CDS II 2015 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

For x + 100/x > 50, the inequality x^2 - 50x + 100 > 0 must hold. The roots of x^2 - 50x + 100 = 0 are x = (50 +/- sqrt(2100))/2 = 25 +/- 5*sqrt(21). Since sqrt(21) is approximately 4.58, the roots are approximately 2.1 and 47.9. The inequality holds for x < 2.1 or x > 47.9. For natural numbers x <= 100, x can be 1, 2 (2 values) or 48, 49, ..., 100 (100 - 48 + 1 = 53 values). Total values = 2 + 53 = 55.

Explanation

The number N satisfies N = LCM(6, 8, 12)k - 1 = 24k - 1. We need the least N that is a multiple of 13. 24k - 1 = 0 (mod 13) implies 11k = 1 (mod 13), which simplifies to -2k = 1 (mod 13). This gives k = 6. Thus, N = 24(6) - 1 = 143. 143 is divisible by 13 (143 = 11 * 13). The sum of the digits of 143 is 1 + 4 + 3 = 8.

Explanation

When a trader sells a mixture at the cost price of the pure ingredient and makes a profit, the profit percentage is equal to the ratio of the added free ingredient (water) to the original pure ingredient (milk). Using the formula Profit % = (Quantity of Water)/(Quantity of Milk) × 100, we substitute the given profit of 20% to get 20 = (Water)/(Milk) × 100. This simplifies to Water/Milk = 20/100 = 1/5. This means for every 5 units of milk, 1 unit of water is added. The total quantity of the mixture is therefore 5 + 1 = 6 units. To find the percentage of milk in the mixture, we use the formula (Quantity of Milk)/(Total Mixture) × 100 = 5/6 × 100 = 500/6% = 250/3%. Thus, option d is correct.

Explanation

Let y = sqrt(4 + sqrt(4 - sqrt(4 + ...))). Then y = sqrt(4 + sqrt(4 - y)). Squaring both sides gives y^2 - 4 = sqrt(4 - y). Squaring again gives (y^2 - 4)^2 = 4 - y, or y^4 - 8y^2 + y + 12 = 0. Testing the option (sqrt(13)+1)/2: y^2 = (13+1+2sqrt(13))/4 = (7+sqrt(13))/2. Then y^2 - 4 = (sqrt(13)-1)/2. Also, 4 - y = (8 - sqrt(13) - 1)/2 = (7-sqrt(13))/2. Since ((sqrt(13)-1)/2)^2 = (13+1-2sqrt(13))/4 = (7-sqrt(13))/2, the equation holds.

Explanation

The expression is (n^3 - n)(n - 2) = n(n - 1)(n + 1)(n - 2) = (n - 2)(n - 1)n(n + 1). This is the product of four consecutive integers. The product of k consecutive integers is always divisible by k!. For k = 4, 4! = 24. For n = 3, the value is 1 2 3 * 4 = 24. Thus, the largest natural number that divides all such expressions is 24.

Explanation

To find the average of (m + n) numbers, we first calculate the total sum of all numbers involved. The sum of m numbers is given by the product of the number of terms and their average, which is m × n⁴. Similarly, the sum of n numbers is n × m⁴. Therefore, the total sum of (m + n) numbers is mn⁴ + nm⁴. This expression can be simplified by factoring out the common term mn, resulting in mn(n³ + m³).

The average of (m + n) numbers is the total sum divided by the total number of terms: Average = (mn(m³ + n³))/(m + n)

Applying the algebraic identity m³ + n³ = (m + n)(m² - mn + n²), the formula becomes: Average = (mn(m + n)(m² - mn + n²))/(m + n)

By canceling the common term (m + n) from the numerator and the denominator, we obtain: Average = mn(m² - mn + n²)

Rearranging the terms within the parentheses, we get mn(m² + n² - mn), which corresponds to option d.

Explanation

The unit digit of 4^n is 6 if n is even and 4 if n is odd. For (234)^100, the unit digit is 6. For (234)^101, the unit digit is 4. The sum of the unit digits is 6 + 4 = 10, so the resulting unit digit is 0.

Explanation

Given x = √3 + √2, we first find the value of 1/x by rationalizing the denominator: 1/x = 1/(√3 + √2) = (√3 - √2) / ((√3 + √2)(√3 - √2)) = (√3 - √2) / (3 - 2) = √3 - √2. Now, calculate the sum x + 1/x: x + 1/x = (√3 + √2) + (√3 - √2) = 2√3. The required expression is x³ + 1/x³ + x + 1/x. Using the algebraic identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x), we can substitute the value of x + 1/x: x³ + 1/x³ = (2√3)³ - 3(2√3) = 8(3√3) - 6√3 = 24√3 - 6√3 = 18√3. Adding the remaining terms x + 1/x to this result gives: 18√3 + 2√3 = 20√3. Thus, the final value is 20√3, which corresponds to option b.

Explanation

This is a telescoping series problem where each term is of the form 1/n(n+d). The general formula for the sum of such a series is S = 1/d (1/a - 1/l), where d is the common difference between the factors in the denominator, a is the first factor of the first term, and l is the last factor of the last term. In the given series 1/(1 × 4) + 1/(4 × 7) + … + 1/(16 × 19), the common difference is d = 4 - 1 = 3. The first factor a = 1 and the last factor l = 19. Substituting these values into the formula, we get S = 1/3 (1/1 - 1/19). Simplifying the expression inside the bracket, we have (19 - 1)/19 = 18/19. Therefore, the final sum is S = 1/3 × 18/19 = 6/19, which corresponds to option B.

Explanation

First, convert the speeds of both trains into a uniform unit of km/h. The speed of the first train is 1.5 km/min, which is 1.5 × 60 = 90 km/h. The speed of the second train is given as 60 km/h. Since the trains are moving in the same direction, the relative speed is the difference between their speeds: Relative Speed = 90 km/h - 60 km/h = 30 km/h.

To find the distance in meters, convert the relative speed to m/s by multiplying by 5/18: Relative Speed = 30 × 5/18 = 25/3 m/s.

The man in the faster train acts as a point object. When a point object crosses another train, the total distance covered is equal to the length of the train being crossed. Using the distance formula Distance = Speed × Time, we have: Length of slower train = 25/3 m/s × 27 s = 25 × 9 = 225 m. Thus, the correct option is (a).

Explanation

Let C be the total capacity of the tin in bottles. The condition is (4/5)C - 6 + 4 = (3/4)C. This simplifies to (4/5 - 3/4)C = 2, which is (1/20)C = 2. Therefore, C = 40.

Explanation

Use the general division formula N = dq + r, where N is the dividend, d is the divisor, q is the quotient, and r is the remainder. From the given conditions, we have N = 7q₁ + 3 and q₁ = 11q₂ + 6. By substituting the expression for q₁ into the formula for N, we get N = 7(11q₂ + 6) + 3, which simplifies to N = 77q₂ + 42 + 3 = 77q₂ + 45. To find the remainder m when N is divided by 11, we rewrite N as N = 11(7q₂) + 44 + 1, which is N = 11(7q₂ + 4) + 1. From this, the remainder m = 1 and the resulting quotient is q₃ = 7q₂ + 4. To find the remainder n when q₃ is divided by 7, we observe that q₃ = 7(q₂) + 4. This clearly shows that the remainder n = 4. Thus, the values of m and n are 1 and 4 respectively, which corresponds to option a.

Explanation

y = sqrt((x-1)^2) / (x-1) = |x-1| / (x-1). If x > 1, y = (x-1)/(x-1) = 1. If x < 1, y = -(x-1)/(x-1) = -1. If x = 1, the expression is undefined because the denominator is zero. Thus, statements 1 and 2 are correct, but 3 is incorrect.

Explanation

If (x + 1) is a factor, then substituting x = -1 must yield 0. For Ax^2 + Bx + C, A(-1)^2 + B(-1) + C = A - B + C = 0. For Bx^2 + Ax + C, B(-1)^2 + A(-1) + C = B - A + C = 0. Subtracting these two equations gives (A - B + C) - (B - A + C) = 0, so 2A - 2B = 0, which means A = B. However, the problem states A is not equal to B. This suggests a flaw in the question, but adding the equations gives 2C = 0, so C = 0. In most standard test keys for this specific problem, C = 0 is the intended answer.

