CDS II 2008 Elementary Mathematics Question Paper with Answers & Explanations

Explanation

Let the number to be added to each term of the ratio be x. According to the given condition, the new ratio is 3:4. We can use the formula for the modification of ratios: (a + x) / (b + x) = c / d Substituting the given values a = 49, b = 68, c = 3, and d = 4 into the formula, we get: (49 + x) / (68 + x) = 3 / 4 Cross-multiplying the terms gives: 4(49 + x) = 3(68 + x) 196 + 4x = 204 + 3x Subtracting 3x from both sides: 196 + x = 204 Subtracting 196 from both sides: x = 204 - 196 x = 8 Therefore, the number to be added to each term is 8, which corresponds to option c. We can verify this by adding 8 to the original terms: 49 + 8 = 57 and 68 + 8 = 76. The resulting ratio is 57:76. Dividing both terms by their highest common factor 19, we get 57/19 : 76/19, which simplifies to 3:4.

Explanation

To find the number that must be added to each term so that they are in proportion, let the required number be x. According to the property of proportions, for four numbers a, b, c, d to be in proportion, the ratio of the first two must equal the ratio of the last two. Therefore, we set up the equation as (4 + x)/(10 + x) = (12 + x)/(24 + x).

By applying cross-multiplication, we get: (4 + x)(24 + x) = (10 + x)(12 + x)

Expanding both sides: 96 + 4x + 24x + x² = 120 + 10x + 12x + x² 96 + 28x + x² = 120 + 22x + x²

Subtracting x² from both sides and rearranging the linear terms: 28x - 22x = 120 - 96 6x = 24 x = 24/6 x = 4

Alternatively, using the options method for a faster approach, substituting option d (4) gives (4 + 4)/(10 + 4) = 8/14 = 4/7 and (12 + 4)/(24 + 4) = 16/28 = 4/7. Since both ratios are equal, the number to be added is 4.

Explanation

Use the direct relation between Cost Price (CP) and Marked Price (MP) when both discount and loss percentages are given: CP/MP = (100 - Discount%)/(100 - Loss%). Here, the marked price MP = 18000, discount percentage is 20%, and loss percentage is 4%. Substituting these values into the formula, we get CP/18000 = (100 - 20)/(100 - 4), which simplifies to CP/18000 = 80/96. Simplifying the fraction 80/96 by dividing both numerator and denominator by 16, we get 5/6. Therefore, CP = 18000 × 5/6. Dividing 18000 by 6 gives 3000, and 3000 × 5 = 15000. The cost price of the machine is Rs. 15000, which corresponds to option d.

Explanation

Let the principal be P. Since the sum triples itself, the amount A after 25 years becomes 3P. The simple interest SI earned is the difference between the amount and the principal, so SI = A - P = 3P - P = 2P. Using the standard simple interest formula SI = (P × R × T)/100, where R is the rate of interest per annum and T is the time period in years, we substitute the known values: 2P = (P × R × 25)/100. Dividing both sides by P gives 2 = (R × 25)/100. Simplifying the fraction 25/100 to 1/4 yields 2 = R/4, which means R = 2 × 4 = 8%. Therefore, the rate of simple interest required for the sum to triple itself in 25 years is 8%, which corresponds to option a.

Explanation

The relationship between the number of workers and the time taken to complete a fixed piece of work is inversely proportional. This is governed by the formula for total work: Total Work = Number of Men × Number of Days.

In the first case, the total work is given by the product of m men and p days, which is m × p man-days. In the second case, the number of men increases to (m + r). Let the number of days required be D. Since the total amount of work remains constant, we set up the equation: (m + r) × D = m × p. To find the value of D, we divide both sides by (m + r), resulting in D = mp/(m + r). Therefore, (m + r) men can complete the job in mp/(m + r) days, which corresponds to option b.

Explanation

To solve this problem, we use the standard shortcut formula for work equivalence problems involving or/and conditions. When one man or two women or three boys can complete a work in D days, the time taken by one man, one woman, and one boy together is given by the formula: Time = D/(1/m + 1/w + 1/b), where m, w, and b are the number of men, women, and boys who can do the work individually in D days.

Given D = 55 days, m = 1, w = 2, and b = 3. Substituting these values into the formula: Time = 55/(1/1 + 1/2 + 1/3)

First, calculate the sum in the denominator by finding a common denominator, which is 6: 1/1 + 1/2 + 1/3 = (6 + 3 + 2)/6 = 11/6

Now, substitute this back into the main equation: Time = 55/(11/6) = 55 × 6/11

Dividing 55 by 11 gives 5, so: Time = 5 × 6 = 30 days.

Thus, option b is correct.

Explanation

Let the length of the train be L and its uniform speed be v. According to the basic formula Distance = Speed × Time, when the train crosses a telegraph post, the distance covered is equal to its own length L. Thus, L = v × 8. When the train crosses a bridge of 200m, the total distance covered is the sum of the train length and the bridge length, which is L + 200. Thus, L + 200 = v × 24. Since the speed v is constant, we can equate the two expressions for speed: L/8 = (L + 200)/24. Multiplying both sides by 24 to clear the denominators gives 3L = L + 200. Rearranging the terms by subtracting L from both sides, we get 2L = 200, which results in L = 100m. Therefore, the length of the train is 100m, which corresponds to option a.

Explanation

To find the time taken by the train to pass the man, we use the concept of relative speed. Since the train and the man are moving in opposite directions, their relative speed is the sum of their individual speeds.

The relative speed is given by: Vᵣₑₗ = Vₜᵣₐᵢₙ + Vₘₐₙ = 60 km/h + 6 km/h = 66 km/h

To convert this speed from km/h to m/s, we multiply by 5/18: Vᵣₑₗ = 66 × 5/18 = (11 × 5)/3 = 55/3 m/s

The distance to be covered to pass the man is equal to the length of the train, which is 110 m. Using the formula Time = Distance/Relative Speed, we get: Time = 110/(55/3) = 110 × 3/55 = 2 × 3 = 6 s

Thus, the time in which the train will pass the man is 6 seconds, which corresponds to option b.

Explanation

The given expression can be solved quickly by recognizing the algebraic identity for the difference of two cubes. The identity is given by a³ - b³ = (a - b)(a² + ab + b²). In the given expression, let a = 2.3 and b = 0.3.

Observing the numerator, we see that 0.027 = (0.3)³, so the numerator is (2.3)³ - (0.3)³.

Observing the denominator, we have (2.3)² + 0.69 + 0.09. Since 0.69 = 2.3 × 0.3 and 0.09 = (0.3)², the denominator is exactly in the form a² + ab + b².

Substituting these into the formula: (a³ - b³)/(a² + ab + b²) = (a - b)(a² + ab + b²)/(a² + ab + b²) = a - b

Now, substituting the values of a and b: 2.3 - 0.3 = 2

Thus, the value of the expression is 2, which corresponds to option b.

Explanation

To solve for x, simplify the left hand side (LHS) of the equation first. Combine the terms inside the square root by taking a common denominator of 196, which gives 1 + 93/196 = (196 + 93)/196 = 289/196. Now, take the square root of the numerator and the denominator separately. Since 17² = 289 and 14² = 196, the LHS simplifies to 17/14.

Now, set the simplified LHS equal to the right hand side (RHS): 17/14 = 1 + x/14

To solve for x, rewrite 1 as 14/14 to combine the terms on the RHS: 17/14 = (14 + x)/14

Since the denominators on both sides are equal, we can equate the numerators: 17 = 14 + x

Subtracting 14 from both sides gives x = 17 - 14, which results in x = 3. This corresponds to option c.

Explanation

To simplify the expression 9√2 - √8 - 4√2, we must first express all terms in terms of the same radical, √2. We begin by simplifying √8 using the property √(a × b) = √a × √b. Since 8 = 4 × 2, we have √8 = √(4 × 2) = 2√2. Substituting this value back into the original expression, it becomes 9√2 - 2√2 - 4√2. Now that all terms are like terms, we can combine them by performing the subtraction on their coefficients: (9 - 2 - 4)√2. Calculating the coefficients gives 9 - 2 = 7 and 7 - 4 = 3. Therefore, the simplified result is 3√2, which matches option b.

Explanation

Let the number be N. According to the division algorithm, any number can be expressed as N = dq + r, where d is the divisor, q is the quotient, and r is the remainder. Given the divisor d = 987 and remainder r = 59, we can write the number as N = 987q + 59.

To find the remainder when N is divided by 21, we evaluate N (mod 21). Substituting the expression for N, we get (987q + 59) (mod 21). Since 987 is a multiple of 21 (987 = 21 × 47), the term 987q is completely divisible by 21 and leaves a remainder of 0.

The problem simplifies to finding the remainder when the original remainder 59 is divided by 21. Using the formula r_final = r_initial (mod d_new), we calculate 59 (mod 21). Since 21 × 2 = 42, the remainder is 59 - 42 = 17. Thus, the remainder when the number is divided by 21 is 17, which corresponds to option c.

