CDS II 2019 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

To find the mean of Series-I, we first determine the unknown frequencies x and y using the total frequency provided for both series. For Series-I, the sum of frequencies is 20 + 15 + 10 + x + y = 100, which simplifies to x + y = 55. For Series-II, the sum is 4 + 8 + 4 + 2x + y = 100, which simplifies to 2x + y = 84. Subtracting the first equation from the second, we get (2x + y) - (x + y) = 84 - 55, resulting in x = 29. Substituting x = 29 back into the first equation gives 29 + y = 55, so y = 26.

The mean for grouped data is calculated using the formula x̄ = Σ fᵢ mᵢ / Σ fᵢ, where fᵢ is the frequency and mᵢ is the midpoint of each class interval. The midpoints for the intervals 10-20, 20-30, 30-40, 40-50, and 50-60 are 15, 25, 35, 45, and 55 respectively. For Series-I, the sum of the products of frequencies and midpoints is calculated as: Σ fᵢ mᵢ = (20 × 15) + (15 × 25) + (10 × 35) + (29 × 45) + (26 × 55) Σ fᵢ mᵢ = 300 + 375 + 350 + 1305 + 1430 = 3760

The mean is therefore x̄ = 3760/100 = 37.6, which corresponds to option C.

Explanation

First, find x and y using the total frequencies. For Series-I: 20 + 15 + 10 + x + y = 100, so x + y = 55. For Series-II: 4 + 8 + 4 + 2x + y = 100, so 2x + y = 84. Subtracting the first equation from the second gives x = 29, and thus y = 26. Frequencies for Series-II are 4, 8, 4, 58, 26. The modal class is 40-50 with frequency f1 = 58, f0 = 4, and f2 = 26. Using the mode formula: Mode = L + [ (f1 - f0) / ((f1 - f0) + (f1 - f2)) ] h = 40 + [ (58 - 4) / ((58 - 4) + (58 - 26)) ] 10 = 40 + [ 54 / (54 + 32) ] 10 = 40 + (54/86)10 = 46.279, which is approximately 46.

Explanation

Showroom C total sales (S1) = 15% of 6400 = 960. Showroom E total sales (S1) = 12% of 6400 = 768. Company X sales by showroom E (S2) = 8% of 3000 = 240. Therefore, Company Y sales by showroom E = 768 - 240 = 528. Percentage = (528 / 960) * 100 = 55%.

Explanation

Number of scooters of both companies sold by showroom B is calculated using the total for S₁ and the percentage given for B in S₁: Value of B (both companies) = 6400 × 21/100 = 64 × 21 = 1344

Number of scooters of company X sold by showroom A is calculated using the total for S₂ and the percentage given for A in S₂: Value of A (company X) = 3000 × 24/100 = 30 × 24 = 720

To find how much percent more B is than A, we use the percentage increase formula: Percentage more = (Value B - Value A)/Value A × 100 Percentage more = (1344 - 720)/720 × 100 Percentage more = 624/720 × 100

Simplifying the fraction 624/720: Dividing both by 24, we get 26/30, and further simplifying by 2 gives 13/15. Percentage more = 13/15 × 100 = 13 × 20/3 = 260/3 = 86 2/3%

Thus, the number of scooters of both companies sold by showroom B is 86 2/3% more than the number of scooters of company X sold by showroom A, which corresponds to option C.

Explanation

The number of scooters of company Y sold by any showroom is calculated as the difference between the total number of scooters sold (S1) and the number of scooters of company X sold (S2). To find the average for showrooms A, C, and E, we first calculate the total scooters sold by these three showrooms for both S1 and S2.

The combined percentage for showrooms A, C, and E in S1 is 19% + 15% + 12% = 46%. Total S1 scooters for A, C, and E = 46/100 × 6400 = 46 × 64 = 2944.

The combined percentage for showrooms A, C, and E in S2 is 24% + 20% + 8% = 52%. Total S2 scooters for A, C, and E = 52/100 × 3000 = 52 × 30 = 1560.

The total number of company Y scooters sold by showrooms A, C, and E is the difference between these two values: Total Y scooters = 2944 - 1560 = 1384.

The average number of scooters of company Y sold by these three showrooms is: Average = 1384/3 = 461 1/3.

This corresponds to option a.

Explanation

Showroom A total sales (S1) = 19% of 6400 = 1216. Company X sales by showroom B (S2) = 18% of 3000 = 540. Company X sales by showroom E (S2) = 8% of 3000 = 240. Total X sales by B and E = 540 + 240 = 780. Difference = 1216 - 780 = 436.

Explanation

Pedestrians killed in 2017 = 20457. Average per day = 20457 / 365 = 56.046, which rounds to 56.

Explanation

To calculate the approximate percentage change in pedestrian fatalities from 2014 to 2017, use the standard percentage change formula:

Percentage Change = (Final Value - Initial Value)/Initial Value × 100

From the given data table, the number of pedestrians killed in 2014 (Initial Value) is 12330 and the number of pedestrians killed in 2017 (Final Value) is 20457.

First, calculate the absolute increase in fatalities: Increase = 20457 - 12330 = 8127

Now, substitute these values into the percentage formula: Percentage Change = 8127/12330 × 100

Performing the division: 8127 ÷ 12330 ≈ 0.6591

Multiplying by 100 to get the percentage: 0.6591 × 100 = 65.91%

Rounding this value to the nearest whole number gives approximately 66%, which corresponds to option a.

Explanation

Bikers killed in 2017 = 48746. Average per day = 48746 / 365 = 133.55, which rounds to 134.

Explanation

Cyclists killed in 2017 = 3559. Average per day = 3559 / 365 = 9.75, which rounds to 10.

Explanation

Let sqrt(a) = x and sqrt(b) = y. Then x^3 + y^3 = 32 and xy(x + y) = 31. Using the identity (x + y)^3 = x^3 + y^3 + 3xy(x + y), we get (x + y)^3 = 32 + 3(31) = 125, so x + y = 5. Then (x + y)^2 = x^2 + y^2 + 2xy = a + b + 2xy. From xy(x + y) = 31, xy = 31/5 = 6.2. Thus, a + b = 5^2 - 2(6.2) = 25 - 12.4 = 12.6. The value of 5(a + b) / 7 = 5(12.6) / 7 = 63 / 7 = 9.

Explanation

x = (1 + sqrt(3))/2 implies 2x - 1 = sqrt(3). Squaring both sides gives 4x^2 - 4x + 1 = 3, or 2x^2 - 2x - 1 = 0. For y = x^3, we use 2x^3 = 2x^2 + x = (2x + 1) + x = 3x + 1, so x = (2y - 1)/3. Substituting this into 2x^2 - 2x - 1 = 0 gives 2((2y - 1)/3)^2 - 2((2y - 1)/3) - 1 = 0. Multiplying by 9: 2(4y^2 - 4y + 1) - 6(2y - 1) - 9 = 0, which simplifies to 8y^2 - 8y + 2 - 12y + 6 - 9 = 0, or 8y^2 - 20y - 1 = 0.

Explanation

The Least Common Multiple (LCM) of two numbers must always be a multiple of their Highest Common Factor (HCF). Here, HCF = 12. Checking the options: 60/12 = 5, 36/12 = 3, 24/12 = 2, but 80/12 = 6.66. Thus, 80 cannot be the LCM.

Explanation

Statement 1 is incorrect because the unit digit of 17¹⁷⁴ is determined by the cyclicity of 7, which is 4. To find the unit digit, we divide the exponent 174 by the cyclicity: 174 (mod 4) = 2. The unit digit is therefore the same as 7², which is 49, ending in 9.