Explanation

For a number to be divisible by 225, it must be divisible by 25 and 9. Divisibility by 25 requires the last two digits (7q) to be 75, so q = 5. Divisibility by 9 requires the sum of digits (8+7+6+p+3+7+5) = 36 + p to be a multiple of 9. This means p can be 0 or 9. Both (0, 5) and (9, 5) are valid. Given the options, (9, 5) is often selected as the intended answer.

Explanation

For a cubic equation of the form Ax³ + Bx² + Cx + D = 0 with roots a, b, and c, the relations between the coefficients and the roots (Vieta's formulas) are as follows: the sum of the roots a + b + c = -B/A, the sum of the roots taken two at a time ab + bc + ca = C/A, and the product of the roots abc = -D/A.

Comparing the given equation x³ - 3x² + 2x + 1 = 0 with the standard form, we have A = 1, B = -3, C = 2, and D = 1.

The value we need to find is 1/a + 1/b + 1/c. By taking the common denominator, this expression can be rewritten as: 1/a + 1/b + 1/c = (bc + ac + ab)/abc

Substituting the values from Vieta's formulas: ab + bc + ca = C/A = 2/1 = 2 abc = -D/A = -1/1 = -1

Therefore, the value is 2/-1 = -2. This corresponds to option c.

Explanation

To find the time required to fill the tank, we use the formula for net work rate: Net Rate = Rate of Tap A + Rate of Tap B - Rate of Leakage. The rate is the reciprocal of the time taken to complete the task.

The rate of Tap A is 1/10 tank per minute, the rate of Tap B is 1/20 tank per minute, and the rate of the leakage is 1/40 tank per minute. Substituting these values into the formula, we get:

Net Rate = 1/10 + 1/20 - 1/40

To solve this, find a common denominator, which is 40:

Net Rate = 4/40 + 2/40 - 1/40 = (4 + 2 - 1)/40 = 5/40 = 1/8 tank per minute.

The total time required to fill the tank is the reciprocal of the net rate:

Time = 1/(1/8) = 8 minutes.

Thus, both taps will take 8 minutes to fill the tank despite the leakage, making option d the correct choice.

Explanation

Compare by raising to the 12th power: (sqrt 2)^12 = 2^6 = 64. (4th root 6)^12 = 6^3 = 216. (3rd root 4)^12 = 4^4 = 256. Since 64 < 216 < 256, the correct order is sqrt 2 < 4th root 6 < 3rd root 4.

Explanation

To find the single equivalent discount for a series of successive discounts, the most efficient method is to use the multiplier formula. The formula for the remaining value multiplier after successive discounts d₁, d₂, ..., dₙ is (1 - d₁)(1 - d₂)...(1 - dₙ). In this problem, the discounts are d₁ = 10%, d₂ = 20%, and d₃ = 40%. Converting these percentages to decimals, we have 0.10, 0.20, and 0.40.

The combined multiplier is calculated as (1 - 0.10)(1 - 0.20)(1 - 0.40) = 0.9 × 0.8 × 0.6. Multiplying these values gives 0.9 × 0.8 = 0.72, and then 0.72 × 0.6 = 0.432. This means the final price is 43.2% of the original price.

The single equivalent discount is the total reduction from the original 100%. Using the formula SED = 1 - Multiplier, we get SED = 1 - 0.432 = 0.568. Converting this back to a percentage, we get 0.568 × 100 = 56.8%. Therefore, the single discount equivalent to these three successive discounts is 56.8%, which matches option C.

Explanation

To find the range of the variable k, we must determine the minimum and maximum possible values it can take based on the given constraints of x, y, and z. The expression is k = x - y + 2z.

To calculate the minimum value of k, we substitute the minimum possible value for terms with positive coefficients and the maximum possible value for terms with negative coefficients. kₘᵢₙ = xₘᵢₙ - yₘₐₓ + 2zₘᵢₙ Substituting the given limits: kₘᵢₙ = (-2) - (2) + 2(3) kₘᵢₙ = -4 + 6 = 2

To calculate the maximum value of k, we substitute the maximum possible value for terms with positive coefficients and the minimum possible value for terms with negative coefficients. kₘₐₓ = xₘₐₓ - yₘᵢₙ + 2zₘₐₓ Substituting the given limits: kₘₐₓ = (1) - (-1) + 2(6) kₘₐₓ = 1 + 1 + 12 = 14

Combining these results, the range of k is 2 ≤ k ≤ 14, which corresponds to option c.

Explanation

The given equation 21x + 48y = 5 is a linear Diophantine equation of the form ax + by = c. According to the fundamental theorem of linear Diophantine equations, the equation ax + by = c has an integer solution if and only if the greatest common divisor (gcd) of a and b divides c.

In this problem, a = 21 and b = 48. To find gcd(21, 48), we look for the largest integer that divides both numbers: 21 = 3 × 7 48 = 3 × 16 Thus, gcd(21, 48) = 3.

Now, we check if gcd(21, 48) divides the constant term c = 5. We observe that 3 does not divide 5 because 5 ÷ 3 leaves a remainder of 2. Since the condition gcd(a, b) | c is not satisfied, there are no integers x and y that can satisfy the equation. Consequently, the number of pairs (x, y) is zero, making option a) the correct choice.

Explanation

For statement 1, the difference (y - x) is an even integer if both x and y have the same parity. Since x is prime, it can be 2 (even) or any odd prime. Since y is composite, it can be 4, 6, 8 (even) or 9, 15, 21 (odd). If we take x = 2 and y = 4, then y - x = 4 - 2 = 2, which is even. Alternatively, if we take x = 3 and y = 9, then y - x = 9 - 3 = 6, which is even. Thus, statement 1 is correct.

For statement 2, the product xy is an even integer if at least one of the variables is even. If x = 2 (the only even prime), xy = 2y, which is always even for any positive integer y. Similarly, if y is any even composite number like 4, then xy = 4x, which is always even. Thus, statement 2 is correct.

For statement 3, 0.5(x + y) can be written as (x + y)/2. For this to be an even integer, x + y must be a multiple of 4, such that (x + y)/2 = 2k, implying x + y = 4k. We can choose x = 3 (prime) and y = 9 (composite), then x + y = 12, and 0.5(12) = 6, which is an even integer. Another example is x = 2 and y = 6, giving 0.5(2 + 6) = 4, which is also even. Thus, statement 3 is correct. Since all three statements are correct, option d is the right choice.

Explanation

We have 3x + 2y = 5k + 2 and 2x + 3y = 5m + 3. Subtracting the second equation from the first: (3x + 2y) - (2x + 3y) = (5k + 2) - (5m + 3), which simplifies to x - y = 5(k - m) - 1. A remainder of -1 modulo 5 is equivalent to a remainder of 4.

Explanation

Let L be the length of the race. The ratio of speeds of two participants in a race is equal to the ratio of the distances they cover in the same time. Since A beats B by 30 m, when A covers L, B covers L - 30. Similarly, since B beats C by 20 m, when B covers L, C covers L - 20. Finally, since A beats C by 48 m, when A covers L, C covers L - 48.

The speed ratios are expressed as: Vₐ / Vᵦ = L / (L - 30) Vᵦ / V_C = L / (L - 20) Vₐ / V_C = L / (L - 48)

Using the relation Vₐ / V_C = (Vₐ / Vᵦ) × (Vᵦ / V_C), we substitute the values: L / (L - 48) = [L / (L - 30)] × [L / (L - 20)] Dividing both sides by L: 1 / (L - 48) = L / [(L - 30)(L - 20)] (L - 30)(L - 20) = L(L - 48) L² - 50L + 600 = L² - 48L 2L = 600 ⇒ L = 300 m. Thus, statement 1 is correct.

To find the speed ratios, substitute L = 300: Vₐ / Vᵦ = 300 / 270 = 10 / 9 Vᵦ / V_C = 300 / 280 = 15 / 14 To combine these, make the Vᵦ term common: Vₐ : Vᵦ = (10 × 5) : (9 × 5) = 50 : 45 Vᵦ : V_C = (15 × 3) : (14 × 3) = 45 : 42 Vₐ : Vᵦ : V_C = 50 : 45 : 42. Thus, statement 2 is also correct. Option c is the correct choice.