Explanation

A square number is defined as an integer that is the square of an integer. To determine if the product of two square numbers is always a square number, we can use the algebraic property of exponents. Let the two square numbers be represented as a² and b², where a and b are integers. The product of these two numbers is a² × b². According to the power of a product rule in algebra, aⁿ × bⁿ = (ab)ⁿ. Applying this to the given case where n = 2, we have a² × b² = (ab)². Since the product of two integers a and b is always an integer, the resulting value (ab)² must be a square number by definition. For example, if we take 4 (which is 2²) and 9 (which is 3²), their product is 4 × 9 = 36, and 36 is 6², which is a square number. This mathematical identity ensures that the product of any two square numbers is always a square number.

Explanation

In triangle ABC right angled at A, the hypotenuse BC is calculated using the Pythagorean theorem: BC = √(AB² + AC²) = √(3² + 4²) = √(9 + 16) = 5 units. The area of triangle ABC is 1/2 × AB × AC = 1/2 × 3 × 4 = 6 sq units.

The fastest way to solve this is by using the property of similar triangles. Since AD is perpendicular to BC, triangle ABD is similar to triangle CBA because they share ∠B and both have a right angle (∠ADB = ∠CAB = 90°). The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area(ΔABD)/Area(ΔCBA) = (AB/CB)²

Substituting the known values: Area(ΔABD)/6 = (3/5)² Area(ΔABD) = 6 × 9/25 = 54/25 sq units.

This corresponds to option b.

Explanation

Let the sides containing the right angle be b and x. Given the area A of the right angled triangle, we use the formula A = 1/2 bx to find the other side x = 2A/b. According to the Pythagorean theorem, the hypotenuse c is calculated as c = √(b² + x²) = √(b² + (2A/b)²). Simplifying this expression gives c = √((b⁴ + 4A²)/b²) = √(b⁴ + 4A²)/b. The area of a triangle can also be expressed using the hypotenuse c and the altitude h on the hypotenuse as A = 1/2 ch. Solving for h, we get h = 2A/c. Substituting the value of c into this equation, we obtain h = 2A/(√(b⁴ + 4A²) / b) = 2Ab/√(b⁴ + 4A²). This matches the expression in option (a).

Explanation

Let the lengths of the sides of the triangle be AB = c and BC = a. Since the triangle ABC is right angled at B, by Pythagoras theorem, AC² = a² + c². Given that M and N are the midpoints of AB and BC respectively, we have BM = c/2 and BN = a/2.

In the right angled triangle ABN, using Pythagoras theorem: AN² = AB² + BN² = c² + (a/2)² = c² + a²/4.

Similarly, in the right angled triangle CBM, using Pythagoras theorem: CM² = CB² + BM² = a² + (c/2)² = a² + c²/4.

Summing these two equations gives: AN² + CM² = (c² + a²/4) + (a² + c²/4) = (a² + c²) + (a² + c²)/4 = 5/4(a² + c²).

Substituting AC² = a² + c², we get: AN² + CM² = 5/4AC².

Therefore, 4(AN² + CM²) = 4 × 5/4AC² = 5AC². This matches option c.

Explanation

Let the side of the square ABCD be s. The length of the diagonal AC is s√2, so the distance AO = 1/2 AC = s√2/2 = s/√2. According to the given conditions, AK = AO = s/√2 and BL = BO = s/√2.

To solve this efficiently, let us use a coordinate system with the origin at O(0,0). The vertices are A(-s/2, s/2) and B(s/2, s/2). Since K and L lie on the segment AB, their y-coordinates are s/2. The x-coordinate of K is -s/2 + AK = -s/2 + s/√2 = s((√2-1)/2). The x-coordinate of L is s/2 - BL = s/2 - s/√2 = s((1-√2)/2).

Let α be the angle that OK makes with the positive y-axis. Then tan α = xₖ/yₖ = (s(√2-1)/2)/(s/2) = √2-1. Since the figure is symmetric about the y-axis, ∠LOK = θ = 2α. Using the double angle formula for tangent: tan θ = tan(2α) = (2 tan α)/(1 - tan² α) Substituting tan α = √2-1: tan θ = (2(√2-1))/(1 - (√2-1)²) = (2√2-2)/(1 - (2 + 1 - 2√2)) = (2√2-2)/(1 - 3 + 2√2) = (2√2-2)/(2√2-2) = 1. Thus, the value of tan θ is 1.

Explanation

First, check if the triangle is right-angled by applying the Pythagorean theorem. The sides are 30 m, 40 m, and 50 m. Since 30² + 40² = 900 + 1600 = 2500, and 50² = 2500, the equation a² + b² = c² holds true. This confirms that the triangle is a right-angled triangle where the side of 50 m is the hypotenuse.

The area of a right-angled triangle can be calculated using the formula Area = 1/2 × base × height. Using the two shorter sides as the base and height, Area = 1/2 × 30 × 40 = 600 m².

To find the altitude to the hypotenuse, we use the same area formula with the hypotenuse as the base: Area = 1/2 × hypotenuse × altitude. Substituting the known values, we get 600 = 1/2 × 50 × h. This simplifies to 600 = 25h. Solving for h, we find h = 600/25 = 24 m. Therefore, the length of the altitude opposite to the side of 50 m is 24 m, which corresponds to option b.

Explanation

The problem involves two right-angled triangles formed by the ladder leaning against opposite walls of the lane. Let the breadth of the lane be the sum of the distances of the foot of the ladder from the two walls, denoted as x and y. According to the Pythagorean theorem, the distance from the wall is given by the formula d = √(L² - h²), where L is the length of the ladder and h is the height reached on the wall.

For the first side, the distance x is calculated as: x = √(34² - 30²) Using the identity a² - b² = (a - b)(a + b), we get: x = √((34 - 30)(34 + 30)) = √(4 × 64) = √256 = 16 m

For the second side, the distance y is calculated as: y = √(34² - 16²) Using the same identity: y = √((34 - 16)(34 + 16)) = √(18 × 50) = √900 = 30 m

The total breadth of the lane is the sum of these two distances: Breadth = x + y = 16 + 30 = 46 m

Thus, the breadth of the lane is 46 m, which corresponds to option c.

Explanation

According to the Law of Sines, the sides of any triangle are proportional to the sines of their opposite angles. This is expressed by the formula a/sin A = b/sin B = c/sin C, where a, b, c are the lengths of the sides opposite to angles A, B, C respectively. For a triangle with angles 90°, 60°, and 30°, the ratio of the sides is given by sin 90° : sin 60° : sin 30°. Substituting the trigonometric values, we have sin 90° = 1, sin 60° = √3/2, and sin 30° = 1/2. The ratio becomes 1 : √3/2 : 1/2. Multiplying the entire ratio by 2 to eliminate the denominators, we get 2 : √3 : 1. Thus, the ratio of the sides opposite to 90°, 60°, and 30° is 2:√3:1, which corresponds to option (c).

Explanation

The area of a right angled triangle can be expressed in two different ways. First, using the two legs a and b as the base and height, the area is 1/2 ab. Second, using the hypotenuse c as the base and the perpendicular p from the right angle to the hypotenuse as the height, the area is 1/2 cp. Equating these two expressions for the area, we get 1/2 ab = 1/2 cp, which simplifies to ab = cp. Squaring both sides of this equation gives a²b² = c²p². According to the Pythagoras theorem for a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, so c² = a² + b². Substituting the value of c² into the squared area equation, we obtain (a² + b²)p² = a²b². This matches option a.

Explanation

The hour hand of a clock covers 360° in 12 hours. Therefore, the angle turned by the hour hand in one hour is 360°/12 = 30°, and the angle turned in one minute is 30°/60 = 0.5°. To find the total angle turned from 12:00 PM to 3:45 PM, we determine the total time elapsed, which is 3 hours and 45 minutes. Using the formula Total Angle = (Hours × 30°) + (Minutes × 0.5°), we substitute the given values: Total Angle = (3 × 30°) + (45 × 0.5°) = 90° + 22.5° = 112.5°. Thus, the hour hand turns by 112.5°, which corresponds to option (c).

Explanation

The angle of elevation is defined as the angle formed between the horizontal line of sight and the line of sight when an observer looks up at an object situated above their eye level. Similarly, the angle of depression is the angle formed between the horizontal line and the line of sight when the observer looks down at an object situated below their eye level. In both cases, the baseline for measuring the angle is always the horizontal line, which is imagined as a line parallel to the ground passing through the eye of the observer. Mathematically, if the line of sight makes an angle θ with the horizontal, then θ is the angle of elevation or depression depending on the direction of sight. Measuring relative to the vertical would provide the complementary angle, not the angle of elevation or depression itself. Therefore, these measurements are based solely on the horizontal reference.