Statement 2 is correct. Let any two odd numbers be 2n+1 and 2m+1. The difference of their squares is calculated as (2n+1)² - (2m+1)² = (4n² + 4n + 1) - (4m² + 4m + 1) = 4n(n+1) - 4m(m+1). Since the product of any two consecutive integers k(k+1) is always even, we can substitute n(n+1) = 2a and m(m+1) = 2b. The expression becomes 4(2a) - 4(2b) = 8(a-b), which is always divisible by 8.

Statement 3 is correct. Two consecutive odd numbers can be represented as 2n-1 and 2n+1. Their product is (2n-1)(2n+1) = (2n)² - 1² = 4n² - 1. Adding 1 to this product gives (4n² - 1) + 1 = 4n², which is (2n)². This result is always a perfect square.

Since only statements 2 and 3 are correct, option c is the correct answer.

Explanation

Let P be the principal. Annual compounding return A1 = P(1 + 0.20)^2 = 1.44P. Half-yearly compounding return A2 = P(1 + 0.10)^4 = 1.4641P. Difference = 1.4641P - 1.44P = 0.0241P. Given 0.0241P = 482, P = 482 / 0.0241 = 20,000.

Explanation

For a polynomial to be resolved into three linear factors, it must have three real roots. If k = 0, the expression is x^3 - 7x + 6. Testing possible integer roots, x = 1 gives 1 - 7 + 6 = 0; x = 2 gives 8 - 14 + 6 = 0; x = -3 gives -27 + 21 + 6 = 0. Since there are three linear factors (x-1)(x-2)(x+3), k = 0 is correct.

Explanation

The time they meet at the starting point is the LCM of their individual lap times: LCM(252, 308, 198). Prime factorizations: 252 = 2^2 3^2 7, 308 = 2^2 7 11, 198 = 2 3^2 11. LCM = 2^2 3^2 7 * 11 = 2772 seconds. 2772 seconds = 46 minutes and 12 seconds.

Explanation

To find the Least Common Multiple (LCM) of fractions, the standard mathematical formula is used: LCM of fractions = LCM of Numerators/HCF of Denominators

In this problem, the fractions are 1/3, 5/6, 2/9, and 4/27. The numerators are 1, 5, 2, and 4. The denominators are 3, 6, 9, and 27.

Step 1: Calculate the LCM of the numerators (1, 5, 2, 4). Since 2 is a factor of 4, we only need the LCM of 5 and 4. LCM(1, 5, 2, 4) = 5 × 4 = 20.

Step 2: Calculate the Highest Common Factor (HCF) of the denominators (3, 6, 9, 27). We observe that 3 is the smallest denominator and it exactly divides 6, 9, and 27. Therefore, HCF(3, 6, 9, 27) = 3.

Substituting these values into the formula: LCM = 20/3

This matches option (d).

Explanation

First, solve the quadratic equation x² + 5x + 6 = 0 to find its roots. Using the factorization method, we look for two numbers that multiply to 6 and add to 5, which are 2 and 3. Thus, the equation becomes (x + 2)(x + 3) = 0, giving the roots x = -2 and x = -3. Since the equations x² + 5x + 6 = 0 and x² + kx + 1 = 0 share a common root, this common root must be either -2 or -3.

Case 1: If the common root is x = -2, substitute this value into the second equation: (-2)² + k(-2) + 1 = 0 4 - 2k + 1 = 0 5 = 2k ⇒ k = 5/2

Case 2: If the common root is x = -3, substitute this value into the second equation: (-3)² + k(-3) + 1 = 0 9 - 3k + 1 = 0 10 = 3k ⇒ k = 10/3

Therefore, the possible values for k are 5/2 or 10/3, which corresponds to option (b).

Explanation

Use the Simple Interest (SI) formula: SI = (P × R × T)/100. First, calculate the interest received from B. Given P = 25000, R = 7%, and T = 4 years, the interest from B is SIᵦ = (25000 × 7 × 4)/100 = 250 × 28 = 7000. The total interest received from both B and C is given as 11200. Therefore, the interest received from C is SI꜀ = 11200 - 7000 = 4200. Now, let the amount lent to C be P꜀. Using the same rate and time, we have 4200 = (P꜀ × 7 × 4)/100. This simplifies to 4200 = (P꜀ × 28)/100. Solving for P꜀, we get P꜀ = (4200 × 100)/28 = 420000/28 = 15000. Thus, A lent ₹15000 to C, which corresponds to option (c).

Explanation

Whenever two items are sold at the same selling price, one at a profit of x% and the other at a loss of x%, the transaction always results in a net loss. The shortcut formula to calculate this net loss percentage is Net Loss % = x²/100 %.

In this problem, the profit and loss percentage is given as x = 30%. Substituting this value into the formula, we get:

Net Loss % = 30²/100 = 900/100 = 9%

Therefore, the trader incurs a net loss of 9% on the entire transaction, making option (c) the correct choice. This formula is the fastest way to solve such problems in competitive exams, bypassing the need to calculate individual cost prices.

Explanation

Let the monthly incomes of A and B be 4x and 3x respectively. According to the basic formula Income = Expenditure + Savings, the expenditure of a person is calculated as Expenditure = Income - Savings. Since both A and B save ₹600, the expenditure of A is 4x - 600 and the expenditure of B is 3x - 600. Given that the ratio of their expenditures is 3:2, we can set up the following equation:

(4x - 600)/(3x - 600) = 3/2

Cross multiplying the terms, we get: 2(4x - 600) = 3(3x - 600) 8x - 1200 = 9x - 1800 9x - 8x = 1800 - 1200 x = 600

Now, substituting the value of x to find the monthly income of A: Monthly income of A = 4x = 4 × 600 = 2400

Thus, the monthly income of A is ₹2400, which corresponds to option (c).

Explanation

Let the initial train fare and bus fare be represented by 3x and 4x respectively, based on the given ratio 3:4. A percentage increase is calculated by multiplying the original value by (1 + r/100). For the train fare, the revised fare is 3x × (1 + 20/100) = 3x × 1.2 = 3.6x. For the bus fare, the revised fare is 4x × (1 + 30/100) = 4x × 1.3 = 5.2x. To find the revised ratio, we divide the revised train fare by the revised bus fare: 3.6x/5.2x = 36/52. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, 4, we get 9/13. Thus, the revised ratio between train fare and bus fare is 9:13, which corresponds to option (a).

Explanation

Quotient = 182. Remainder = Quotient - 175 = 182 - 175 = 7. The number N = (Divisor Quotient) + Remainder = (17 182) + 7 = 3094 + 7 = 3101.

Explanation

The total food stock is 240 men * 48 days = 11520 man-days. For 160 men, the stock will last 11520 / 160 = 72 days.

Explanation

Volume of cylinder V = pi (d/2)^2 h. Since d = h, V = pi (h^2 / 4) h = pi h^3 / 4. The sphere of maximum volume has radius r = h/2. Volume of sphere Vs = (4/3) pi (h/2)^3 = (4/3) pi (h^3 / 8) = pi h^3 / 6. Remaining volume = V - Vs = (pi h^3 / 4) - (pi h^3 / 6) = (pi h^3 / 12). Expressed as a fraction of V: (pi h^3 / 12) / (pi * h^3 / 4) = 4 / 12 = 1/3.

Explanation

According to the Intercept Theorem (or the proportionality theorem for parallel lines), when three or more parallel lines are intersected by two transversals, the lengths of the segments intercepted on one transversal are proportional to the corresponding segments intercepted on the other transversal.

The applicable formula for this geometry problem is PQ/QR = LM/MN.

Substituting the given values PQ = 3 cm, QR = 9 cm, and MN = 10.5 cm into the formula: 3/9 = LM/10.5

Simplifying the ratio on the left side gives: 1/3 = LM/10.5

Multiplying both sides by 10.5 to solve for LM: LM = 10.5/3 = 3.5 cm.