Explanation

For the roots of x^2 - 4x - log10(N) = 0 to be real, the discriminant D must be >= 0. D = (-4)^2 - 4(1)(-log10(N)) = 16 + 4log10(N) >= 0. This implies log10(N) >= -4, so N >= 10^(-4) = 1/10000. The minimum value of N is 1/10000.

Explanation

The speed of the boat in still water is u = 15 km/h and the distance traveled one way is d = 30 km. Let the speed of the stream be v km/h. The speed downstream is u+v and the speed upstream is u-v. The total time taken for the round trip is 4 hours 30 minutes, which is 4.5 hours. Using the total time formula T = d/(u+v) + d/(u-v), we substitute the values: 4.5 = 30/(15+v) + 30/(15-v). Simplifying the right side by taking a common denominator: 4.5 = 30 ((15-v+15+v)/((15+v)(15-v))) = (30 × 30)/(225-v²) = 900/(225-v²). Solving for v², we get 225-v² = 900/4.5 = 200, which simplifies to v² = 225-200 = 25. Taking the square root, v = 5 km/h. This corresponds to option (b). For a faster competitive exam approach, substituting v=5 from the options into the time equation yields 30/20 + 30/10 = 1.5 + 3 = 4.5 hours, which perfectly matches the given time.

Explanation

This problem can be solved directly using the chain rule formula for work and time: (M₁ D₁ H₁)/W₁ = (M₂ D₂ H₂)/W₂, where M represents the number of men, D the number of days, H the hours worked per day, and W the units of work completed.

Given the first scenario: M₁ = 4, D₁ = 4, H₁ = 4, and W₁ = 4. Given the second scenario: M₂ = 2, D₂ = 2, H₂ = 2, and we need to find W₂.

Substituting these values into the formula: (4 × 4 × 4)/4 = (2 × 2 × 2)/W₂

Simplifying the left side: 16 = 8/W₂

Rearranging to solve for W₂: W₂ = 8/16 W₂ = 1/2

Therefore, 2 men working 2 hours per day for 2 days will complete 1/2 units of work, which corresponds to option (c).

Explanation

Let the original speed of the car be v km/hr. The relationship between distance D, original speed v, change in speed Δv, and the resulting change in time Δt can be expressed by the formula D = v(v + Δv)/Δv × Δt. Substituting the given values D = 300 km, Δv = 15 km/hr, and Δt = 1 hour, we obtain the equation 300 = v(v + 15)/15 × 1. Multiplying both sides by 15 gives 4500 = v² + 15v, which rearranges to the quadratic equation v² + 15v - 4500 = 0. Factoring this equation, we get v² + 75v - 60v - 4500 = 0, which simplifies to (v + 75)(v - 60) = 0. Since speed cannot be negative, the original speed v must be 60 km/hr. Verifying with option (c), the original time taken is 300/60 = 5 hours and the new time taken at 75 km/hr is 300/75 = 4 hours, which perfectly matches the given 1 hour difference.

Explanation

This problem is based on the concept of inverse proportion, where the number of persons is inversely proportional to the time taken to complete a fixed amount of work. The standard formula used for such problems is M₁D₁ = M₂D₂, where M represents the number of persons and D represents the number of days.

In this case, the initial parameters are M₁ = m and D₁ = d. The new number of persons is M₂ = m + 2, and we need to find the new number of days, D₂.

Substituting the values into the formula: m × d = (m + 2) × D₂

To solve for D₂, we divide both sides by (m + 2): D₂ = md/(m + 2)

This represents the total man-days required to paint the house divided by the new total number of workers. Thus, the time taken by (m + 2) persons is md/(m + 2), which corresponds to option d.

Explanation

To find the combined ratio a:b:c, we equalize the common term b in both given ratios. Given a:b = 3:5 and b:c = 7:8, the LCM of the b values (5 and 7) is 35. Multiplying the first ratio by 7 and the second ratio by 5, we get a:b = 21:35 and b:c = 35:40. Thus, the combined ratio is a:b:c = 21:35:40.

Now, we substitute these values into the required expression 2a:3b:7c. Let a = 21k, b = 35k, c = 40k. The required ratio becomes 2(21k) : 3(35k) : 7(40k), which simplifies to 42k : 105k : 280k. To simplify the ratio 42:105:280, we divide each term by their highest common factor, which is 7. Dividing by 7, we get 42/7 : 105/7 : 280/7, which results in 6:15:40. This corresponds to option c.

Explanation

To find the time taken for two trains to clear each other when running in opposite directions, we use the concept of relative speed and total distance. The total distance to be covered is the sum of the lengths of the two trains, as they must pass each other completely.

Total distance D = 121 m + 99 m = 220 m.

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds. Relative speed Vᵣₑₗ = 40 km/h + 32 km/h = 72 km/h.

To convert the relative speed from km/h to m/s, multiply by 5/18: Vᵣₑₗ = 72 × 5/18 = 4 × 5 = 20 m/s.

The formula for time is Time = Total Distance / Relative Speed. Substituting the values: t = 220 m / 20 m/s = 11 s.

Thus, the trains will clear each other in 11 seconds, which corresponds to option b.

Explanation

Since the time taken by all three athletes to reach the point where the winner finishes is the same, the distance covered by them is directly proportional to their speeds. Given the ratio of speeds as 16:15:11, the ratio of distances covered in the same time is also 16:15:11.

The total race distance is 4 km, which is 4000 m. This distance represents the 16 units covered by the winner. Therefore, the value of one unit of distance is calculated as: 1 unit = 4000/16 = 250 m

The distance covered by the athlete in the second position (15 units) is 15 × 250 = 3750 m. The distance covered by the athlete in the third position (11 units) is 11 × 250 = 2750 m.

The distance between the second and third athletes is the difference between their covered distances: Distance = 3750 m - 2750 m = 1000 m

Alternatively, the difference in the ratio units between the second and third athlete is 15 - 11 = 4 units. Using the unit value, the distance is 4 × 250 = 1000 m, which matches option (a).

Explanation

The given series is periodic with a repeating cycle of 6 terms. Let the terms be T₁, T₂, T₃, T₄, T₅, T₆, …. The first six terms are 1/4, 1/5, -1/6, -1/4, -1/5, 1/6. Summing these six terms, we get: S₆ = 1/4 + 1/5 - 1/6 - 1/4 - 1/5 + 1/6 = 0 Since the sum of every block of 6 terms is 0, we can find the sum of the first 47 terms by dividing 47 by the period length 6. 47 = 6 × 7 + 5 This indicates there are 7 complete cycles and 5 remaining terms. The total sum is: S₄₇ = 7 × S₆ + (T₁ + T₂ + T₃ + T₄ + T₅) Substituting the values: S₄₇ = 7 × 0 + (1/4 + 1/5 - 1/6 - 1/4 - 1/5) S₄₇ = 0 + (1/4 - 1/4) + (1/5 - 1/5) - 1/6 S₄₇ = -1/6 Thus, the sum of the first 47 terms is -1/6, which corresponds to option b.

Explanation

According to the Factor Theorem, a polynomial P(x) has a factor (x - a) if and only if P(a) = 0. To check if (x - 2) is a factor of the expression x⁴ - 6x³ + 12x² - 24x + 32 mentioned in option (c), we substitute x = 2 into the polynomial. Let f(x) = x⁴ - 6x³ + 12x² - 24x + 32. The calculation is as follows: f(2) = 2⁴ - 6(2³) + 12(2²) - 24(2) + 32. This simplifies to 16 - 6(8) + 12(4) - 48 + 32, which equals 16 - 48 + 48 - 48 + 32. Further simplification leads to 16 - 48 + 32 = 0. Since f(2) = 0, the Factor Theorem confirms that (x - 2) is a factor of the polynomial. Therefore, option (c) is correct. For option (d), substituting x = 2 into x⁴ + 6x³ - 12x² + 24x - 32 gives 16 + 48 - 48 + 48 - 32 = 32, which is not zero, meaning (x - 2) is not a factor there. Similarly, substituting x = -2 for options (a) and (b) does not result in zero.

Explanation

Let the number be x. The equation is 0.20x + 20 = x. This simplifies to 0.80x = 20, so x = 20 / 0.80 = 25.

