triangle, we have angle OAP + angle OPA + angle POA = 180°. Substituting the known values, we get 2 × angle OAP + 90° = 180°, which simplifies to 2 × angle OAP = 90°. Therefore, angle OAP = 45°. Since the point O lies on the line segment AB, angle BAP is the same as angle OAP. Thus, angle BAP = 45°, which corresponds to option c.

Explanation

The angle ∠AQP is an inscribed angle that subtends the arc AP. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of its corresponding central angle that subtends the same arc. In this case, the central angle subtending arc AP is ∠AOP. Since it is given that the radius OP is perpendicular to the diameter AB, the measure of the central angle ∠AOP is exactly 90°. Using the formula ∠AQP = 1/2 ∠AOP, we substitute the value to get ∠AQP = 1/2 × 90° = 45°. This result remains constant regardless of where the point Q is located on the arc PB, as all inscribed angles subtending the same arc are equal. Thus, the measure of angle ∠AQP is 45°, which corresponds to option c.

Explanation

Let the five consecutive integers be n, n+1, n+2, n+3, n+4. The product of these integers is P = n(n+1)(n+2)(n+3)(n+4). The condition given is that P must be equal to one of the integers in the sequence, meaning P ∈ {n, n+1, n+2, n+3, n+4}.

If the sequence contains the integer 0, the product P will be 0. Since 0 is one of the integers in the sequence, the condition P = 0 is automatically satisfied. There are 5 such sequences of five consecutive integers that include 0: {0, 1, 2, 3, 4}, {-1, 0, 1, 2, 3}, {-2, -1, 0, 1, 2}, {-3, -2, -1, 0, 1}, and {-4, -3, -2, -1, 0}.

If the sequence does not contain 0, the absolute value of the product of any five consecutive integers is at least 1 × 2 × 3 × 4 × 5 = 120. However, the maximum absolute value of any single integer in such a sequence is far less than 120. For example, in the sequence {1, 2, 3, 4, 5}, the product is 120 while the maximum element is 5. Thus, no sequence without 0 can satisfy the condition. Therefore, there are exactly 5 possible sequences, which corresponds to option d.

Explanation

Let the five consecutive integers be n, n+1, n+2, n+3, n+4. The condition requires their product Pr = n(n+1)(n+2)(n+3)(n+4) to be equal to one of the integers k in the sequence. If k ≠ 0, the product of the remaining four integers must be 1. However, it is impossible for four distinct integers to have a product of 1 because the only integers with an absolute value of 1 are 1 and -1, and we cannot have four distinct integers from that set. Consequently, the only way the product can equal one of the integers in the sequence is if the product is 0, which implies that 0 must be one of the five consecutive integers.

The possible sequences of five consecutive integers that contain 0 are: {-4, -3, -2, -1, 0}, {-3, -2, -1, 0, 1}, {-2, -1, 0, 1, 2}, {-1, 0, 1, 2, 3}, and {0, 1, 2, 3, 4}.

By examining these sequences, the smallest integer occurring is P = -4 and the largest integer occurring is Q = 4. The arithmetic mean of P and Q is calculated using the formula: (P + Q)/2 = (-4 + 4)/2 = 0/2 = 0. This matches option (a).