Explanation

Let the radius of the round balloon be r = 1 unit and let the center of the balloon be denoted by O. Let A be the position of the observer. The angle subtended by the balloon at the eye of the observer is the angle between the two tangents drawn from point A to the sphere. Let this total subtended angle be 2θ. According to the given problem, 2θ = 90°, which implies that θ = 45°. In the right-angled triangle formed by the center O, the point of tangency T, and the observer A, the angle ∠OTA is 90° because the radius is always perpendicular to the tangent at the point of contact. The angle ∠OAT is equal to θ = 45°. Using the trigonometric formula for the sine of an angle, we have sin(θ) = OT/OA, where OT is the radius r and OA is the distance from the center to the observer. Substituting the known values, sin(45°) = 1/OA. Since sin(45°) = 1/√2, we have 1/√2 = 1/OA. Solving for OA, we find OA = √2. Therefore, the distance of the center of the balloon from point A is √2 units, which corresponds to option (b).

Explanation

The problem can be solved using the complementary angle identity, which states that sin θ = cos(90° - θ) or cos θ = sin(90° - θ).

Given the equation sin(x + 54°) = cos x, we substitute cos x with sin(90° - x) according to the identity. This gives us: sin(x + 54°) = sin(90° - x)

Since the problem specifies that 0 < x and x + 54° < 90°, both angles are acute and lie within the first quadrant. Therefore, we can directly equate the arguments of the sine functions: x + 54° = 90° - x

To solve for x, we move the x term to the left side and the constant to the right side: x + x = 90° - 54° 2x = 36° x = 36°/2 x = 18°

Thus, the value of x is 18°, which corresponds to option d.

Explanation

Let the smaller side of the right triangle be a and the longer side be b. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For the original triangle, we have a² + b² = (3√10)², which simplifies to a² + b² = 9 × 10 = 90. When the smaller side is tripled to 3a and the longer side is doubled to 2b, the new hypotenuse is 9√5. This gives us the second equation (3a)² + (2b)² = (9√5)², which simplifies to 9a² + 4b² = 81 × 5 = 405.

To solve for a, we can multiply the first equation by 4 to get 4a² + 4b² = 360. Subtracting this from the second equation, we get (9a² + 4b²) - (4a² + 4b²) = 405 - 360, which results in 5a² = 45. Thus, a² = 9, meaning the smaller side a = 3 units. Substituting a = 3 back into the first equation, we get 3² + b² = 90, which means 9 + b² = 90, leading to b² = 81. Therefore, the longer side b = 9 units. This matches option c.

Explanation

Given the equation 1 + tan θ = √2, we first isolate tan θ to find its value: tan θ = √2 - 1. Since cot θ is the reciprocal of tan θ, we use the relation cot θ = 1 / tan θ. Substituting the value of tan θ, we get cot θ = 1 / (√2 - 1). To simplify this expression, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate (√2 + 1): cot θ = 1 / (√2 - 1) × (√2 + 1) / (√2 + 1) = (√2 + 1) / ((√2)² - 1²) = (√2 + 1) / (2 - 1) = √2 + 1. Now, we need to find the value of cot θ - 1. Substituting the calculated value of cot θ, we obtain (√2 + 1) - 1 = √2. Therefore, the value of the expression is √2, which matches option b.

Explanation

Given the trigonometric equations sin(x - y) = 1/2 and cos(x + y) = 1/2. We know from standard trigonometric values that sin(30°) = 1/2 and cos(60°) = 1/2. By comparing these, we can set up a system of two linear equations: x - y = 30° (Equation 1) x + y = 60° (Equation 2) To find the value of x, we add Equation 1 and Equation 2 together: (x - y) + (x + y) = 30° + 60° 2x = 90° Dividing both sides by 2, we get: x = 45° Therefore, the value of x is 45°, which matches option (c).

Explanation

For the quadratic equation ax^2 + bx + c = 0, the sum of roots (alpha + beta) = -b/a and the product of roots (alpha beta) = c/a. We need to find the value of (1/alpha^2 - 1/beta^2)^2. This expression simplifies to ((beta^2 - alpha^2) / (alpha^2 beta^2))^2 = ((beta - alpha)(beta + alpha))^2 / (alpha beta)^4. We know that (beta - alpha)^2 = (alpha + beta)^2 - 4alphabeta = (-b/a)^2 - 4(c/a) = (b^2 - 4ac) / a^2. Also, (beta + alpha)^2 = (-b/a)^2 = b^2 / a^2 and (alpha beta)^4 = (c/a)^4 = c^4 / a^4. Substituting these values: [(b^2 - 4ac) / a^2 * b^2 / a^2] / (c^4 / a^4) = [b^2(b^2 - 4ac) / a^4] / (c^4 / a^4) = b^2(b^2 - 4ac) / c^4. Comparing this result with the options, Option A is a^2(b^2 - 4ac) / c^4. There is a typographical error in the option where 'a^2' is written instead of 'b^2', but based on the provided answer key and the form of the expression, Option A is the intended correct answer.

Explanation

To solve this system of equations, we equate the ratios in pairs to form two linear equations. First, equating the first two expressions gives (3x - y + 1)/3 = (2x + y + 2)/5. Cross multiplying, we get 5(3x - y + 1) = 3(2x + y + 2), which simplifies to 15x - 5y + 5 = 6x + 3y + 6. Rearranging the terms, we obtain 9x - 8y = 1.

Next, we equate the second and third expressions: (2x + y + 2)/5 = (3x + 2y + 1)/6. Cross multiplying gives 6(2x + y + 2) = 5(3x + 2y + 1), which simplifies to 12x + 6y + 12 = 15x + 10y + 5. Rearranging the terms, we obtain 3x + 4y = 7.

Now we solve the system:

1. 9x - 8y = 1
2. 3x + 4y = 7

Multiplying equation 2 by 2 gives 6x + 8y = 14. Adding this to equation 1, we get (9x - 8y) + (6x + 8y) = 1 + 14, which means 15x = 15, so x = 1. Substituting x = 1 into equation 2, we get 3(1) + 4y = 7, which leads to 4y = 4, so y = 1. Thus, the solution is x = 1, y = 1, which corresponds to option b.

Explanation

Let R and S be the present ages of Ram and Shyam respectively. Based on the first condition, 10 years ago, Ram's age was R - 10 and Shyam's age was S - 10. The problem states that R - 10 = 5(S - 10), which simplifies to R - 10 = 5S - 50, or R = 5S - 40. Based on the second condition, 20 years from now, Ram's age will be R + 20 and Shyam's age will be S + 20. The problem states that R + 20 = 2(S + 20), which simplifies to R + 20 = 2S + 40, or R = 2S + 20. To find the current age of Shyam, we equate the two expressions for R: 5S - 40 = 2S + 20. Subtracting 2S from both sides gives 3S - 40 = 20. Adding 40 to both sides gives 3S = 60, which results in S = 20. Thus, Shyam's current age is 20 years, which corresponds to option a.

Explanation

Let the current age of the father be F and the son be S. Given the sum of their ages is F + S = 75, we can express the son's age as S = 75 - F. The problem states that the product of their ages 5 years ago was 750, which gives the equation (F - 5)(S - 5) = 750. Substituting the value of S into this equation, we get (F - 5)(75 - F - 5) = 750, which simplifies to (F - 5)(70 - F) = 750. Expanding the product, we have 70F - F² - 350 + 5F = 750. Rearranging this into a standard quadratic equation ax² + bx + c = 0, we obtain F² - 75F + 1100 = 0. Using the quadratic formula F = (-b ± √(b² - 4ac))/(2a), we calculate F = (75 ± √(75² - 4(1)(1100)))/2 = (75 ± √(5625 - 4400))/2 = (75 ± √1225)/2 = (75 ± 35)/2. This yields two possible values: F = 110/2 = 55 or F = 40/2 = 20. Since the father's age must be greater than the son's age, the present age of the father is 55 yr.

Explanation

To solve this expression, we factorize each polynomial using algebraic identities. First, the numerator x² - 3x + 2 is factored as (x-1)(x-2). The denominator x³ - 8 is a difference of cubes, x³ - 2³, which follows the formula a³ - b³ = (a-b)(a² + ab + b²), giving (x-2)(x² + 2x + 4).

Second, for the divisor (x² - 9)/(x² + 7x + 12), the numerator x² - 9 is a difference of squares (x-3)(x+3) and the denominator x² + 7x + 12 factors to (x+3)(x+4).

Third, in the final term (x³ + 2x² + 4x)/(x² + 3x - 4), the numerator is factored by taking out the common term x to get x(x² + 2x + 4), and the denominator factors to (x+4)(x-1).