Explanation

For statement 1, we use the trigonometric identities 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. The left hand side (LHS) becomes sec²θ/csc²θ = (1/cos²θ)/(1/sin²θ) = sin²θ/cos²θ = tan²θ. For the right hand side (RHS), we substitute cotθ = 1/tanθ to get ((1 - tanθ)/(1 - 1/tanθ))² = ((1 - tanθ)/((tanθ - 1)/tanθ))² = ((1 - tanθ)tanθ/(-(1 - tanθ)))² = (-tanθ)² = tan²θ. Since LHS = RHS, statement 1 is correct for the given range.

For statement 2, the relation cotθ = 1/tanθ is a fundamental reciprocal identity in trigonometry. It is valid for all values of θ except where tanθ is zero or undefined, such as θ = nπ or θ = (2n+1)π/2. It is not restricted to θ = 45° only. Therefore, statement 2 is incorrect. Thus, only option a is the correct choice.

Explanation

Given the values x = a cos θ and y = b cot θ, we first find the terms ax⁻¹ and by⁻¹. By substituting the value of x, we have ax⁻¹ = a(1 / (a cos θ)) = 1 / cos θ = sec θ. Similarly, substituting the value of y gives by⁻¹ = b(1 / (b cot θ)) = 1 / cot θ = tan θ.

The required expression (ax⁻¹ - by⁻¹)(ax⁻¹ + by⁻¹) follows the algebraic identity (A - B)(A + B) = A² - B². Substituting the derived trigonometric terms into this identity, the expression becomes sec² θ - tan² θ. According to the standard trigonometric identity sec² θ = 1 + tan² θ, it follows that sec² θ - tan² θ = 1. Therefore, the final result is 1, which corresponds to option b.

Explanation

Multiply numerator and denominator by (1 + sin theta): cos(theta)(1 + sin theta) / (1 - sin^2 theta) = cos(theta)(1 + sin theta) / cos^2 theta = (1 + sin theta) / cos theta.

Explanation

If x = 15: tan(15+40) = tan 55, tan(15+20) = tan 35, tan(315) = tan 45, tan(70-15) = tan 55, tan(50-15) = tan 35. The product is tan 55 tan 35 tan 45 tan 55 tan 35 = (tan 55 tan 35)^2 tan 45 = 1^2 1 = 1. Thus, x = 15 is correct.

Explanation

Given the equation sin θ cos θ = 2 cos³ θ - 1.5 cos θ, we first express 1.5 as 3/2. Since θ is an acute angle, cos θ ≠ 0, so we divide the entire equation by cos θ to simplify it to sin θ = 2 cos² θ - 3/2. To solve for sin θ, we use the fundamental trigonometric identity cos² θ = 1 - sin² θ. Substituting this into the equation gives sin θ = 2(1 - sin² θ) - 3/2. Simplifying the right side, we get sin θ = 2 - 2 sin² θ - 3/2, which reduces to sin θ = 1/2 - 2 sin² θ. Rearranging the terms into a standard quadratic equation form, we obtain 2 sin² θ + sin θ - 1/2 = 0. Multiplying the entire equation by 2 to eliminate the fraction results in 4 sin² θ + 2 sin θ - 1 = 0. Using the quadratic formula sin θ = (-b ± √(b² - 4ac))/2a with a=4, b=2, c=-1, we calculate sin θ = (-2 ± √(4 - 4(4)(-1)))/8 = (-2 ± √20)/8 = (-2 ± 2√5)/8 = (-1 ± √5)/4. Since θ is an acute angle, sin θ must be positive, which leads to sin θ = (√5 - 1) / 4. This corresponds to option a.

Explanation

In the first quadrant, where the angle θ is between 0° and 90°, the relationship between the sine and cosine functions is determined by the critical angle of 45°. At θ = 45°, we have sin 45° = cos 45° = 1/√2. For any angle 0° < θ < 45°, the value of sin θ is less than cos θ. Conversely, for any angle 45° < θ < 90°, the value of sin θ is greater than cos θ.

Applying this rule to the given statements, for statement 1, the angle is 66°. Since 66° > 45°, it follows that sin 66° > cos 66°, making statement 1 incorrect. For statement 2, the angle is 26°. Since 26° < 45°, it follows that sin 26° < cos 26°, which makes statement 2 correct. Therefore, only statement 2 is correct, which corresponds to option b.

Explanation

The sine function sin θ is defined such that its value always lies within the range [-1, 1] for any real angle θ. This implies that the maximum possible value sin θ can achieve is 1.

Given the expression sin θ = (2a + 3b)/3b, we can simplify the right hand side by dividing the terms in the numerator by the denominator: sin θ = 2a/3b + 3b/3b = 2a/3b + 1.

According to the problem, a and b are positive numbers (a > 0 and b > 0). Consequently, the term 2a/3b must be strictly greater than zero. Adding 1 to a positive number results in a value that is strictly greater than 1.

Since sin θ > 1 contradicts the fundamental property of the sine function where sin θ ≤ 1, the given relation is not possible. Thus, option a is correct.

Explanation

Let the height of the higher plane be H₁ = 3000m and the height of the lower plane be H₂. Let x be the horizontal distance from the point of observation on the ground to the vertical line connecting the two planes.

Using the formula tan(θ) = Perpendicular/Base, we analyze the two triangles formed. For the higher plane with an angle of elevation of 60°: tan(60°) = 3000/x √3 = 3000/x x = 3000/√3 = 1000√3m

For the lower plane with an angle of elevation of 45°: tan(45°) = H₂/x 1 = H₂/(1000√3) H₂ = 1000√3m

The vertical distance between the two planes is the difference between their heights: Vertical Distance = H₁ - H₂ = 3000 - 1000√3 = 1000(3 - √3)m

Thus, the vertical distance is 1000(3 - √3)m, which matches option (c).

Explanation

To find the height of the pole, we use the trigonometric ratio for a right-angled triangle formed by the pole, the ground, and the rope. In this scenario, the pole represents the opposite side, the rope represents the hypotenuse, and the angle with the horizontal is given as 30°.

The formula relating the opposite side and the hypotenuse is: sin(θ) = Opposite/Hypotenuse

Substituting the given values: sin(30°) = height of the pole/√12

Since sin(30°) = 1/2, the equation becomes: 1/2 = h/√12

Solving for h: h = √12/2

Simplifying √12 as √(4 × 3) = 2√3, we get: h = 2√3/2 = √3m

Thus, the height of the pole is √3m, which corresponds to option (d).

Explanation

Use the fundamental trigonometric identity sec² θ - tan² θ = 1. This identity can be factored as a difference of squares: (sec θ - tan θ)(sec θ + tan θ) = 1. Given that tan θ + sec θ = 2, substitute this value into the equation: (sec θ - tan θ)(2) = 1, which implies sec θ - tan θ = 1/2. Now, we have a system of two linear equations:

1. sec θ + tan θ = 2
2. sec θ - tan θ = 1/2 To find tan θ, subtract equation 2 from equation 1: (sec θ + tan θ) - (sec θ - tan θ) = 2 - 1/2 2 tan θ = 3/2 tan θ = 3/4 This result matches option a.

Explanation

For problems involving successive percentage increases or decreases, the fastest method is to use the net percentage change formula: Net % change = x + y + (xy)/100, where x and y are the percentage changes. In this case, the first change is an increase of 40 percent, so x = 40. The second change is a decrease of 20 percent, so y = -20. Substituting these values into the formula, we get:

Net change = 40 + (-20) + (40 × (-20))/100 Net change = 20 - 800/100 Net change = 20 - 8 = 12%

Since the result is positive, it represents a net increase. Thus, the overall salary of A increased by 12 percent, which corresponds to option d. Alternatively, using the multiplier method, the final salary is 1.40 × 0.80 = 1.12 times the original salary, which indicates a total increase of (1.12 - 1) × 100 = 12%.

Explanation

Let the two-digit number be 10a + b where a + b = 7. Reversing increases the number by 27: (10b + a) - (10a + b) = 27 ⇒ 9(b - a) = 27 ⇒ b - a = 3. Solving a + b = 7 and b - a = 3: a = 2, b = 5. Product = 2 × 5 = 10.

Explanation

In a race, the ratio of distances covered by two runners in the same amount of time is equal to the ratio of their speeds. From the given data, when A covers 100m, B covers 100 - 4 = 96m and C covers 100 - 2 = 98m. Therefore, the ratio of the speed of C to the speed of B is V_C : V_B = 98 : 96.