Thus, the length of LM is 3.5 cm, which corresponds to option (b).

Explanation

Area of sector = (1/2) r^2 theta. Given 25.6 = (1/2) 4^2 theta, then 25.6 = 8 * theta. Thus, theta = 25.6 / 8 = 3.2 radians.

Explanation

The orthocentre of a triangle is the point where all three altitudes intersect. An altitude is a perpendicular line segment drawn from a vertex to the opposite side. In a right angled triangle, two of the sides are already perpendicular to each other. If we consider a triangle with vertices A, B, and C, where the right angle is at vertex B, then the altitude from vertex A to the opposite side BC is the side AB itself. Similarly, the altitude from vertex C to the opposite side AB is the side CB. Since both these altitudes meet at vertex B, the point of intersection, or the orthocentre, is the vertex containing the right angle. Because the vertex is a part of the triangle, the orthocentre lies on the triangle, specifically at the right angle vertex. This differs from an acute angled triangle where the orthocentre lies inside and an obtuse angled triangle where it lies outside. Therefore, option c is the correct statement.

Explanation

In triangle ABC, let ∠A be the angle at vertex A and ∠B be the angle at vertex B. Since AX is the angle bisector of ∠A, the angle ∠BAX = 1/2∠A. In triangle ABX, the side AB is opposite to ∠AXB and the side BX is opposite to ∠BAX.

Using the angle sum property of triangle ABX, we have ∠AXB = 180° - (∠B + 1/2∠A). To determine the relationship between AB and BX, we compare their opposite angles: ∠AXB - ∠BAX = 180° - (∠B + 1/2∠A) - 1/2∠A = 180° - (∠A + ∠B).

According to the angle sum property of triangle ABC, 180° - (∠A + ∠B) = ∠C. Since the measure of any angle in a triangle must be positive, ∠C > 0, which implies ∠AXB > ∠BAX. By the geometric theorem stating that the side opposite the larger angle in a triangle is longer, it follows that AB > BX. Thus, option b is correct.

Explanation

The expression is log10(cos theta sin theta tan theta cot theta sec theta cosec theta). Since tan theta cot theta = 1, cos theta sec theta = 1, and sin theta cosec theta = 1, the argument of the log is 1. log10(1) = 0.

Explanation

Given cos^2 x + cos x = 1, we have cos x = 1 - cos^2 x = sin^2 x. The expression sin^12 x + 3sin^10 x + 3sin^8 x + sin^6 x is the expansion of (sin^4 x + sin^2 x)^3. Substituting sin^2 x = cos x, we get (cos^2 x + cos x)^3. Since cos^2 x + cos x = 1, the expression equals 1^3 = 1.

Explanation

sin theta = 3/5 implies cos theta = 4/5, so x = cot theta = (4/5) / (3/5) = 4/3. The expression 1 + 3x + (3x)^2 + (3x)^3 + (3x)^4 + (3x)^5 is a geometric series with a = 1, r = 3x = 4, and n = 6. Sum = (r^n - 1) / (r - 1) = (4^6 - 1) / (4 - 1) = (4096 - 1) / 3 = 4095 / 3 = 1365.

Explanation

Let theta be the angle of elevation of the first pillar. Then tan theta = h/x. The angle of elevation of the second pillar is (90 - theta), so tan(90 - theta) = 2h/y. This means cot theta = 2h/y, or tan theta = y/2h. Equating the two: h/x = y/2h, which gives 2h^2 = xy.

Explanation

sin 19 / cos 71 + cos 73 / sin 17. Since cos 71 = sin(90 - 71) = sin 19 and sin 17 = cos(90 - 17) = cos 73, the expression becomes sin 19 / sin 19 + cos 73 / cos 73 = 1 + 1 = 2.

Explanation

When a line is drawn through each vertex of a triangle parallel to the opposite side, the original triangle becomes the medial triangle of the newly formed larger triangle. In this configuration, the vertices of the original triangle serve as the midpoints of the sides of the larger triangle. According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half its length. Therefore, each side of the larger triangle is exactly twice the length of the corresponding side of the original triangle. If the sides of the original triangle are a, b, and c, the sides of the new triangle are 2a, 2b, and 2c. The perimeter of the new triangle is calculated as 2a + 2b + 2c = 2(a + b + c). Substituting the given perimeter of the original triangle, we get 2 × 22 cm = 44 cm. This corresponds to option b.

Explanation

In a trapezium ABCD with AD ∥ BC, the triangles ΔABC and ΔDBC share the same base BC and the same height, which is the perpendicular distance between the parallel lines AD and BC. Therefore, Area(ΔABC) = Area(ΔDBC). Since Area(ΔABC) = Area(ΔAOB) + Area(ΔBOC) and Area(ΔDBC) = Area(ΔDOC) + Area(ΔBOC), it follows that Area(ΔAOB) = Area(ΔDOC). Given Area(ΔAOB) = 3 cm², we have Area(ΔDOC) = 3 cm².

The area of ΔBOC is calculated as Area(ΔBDC) - Area(ΔDOC) = 8 cm² - 3 cm² = 5 cm².

In any quadrilateral where diagonals intersect at O, the product of the areas of the opposite triangles is equal. The relation is Area(ΔAOD) × Area(ΔBOC) = Area(ΔAOB) × Area(ΔDOC). Substituting the values, we get Area(ΔAOD) × 5 = 3 × 3, which simplifies to Area(ΔAOD) = 9/5 = 1.8 cm². This matches option d.

Explanation

For a point P to lie on a circle where AB is the diameter, the angle ∠APB must be a right angle (90°) according to Thales's theorem. This means that the triangle APB must satisfy the Pythagorean theorem, where the square of the hypotenuse AB is equal to the sum of the squares of the other two sides AP and BP. The formula is x² + y² = D², where x = AP, y = BP, and D is the diameter of the circle.

Given the radius is 6.5 cm, the diameter D = 2 × 6.5 = 13 cm. Therefore, for P to lie on the circle, the condition x² + y² = 13² = 169 must be satisfied.

Checking the options: In option (b), 12² + 5² = 144 + 25 = 169, so P lies on the circle. In option (c), 5² + 12² = 25 + 144 = 169, so P lies on the circle. In option (d), 0² + 13² = 0 + 169 = 169, so P lies on the circle (at point A). In option (a), x = 6.5 cm and y = 6.5 cm, we get x² + y² = (6.5)² + (6.5)² = 42.25 + 42.25 = 84.5. Since 84.5 ≠ 169, the point P does not lie on the circle.

Explanation

For two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. This is a fundamental property of similarity in geometry. Let the length of the corresponding side of triangle ABC be x. Using the given values, the relationship can be expressed using the formula:

Perimeter of ΔABC / Perimeter of ΔPQR = Corresponding side of ΔABC / Corresponding side of ΔPQR

Substituting the given values into the formula:

75 cm / 50 cm = x / 20 cm

Simplifying the ratio of the perimeters:

3/2 = x/20

To find x, multiply both sides by 20:

x = 3/2 × 20 x = 3 × 10 x = 30 cm

Therefore, the length of the corresponding side of triangle ABC is 30 cm, which corresponds to option b.

Explanation

In a parallelogram PORS, the vertices are P, O, R, S and the diagonals are PR and OS. Let the intersection point of these diagonals be Q. According to the properties of a parallelogram, the diagonals bisect each other, meaning Q is the midpoint of both PR and OS.

We are given that triangle ORS is an equilateral triangle with a side length a = 10 cm. In triangle ORS, the segment RQ is the median drawn from vertex R to the side OS. In an equilateral triangle, the median is also the altitude of the triangle.