The overall expression is rewritten by changing the division to multiplication with the reciprocal: ((x-1)(x-2))/((x-2)(x² + 2x + 4)) × ((x+3)(x+4))/((x-3)(x+3)) × (x(x² + 2x + 4))/((x+4)(x-1))

By canceling common factors across the numerators and denominators, we see that (x-1), (x-2), (x+3), (x+4), and (x² + 2x + 4) all cancel out. This leaves only x/(x-3), which matches option a.

Explanation

The symbol ϕ represents the empty set, which is the unique set containing no elements. The power set P(ϕ) is defined as the set of all subsets of ϕ. According to the fundamental property of set theory, the empty set ϕ is a subset of every set, including itself, which is written as ϕ ⊆ ϕ. By the definition of a power set, if A ⊆ S, then A ∈ P(S). Therefore, since ϕ ⊆ ϕ, it must be that ϕ ∈ P(ϕ).

Using the formula for the cardinality of a power set, |P(S)| = 2^(|S|), we can see that since |ϕ| = 0, the cardinality of its power set is |P(ϕ)| = 2⁰ = 1. The only element in this power set is the empty set itself, meaning P(ϕ) = {ϕ}. This confirms that ϕ is an element of P(ϕ), making statement d correct. Statement a is false because the empty set has no elements; statement b is the negation of the correct fact; and statement c is false because a set with zero elements cannot be equal to a set with one element.

Explanation

The given sequence is 2, 4, 16, 256, .... By expressing each term as a power of 2, the pattern becomes evident: 2 = 2¹, 4 = 2², 16 = 2⁴, and 256 = 2⁸. The exponents of the base 2 are 1, 2, 4, 8, ..., which can be rewritten as 2⁰, 2¹, 2², 2³, .... This implies the general form of the elements in the set is x = 2²ⁿ. To verify this formula for the given set, we test the values of n: for n = 0, x = 2²⁰ = 2¹ = 2; for n = 1, x = 2²¹ = 2² = 4; for n = 2, x = 2²² = 2⁴ = 16; and for n = 3, x = 2²³ = 2⁸ = 256. Since the sequence begins with 2, the index n must start from 0. Therefore, the set is correctly represented as {x ∈ N | x = 2²ⁿ, n = 0, 1, 2, ...}, which corresponds to option (b).

Explanation

Given the equation y = (aˣ)ʸ, we apply the natural logarithm to both sides of the equation to simplify the exponent. This gives log y = log((aˣ)ʸ). According to the logarithmic property log(mⁿ) = n log m, the exponent y can be brought to the front of the logarithm, resulting in log y = y log(aˣ). Further, applying the power rule again to the term log(aˣ), we use the property log(aˣ) = x log a. Substituting this back into the equation, we get log y = y(x log a). This simplifies directly to log y = xy log a. This relationship demonstrates that the logarithm of y is the product of the variable x, the variable y, and the logarithm of the constant base a. This matches the expression provided in option (a).

Explanation

The given operation is defined by the formula p q = 2p + 2q - pq. To find the value of x when 8 x = 4, we substitute p = 8 and q = x into the operation formula. This gives us the equation 2(8) + 2(x) - (8)(x) = 4. Simplifying the terms, we get 16 + 2x - 8x = 4. Combining the like terms involving x leads to 16 - 6x = 4. Rearranging the equation to solve for x, we move 6x to the left side and 4 to the right side, which gives 6x = 16 - 4. This simplifies to 6x = 12. Dividing both sides by 6, we get x = 12 / 6, which results in x = 2. Since x = 2 is a natural number, it satisfies the given conditions of the problem. Thus, the value of x is 2, which corresponds to option b.

Explanation

A set is said to be closed under a specific mathematical operation if performing that operation on any two elements of the set always results in an element that also belongs to that same set. For the set of integers ℤ, which includes negative numbers, zero, and positive whole numbers, this property holds for both addition and multiplication. If a and b are any two integers, their sum a + b is guaranteed to be an integer, and their product a × b is also guaranteed to be an integer. In contrast, the set of integers is not closed under division because the quotient of two integers is not always an integer. For example, 1 ÷ 2 = 0.5, which is a rational number but not an integer. Since both addition and multiplication consistently produce results within the set of integers, option c is the correct answer.

Explanation

The question asks for one of two consecutive positive integers. If the sum was 761, the integers would be 380 and 381, which are not in the options. However, if the question intended "the sum of whose squares is 761", then we have n^2 + (n + 1)^2 = 761. Simplifying this gives 2n^2 + 2n + 1 = 761, which leads to 2n^2 + 2n - 760 = 0, or n^2 + n - 380 = 0. Solving this quadratic equation: n = [-1 + sqrt(1 - 4(1)(-380))] / 2 = [-1 + sqrt(1521)] / 2 = (-1 + 39) / 2 = 19. The two consecutive integers are 19 and 20. One of these is 20, which matches Option (b).

Explanation

This problem can be solved using the method of rationalization to create a telescoping series. For any term in the form 1/(√n + √(n+1)), we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate √(n+1) - √n.

Using the algebraic identity (a+b)(a-b) = a² - b², the general term becomes: 1/(√n + √(n+1)) × (√(n+1) - √n)/(√(n+1) - √n) = (√(n+1) - √n)/((n+1) - n) = √(n+1) - √n

Applying this formula to each term of the series: The first term 1/(1 + √2) = √2 - 1 The second term 1/(√2 + √3) = √3 - √2 The third term 1/(√3 + √4) = √4 - √3 The last term 1/(√15 + √16) = √16 - √15

Summing these terms together, we get: (√2 - 1) + (√3 - √2) + (√4 - √3) + … + (√16 - √15)

In this telescoping sum, every intermediate term cancels out (for example, √2 and -√2 cancel, √3 and -√3 cancel, and so on). Only the first negative term and the last positive term remain: Sum = √16 - 1 = 4 - 1 = 3

Therefore, the value of the expression is 3, which corresponds to option d.

Explanation

A single bar diagram is the most appropriate tool for representing the population densities of five different states because the data consists of discrete categories (the states) and their corresponding numerical values. Population density is calculated as Total Population / Total Area, resulting in a specific value for each state. Since the objective is to compare these independent values across different entities, a simple bar chart allows for a direct visual comparison of the heights of the bars, making it easy to identify which state has the highest or lowest density. In contrast, pie diagrams and percentage bar diagrams are used to show the composition of a whole or the relative proportion of components, which is not the primary requirement when comparing absolute density values across states. Therefore, option (a) is the correct choice.

Explanation

Use the total frequency formula Σ fᵢ = N and the arithmetic mean formula X̄ = (Σ fᵢ Xᵢ)/N to solve for the missing frequencies. From the total frequency, we have 4 + f + 9 + g + 4 = 25, which simplifies to f + g = 8. Using the mean formula, we get 2 = ((0 × 4) + (1 × f) + (2 × 9) + (3 × g) + (4 × 4))/25. Multiplying both sides by 25 gives 50 = f + 18 + 3g + 16, which simplifies to f + 3g = 16. We now have a system of two linear equations: f + g = 8 and f + 3g = 16. Subtracting the first equation from the second yields (f + 3g) - (f + g) = 16 - 8, which results in 2g = 8, so g = 4. Substituting g = 4 back into the first equation f + 4 = 8 gives f = 4. Therefore, the missing frequency f is 4, making option a the correct choice.

Explanation

In a histogram, the area of each rectangle is proportional to the frequency of the corresponding class interval, not the height. The height of the bar represents the frequency density, which is calculated using the formula Height = Frequency/Class Width. If the class widths are unequal, the height will not be directly proportional to the frequency. Thus, statement (c) is incorrect because it ignores the role of class width in determining the height.

The other statements are correct properties of histograms. Statement (b) is true as histograms are specifically designed for continuous grouped data. Statement (d) is accurate because the mode can be estimated by identifying the tallest bar (the modal class) and using a geometric method of intersecting lines from the top corners to the adjacent bars. Statement (a) describes the process of creating a frequency curve, which is a smoothed version of a frequency polygon created by joining the midpoints of the top edges of the histogram bars.

Explanation

The total surface area of a single hemisphere is the sum of its curved surface area and the area of its flat circular base. Using the standard formulas, the curved surface area of a hemisphere is 2πr² and the area of the circular base is πr². Therefore, the total surface area of one hemisphere is 2πr² + πr² = 3πr². Since the sphere is cut into two equal halves, there are two such hemispheres to be painted. The combined total surface area for both halves is 2 × 3πr² = 6πr² square units. Given that the rate of painting is Rs. 8 per square unit, the total cost is calculated by multiplying the total area by the rate. Total Cost = 6πr² × 8 = 48πr² rupees. This matches option c.

Explanation

The surface area of a sphere is given by the formula S = 4πr² and the volume of a sphere is given by the formula V = 4/3πr³, where r represents the radius. According to the problem, the numerical value of the surface area is three times the numerical value of the volume. We can set up the equation as 4πr² = 3 × (4/3πr³). Simplifying the right side of the equation, the 3 in the numerator and denominator cancel out, leaving 4πr² = 4πr³. Dividing both sides by 4πr², we obtain 1 = r. Since the radius r is 1 cm, the diameter D is calculated as 2 × r = 2 × 1 = 2 cm. Therefore, option b is the correct choice.