To find how much C beats B in a 100m race, we calculate the distance B covers when C completes 100m using the formula: Distance covered by B = V_B/V_C × Total Distance = 96/98 × 100.

Performing the calculation: Distance covered by B = 48/49 × 100 = 4800/49 ≈ 97.959m.

The distance by which C beats B is the difference between the total race distance and the distance covered by B: Beating distance = 100 - 97.959 = 2.041m.

This value is approximately 2.04m, which corresponds to option b.

Explanation

Let V be the total number of voters. Valid votes cast = 0.90V - 60. Winner's votes = 0.47V. Rival's votes = (0.90V - 60) - 0.47V = 0.43V - 60. The difference is 0.47V - (0.43V - 60) = 308, which simplifies to 0.04V + 60 = 308. Thus, 0.04V = 248, so V = 6200.

Explanation

Numbers are of the form 7k + 3. For two-digit numbers: 10 <= 7k + 3 <= 99, so 7 <= 7k <= 96, which means 1 <= k <= 13. The numbers are 10, 17, ..., 94. This is an arithmetic progression with n = 13 terms. Sum = (13/2)(10 + 94) = (13/2)(104) = 13 * 52 = 676.

Explanation

The volume of the cistern is calculated using the formula for the volume of a rectangular prism: V = length × width × height. Substituting the given values, V = 10 m × 8 m × 6 m = 480 m³. The cross-sectional area of the pipe is given as 4 cm², which must be converted to square meters for consistency: A = 4 × (10⁻² m)² = 4 × 10⁻⁴ m². The volumetric flow rate Q is the product of the cross-sectional area and the nozzle velocity v: Q = A × v = (4 × 10⁻⁴ m²) × (40 m/s) = 0.016 m³/s. The total time t required to fill the cistern is the total volume divided by the flow rate: t = V / Q = 480 m³ / 0.016 m³/s = 30,000 seconds. To convert this time into hours and minutes, divide by 3600 seconds per hour: 30,000 / 3600 = 25 / 3 = 8.333 hours. Since 0.333 hours is equal to 1/3 of an hour, it corresponds to 20 minutes. Therefore, the cistern will be full in 8 hours 20 minutes, which matches option c.

Explanation

In a circle of diameter 30, the maximum number of circles of diameter 10 that can fit is 7 (one center, six around). The height of the drum is 100 cm and the height of each box is 10 cm, allowing for 100/10 = 10 layers. Total boxes = 7 * 10 = 70.

Explanation

When the distance traveled is constant, the time taken is inversely proportional to the speed of the vehicle. This relationship is given by the formula T = D/S, where T is time, D is distance, and S is speed. Given that the speeds of the three buses are in the ratio 2:3:4, the ratio of the time taken by them will be the reciprocal of their speed ratios.

Therefore, the ratio of time taken is 1/2 : 1/3 : 1/4. To convert this fraction ratio into a whole number ratio, we multiply each term by the least common multiple (LCM) of the denominators 2, 3, and 4, which is 12.

Calculating the values: First bus: 1/2 × 12 = 6 Second bus: 1/3 × 12 = 4 Third bus: 1/4 × 12 = 3

The resulting ratio of time taken is 6:4:3, which corresponds to option (d).

Explanation

The side of the square is calculated using the formula for the diagonal, d = a√2, where a is the side length. Given d = 12√2 cm, we have a√2 = 12√2, which gives a = 12 cm. The perimeter of the square is P = 4a = 4 × 12 = 48 cm. Since the perimeter of the equilateral triangle is equal to that of the square, 3s = 48, where s is the side of the triangle. This gives s = 48/3 = 16 cm. The area of an equilateral triangle is given by the formula A = (√3/4)s². Substituting the value of s, we get A = (√3/4) × 16² = (√3/4) × 256 = 64√3 cm². Therefore, the area of the triangle is 64√3 cm², which corresponds to option d.

Explanation

To calculate the speed of the cycle, we use the formula for the distance covered by a wheel in terms of its circumference and the number of revolutions. The distance covered in one revolution is equal to the circumference of the wheel, 2πr. The total speed in km/h is found by multiplying the distance covered per minute by 60 to get the hourly distance and dividing by 1000 to convert meters into kilometers.

The applicable formula is Speed = (n × 2πr × 60)/1000. Substituting the given values where n = 140 revolutions per minute, r = 30 cm = 0.3 m, and using π = 22/7:

Speed = (140 × 2 × 22/7 × 0.3 × 60)/1000 Speed = (20 × 2 × 22 × 0.3 × 60)/1000 Speed = (880 × 18)/1000 Speed = 15840/1000 = 15.84 km/hour

The calculated speed of the cycle is 15.84 km/hour, which corresponds to option (b).

Explanation

To find the total cost of fencing, we first determine the dimensions of the orchard. The total number of plants is 437, and since they are planted in rows with the same number of plants per row, we find the factors of 437. Since 437 is the product of two prime numbers, 19 × 23, the grid arrangement is either 19 rows of 23 plants or 23 rows of 19 plants.

Let the number of rows be R = 19 and plants per row be P = 23. The length of the area occupied by plants is (P - 1) × 2 = (23 - 1) × 2 = 44 m, and the width is (R - 1) × 2 = (19 - 1) × 2 = 36 m. Adding the 1 m clearance on all four sides, the total length L becomes 44 + 1 + 1 = 46 m and the total width W becomes 36 + 1 + 1 = 38 m.

The total perimeter of the orchard is calculated using the formula: Perimeter = 2(L + W) = 2(46 + 38) = 2(84) = 168 m

The total cost of fencing at Rs. 100 per metre is: Total Cost = 168 m × 100 Rs/m = Rs. 16,800

Thus, option b is correct.

Explanation

In triangle ABD, we have two sides AB and AD and the included angle A = 60 degrees. To find the length of the diagonal BD, we use the Law of Cosines: BD² = AB² + AD² - 2(AB)(AD) cos(∠A)

Substitute the given values AB = 2AD and ∠A = 60° into the formula: BD² = (2AD)² + AD² - 2(2AD)(AD) cos(60°)

Since cos(60°) = 1/2, the equation becomes: BD² = 4AD² + AD² - 2(2AD²) (1/2) BD² = 5AD² - 2AD² BD² = 3AD²

Taking the square root on both sides: BD = √3 AD

Therefore, the length of the diagonal BD is √3 AD, which corresponds to option b.

Explanation

The point O is equidistant from the three sides of triangle ABC, which identifies O as the incenter of the triangle. The incenter is the point of intersection of the internal angle bisectors of the triangle.

For statement 1, since OA, OB, and OC are angle bisectors, we have ∠OAC = ∠A/2, ∠OBA = ∠B/2, and ∠OCB = ∠C/2. The sum of these angles is ∠OAC + ∠OBA + ∠OCB = ( ∠A + ∠B + ∠C ) / 2. Using the angle sum property of a triangle, ∠A + ∠B + ∠C = 180°, so the sum is 180° / 2 = 90°. Thus, statement 1 is correct.

For statement 2, the angle subtended by a side at the incenter is given by the formula ∠BOC = 90° + ∠BAC/2. This is different from the statement ∠BOC = 2∠BAC, making statement 2 incorrect.

For statement 3, the line OA is the angle bisector of ∠BAC. By the property of angle bisectors, any point lying on the bisector of an angle is equidistant from the two arms of that angle. Therefore, any point on the segment OA will have equal perpendicular distances to the sides AB and AC. Thus, statement 3 is correct.

Since statements 1 and 3 are correct, option c is the right choice.

Explanation

The curved surface area (CSA) of a cylinder is given by the formula CSA = 2πrh. Since the area is directly proportional to the height h, doubling the height while keeping the radius r constant results in a new area CSA' = 2πr(2h) = 2(2πrh) = 2 × CSA. Thus, statement 1 is correct as the curved surface area is doubled.

The total surface area (TSA) of a hemispherical solid is the sum of its curved surface area 2πr² and the area of its circular base πr², which gives TSA = 3πr². The area is proportional to the square of the radius r. If the radius is doubled, the new total surface area becomes TSA' = 3π(2r)² = 3π(4r²) = 4(3πr²) = 4 × TSA. Thus, statement 2 is correct as the total surface area becomes fourfold.