The formula for the altitude h of an equilateral triangle with side a is: h = (√3/2) a

Substituting the given side length a = 10 cm: RQ = (√3/2) × 10 = 5√3 cm

Since Q is the midpoint of the diagonal PR, the total length of PR is twice the length of RQ: PR = 2 × RQ = 2 × 5√3 = 10√3 cm

Thus, the length of the diagonal PR is 10√3 cm, which corresponds to option b).

Explanation

Let the length, width, and height of the cuboid be l, w, and h respectively. The volume V of a cuboid is given by the formula V = l · w · h. The areas of three adjacent faces are the products of these dimensions taken two at a time, which are x = l · w, y = w · h, and z = l · h. To find the relation between the volume and these areas, multiply x, y, and z together: xyz = (l · w) · (w · h) · (l · h) = l² · w² · h². This product can be rewritten as (l · w · h)². Substituting the volume formula V = l · w · h into this expression, we get xyz = V². Therefore, the correct relationship is V² = xyz, which corresponds to option (b).

Explanation

In an equilateral triangle, the median, altitude, and angle bisector are all the same segment. Let a be the side length of the equilateral triangle and l be the length of the median. The formula relating the altitude (median) to the side length is l = (√3/2)a. To express the area in terms of l, we first solve for a: a = 2l/√3. The standard formula for the area A of an equilateral triangle is A = (√3/4)a². Substituting the value of a into this area formula, we get A = (√3/4)(2l/√3)². Squaring the term inside the parentheses yields A = (√3/4) · (4l²/3). Canceling the 4 from the numerator and denominator results in A = √3l²/3. This matches option (a).

Explanation

The length of the wire is equal to the perimeter of the sector. The formula for the perimeter of a sector is given by P = θ/360 × 2πR + 2R, where R is the radius and θ is the central angle. Substituting the given values R = 20 cm and θ = 150°, we get:

P = 150/360 × 2π(20) + 2(20) = 5/12 × 40π + 40 = 50π/3 + 40 cm.

When this wire is bent into the form of a circle, its total length becomes the circumference of the new circle. Let r be the radius of the new circle. The formula for the circumference is C = 2πr. Equating the perimeter of the sector to the circumference of the circle:

2πr = 50π/3 + 40

Dividing both sides by 2π to find r:

r = 50π/(3(2π)) + 40/(2π) = 25/3 + 20/π

Using the approximate value of π ≈ 3.14159:

r ≈ 8.333 + 6.366 = 14.699 cm.

Comparing this result with the given options, 14.699 cm does not match 19/3 cm, 7 cm, or 8 cm. Therefore, the correct option is d.

Explanation

The area of a triangle formed by joining the mid-points of the sides of a given triangle is exactly one-fourth of the area of the original triangle. This is based on the property that the mid-segment of a triangle is parallel to the third side and half its length, creating four smaller congruent triangles within the original.

Let the area of the original triangle be A₁ = 128 cm². Since triangle PQR is formed by joining the mid-points of the sides of the original triangle, its area A₂ is: A₂ = 1/4 × A₁ = 1/4 × 128 = 32 cm²

Similarly, triangle ABC is formed by joining the mid-points of the sides of triangle PQR. Therefore, its area A₃ is one-fourth of the area of triangle PQR: A₃ = 1/4 × A₂ = 1/4 × 32 = 8 cm²

Alternatively, since the process is repeated twice, the final area is A₁ × (1/4)² = 128 × 1/16 = 8 cm². This corresponds to option b.

Explanation

Since line p is parallel to line q, the alternate interior angles formed by the transversals BE and CD are equal. Specifically, angle ACB equals angle ADE and angle ABC equals angle AED. Additionally, angle BAC equals angle EAD because they are vertically opposite angles. By the AA similarity criterion, triangle ABC is similar to triangle ADE.

The standard geometric theorem for similar triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Given that AC and AD are corresponding sides of triangle ABC and triangle ADE respectively, the ratio of their areas can be calculated using the formula: Area(△ABC)/Area(△ADE) = (AC/AD)² Substituting the given ratio AC:AD = 4:9 into the formula, we get: Area(△ABC)/Area(△ADE) = (4/9)² = 16/81 Therefore, the ratio of the area of triangle ABC to that of triangle ADE is 16:81, which corresponds to option c.

Explanation

Let the total length of the wire be L. Since the wire is used to form an equilateral triangle, its perimeter is L, meaning the side of the triangle is sₜ = L/3. Similarly, the perimeter of the square is L, so the side of the square is sₛ = L/4.

Using the area formulas, the area of the equilateral triangle is Aₜ = (√3/4) sₜ² and the area of the square is Aₛ = sₛ². Substituting the side values, we get Aₜ = (√3/4) (L/3)² = √3L²/36 and Aₛ = (L/4)² = L²/16.

The ratio of the area of the triangle to the area of the square is Aₜ : Aₛ = √3L²/36 : L²/16. Simplifying this expression by canceling L² and multiplying by the reciprocal of the second term, we get (√3/36) × 16/1 = 16√3/36. Reducing the fraction by dividing both numerator and denominator by 4, we obtain 4√3/9. Therefore, the required ratio is 4√3 : 9, which matches option (c).

Explanation

A parallelogram with all four sides of equal length is defined as a rhombus. The area of a rhombus is calculated using the formula Area = 1/2 × d₁ × d₂, where d₁ and d₂ are the lengths of the two diagonals. Given that the diagonals are in the ratio 1:2, we can express them as d₁ = x and d₂ = 2x. The problem states that the sum of the lengths of the diagonals is 12 cm, which gives the equation x + 2x = 12. Solving for x, we find 3x = 12, resulting in x = 4 cm. Consequently, the lengths of the diagonals are d₁ = 4 cm and d₂ = 8 cm. Substituting these values into the area formula, we get Area = 1/2 × 4 × 8 = 16 cm². This matches option c.

Explanation

In a right angled triangle, the length of the perpendicular drawn from the vertex containing the right angle to the hypotenuse is given by the formula h = (a × b)/c, where a and b are the lengths of the two legs and c is the length of the hypotenuse. First, we calculate the hypotenuse AC using the Pythagoras theorem: AC = √(AB² + BC²). Substituting the given values, AC = √(5² + 10²) = √(25 + 100) = √125 = 5√5 cm. Now, substituting AB = 5, BC = 10, and AC = 5√5 into the altitude formula, we get h = (5 × 10)/(5√5) = 50/(5√5) = 10/√5. To rationalize the denominator, multiply the numerator and denominator by √5, which gives h = 10√5/5 = 2√5 cm. Thus, option b is correct.

Explanation

V1 = V2 implies pi r1^2 h1 = pi r2^2 h2. Thus, r1^2 / r2^2 = h2 / h1. Given h1:h2 = 2:3, then r1^2 / r2^2 = 3/2. Taking the square root, r1 / r2 = sqrt(3) / sqrt(2).

Explanation

The net percentage increase in the area of a rectangle when its length and breadth are increased by x% and y% respectively can be calculated using the formula for successive percentage increase:

Net Increase = x + y + xy/100

Substituting the given values where the increase in length x = 20 and the increase in breadth y = 10:

Net Increase = 20 + 10 + (20 × 10)/100 = 30 + 200/100 = 30 + 2 = 32%

Alternatively, this can be solved using the multiplier method. If the original length is L and breadth is B, the original area is L × B. The new length becomes 1.20L and the new breadth becomes 1.10B. The new area is 1.20L × 1.10B = 1.32LB. The fractional increase is 1.32 - 1 = 0.32, which is equivalent to 32%. Therefore, option a is the correct choice.

Explanation

For a right triangle with a fixed hypotenuse, the area is maximized when the triangle is isosceles (legs are equal). Let legs be a. a^2 + a^2 = 10^2 = 100, so 2a^2 = 100, and a^2 = 50. Max Area = (1/2) a a = (1/2) * 50 = 25 cm^2.