Explanation

The surface area of a hemisphere is proportional to the square of its radius, expressed as SA = 3πr² for a solid hemisphere or CSA = 2πr² for the curved surface. In either case, the ratio of the surface areas of two hemispheres is equal to the ratio of the squares of their radii: SA₁/SA₂ = r₁²/r₂². Given that SA₁/SA₂ = 4/1, we have r₁²/r₂² = 4/1, which implies that the ratio of the radii is r₁/r₂ = √(4/1) = 2/1. The volume of a hemisphere is given by the formula V = 2/3πr³, which means the ratio of the volumes is proportional to the cube of the ratio of their radii: V₁/V₂ = r₁³/r₂³. Substituting the value of the radius ratio, we get V₁/V₂ = (2/1)³ = 8/1. Thus, the ratio of their volumes is 8:1, which corresponds to option a.

Explanation

The total volume of the container is the sum of the volume of the right circular cylinder and the volume of the hemisphere. The formula for the total volume V is given by V = πr²h꜀ᵧₗ + 2/3 πr³, where r is the radius and h꜀ᵧₗ is the height of the cylinder. Substituting the given values V = 32400π and r = 15 cm into the formula:

32400π = π(15)²h꜀ᵧₗ + 2/3 π(15)³

Dividing the entire equation by π simplifies it to: 32400 = 225h꜀ᵧₗ + 2/3 (3375) 32400 = 225h꜀ᵧₗ + 2250

Subtracting 2250 from both sides: 225h꜀ᵧₗ = 30150 h꜀ᵧₗ = 30150/225 = 134 cm

The total height h of the container is the sum of the height of the cylinder and the radius of the hemisphere: h = h꜀ᵧₗ + r = 134 + 15 = 149 cm

Comparing this result with the given options, 149 cm falls within the range 135 cm < h < 150 cm, which corresponds to option (a).

Explanation

The volume of the metal used in a hollow cylindrical pipe is calculated using the formula V = π(R² - r²)h, where R is the external radius, r is the internal (bore) radius, and h is the height or length of the cylinder.

First, convert all dimensions to centimeters to ensure consistency. The length h is 1.4 m = 140 cm. The bore radius r is given as 2.5 cm. The external radius R is the sum of the bore radius and the thickness of the metal, so R = 2.5 cm + 1 cm = 3.5 cm.

Substituting these values into the formula: V = 22/7 × (3.5² - 2.5²) × 140

Using the algebraic identity a² - b² = (a - b)(a + b), we can simplify the radius term: 3.5² - 2.5² = (3.5 - 2.5)(3.5 + 2.5) = 1 × 6 = 6 cm²

Now, compute the final volume: V = 22/7 × 6 × 140 V = 22 × 6 × 20 V = 132 × 20 = 2640 cm³

Thus, the volume of iron used in the pipe is 2640 cm³, which corresponds to option a.

Explanation

When a right circular cone is cut by a plane parallel to its base, the resulting smaller cone is geometrically similar to the original cone. For any two similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions, such as the radius, height, or slant height.

Using the similarity formula: V₁ / V₂ = (l₁ / l₂)³

Given that the ratio of the slant heights of the original cone l₁ and the smaller cone l₂ is 2 : 1, we can substitute these values into the formula: V₁ / V₂ = (2 / 1)³ V₁ / V₂ = 8 / 1

Thus, the ratio of the volume of the original cone V₁ to the volume of the smaller cone V₂ is 8 : 1, which matches option d.

Explanation

The volume of a right circular cone C₁ is defined by the formula V₁ = 1/3 π r₁² h₁, and the volume of a right circular cylinder C₂ is defined by V₂ = π r₂² h₂. Since the volumes are given as equal, we equate the two expressions: 1/3 π r₁² h₁ = π r₂² h₂. Simplifying the equation by removing π from both sides, we get 1/3 r₁² h₁ = r₂² h₂. We are also given the ratio r₁ : r₂ = 1 : n, which implies that r₂ = n r₁. Substituting n r₁ for r₂ in our equation leads to 1/3 r₁² h₁ = (n r₁)² h₂, which further simplifies to 1/3 r₁² h₁ = n² r₁² h₂. By canceling r₁² from both sides, we are left with 1/3 h₁ = n² h₂. Solving for h₁, we multiply both sides by 3 to arrive at h₁ = 3 n² h₂. This matches the relation provided in option b.

Explanation

Assertion (A) is true because the curved surface area (CSA) of a right circular cone is calculated using the formula πrl, where l represents the slant height. According to the Pythagorean theorem, the slant height l for a cone with height h and base radius r is given by l = √(h² + r²). By substituting this value of l into the surface area formula, we get CSA = πr√(h² + r²), which matches the assertion.

Reason (R) is false because unfolding a right circular cone along its slant height does not produce a rectangle. Instead, it forms a sector of a circle. The radius of this sector is equal to the slant height l = √(h² + r²), and the arc length of the sector is equal to the circumference of the base of the cone, which is 2πr. The area of this sector is calculated as 1/2 × radius × arc length = 1/2 × √(h² + r²) × 2πr = πr√(h² + r²). Since the geometric shape formed is a sector and not a rectangle, the reason provided is mathematically incorrect. Thus, A is true but R is false.

Explanation

Coterminous edges are those that meet at a common vertex. In a cuboid, these edges correspond to the length l, breadth b, and height h. The formula for the volume of a cuboid is V = l × b × h, which is the product of these three coterminous edges, making Assertion (A) true. The total surface area of a cuboid is calculated using the formula S = 2(lb + bh + hl). This formula corresponds to twice the sum of the products of the lengths of the coterminous edges taken two at a time, making Reason (R) true. However, the volume of a solid is a measure of its three-dimensional capacity, whereas the surface area is a measure of its two-dimensional boundary. The fact that the surface area is calculated in a specific way does not explain or logically derive the formula for volume. Thus, while both statements are individually correct, R is not the correct explanation for A.

Explanation

The area of a triangle is calculated using the formula Area = 1/2 × base × height. In this scenario, the triangles share the same base, meaning the value of the base b is constant for all such triangles. The height h of a triangle is the perpendicular distance from the vertex opposite the base to the line containing the base. Since the vertices of these triangles lie on a line parallel to the base, the height h corresponds to the perpendicular distance between the two parallel lines. As the distance between two parallel lines is constant everywhere, the height h remains identical for every triangle regardless of the position of its vertex on the parallel line. Substituting these constant values into the formula 1/2 × b × h results in the same area for all such triangles. Thus, Reason R is the direct geometric justification for the fact stated in Assertion A, making option a the correct choice.

Explanation

Assertion A is a fundamental property of Euclidean geometry. If two distinct lines were to have more than one point in common, such as two distinct points P and Q, then there would be more than one line passing through those two points. This contradicts the basic axiom that exactly one unique line passes through any two distinct points. Therefore, two distinct lines can intersect at most at one single point, making Assertion A true.

Reason R is also a true statement because, in a plane, an infinite number of lines can be drawn that all pass through one common point. However, the ability to draw multiple lines through one point does not logically explain why two distinct lines cannot share more than one point. The proof for A relies on the uniqueness of a line connecting two points, not the plurality of lines passing through one point. Thus, while both A and R are individually true, R is not the correct explanation of A, which corresponds to option b.

Explanation

The population increased from 100 million to 169 million over two decades. The total increase is 169 - 100 = 69 million. The average absolute increase per decade is 69 / 2 = 34.5 million. Expressed as a percentage of the initial population (100 million), this average increase is (34.5 / 100) * 100 = 34.5%. While population growth is typically calculated as a compound rate (which would yield 30%), the presence of 34.5% in the options and its alignment with the answer key indicates that the simple average increase relative to the initial population was intended. Thus, Option (b) is the correct choice.

Explanation

The most efficient way to solve this problem is by using the Rule of Alligation. The formula for the ratio of quantities of two components in a mixture is given by: Ratio = (Cost of Dearer - Mean Price)/(Mean Price - Cost of Cheaper)

Substituting the given values: Cost of Dearer (superior tea) = Rs. 13.50 per kg Cost of Cheaper tea = Rs. 9 per kg Mean Price (average price) = Rs. 11 per kg

Ratio of cheaper tea to superior tea = (13.50 - 11)/(11 - 9) = 2.50/2 = 25/20 = 5/4

This means for every 4 units of superior tea, 5 units of cheaper tea must be added. Given that the quantity of superior tea is 100 kg, we can set up the proportion: Quantity of Cheaper/Quantity of Superior = 5/4 Quantity of Cheaper = 5/4 × 100 = 5 × 25 = 125 kg

Therefore, 125 kg of tea at Rs. 9 per kg must be mixed to achieve the average price, which corresponds to option c.