Since both statements are mathematically accurate, option (c) is the correct choice.

Explanation

The volume of water pumped out of a cubical tank forms a rectangular prism where the base is the square base of the cube and the height is the drop in the water level. Let the side of the cube be s and the drop in water level be h. The formula for the volume of the water removed is V = s² × h. Substituting the given values, we have 128 = s² × 2. Solving for s², we get s² = 128/2 = 64. Taking the square root of both sides, the side of the cube s = √64 = 8 m. The maximum capacity of the tank is the total volume of the cube, which is calculated using the formula Vₜₒₜₐₗ = s³. Therefore, the capacity is 8³ = 8 × 8 × 8 = 512 m³. This corresponds to option a.

Explanation

The circumference of a circle is defined by the formula C = 2πr. Given that the circumference C = 100 cm, we solve for the radius r: r = 100/2π = 50/π cm. The diameter of the circle d is twice the radius, so d = 2 × 50/π = 100/π cm. In a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. If a represents the side of the square, the relationship is d = a√2. Substituting the known value of the diameter, we get a√2 = 100/π. Dividing both sides by √2 gives a = 100/(π√2). Rationalizing the denominator by multiplying both numerator and denominator by √2, we obtain a = 100√2/2π = 50√2/π cm, which corresponds to option c.

Explanation

Volume of the cube is 1^3 = 1 cm^3. For the spherical balls, surface area 4pir^2 = pi^(1/3), so r^2 = pi^(1/3) / (4pi) = 1 / (4 pi^(2/3)). Thus, r = 1 / (2 pi^(1/3)). The volume of one ball is (4/3)pir^3 = (4/3)pi(1 / (8pi)) = 1/6 cm^3. The number of balls is 1 / (1/6) = 6.

Explanation

The volume of water in the container follows an exponential growth pattern where the volume doubles every 5 minutes. This can be represented by the formula V(t) = V₀ · 2ᵗ/⁵, where V(t) is the volume at time t and V₀ is the initial volume. To solve this efficiently, it is best to work backward from the time the container is full.

Given that the container is full at t = 30 minutes, let the full capacity be C. Since the volume doubles every 5 minutes, it must have been half-full 5 minutes prior to being full. Therefore, at t = 30 - 5 = 25 minutes, the volume was C/2. To find when it was one-fourth full, we subtract another 5-minute interval, as half of C/2 is C/4. Thus, at t = 25 - 5 = 20 minutes, the container was 1/4 full. Mathematically, V(20) = C · 2⁽²⁰⁻³⁰⁾/⁵ = C · 2⁻² = C/4. Option C is the correct result.

Explanation

The volume of a cylinder with radius r and height h is given by the formula V꜀ᵧₗᵢₙ꜡ₑᵣ = πr²h. Since there are 30 such cylinders, the total volume of metal available is 30 × πr²h. The volume of a cone with the same radius and height is given by the formula V꜀ₒₙₑ = 1/3 πr²h. To find the number of cones, we divide the total volume by the volume of one cone: n = (30πr²h)/(1/3 πr²h) = 30 × 3 = 90. Thus, Statement I is correct.

Regarding Statement II, the curved surface of a cylinder is indeed a rectangle when flattened, with dimensions 2πr by h. However, the curved surface of a cone, when flattened, forms a sector of a circle with a radius equal to the slant height l of the cone, not a triangle. Therefore, Statement II is incorrect. Consequently, Statement I is correct but Statement II is not correct, making option c the right choice.

Explanation

Let the side of the square ABCD be s. In an equilateral triangle BOC with side s, the altitude from O to BC is given by the formula h = (√3/2)s. Since B and C are vertices of the square, BC is one of the sides. The distance from vertex O to the side BC is (√3/2)s, and therefore the distance from O to the opposite side AD is s - (√3/2)s = s(1 - √3/2).

Let M be the midpoint of AD. O lies on the perpendicular bisector of AD, so OM is the distance from O to AD. In the right triangle OMD, we have MD = s/2 and OM = s(1 - √3/2). The angle ∠ODM can be found using the tangent ratio: tan(∠ODM) = OM/MD = s(1 - √3/2)/(s/2) = 2(1 - √3/2) = 2 - √3. Knowing that tan 15° = 2 - √3, we find ∠ODM = 15°.

Because ΔAOD is an isosceles triangle with OA = OD, we have ∠OAD = ∠ODA = 15°. The sum of angles in ΔAOD is 180°, so ∠AOD = 180° - (15° + 15°) = 150°. Given that ∠AOD = 2θ, we have 2θ = 150°, which implies θ = 75°. The value of tan θ is tan 75° = tan(45° + 30°) = (tan 45° + tan 30°)/(1 - tan 45° tan 30°) = (1 + 1/√3)/(1 - 1/√3) = (√3 + 1)/(√3 - 1). Rationalizing the denominator gives (√3 + 1)²/(3 - 1) = (4 + 2√3)/2 = 2 + √3, which corresponds to option d.

Explanation

The distance covered by a wheel in one revolution is equal to its circumference, given by the formula πd, where d is the diameter. The total distance D covered in n revolutions is D = n × πd.

First, convert the total distance into a single unit (meters). Since 1 km = 1000 m and 1 decametre = 10 m, the total distance D = (2 × 1000) + (26 × 10) = 2000 + 260 = 2260 m.

Given that the number of revolutions n = 452, we substitute the values into the formula: 2260 = 452 × 22/7 × d

Rearranging the equation to solve for d: d = (2260 × 7)/(452 × 22)

Simplifying the expression, we observe that 2260 ÷ 452 = 5. d = (5 × 7)/22 = 35/22

Converting the improper fraction to a mixed fraction: d = 1 13/22 m

Thus, the diameter of the wheel is 1 13/22 m, which corresponds to option b.

Explanation

This problem can be solved using the concept of rotation in geometry. In square ABCD, we have AB = AD and ∠BAD = 90°. Similarly, in square AXYZ, we have AX = AZ and ∠XAZ = 90°.

Consider a rotation of 90° about point A. This rotation maps point B to point D because AB = AD and ∠BAD = 90°. Depending on the orientation of the square AXYZ (whether it is described clockwise or counter-clockwise), the rotation of 90° about A will map point X to point Z or point Z to point X.

Case 1: If the rotation maps X to Z, then triangle ABX is congruent to triangle ADZ by the SAS (Side-Angle-Side) criterion, as AB = AD, AX = AZ, and ∠BAX = ∠DAZ. Consequently, the corresponding parts of congruent triangles are equal, giving BX = DZ and ∠ABX = ∠ADZ.

Case 2: If the rotation maps Z to X, then triangle ABZ is congruent to triangle ADX by the SAS criterion, as AB = AD, AZ = AX, and ∠BAZ = ∠DAX. This results in BZ = DX and ∠ABZ = ∠ADX.

Since one of these two cases must occur, both statement 1 (BX = DZ or BZ = DX) and statement 2 (∠ABX = ∠ADZ or ∠ADX = ∠ABZ) are correct. Therefore, option c is the correct choice.

Explanation

To find the fourth side of the cyclic quadrilateral, we use the Law of Cosines. Let the sides be AB = 2 cm, BC = 5 cm, CD = 3 cm, and DA = d cm, with the angle ∠B = 60°. First, calculate the length of the diagonal AC using the Law of Cosines in ΔABC: AC² = AB² + BC² - 2(AB)(BC) cos B AC² = 2² + 5² - 2(2)(5) cos 60° AC² = 4 + 25 - 20(1/2) AC² = 29 - 10 = 19

In a cyclic quadrilateral, opposite angles are supplementary. Therefore, the angle opposite to ∠B is ∠D = 180° - 60° = 120°. Now, apply the Law of Cosines in ΔADC to find the side d: AC² = CD² + DA² - 2(CD)(DA) cos D 19 = 3² + d² - 2(3)(d) cos 120° Since cos 120° = -1/2, the equation becomes: 19 = 9 + d² - 6d(-1/2) 19 = 9 + d² + 3d d² + 3d - 10 = 0

Solving this quadratic equation by factoring: (d + 5)(d - 2) = 0 This gives d = -5 or d = 2. Since the length of a side cannot be negative, the fourth side is d = 2 cm, which corresponds to option a).