Explanation

Let the radius of the circle be r = 201 mm. For a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Therefore, the diagonal d = 2r. The area of the circle is given by the formula A꜀ᵢᵣ꜀ₗₑ = πr². The area of a square can be expressed in terms of its diagonal as Aₛᵩᵤₐᵣₑ = d²/2. Substituting d = 2r into this formula, we get Aₛᵩᵤₐᵣₑ = (2r)²/2 = 4r²/2 = 2r². To find the ratio of the area of the circle to that of the square, we divide the two expressions: A꜀ᵢᵣ꜀ₗₑ/Aₛᵩᵤₐᵣₑ = πr²/2r² = π/2. Substituting the standard value of π ≈ 22/7, the ratio becomes (22/7)/2 = 22/14 = 11/7. This results in the ratio 11:7, which corresponds to option (a).

Explanation

Given the diameter of the cylinder is 20 cm, the radius r is 20/2 = 10 cm. The formula for the curved surface area (CSA) of a right circular cylinder is CSA = 2πrh. Substituting the given values, we have 1000 = 2π(10)h, which simplifies to 1000 = 20πh. Solving for the height h, we get h = 1000/(20π) = 50/π cm.

The formula for the volume V of a cylinder is V = πr²h. By substituting the radius r = 10 cm and the calculated height h = 50/π cm into the volume formula, we get V = π(10)²(50/π). The π terms in the numerator and denominator cancel out, leaving V = 100 × 50 = 5000 cm³. Thus, option (c) is the correct answer.

Explanation

The length of an arc of a circle is given by the formula l = θ/360° × 2πr, where l is the arc length, θ is the angle subtended at the center in degrees, r is the radius of the circle, and π is approximately 22/7.

Given values are: Arc length l = 33 cm Radius r = 14 cm

Substituting these values into the formula: 33 = θ/360° × 2 × 22/7 × 14

Simplifying the right side: 33 = θ/360° × 2 × 22 × 2 33 = θ/360° × 88

Now, solve for θ: θ = (33 × 360°)/88

Dividing both 33 and 88 by their greatest common divisor, 11: θ = (3 × 360°)/8 θ = 3 × 45° θ = 135°

Thus, the angle subtended by the arc at the centre of the circle is 135°, which corresponds to option c.

Explanation

Let the side of the square inscribed in the semicircle be s₁. The base of this square lies on the diameter of the semicircle, and its upper vertices touch the arc. The distance from the center of the circle to the upper vertex is the radius r. Forming a right triangle with the radius as the hypotenuse, the height as s₁, and the base as s₁/2, we apply the Pythagorean theorem: (s₁/2)² + s₁² = r². This simplifies to s₁²/4 + s₁² = r², which means 5s₁²/4 = r². Therefore, the area of the square in the semicircle is A₁ = s₁² = 4r²/5.

Now, let the side of the square inscribed in the circle be s₂. For a square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Thus, s₂√2 = 2r. Squaring both sides gives 2s₂² = 4r², which simplifies to s₂² = 2r². Therefore, the area of the square in the circle is A₂ = 2r².

The required ratio is A₁/A₂ = (4r²/5) / (2r²) = 4/10 = 2/5. This matches option b.

Explanation

Using synthetic division to divide x^4 + 0x^3 - x^2 + 7x + 5 by (x + 2): The coefficients are 1, 0, -1, 7, 5. Bringing down 1: 1. 1 * (-2) = -2; 0 + (-2) = -2. -2 * (-2) = 4; -1 + 4 = 3. 3 * (-2) = -6; 7 + (-6) = 1. 1 * (-2) = -2; 5 + (-2) = 3 (remainder). The quotient coefficients are 1, -2, 3, 1. Thus a = 1, b = -2, c = 3, d = 1.

Explanation

Using Heron's formula, semi-perimeter s = (30 + 28 + 16) / 2 = 37. Area = sqrt(s (s-a) (s-b) (s-c)) = sqrt(37 (37-30) (37-28) (37-16)) = sqrt(37 7 9 * 21). The quantities for which logarithms are needed are 37, 7, 9, and 21.

Explanation

Assume log10(1995) = 3.3 (the expression 3 3000 is likely a typo for 3.3000). Let y = (0.001995)^(1/8). Then log10(y) = (1/8) log10(1995 10^-6) = (1/8) (log10(1995) - 6). Substituting 3.3 for log10(1995), log10(y) = (1/8) * (3.3 - 6) = -2.7 / 8 = -0.3375. Thus, y = 10^(-0.3375) = 1 / (10^0.3375).

Explanation

The expansion of (x - a)(x - b)(x - c) is calculated as follows: (x - a)(x - b) = x^2 - (a + b)x + ab. Then (x^2 - (a + b)x + ab)(x - c) = x^3 - cx^2 - (a + b)x^2 + (a + b)cx + abx - abc = x^3 - (a + b + c)x^2 + (ab + bc + ca)x - abc.

Explanation

The area A of an equilateral triangle with side s is given by the formula A = (√3/4)s². Substituting s = 7 cm, we get A = (√3/4) × 7² = 49√3/4. We need to find the value of log₁₀ A⁴, which can be written as log₁₀ (A⁴) = log₁₀ (49√3/4)⁴ = log₁₀ (49⁴ × (√3)⁴)/(4⁴) = log₁₀ (7⁸ × 3²)/(2⁸).

Using logarithm properties, this expression expands to log₁₀ A⁴ = 8 log₁₀ 7 + 2 log₁₀ 3 - 8 log₁₀ 2, which can be rearranged as 8(log₁₀ 7 - log₁₀ 2) + 2 log₁₀ 3. From the given values, log₁₀ 35 = log₁₀ (7 × 5) = log₁₀ 7 + log₁₀ 5 = 1.5441. Since log₁₀ 5 = 1 - log₁₀ 2, we have log₁₀ 7 + 1 - log₁₀ 2 = 1.5441, which simplifies to log₁₀ 7 - log₁₀ 2 = 0.5441.

Additionally, log₁₀ 1050 = log₁₀ (3 × 35 × 10) = log₁₀ 3 + log₁₀ 35 + log₁₀ 10 = 3.0212. Substituting log₁₀ 35 = 1.5441 and log₁₀ 10 = 1, we get log₁₀ 3 + 1.5441 + 1 = 3.0212, so log₁₀ 3 = 3.0212 - 2.5441 = 0.4771.

Substituting these values into the expression for log₁₀ A⁴: log₁₀ A⁴ = 8(0.5441) + 2(0.4771) = 4.3528 + 0.9542 = 5.3070. Thus, option a) is correct.

Explanation

The volume of the material in the hollow sphere remains constant when it is melted into a cone. The volume of a hollow sphere is calculated using the formula V = 4/3 π (R³ - r³), where R is the external radius and r is the internal radius. Given the external diameter is 6 cm and internal diameter is 4 cm, we have R = 3 cm and r = 2 cm. Substituting these values, the volume of the hollow sphere is V = 4/3 π (3³ - 2³) = 4/3 π (27 - 8) = 4/3 π × 19 = 76π/3 cm³. The volume of a cone is given by the formula V = 1/3 π r꜀ₒₙₑ² h. Given the base diameter of the cone is 8 cm, the radius r꜀ₒₙₑ is 4 cm. Equating the two volumes, we get 1/3 π (4²) h = 76π/3. Simplifying the equation, 16πh/3 = 76π/3, which leads to 16h = 76. Solving for h, we find h = 76/16 = 4.75 cm. Thus, option a is the correct answer.