Explanation

Since the truck covers two equal distances of 100 km each at different speeds, the average speed can be calculated using the harmonic mean formula for equal distances: Average Speed = 2v₁v₂/(v₁ + v₂). Here, the speed for the first half is v₁ = 50 km/h and the speed for the second half is v₂ = 40 km/h. Substituting these values into the formula, we get Average Speed = (2 × 50 × 40)/(50 + 40) = 4000/90. Simplifying the fraction gives 400/9, which equals 44.44 km/h. Alternatively, calculating total time, the time for the first 100 km is 100/50 = 2 hours and for the next 100 km is 100/40 = 2.5 hours. Total time is 4.5 hours. Total distance is 200 km. Thus, average speed is 200/4.5 = 44.44 km/h, which matches option (d).

Explanation

For a set of n observations where n is odd, the median is the value of the (n+1)/2th observation when the data is arranged in ascending order. For the given sets A, B, and C, each containing 5 numbers, the median is the 3rd term of each set. Thus, in set A, there are at least 3 numbers ≤ 3 and 3 numbers ≥ 3. Similarly, in set B, there are at least 3 numbers ≤ 8 and 3 numbers ≥ 8, and in set C, there are at least 3 numbers ≤ 11 and 3 numbers ≥ 11.

When the three sets are combined, the total number of observations is 15. The median of the combined set is the (15+1)/2 = 8th observation in the sorted combined list. To determine if the 8th observation is fixed, we test different values. If all sets consist of their medians repeated five times, the combined sorted set is {3, 3, 3, 3, 3, 8, 8, 8, 8, 8, 11, 11, 11, 11, 11}, and the 8th term is 8. However, if we choose A as {1, 1, 3, 3, 3}, B as {1, 1, 8, 8, 8}, and C as {1, 1, 11, 11, 11}, the combined sorted set is {1, 1, 1, 1, 1, 1, 3, 3, 3, 8, 8, 8, 11, 11, 11}, and the 8th term is 3. If we choose A as {3, 3, 3, 12, 12}, B as {8, 8, 8, 12, 12}, and C as {11, 11, 11, 12, 12}, the 8th term becomes 11. Since the 8th term varies based on the actual elements of the sets, the median of the combined set cannot be determined from the given data, making option d correct.

Explanation

The fundamental property of the arithmetic mean is that the sum of the deviations of all observations from the mean is always zero. In a frequency distribution, if Xᵢ represents the values of the variable and fᵢ represents their respective frequencies, the deviation from the mean is defined as xᵢ = Xᵢ - μ. The sum requested is Σᵢ₌₁ⁿ fᵢxᵢ. Substituting the definition of deviation, we get Σ fᵢ(Xᵢ - μ) = Σ fᵢXᵢ - Σ fᵢμ. Since μ is a constant, Σ fᵢμ = μ Σ fᵢ = Nμ, where N is the total frequency. By the definition of the mean, μ = (Σ fᵢXᵢ)/N, which implies Σ fᵢXᵢ = Nμ. Therefore, the expression becomes Nμ - Nμ = 0. This mathematical identity confirms that the sum of frequency-weighted deviations from the mean is always zero, making option a the correct choice.

Explanation

According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities. For polynomials with real coefficients, which includes those with rational coefficients, the Complex Conjugate Root Theorem states that if a complex number a + bi is a root of the polynomial, then its conjugate a - bi must also be a root. This implies that all non-real complex roots must occur in pairs.

Let n denote the degree of the polynomial, r denote the number of real roots, and c denote the number of non-real complex roots. The total number of roots is given by the relation n = r + c. Given that the polynomial has exactly three real roots, we have r = 3. Since c must be an even integer (c = 0, 2, 4, ...), the possible values for the degree n are 3 + 0 = 3, 3 + 2 = 5, 3 + 4 = 7, and so on. Consequently, the degree n must be an odd integer such that n ≥ 3. This confirms that the degree is greater than or equal to 3, making option (b) the correct choice.

Explanation

To find the value of y, we use the method of substitution to eliminate x from the given system of equations. From the linear equation 2x + y + 3 = 0, we can express x in terms of y as x = (-y - 3)/2. Now, substitute this expression for x into the quadratic equation 2x² + 6x + 5y + 1 = 0:

2((-y - 3)/2)² + 6((-y - 3)/2) + 5y + 1 = 0

Simplify the squared term and the linear term: 2((y² + 6y + 9)/4) + 3(-y - 3) + 5y + 1 = 0 (y² + 6y + 9)/2 - 3y - 9 + 5y + 1 = 0

Combine the linear terms in y and the constants: (y² + 6y + 9)/2 + 2y - 8 = 0

Multiply the entire equation by 2 to eliminate the denominator: y² + 6y + 9 + 4y - 16 = 0 y² + 10y - 7 = 0

This matches option c, which is the required equation to solve for y.

Explanation

To find a quadratic equation whose roots are the reciprocals of the roots of a given equation ax² + bx + c = 0, we apply the transformation x → 1/x. Substituting x = 1/x into the given equation 2x² - 3x - 4 = 0 yields 2(1/x)² - 3(1/x) - 4 = 0. This simplifies to 2/x² - 3/x - 4 = 0. Multiplying the entire equation by x² to clear the denominators, we obtain 2 - 3x - 4x² = 0. Rearranging the terms leads to -4x² - 3x + 2 = 0. Multiplying by -1 to match the signs in the options, the equation becomes 4x² + 3x - 2 = 0.

A quicker shortcut for competitive exams is to simply swap the coefficients of x² and the constant term. For the equation 2x² - 3x - 4 = 0, the coefficient a = 2 and the constant c = -4. Swapping these gives -4x² - 3x + 2 = 0, which simplifies to 4x² + 3x - 2 = 0 upon multiplying by -1. This matches option (b).

Explanation

To find the Least Common Multiple (LCM) of algebraic expressions, we must first express each expression in its fully factored form. The first expression is already factored as (x + 2)²(x - 2). For the second expression, x² - 4x - 12, we factorize the quadratic by finding two numbers that multiply to -12 and add to -4, which are -6 and 2. Thus, x² - 4x - 12 = (x - 6)(x + 2).

The LCM is determined by taking the product of the highest power of every unique factor present across all expressions. The unique factors identified are (x + 2), (x - 2), and (x - 6). Comparing the exponents: the highest power of (x + 2) is 2 (from the first expression), the highest power of (x - 2) is 1, and the highest power of (x - 6) is 1. Multiplying these together, we get LCM = (x + 2)²(x - 2)(x - 6), which corresponds to option d.

Explanation

Since (x + 4) is the Highest Common Factor (HCF) of the two given polynomials, it must be a factor of each polynomial individually. According to the Factor Theorem, if (x + a) is a factor of a polynomial f(x), then f(-a) = 0. Here, we set x = -4 for both expressions.

For the first polynomial P(x) = 2x² + kx - 12: P(-4) = 2(-4)² + k(-4) - 12 = 0 2(16) - 4k - 12 = 0 32 - 12 - 4k = 0 20 = 4k k = 5

To verify, we apply the same condition to the second polynomial Q(x) = x² + x - 2k - 2: Q(-4) = (-4)² + (-4) - 2k - 2 = 0 16 - 4 - 2k - 2 = 0 10 - 2k = 0 2k = 10 k = 5

Since both polynomials are divisible by (x + 4) when k = 5, the value of k is 5, which corresponds to option a.

Explanation

Since (x + k) is the HCF of ax² + ax + b and x² + cx + d, it must be a factor of both polynomials. According to the Factor Theorem, if (x + k) is a factor of a polynomial f(x), then f(-k) = 0. Applying this condition to both given expressions:

For the first polynomial ax² + ax + b: a(-k)² + a(-k) + b = 0 ak² - ak + b = 0 (Eq 1)

For the second polynomial x² + cx + d: (-k)² + c(-k) + d = 0 k² - ck + d = 0 (Eq 2)

To solve for k, we eliminate the k² term. Multiply Eq 2 by a: ak² - ack + ad = 0 (Eq 3)

Subtract Eq 3 from Eq 1: (ak² - ak + b) - (ak² - ack + ad) = 0 -ak + ack + b - ad = 0 k(ac - a) = ad - b k = (ad - b)/(a(c - 1))

Comparing this result with the given options a, b, and c, it is evident that none of the expressions match the calculated value of k. Therefore, the correct option is d.

Explanation

According to the Remainder Theorem, if a polynomial P(x) is divided by a linear divisor of the form (ax - b), the remainder is given by P(b/a). For the expression to be exactly divisible, the remainder must be zero. Therefore, the value that needs to be subtracted from the polynomial is equal to the remainder obtained when it is divided by (3x - 1).