Explanation

A quadrilateral whose diagonals bisect each other is a parallelogram. When the diagonals are also equal, as specified in set A, the parallelogram is a rectangle. Thus, set A represents the set of all rectangles. Similarly, a parallelogram whose diagonals bisect each other at 90° is a rhombus. Therefore, set B represents the set of all rhombuses. The intersection A ∩ B consists of quadrilaterals that satisfy both conditions: they must be both rectangles and rhombuses. A rectangle with all sides equal or a rhombus with all angles equal to 90° is a square. Hence, the set of quadrilaterals that have equal diagonals bisecting each other and diagonals bisecting each other at 90° is the set of squares, making option (c) the correct choice.

Explanation

Let the side of the square be s. The hypotenuse of the triangle connects (x, 0) and (0, y), so its equation is X/x + Y/y = 1. The vertex of the square (s, s) lies on this line, so s/x + s/y = 1. Solving for s gives s = xy / (x + y). The perimeter is 4s = 4xy / (x + y).

Explanation

To find the area of the remaining sheet, we subtract the area of the semicircular portion from the total area of the rectangular sheet. The area of the rectangle is calculated using the formula Area = length × width, so Area of rectangle = 18 cm × 14 cm = 252 cm². Since the semicircle is taken out using the smaller side as the diameter, the diameter d = 14 cm, which means the radius r = 7 cm. The area of a semicircle is given by the formula 1/2 π r². Substituting the values, Area of semicircle = 1/2 × 22/7 × (7)² = 1/2 × 22/7 × 49 = 11 × 7 = 77 cm². Finally, the area of the remaining sheet = Area of rectangle - Area of semicircle = 252 cm² - 77 cm² = 175 cm². This result matches option (d).

Explanation

The fastest way to solve this problem is by using the formula for successive percentage change. When two dimensions of a rectangle are increased by x% and y%, the net percentage increase in area is given by the formula: Net Increase = x + y + xy/100. Substituting the given values where x = 30 and y = 20, we get: Net Increase = 30 + 20 + (30 × 20)/100 = 50 + 600/100 = 50 + 6 = 56%. Alternatively, let the side of the square be 10 units. The original area is 10 × 10 = 100 square units. The new dimensions after the increases are 10 × 1.3 = 13 units and 10 × 1.2 = 12 units. The area of the resulting rectangle is 13 × 12 = 156 square units. The increase in area is 156 - 100 = 56 square units, which corresponds to an increase of 56% over the original area of 100. Thus, option d is correct.

Explanation

The total volume of the tank is the sum of the volumes of the hemispherical bottom and the cylindrical top. Using the formula V = 2/3 πr³ + πr²h, we substitute the given values V = 3312 π and r = 12:

3312 π = 2/3 π(12)³ + π(12)²h

Dividing the entire equation by π:

3312 = 2/3 (1728) + 144h 3312 = 1152 + 144h 144h = 3312 - 1152 144h = 2160 h = 2160/144 = 15 m

The surface area of the hemispherical portion is the curved surface area Sₕₑₘᵢ = 2 πr² and the surface area of the cylindrical portion is the curved surface area S꜀ᵧₗ = 2 πrh. The ratio of these surface areas is:

Ratio = 2πr²/2πrh = r/h

Substituting the values of r = 12 and h = 15:

Ratio = 12/15 = 4/5

Therefore, the ratio of the surface areas of the spherical and cylindrical portions is 4:5, which corresponds to option b.

Explanation

Let the length, width, and height of the cuboid be denoted by l, w, and h respectively. The areas of three mutually perpendicular faces of a cuboid are the products of its dimensions taken two at a time. Therefore, we can represent these areas as x = l × w, y = w × h, and z = h × l.

The formula for the volume V of a cuboid is given by the product of its three dimensions: V = l × w × h. To find the value of xyz, we multiply the three given areas together: xyz = (l × w) × (w × h) × (h × l) xyz = l² × w² × h² xyz = (l × w × h)²

Substituting the volume V into the expression, we get xyz = V². Thus, option b is the correct choice.

Explanation

The volume of a cone is given by the formula V = 1/3 π r² h, where r is the radius of the base and h is the height. In this problem, the water forms a smaller cone inside the larger inverted cone. Because the water fills the cone up to half of its height, the height of the water cone is hᵥᵥₐₜₑᵣ = 1/2 hₜₒₜₐₗ. Since the water cone is similar to the original cone, the ratio of their radii is equal to the ratio of their heights, meaning rᵥᵥₐₜₑᵣ = 1/2 rₜₒₜₐₗ.

The volume of the water is Vᵥᵥₐₜₑᵣ = 1/3 π (rᵥᵥₐₜₑᵣ)² hᵥᵥₐₜₑᵣ. Substituting the ratios, we get Vᵥᵥₐₜₑᵣ = 1/3 π (rₜₒₜₐₗ/2)² (hₜₒₜₐₗ/2). This simplifies to Vᵥᵥₐₜₑᵣ = 1/3 π (rₜₒₜₐₗ²/4) (hₜₒₜₐₗ/2) = 1/8 (1/3 π rₜₒₜₐₗ² hₜₒₜₐₗ). Since the term in the parentheses is the total volume V, the volume of water is Vᵥᵥₐₜₑᵣ = V/8, which corresponds to option (a).

Explanation

Let the common area of the three rectangles be A. For each rectangle Rᵢ with length xᵢ and breadth yᵢ, we have A = xᵢyᵢ, which implies yᵢ = A/xᵢ. When the parallel sides along the breadth are joined to form a cylinder, the length xᵢ becomes the circumference of the circular base and the breadth yᵢ becomes the height of the cylinder.

The circumference is given by 2πrᵢ = xᵢ, so the radius is rᵢ = xᵢ/2π. The height of the cylinder is hᵢ = yᵢ = A/xᵢ. The volume Vᵢ of the cylinder is calculated using the formula V = πr²h. Substituting the values for rᵢ and hᵢ:

Vᵢ = π(xᵢ/2π)²(A/xᵢ) = π (xᵢ²/4π²) (A/xᵢ) = Axᵢ/4π

From this expression, it is evident that the volume V is directly proportional to the length x since A and 4π are constants. Given the condition x₁ < x₂ < x₃, it follows directly that V₁ < V₂ < V₃. Thus, option (c) is the correct relationship.

Explanation

Surface area of cube 6a^2 = 13254, so a^2 = 2209. Thus, a = 47 cm. The diagonal of the cube is asqrt(3) = 47sqrt(3) cm.

Explanation

The area of a trapezium is calculated using the formula Area = 1/2 × (sum of parallel sides) × height. Given that the parallel sides are in the ratio 5:7, we can let the smaller side be 5x and the larger side be 7x. The height (perpendicular distance between them) is given as 14 cm and the total area is 336 cm².

Substituting these values into the formula: 336 = 1/2 × (5x + 7x) × 14 336 = 7 × (12x) 336 = 84x

To find the value of x, we divide the area by the coefficient: x = 336/84 = 4

The smaller parallel side is 5x. Substituting the value of x: Smaller side = 5 × 4 = 20 cm

Thus, option (a) is the correct answer.

Explanation

Volume of the cube = 44^3 = 85184 cm^3. Volume of one spherical bullet = (4/3)pir^3 = (4/3)(22/7)2^3 = 704/21 approx 33.5238 cm^3. Number of bullets = 85184 / (704/21) = (85184 21) / 704 = 121 21 = 2541.

Explanation

To find the volume of water flowing into the sea per minute, we use the formula for discharge: Volume = Cross-sectional Area × Velocity × Time.

First, calculate the cross-sectional area of the river: Area = width × depth = 45 m × 2.5 m = 112.5 m²

Next, convert the speed of the river from km/hour to m/second using the shortcut factor of 5/18: Velocity = 3.6 km/h × 5/18 = 18/18 m/s = 1 m/s

The time interval given is one minute, which is equal to 60 seconds.

Now, substitute these values into the volume formula: Volume = 112.5 m² × 1 m/s × 60 s Volume = 112.5 × 60 = 6750 m³

Thus, the amount of water that runs into the sea per minute is 6750 m³, which corresponds to option (b).

Explanation

Using the Intersecting Chords Theorem, if two chords of a circle intersect, the product of the segments of one chord is equal to the product of the segments of the other chord. Here, diameter AB and chord CD intersect at point P. Therefore, CP ⋅ PD = AP ⋅ PB.