Explanation

The initial flat surface area of the cylinder consists of the top and bottom circles, calculated as Aᵢₙᵢₜᵢₐₗ = 2πr². Substituting r = 6cm, we get Aᵢₙᵢₜᵢₐₗ = 2π(6)² = 72π cm². The total volume of the metallic cylinder is V꜀ᵧₗ = πr²h = π(6)²(10) = 360π cm³. This volume is divided between two cones in the ratio 1:2, so the volumes are V₁ = 1/3(360π) = 120π cm³ and V₂ = 2/3(360π) = 240π cm³. Using the cone volume formula V = 1/3πr²h with h = 10cm, we find the squared radii of the cones: r₁² = 3V₁/πh = 3(120π)/π(10) = 36 and r₂² = 3V₂/πh = 3(240π)/π(10) = 72. The total flat surface area of the two cones is the sum of their base areas, A_final = πr₁² + πr₂² = π(36) + π(72) = 108π cm². The increase in area is 108π - 72π = 36π cm². The percentage increase is 36π/72π × 100 = 50%, which corresponds to option b.

Explanation

To find the maximum perimeter of a right-angled triangle with integer sides where one side is 15 cm, we use the Pythagorean theorem a² + b² = c². To maximize the perimeter, the given side 15 must be a leg rather than the hypotenuse, as the hypotenuse is always the longest side. Let a = 15. We then have c² - b² = 15², which simplifies to (c - b)(c + b) = 225.

To maximize the sum b + c, we need the smallest possible positive integer value for the factor (c - b). The smallest factor of 225 is 1. Setting c - b = 1 and c + b = 225, we can solve for b and c. Adding the two equations gives 2c = 226, so c = 113. Subtracting the first from the second gives 2b = 224, so b = 112. The three integer sides are thus 15, 112, and 113. The perimeter P is calculated as P = a + b + c = 15 + 112 + 113 = 240 cm. This corresponds to option a.

Explanation

A skeleton cube has 12 edges. Rod of 24 ft cut into 12 equal pieces: each edge = 24 ÷ 12 = 2 ft. Area of one face = 2² = 4 square feet.

Explanation

In a trapezium ABCD with an incircle, the length of the side BC is equal to the sum of the tangents from vertices B and C to the circle. Given EB = 25 cm and FC = 16 cm, the length of BC is calculated as BC = BE + CF = 25 + 16 = 41 cm. Since AD is perpendicular to the parallel bases AB and CD, the height AD is equal to the diameter d of the incircle.

Let r be the radius of the incircle. The tangents from vertex A to the circle are equal, so AE = r. Similarly, the tangents from vertex D to the circle are equal, so DF = r. Therefore, the lengths of the bases are AB = r + 25 and CD = r + 16.

By dropping a perpendicular from C to AB, we form a right triangle where the height is d, the hypotenuse is BC = 41 cm, and the base is the difference between the two parallel sides: AB - CD = (r + 25) - (r + 16) = 9 cm. Applying the Pythagorean theorem:

d² + 9² = 41² d² + 81 = 1681 d² = 1600 d = √1600 = 40 cm.

Explanation

The volume of the large sphere is the sum of the volumes of the three smaller spheres: (4/3)πR^3 = (4/3)π(3^3 + 4^3 + 5^3) = (4/3)π(27 + 64 + 125) = (4/3)π(216). Therefore, R^3 = 216, which gives R = 6 cm.

Explanation

The volume of a hemisphere is (2/3)πr^3. Given (2/3)πr^3 = 155232. Using π = 22/7, we get (2/3) (22/7) r^3 = 155232, which simplifies to (44/21)r^3 = 155232. Solving for r^3: r^3 = (155232 21) / 44 = 3528 21 = 74088. The cube root of 74088 is 42. Thus, r = 42 cm.

Explanation

The capacity of a bucket in the form of a truncated cone is calculated using the formula for the volume of a frustum of a cone: V = 1/3 πh(R² + r² + Rr). Given the diameters of the top and base are 12 cm and 6 cm respectively, we first find the radii: R = 12/2 = 6 cm and r = 6/2 = 3 cm. The height h is given as 7 cm. Substituting these values and π = 22/7 into the formula, we get V = 1/3 × 22/7 × 7 × (6² + 3² + 6 × 3). Simplifying the expression, the 7 in the numerator and denominator cancel out, leaving V = 22/3 × (36 + 9 + 18). This becomes V = 22/3 × 63. Further simplification gives V = 22 × 21, which equals 462 cm³. Therefore, option (b) is the correct answer.

Explanation

The slant height l of the cone is sqrt(r^2 + h^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = 10 cm. The total surface area of a right circular cone is πrl + πr^2 = π(6)(10) + π(6^2) = 60π + 36π = 96π cm^2.

Explanation

When six cubes of edge a = 12cm are joined end to end, they form a cuboid with length l = 6 × 12 = 72cm, width w = 12cm, and height h = 12cm. The total surface area (TSA) of a cuboid is given by the formula TSA = 2(lw + wh + lh). Substituting the values, we get TSA = 2(72 × 12 + 12 × 12 + 12 × 72). This simplifies to TSA = 2(864 + 144 + 864) = 2(1872) = 3744cm². Alternatively, a faster way is to count the exposed faces. Six separate cubes have 6 × 6 = 36 faces. When joined end to end, there are 5 joints, and each joint hides 2 faces. Therefore, the number of exposed faces is 36 - (5 × 2) = 26. Since the area of one face is 12 × 12 = 144cm², the total surface area is 26 × 144 = 3744cm², which corresponds to option (c).

Explanation

Let the required real number be x. According to the given condition, the sum of the number and its reciprocal is expressed by the equation x + 1/x = 26/5. To solve for x, multiply the entire equation by 5x to eliminate the denominators, which results in 5x² + 5 = 26x. Rearranging this into the standard quadratic form ax² + bx + c = 0, we obtain 5x² - 26x + 5 = 0.

To determine the number of possible real solutions, we use the discriminant formula D = b² - 4ac. Substituting the values a = 5, b = -26, and c = 5, we calculate D = (-26)² - 4(5)(5) = 676 - 100 = 576. Since the discriminant is positive (D > 0), the quadratic equation must have two distinct real roots. We can find these roots by factoring the equation: 5x² - 25x - x + 5 = 0, which simplifies to 5x(x - 5) - 1(x - 5) = 0, leading to (5x - 1)(x - 5) = 0. This gives x = 5 and x = 1/5. Thus, there are two such real numbers, making option c the correct choice.

Explanation

Statement 1 is correct. Two numbers are relatively prime if their greatest common divisor (GCD) is 1. If gcd(p, q) = 1 and gcd(p, r) = 1, then p shares no prime factors with q and no prime factors with r. Since the prime factors of the product qr are simply the union of the prime factors of q and r, p cannot share any prime factor with qr. Thus, gcd(p, qr) = 1, making p relatively prime to the product qr.

Statement 2 is incorrect. For p to divide r given p divides qr, the necessary condition is that p must be relatively prime to q, not that p must divide q. This is known as Euclid's Lemma. To disprove Statement 2, consider p = 2, q = 2, and r = 3. Here, p divides q since 2 divides 2, and p divides the product qr since 2 divides 6. However, p does not divide r because 2 does not divide 3. Therefore, the condition that p divides q does not force p to divide r. Thus, only option a is correct.

Explanation

Let Radha's current age be Rd and Rani's be Rn. Five years ago: Rd - 5 = 3(Rn - 5) => Rd - 3Rn = -10. One year ago: Rd - 1 = 2(Rn - 1) => Rd - 2Rn = -1. Subtracting the first equation from the second gives Rn = 9. Then Rd = 2(9) - 1 = 17. The age difference is 17 - 9 = 8.