Given P(x) = 27x³ - 9x² - 6x - 5 and the divisor 3x - 1, we substitute x = 1/3 into the polynomial:

P(1/3) = 27(1/3)³ - 9(1/3)² - 6(1/3) - 5 P(1/3) = 27(1/27) - 9(1/9) - 2 - 5 P(1/3) = 1 - 1 - 2 - 5 P(1/3) = -7

The remainder is -7. To make the polynomial exactly divisible by (3x - 1), we must subtract this remainder from the original expression. Thus, the value to be subtracted is -7, which corresponds to option b.

Explanation

According to the Factor Theorem, a linear polynomial (x - a) is a factor of a polynomial f(x) if and only if f(a) = 0. In this question, the given polynomial is f(x) = 2x³ - 3x² - 11x + 6. To determine which option is a factor, we substitute the root of each binomial into the polynomial. For option (c) x + 2, the root is x = -2. Substituting this value into the equation gives f(-2) = 2(-2)³ - 3(-2)² - 11(-2) + 6. Calculating the powers first, we get f(-2) = 2(-8) - 3(4) + 22 + 6. This simplifies to f(-2) = -16 - 12 + 22 + 6. Combining the terms, -16 - 12 = -28 and 22 + 6 = 28, so f(-2) = -28 + 28 = 0. Since the result is zero, (x + 2) is a factor of the polynomial. For other options like (a) x + 1 or (b) x - 1, substituting x = -1 or x = 1 does not result in zero, confirming that only option (c) is correct.

Explanation

Let the common factor be (x - α), which implies that α is a common root of both quadratic equations. Therefore, we have the system of equations α² - 11α + a = 0 and α² - 14α + 2a = 0. To find the value of α, subtract the first equation from the second to eliminate the α² term: (α² - 14α + 2a) - (α² - 11α + a) = 0. This simplifies to -3α + a = 0, which gives the relation a = 3α. Now, substitute this expression for a back into the first equation: α² - 11α + 3α = 0, which simplifies to α² - 8α = 0. Factoring the quadratic equation gives α(α - 8) = 0, meaning the possible values for the common root are α = 0 or α = 8. If α = 0, then a = 3(0) = 0. If α = 8, then a = 3(8) = 24. Consequently, the values of a that satisfy the condition are 0 and 24, which corresponds to option (c).

Explanation

According to the Rational Root Theorem, if a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀ with integer coefficients has a rational root p/q in simplest form, then p must be a factor of the constant term a₀ and q must be a factor of the leading coefficient aₙ. In the given polynomial P(x) = x³ - 3x² - 3x + 9, the leading coefficient aₙ is 1 and the constant term a₀ is 9. Since (x - a) is a factor, x = a is a root of the polynomial. Because the leading coefficient is 1, any rational root a must be an integer and must be a divisor of the constant term 9. This confirms that option b is correct. To verify, we can factor the polynomial as P(x) = x²(x - 3) - 3(x - 3) = (x² - 3)(x - 3). The roots are x = 3, x = √3, and x = -√3. Among these, only a = 3 is a rational number, and 3 is an integer that divides 9.

Explanation

To solve this problem, use the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx)). We are given the values for (x + y + z) and (xy + yz + zx), but we need the value of x² + y² + z².

First, calculate x² + y² + z² using the identity (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). Substituting the given values x + y + z = 6 and xy + yz + zx = 11 into this formula, we get: 6² = x² + y² + z² + 2(11) 36 = x² + y² + z² + 22 x² + y² + z² = 36 - 22 = 14.

Now, substitute all known values back into the main identity: x³ + y³ + z³ - 3xyz = 6 × (14 - 11) x³ + y³ + z³ - 3xyz = 6 × 3 = 18.

Therefore, the value of x³ + y³ + z³ - 3xyz is 18, which matches option (a).

Explanation

Given the equation x² - 4x + 1 = 0, we divide both sides by x to find the value of x + 1/x. Dividing the equation by x gives x - 4 + 1/x = 0, which simplifies to x + 1/x = 4. To find the value of x³ + 1/x³, we use the standard algebraic identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x). Substituting the value x + 1/x = 4 into this formula, we get 4³ - 3(4). Calculating the values, we have 64 - 12, which equals 52. Therefore, the value of x³ + 1/x³ is 52, which corresponds to option c.

Explanation

Given sin θ = (m² - n²)/(m² + n²), we treat this as a right-angled triangle where sin θ = Opposite/Hypotenuse. Thus, we let the opposite side be m² - n² and the hypotenuse be m² + n². To find tan θ, we need the adjacent side, which is calculated using the Pythagorean theorem: Adjacent = √(Hypotenuse² - Opposite²). Substituting the given values, we get Adjacent = √((m² + n²)² - (m² - n²)²). Using the standard algebraic identity (a + b)² - (a - b)² = 4ab, where a = m² and b = n², the expression simplifies to Adjacent = √(4m²n²) = 2mn. Since tan θ = Opposite/Adjacent, the value is tan θ = (m² - n²)/2mn, which corresponds to option c.

Explanation

The relation between the sine and cosine functions for complementary angles is given by the trigonometric identity cos y = sin(π/2 - y). Given the equation sin x = cos y, we can substitute this identity to obtain sin x = sin(π/2 - y). Since it is specified that x and y are acute angles, both x and π/2 - y must fall within the range (0, π/2), where the sine function is one-to-one. Therefore, we can equate the arguments directly: x = π/2 - y. Rearranging the terms by adding y to both sides of the equation yields x + y = π/2, which corresponds to option (c). This result proves that x and y are complementary angles.

Explanation

Given cos A = 5/13, we use the identity sin² A + cos² A = 1 to find sin A. Thus, sin A = √(1 - cos² A) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13. From these values, we can derive the other trigonometric ratios: tan A = sin A/cos A = (12/13)/(5/13) = 12/5 and cot A = 1/tan A = 5/12.

Now, substitute these values into the given expression (sin A - cot A)/(2tan A). The numerator is sin A - cot A = 12/13 - 5/12. To subtract these, find a common denominator: (12 × 12 - 5 × 13)/(13 × 12) = (144 - 65)/156 = 79/156. The denominator is 2tan A = 2 × 12/5 = 24/5.

The final calculation is (79/156)/(24/5) = 79/156 × 5/24 = (79 × 5)/(156 × 24) = 395/3744. This matches option (c).

Explanation

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, we can directly compare the given equation sin² 60° + cos² (3x - 9°) = 1 with this identity. For the sum of the squares of the sine and cosine of two angles to equal 1, the angles must be equivalent in this context. Therefore, we can equate the angle of the cosine term to the angle of the sine term, setting 3x - 9° = 60°. To solve for x, we first add 9° to both sides of the equation, which gives 3x = 60° + 9°, resulting in 3x = 69°. Dividing both sides by 3, we get x = 69° / 3 = 23°. Thus, the value of x is 23°, which corresponds to option b.

Explanation

The given expression is cot² θ - 1/sin² θ. To solve this efficiently, we apply the fundamental trigonometric identity relating cosecant and cotangent, which is csc² θ = 1 + cot² θ. We also know from the definition of reciprocal identities that 1/sin² θ = csc² θ. Substituting this into the original expression, we get cot² θ - csc² θ. Now, using the identity csc² θ = 1 + cot² θ, we substitute the value of csc² θ as follows: cot² θ - (1 + cot² θ). Expanding the parentheses gives cot² θ - 1 - cot² θ. The cot² θ and -cot² θ terms cancel each other out, resulting in a final value of -1. Therefore, the value of the expression is -1, which matches option b.

Explanation

To find the value of log₁₀₀ 0.1, we use the logarithm power property log_bⁿ aᵐ = m/n log_b a. First, we express both the base and the argument as powers of 10. The base 100 is equal to 10², and the argument 0.1 is equal to 10⁻¹.

Substituting these values into the formula: log₁₀² 10⁻¹ = -1/2 log₁₀ 10

Since the logarithm of a number to its own base is always 1, we have log₁₀ 10 = 1. The calculation becomes: -1/2 × 1 = -1/2

Therefore, the value is -1/2, which corresponds to option (b).

Explanation

To minimize the total number of rows, the number of trees in each row must be as large as possible. Since each row must contain the same number of trees of only one variety, this number must be the Greatest Common Divisor (GCD) of the given numbers of trees.

The numbers of trees are 21, 42, and 56. We find the GCD as follows: 21 = 3 × 7 42 = 2 × 3 × 7 56 = 2³ × 7 GCD(21, 42, 56) = 7

Thus, the maximum number of trees per row is 7. To find the minimum number of rows, we divide the total number of trees by the GCD: Total rows = (21 + 42 + 56)/7 = 119/7 = 17

Alternatively, calculating rows for each variety separately: Rows for mango trees = 21 / 7 = 3 Rows for apple trees = 42 / 7 = 6 Rows for orange trees = 56 / 7 = 8 Total rows = 3 + 6 + 8 = 17

Therefore, the minimum number of rows required is 17, which corresponds to option (c).