Given that the diameter AB = d is divided by P in the ratio 1:2, we calculate the lengths of the segments AP and PB as: AP = 1/(1+2) d = d/3 PB = 2/(1+2) d = 2d/3

Since the diameter AB is perpendicular to the chord CD, P must be the midpoint of CD. Let CP = PD = x. Substituting these values into the theorem formula: x ⋅ x = d/3 ⋅ 2d/3 x² = 2d²/9 x = √2d/3

The total length of chord CD is 2x: CD = 2 ⋅ √2d/3 = 2√2d/3

Thus, the length of chord CD is 2√2d/3, which corresponds to option (c).

Explanation

The area of any regular polygon can be calculated by dividing it into n identical isosceles triangles, where n is the number of sides. The center of the inscribed circle serves as the common vertex for these triangles. The height (altitude) of each triangle corresponds to the radius r of the inscribed circle, and the base of each triangle corresponds to the side length s of the polygon.

Using the standard formula for the area of a triangle, the area of one such triangle is 1/2 × base × height = 1/2 × s × r. Since the polygon consists of n such congruent triangles, the total area A of the polygon is given by the formula A = n × (1/2 × s × r).

Rearranging the terms, we get A = 1/2 × (n × s) × r. The perimeter p of a regular polygon is the total length of its boundary, which is the product of the number of sides and the length of each side, so p = n × s. Substituting p into the area equation yields A = pr/2. This corresponds to option c.

Explanation

Let H be the HCF and L be the LCM. L = 12H and L + H = 403. So 13H = 403, which means H = 31 and L = 12 31 = 372. The product of two numbers is L H, so 93 X = 372 31. X = (372 * 31) / 93 = 372 / 3 = 124.

Explanation

To find the value of the given expression 0.63 + 0.37, we perform a simple decimal addition. Aligning the decimal points, we add the hundredths place first: 3 + 7 = 10, so we write 0 and carry over 1 to the tenths place. Adding the tenths place gives 6 + 3 + 1 = 10, so we write 0 and carry over 1 to the units place. Thus, the sum is 1.00, which simplifies to 1. Alternatively, converting the decimals to fractions gives 63/100 + 37/100 = (63 + 37)/100 = 100/100 = 1. This matches option a.

Explanation

For compound interest calculated half-yearly, the formula for the total amount A is given by A = P(1 + R/2n)²ᵗ, where P is the principal, R is the annual rate of interest, n is 100, and t is the time in years. Here, P = 10000, R = 10, and t = 1.

Substituting the values into the formula: A = 10000(1 + 10/(2 × 100))²ˣ¹ A = 10000(1 + 5/100)² A = 10000(1.05)² A = 10000 × 1.1025 A = 11025

The compound interest CI is calculated by subtracting the principal from the total amount: CI = A - P CI = 11025 - 10000 CI = 1025

Alternatively, using the shortcut for two periods at 5% each, the effective interest rate is 5 + 5 + (5 × 5)/100 = 10.25%. Interest = 10.25% of 10000 = 1025. Thus, option b is correct.

Explanation

Initial milk = (7/10) 30 = 21 litres. Initial water = (3/10) 30 = 9 litres. Let x be the water added. The new ratio is 21/(9 + x) = 1/2. Solving for x: 42 = 9 + x, which gives x = 33 litres.

Explanation

For the expression 6 + 5x - x^2, the maximum value occurs at x = -b/2a = -5/(-2) = 2.5. Max value = 6 + 5(2.5) - (2.5)^2 = 6 + 12.5 - 6.25 = 12.25. For the expression y - 6 - y^2, the maximum value occurs at y = -b/2a = -1/(-2) = 0.5. Max value = 0.5 - 6 - (0.5)^2 = 0.5 - 6 - 0.25 = -5.75. The difference is 12.25 - (-5.75) = 18.

Explanation

The hour hand of a clock completes one full rotation of 360° in 12 hours. Therefore, the angular speed of the hour hand is 360° / 12 = 30° per hour. To find the speed per minute, we divide by 60, giving 30° / 60 = 0.5° per minute.

The time elapsed from noon to 10 minutes past 5 is 5 hours and 10 minutes. Using the formula for the total angle moved by the hour hand: Total Angle = (Hours × 30°) + (Minutes × 0.5°)

Substituting the values into the formula: Total Angle = (5 × 30°) + (10 × 0.5°) Total Angle = 150° + 5° = 155°

Thus, the hour hand moves through an angle of 155°, making option d the correct choice.

Explanation

The range of the square of a cosine function, cos² θ, for any real value of θ is [0, 1]. To find the minimum value of the expression cos² x + cos² y - cos² z, we must minimize the positive terms and maximize the negative term. The minimum possible value for cos² x and cos² y is 0, which occurs when x and y are odd multiples of π/2 (for example, x = π/2 and y = π/2). Simultaneously, to make the overall value as small as possible, we must maximize the term being subtracted, cos² z. The maximum possible value for cos² z is 1, which occurs when z is a multiple of π (for example, z = 0). Substituting these values into the expression, we get 0 + 0 - 1 = -1. Thus, the minimum value of the given trigonometric expression is -1, which corresponds to option a.

Explanation

cot(pi/4) = 1, sec(pi/3) = 2, and cos(pi/6) = sqrt(3)/2. Plugging these into the expression: 32(1)^2 - 8(2)^2 + 8(sqrt(3)/2)^3 = 32 - 32 + 8(3sqrt(3)/8) = 3sqrt(3).

Explanation

First, calculate the value of the expression inside the cube root: 1 - x = 1 - 91/216 = (216 - 91)/216 = 125/216.

Now, apply the cube root to this result: (1 - x)¹/³ = (125/216)¹/³ = ³√125/³√216 = 5/6.

Substitute this value back into the main expression: 3 - 1/(1 - x)¹/³ = 3 - 1/(5/6).

The reciprocal of 5/6 is 6/5. Therefore, the expression becomes: 3 - 6/5 = (3 × 5 - 6)/5 = (15 - 6)/5 = 9/5.

The final result matches option a.

Explanation

A cuboid has 8 vertices. At each vertex, there are 6 right-angled triangles that can be formed: 3 using the edges meeting at that vertex and 3 using one edge and a face diagonal from that vertex. This gives 8 * 6 = 48 right-angled triangles. Since 48 is not among options a, b, or c, the correct option is d.

Explanation

Let n be the number of students. Total initial weight = 43n. Sum of weights of new students = 42 + 36.5 + 39 + 42.5 = 160. New average = (43n + 160) / (n + 4) = 42.5. Solving for n: 43n + 160 = 42.5n + 170, which simplifies to 0.5n = 10, so n = 20.

Explanation

Use the basic average formula: Sum = Average × Number of items.

Four years ago, the sum of the ages of A and B was 2 × 18 = 36 years. Since four years have passed for both individuals, their current combined age is 36 + (2 × 4) = 44 years.

The current total sum of the ages of A, B, and C is 3 × 24 = 72 years. By subtracting the combined current age of A and B from the total sum of all three, we find the current age of C: 72 - 44 = 28 years.

To find the age of C after 8 years, add 8 to the current age: 28 + 8 = 36 years. Thus, option c is the correct value.

Explanation

Sorting the data set in ascending order: a - 3.5, a - 3, a - 2.5, a - 2, a - 0.5, a + 0.5, a + 4, a + 5. There are 8 values. The median is the average of the 4th and 5th terms: ((a - 2) + (a - 0.5)) / 2 = (2a - 2.5) / 2 = a - 1.25.

Explanation

The new values are y_i = (i + 1)x_i = (i + 1)i = i^2 + i. The sum of the first n such values is Sum(i^2) + Sum(i) = [n(n+1)(2n+1)/6] + [n(n+1)/2] = [n(n+1)/2] [(2n+1)/3 + 1] = [n(n+1)/2] [(2n+4)/3] = [n(n+1)(n+2)/3]. The new mean is the sum divided by n, which is (n+1)(n+2)/3.

Explanation

The weighted arithmetic mean is Sum(w_i * x_i) / Sum(w_i). Here, x_i = i and w_i = i for i = 1 to 10. Weighted mean = Sum(i^2) / Sum(i) = [10(11)(21)/6] / [10(11)/2] = 385 / 55 = 7.