Explanation

The equation is 5x + 7y = 500, where x and y are positive integers. 7y = 5(100 - x), which implies 7y must be a multiple of 5, so y must be a multiple of 5 (y = 5k). Substituting y = 5k: 7(5k) = 5(100 - x) => 7k = 100 - x => x = 100 - 7k. For x >= 1, 100 - 7k >= 1 => 7k <= 99 => k <= 14. For y >= 1, k >= 1. Thus, k can be any integer from 1 to 14, giving 14 ways.

Explanation

y = k / sqrt(x). Given y = 36 and x = 36, 36 = k / sqrt(36) = k / 6, so k = 216. When y = 54, 54 = 216 / sqrt(x) => sqrt(x) = 216 / 54 = 4, so x = 16.

Explanation

We look for a value (a + b)^2 = a^2 + b^2 + 2ab = 16 + 6sqrt(7). Setting 2ab = 6sqrt(7) gives ab = 3sqrt(7). Testing a = 3 and b = sqrt(7), we get a^2 + b^2 = 9 + 7 = 16. Thus, 16 + 6sqrt(7) = (3 + sqrt(7))^2, and its square root is 3 + sqrt(7).

Explanation

Number of digits = floor(log10(N)) + 1. For 7^25: 25 0.845 = 21.125, digits = 22. For 8^23: 23 log10(2^3) = 69 0.301 = 20.769, digits = 21. For 9^20: 20 log10(3^2) = 40 * 0.477 = 19.08, digits = 20. The number of digits are 22, 21, and 20.

Explanation

The problem asks for the smallest positive integer x that satisfies two conditions: x ≡ 7 (mod 14) and x ≡ 5 (mod 15). Using the first condition, we can express x as x = 14k + 7, where k is a non-negative integer. We substitute this expression into the second condition to find k: 14k + 7 ≡ 5 (mod 15) 14k ≡ -2 (mod 15) Since 14 ≡ -1 (mod 15), we can simplify the congruence to: -1k ≡ -2 (mod 15) Multiplying both sides by -1 gives: k ≡ 2 (mod 15) The smallest non-negative integer value for k is k = 2. Substituting this value back into the expression for x: x = 14(2) + 7 = 28 + 7 = 35 Verifying with the second condition, 35 ÷ 15 gives a quotient of 2 and a remainder of 5, which is correct. Since x = 35, it satisfies the range 30 < x < 40, making option b the correct choice.

Explanation

Let the total capacity of the water tank be 1 unit. The rate of drainage for tap X is Rₓ = 1/20 tanks per minute. When both taps X and Y are opened, the combined rate of drainage is Rₓ₊ᵧ = 1/15 tanks per minute. According to the work-rate formula for combined efforts, Rₓ₊ᵧ = Rₓ + Rᵧ. To find the rate of tap Y alone, we rearrange the formula to Rᵧ = Rₓ₊ᵧ - Rₓ. Substituting the given values, we have Rᵧ = 1/15 - 1/20. Finding a common denominator of 60, the calculation becomes Rᵧ = 4/60 - 3/60 = 1/60 tanks per minute. The time required for tap Y to drain the full tank is the reciprocal of its rate: Time = 1 / Rᵧ = 1 / (1/60) = 60 minutes. Therefore, option (c) is the correct answer.

Explanation

Statement 1 is incorrect because √75 can be simplified as √(25 × 3) = 5√3. Since √3 is a well known irrational number, any non zero rational multiple of it is also irrational.

Statement 2 is correct. To check the existence of a positive integer x for the inequality -4x/5 < -7/8, we multiply both sides by -1 and reverse the inequality sign to get 4x/5 > 7/8. Multiplying by 5/4 gives x > 35/32, which is approximately x > 1.09375. Any positive integer x ≥ 2 satisfies this condition.

Statement 3 is incorrect because it does not hold for all real values of x. Simplifying the expression gives (x - 2)/x = 1 - 2/x. The inequality 1 - 2/x < 1 simplifies to -2/x < 0, which implies 2/x > 0. This is only true when x is positive. If x is negative, say x = -1, the expression becomes (-1 - 2)/-1 = 3, and 3 is not less than 1.

Statement 4 is correct because 4.232323.... is a non terminating repeating decimal. Any repeating decimal is a rational number and can be expressed as p/q. Specifically, if n = 4.2323..., then 100n = 423.2323..., so 99n = 419, which means n = 419/99.

Therefore, only statements 2 and 4 are correct, making option d the correct choice.

Explanation

A 30-day month beginning with Saturday has 5 Saturdays and 5 Sundays (Sundays are on 2, 9, 16, 23, 30). Other days = 30 - 5 = 25. Total visitors = (5 510) + (25 240) = 2550 + 6000 = 8550. Average per day = 8550 / 30 = 285.

Explanation

36/11 = 3 + 3/11. So 1/(x + 1/(y + 1/z)) = 3/11, which means x + 1/(y + 1/z) = 11/3 = 3 + 2/3. Thus x = 3. Then 1/(y + 1/z) = 2/3, which means y + 1/z = 3/2 = 1 + 1/2. Thus y = 1 and z = 2. The sum x + y + z = 3 + 1 + 2 = 6.

Explanation

When two items are sold at the same selling price, where one is sold at a profit of x% and the other at a loss of x%, there is always a net loss. The shortcut formula to calculate the overall loss percentage is given by Loss % = x²/100. In this problem, the selling price of each item is Rs. 990 and the common percentage value x is 10. Substituting the value of x into the formula, we get Loss % = 10²/100 = 100/100 = 1%. Therefore, the combined result for the two items is a 1% loss, which matches option a. This method is the fastest way to solve such problems in competitive exams as it eliminates the need to calculate the individual cost prices of the two items.

Explanation

Let v_c and v_t be the speeds of the car and the train in km/h respectively. Using the formula Time = Distance/Speed, we can set up two equations based on the given conditions.

In the first case, 120 km is covered by train and 480 km by car in 11 hours: 120/v_t + 480/v_c = 11 (Equation 1)

In the second case, 200 km is covered by train and 400 km by car in 11 hours 40 minutes. Since 40 minutes is 40/60 = 2/3 hours, the total time is 11 + 2/3 = 35/3 hours: 200/v_t + 400/v_c = 35/3 (Equation 2)

Subtracting Equation 1 from Equation 2: (200 - 120)/v_t + (400 - 480)/v_c = 35/3 - 11 80/v_t - 80/v_c = 2/3 Dividing by 80, we get 1/v_t - 1/v_c = 2/(3 × 80) = 1/120 (Equation 3)

Now, divide Equation 1 by 120: 1/v_t + 4/v_c = 11/120 (Equation 4)

Subtracting Equation 3 from Equation 4: (1/v_t + 4/v_c) - (1/v_t - 1/v_c) = 11/120 - 1/120 5/v_c = 10/120 = 1/12 v_c = 60 km/h

Substitute v_c = 60 into Equation 3: 1/v_t - 1/60 = 1/120 1/v_t = 1/120 + 2/120 = 3/120 = 1/40 v_t = 40 km/h

The ratio of the speed of the car to that of the train is v_c : v_t = 60 : 40 = 3 : 2, which matches option a.

Explanation

Let f(x) = x - x^2. To find the maximum, we take the derivative f'(x) = 1 - 2x and set it to 0, giving x = 0.5. The second derivative f''(x) = -2 is negative, confirming a maximum.

Explanation

To find the largest n such that 10ⁿ divides the product, factorise into primes (10ⁿ = 2ⁿ × 5ⁿ, so n = min(power of 2, power of 5)).