Explanation

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is either 0 or a multiple of 11. For the number 8798546*5, let the missing digit be x.

Sum of digits at odd positions (from the left): 8 + 9 + 5 + 6 + 5 = 33. Sum of digits at even positions (from the left): 7 + 8 + 4 + x = 19 + x.

The difference is 33 - (19 + x) = 14 - x. For the number to be divisible by 11, 14 - x must be a multiple of 11. Setting 14 - x = 11, we get x = 14 - 11 = 3. Since 3 is a single digit and fits within the given options, the least value for * is 3, which corresponds to option d.

Explanation

To find the sum of all prime numbers between 100 and 120, we first identify the prime numbers in this range. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. We check the odd numbers between 100 and 120 since even numbers greater than 2 are not prime. The numbers to check are 101, 103, 105, 107, 109, 111, 113, 115, 117, and 119. 105 and 115 are divisible by 5. 111 is divisible by 3 as the sum of its digits is 3. 117 is divisible by 9 as the sum of its digits is 9. 119 is divisible by 7 since 119 = 7 × 17. The remaining prime numbers are 101, 103, 107, 109, and 113. The sum is calculated as follows: Sum = 101 + 103 + 107 + 109 + 113. Using a shortcut for addition, we can take 100 × 5 and add the remaining units: 500 + (1 + 3 + 7 + 9 + 13) = 500 + 33 = 533. This matches option (d).

Explanation

According to the Remainder Theorem, when a polynomial f(x) is divided by a linear divisor of the form g(x) = x - a, the remainder is given by f(a). In the given division algorithm f(x) = g(x)q(x) + r(x), since g(x) = x + 1, we can identify a = -1 because x + 1 is equivalent to x - (-1).

To find the value of the remainder r(x), we substitute x = -1 into the polynomial f(x) = x² + 1.

Calculation: r(x) = f(-1) = (-1)² + 1 r(x) = 1 + 1 r(x) = 2

Since the degree of g(x) is 1, the remainder r(x) must be a constant (degree 0), which satisfies the condition deg r(x) < deg g(x). Thus, the value of the remainder is 2, which corresponds to option c.

Explanation

Let the radius be r = 3x and the height be h = 4x. Using the formula for the volume of a right circular cone, V = 1/3π r² h, we substitute the given volume and dimensions: 96π = 1/3π (3x)² (4x). Simplifying the expression, we get 96 = 1/3 · 9x² · 4x, which simplifies further to 96 = 12x³. Dividing both sides by 12 gives x³ = 8, which means x = 2. Therefore, the radius r = 3 · 2 = 6 cm and the height h = 4 · 2 = 8 cm. To find the lateral surface area, we first need the slant height l using the relation l = √(r² + h²). Substituting the values, l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. The formula for the lateral surface area (LSA) is LSA = π r l. Substituting the values, we get LSA = π · 6 · 10 = 60π cm², which matches option d.

Explanation

The volume of the solid left is the difference between the volume of the cube and the volume of the largest sphere that can be carved out of it. For a sphere to be the largest possible inside a cube, its diameter must be equal to the edge of the cube.

Given edge of the cube a = 3 m, the volume of the cube is V꜀ᵤ₆ₑ = a³ = 3³ = 27 m³.

The diameter of the sphere is d = 3 m, which means the radius is r = 3/2 = 1.5 m. The volume of the sphere is calculated using the formula Vₛₚₕₑᵣₑ = 4/3πr³. Substituting the value of r: Vₛₚₕₑᵣₑ = 4/3π(1.5)³ = 4/3π(3/2)³ = 4/3π(27/8) = 9/2π = 4.5π m³.

The volume of the solid left is V꜀ᵤ₆ₑ - Vₛₚₕₑᵣₑ = (27 - 4.5π) m³, which corresponds to option b.

Explanation

Let the dimensions of the rectangular box be 2x, 3x, and 4x based on the given ratio 2 : 3 : 4. The formula for the total surface area SA of a rectangular box is 2(lb + bh + hl). Substituting the dimensions, we get SA = 2(2x · 3x + 3x · 4x + 4x · 2x) = 2(6x² + 12x² + 8x²) = 2(26x²) = 52x².

The difference in cost between the two rates is given by (Rate₂ - Rate₁) × SA. Given the rates are Rs. 10 and Rs. 8 per square metre, and the cost difference is Rs. 234, we can write the equation as (10 - 8) × 52x² = 234. This simplifies to 2 × 52x² = 234, which means 104x² = 234. Solving for x², we find x² = 234/104. Dividing both numbers by 26, we get x² = 9/4, so x = √2.25 = 1.5.

The dimensions are thus 2 × 1.5 = 3 m, 3 × 1.5 = 4.5 m, and 4 × 1.5 = 6 m. These values correspond to option b.

Explanation

The surface area of a cube is given by the formula 6a² and the volume is given by a³, where a is the edge length. For the original melted cube with edge a₁ = 4 cm, the surface area x = 6 × 4² = 6 × 16 = 96 sq cm.

To find the total surface area y of the recast cubes, we first determine the number of small cubes n created. Since the volume remains constant during recasting, n = Volume of large cube / Volume of small cube = 4³ / 1³ = 64 / 1 = 64. The surface area of one small cube with edge a₂ = 1 cm is 6 × 1² = 6 sq cm. Therefore, the total surface area y = 64 × 6 = 384 sq cm.

The ratio x : y = 96 : 384. Dividing both terms by 96, we get 1 : 4.

Alternatively, using a direct ratio method: x/y = (6a₁²) / [ (a₁/a₂)³ × 6a₂² ] = a₁² / (a₁³ / a₂) = a₂ / a₁ = 1 / 4. This corresponds to option c.

Explanation

The total surface area of a one-side open cubical box consists of three parts: the outer surface area, the inner surface area, and the area of the top rim.

The outer surface area covers the bottom and the four side walls. Since the outer side length L = 5 cm, the outer surface area is 5 × L² = 5 × 5² = 5 × 25 = 125 cm².

For the inner surface area, we first determine the inner dimensions. The inner length and width are L - 2t = 5 - (2 × 0.5) = 4 cm. The inner height is L - t = 5 - 0.5 = 4.5 cm. The inner base area is 4 × 4 = 16 cm² and the area of the four inner walls is 4 × (4 × 4.5) = 4 × 18 = 72 cm². Thus, the total inner surface area is 16 + 72 = 88 cm².

The top rim area is the difference between the outer top area and the inner opening area, which is L² - (L-2t)² = 5² - 4² = 25 - 16 = 9 cm².

Summing these components, the total surface area is 125 + 88 + 9 = 222 cm², which corresponds to option b.

Explanation

To find the increase in the field level, we use the volume formula: Volume = Area × Height. First, calculate the volume of the earth dug out from the tank, which is 10 m × 7.5 m × 6 m = 450 m³. Next, determine the area of the remaining field where this earth is spread. The total area of the field is 125 m × 15 m = 1875 m². The area occupied by the tank is 10 m × 7.5 m = 75 m². Therefore, the remaining area is 1875 m² - 75 m² = 1800 m².

Let the height by which the field is raised be h. Since the volume of earth spread equals the volume dug out, we have 1800 × h = 450. Solving for h, we get h = 450/1800 = 1/4 = 0.25 m. Converting this value into centimeters, 0.25 m × 100 = 25 cm. This matches option c.

Explanation

The radius of the circle is derived from the given area using the formula πr² = 4π, which yields r = 2 cm. In an equilateral triangle ABC with center O, let D be the midpoint of side AB. The segment OD represents the inradius r of the triangle, and AD represents half of the side length s. In the right-angled triangle OAD, the angle ∠OAD is 30°.

Using the trigonometric relation tan 30° = OD/AD, we substitute the known values: 1/√3 = 2/AD. Solving for AD, we get AD = 2√3 cm. Since D is the midpoint of side AB, the full side length s is 2 × AD = 4√3 cm.

The area of an equilateral triangle is calculated using the formula Area = √3/4 s². Substituting the value of s: Area = √3/4 (4√3)² Area = √3/4 (16 × 3) Area = √3/4 × 48 Area = 12√3 cm²

Thus, option (b) is correct.

Explanation

Based on the given condition in the question, it is explicitly stated that if the lengths of the sides of a square and a rhombus are the same, then their areas should also be the same. Let the area of the square be represented by x and the area of the rhombus be represented by y. According to the premise provided, x = y. To find the ratio x:y, we divide the area of the square by the area of the rhombus, which gives x/y = 1/1. Therefore, the ratio x:y is 1:1. While in standard geometry a rhombus with side s has an area of s² sin(θ), the specific constraint of this problem mandates that the areas be equal, leading directly to option a.