6²³ = 2²³ × 3²³ 75⁹ = (3 × 5²)⁹ = 3⁹ × 5¹⁸ 100 = 2² × 5²

Combined: 2²³⁺² × 5¹⁸⁺² × 3³² = 2²⁵ × 5²⁰ × 3³²

n = min(25, 20) = 20

Explanation

The units digit of 3^k follows a cycle of 4: 3, 9, 7, 1. For 3^98, 98 mod 4 = 2, so the units digit is 9. For 3^89, 89 mod 4 = 1, so the units digit is 3. The units digit of 3^98 - 3^89 is 9 - 3 = 6.

Explanation

Let the four consecutive natural numbers be x, x+1, x+2, x+3. Their sum of squares is x^2 + (x+1)^2 + (x+2)^2 + (x+3)^2 = 4x^2 + 12x + 14 = 294. This simplifies to 4x^2 + 12x - 280 = 0, or x^2 + 3x - 70 = 0. Factoring gives (x+10)(x-7) = 0. Since x must be natural, x = 7. The numbers are 7, 8, 9, 10, and their sum is 34.

Explanation

For a quadratic equation ax² + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a. In the given equation x² + px + q = 0, the coefficients are a = 1, b = p, and c = q. Since the roots are stated to be p and q, we can set up the following system of equations: Sum of roots: p + q = -p/1 = -p Product of roots: p ⋅ q = q/1 = q From the product equation pq = q, we can divide both sides by q because the problem specifies q ≠ 0, which directly gives p = 1. Now, substitute p = 1 into the sum equation: 1 + q = -1. Solving for q yields q = -1 - 1 = -2. Therefore, the values of p and q are 1 and -2 respectively, making option a the correct choice.

Explanation

x^2 - y^2 = 35 implies (x-y)(x+y) = 35. The factor pairs of 35 are (1, 35) and (5, 7). Case 1: x-y=1, x+y=35 => 2x=36, x=18, y=17. Case 2: x-y=5, x+y=7 => 2x=12, x=6, y=1. There are 2 such pairs.

Explanation

Since (b-6) is a root of x^2 - 6x + b = 0, substituting x = b-6 gives (b-6)^2 - 6(b-6) + b = 0. Expanding this: b^2 - 12b + 36 - 6b + 36 + b = 0 => b^2 - 17b + 72 = 0. Factoring gives (b-8)(b-9) = 0, so b = 8 or b = 9. The maximum value of b^2 is 9^2 = 81.

Explanation

a = sqrt(7 + 4sqrt(3)). Note that 7 + 4sqrt(3) = (2 + sqrt(3))^2. So a = 2 + sqrt(3). Then 1/a = 1 / (2 + sqrt(3)) = 2 - sqrt(3). The sum a + 1/a = (2 + sqrt(3)) + (2 - sqrt(3)) = 4.

Explanation

To maximize 1 / (x^2 + 5x + 10), we minimize the denominator g(x) = x^2 + 5x + 10. The minimum occurs at x = -b/2a = -5/2 = -2.5. g(-2.5) = (-2.5)^2 + 5(-2.5) + 10 = 6.25 - 12.5 + 10 = 3.75 = 15/4. The maximum value is 1 / (15/4) = 4/15.

Explanation

The work done is (workers * days). The ratio is [(x + 2)(x - 3)] / [(x + 4)(x - 2)] = 3/4. Cross-multiplying: 4(x^2 - x - 6) = 3(x^2 + 2x - 8) => 4x^2 - 4x - 24 = 3x^2 + 6x - 24 => x^2 - 10x = 0. Since x must be greater than 3, x = 10.

Explanation

Statement (a) is correct because a prime number is defined as a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Since 1 has only one divisor, it cannot be prime, nor can it be composite as composite numbers require more than two divisors. Statement (b) is mathematically accurate since 0 is the additive identity and serves as the boundary between positive and negative numbers, meaning it belongs to neither category. Statement (c) is incorrect based on the properties of parity. For the product p × q to be even, it is only required that at least one of the integers p or q be even. A simple counter-example is p = 2 and q = 3; here, p × q = 6, which is even, yet q is odd. Thus, the claim that both must always be even is false. Statement (d) is a fundamental fact of number theory, as √2 is a non-terminating, non-repeating decimal that cannot be written as a simple fraction a/b where a and b are integers.

Explanation

Let n^2 + 19n + 92 = k^2. Multiplying by 4: 4n^2 + 76n + 368 = 4k^2 => (2n + 19)^2 + 7 = (2k)^2 => (2k)^2 - (2n + 19)^2 = 7. This is a difference of squares: (2k - (2n + 19))(2k + (2n + 19)) = 7. The factor pairs of 7 are (1, 7), (7, 1), (-1, -7), and (-7, -1). Solving (2k + 2n + 19) - (2k - 2n - 19) = 4n + 38 = 7 - 1 = 6 gives n = -8. Solving 4n + 38 = 1 - 7 = -6 gives n = -11. The sum of integer values is -8 + (-11) = -19.

Explanation

The first polynomial x³ + 3x² + 3x + 1 follows the algebraic identity (a+b)³ = a³ + 3a²b + 3ab² + b³, allowing it to be written as (x + 1)³. The third polynomial x² + 5x + 4 is a quadratic expression that can be factored by splitting the middle term 5x into 4x + x, resulting in (x + 1)(x + 4). For the second polynomial x³ + 5x² + 5x + 4, we use the factor theorem. Testing x = -4 gives (-4)³ + 5(-4)² + 5(-4) + 4 = -64 + 80 - 20 + 4 = 0, which confirms that (x + 4) is a factor. Dividing x³ + 5x² + 5x + 4 by (x + 4) using polynomial division yields the quotient x² + x + 1. Thus, the second polynomial is factored as (x + 4)(x² + x + 1). The LCM of polynomials is the product of the highest powers of all distinct factors present across the expressions. The distinct factors are (x + 1) with the highest power (x + 1)³, (x + 4) with the highest power (x + 4)¹, and (x² + x + 1) with the highest power (x² + x + 1)¹. Multiplying these together, we get (x + 1)³(x + 4)(x² + x + 1), which matches option (a).

Explanation

Let a = x - y, b = y - z, and c = z - x. First, observe the sum of these variables: a + b + c = (x - y) + (y - z) + (z - x) = 0. According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc. Substituting the original terms back into this identity, we find that the numerator (x - y)³ + (y - z)³ + (z - x)³ = 3(x - y)(y - z)(z - x). Now, substitute this result into the given expression: 3(x - y)(y - z)(z - x) / 9(x - y)(y - z)(z - x). By canceling the common terms (x - y)(y - z)(z - x) from the numerator and denominator, the expression simplifies to 3/9, which further reduces to 1/3. Therefore, option (b) is the correct value.

Explanation

X = {a, {b}, c} and Y = {{a}, b, c}. The intersection X cap Y = {c}. Then (X cap Y) cap Z = {c} cap {a, b, {c}}. Since c is not an element of Z (only {c} is), the intersection is empty (phi).

Explanation

For a quadratic equation of the form ax² + bx + c = 0, the sum of the roots is given by the formula -b/a and the product of the roots is given by c/a. Comparing the given equation px² + 3x + 2q = 0 with the standard form, we identify a = p, b = 3, and c = 2q.

According to the problem, the sum of the roots is -6. Using the formula -b/a = -6, we substitute the values to get -3/p = -6. Solving for p, we find 6p = 3, which gives p = 1/2.

Next, the product of the roots is also given as -6. Using the formula c/a = -6, we substitute the values to get 2q/p = -6. Now, substituting the value of p = 1/2 into this equation, we have 2q / (1/2) = -6, which simplifies to 4q = -6. Solving for q, we get q = -6/4 = -3/2.

To find the value of (p - q), we substitute the values of p and q: (p - q) = 1/2 - (-3/2) = 1/2 + 3/2 = 4/2 = 2.

Thus, the value is 2, which corresponds to option (c).