Question 1

The square of the sum of three positive consecutive natural numbers exceeds the sum of their squares by 292. What is the largest of these three numbers?

Accepted Answer

Introduction / Context: This problem combines algebra and properties of consecutive natural numbers. You are told that the square of the sum of three consecutive natural numbers is greater than the sum of their individual squares by 292. From this relationship you must determine the largest of the three numbers. This tests your ability to model consecutive numbers with variables and manipulate algebraic expressions involving squares and sums. Given Data / Assumptions: - The three numbers are positive consecutive natural numbers. - Let them be represented in terms of a central variable. - The square of their sum minus the sum of their squares equals 292. - We need to find the largest of these three numbers. Concept / Approach: For three consecutive integers, a convenient representation is n - 1, n and n + 1, where n is the middle number. This choice simplifies the algebra because the deviations from the middle are symmetric. We first compute the square of their sum, then the sum of their squares and form an equation using the given difference of 292. Solving the resulting quadratic equation in n produces the middle number, and hence the largest number n + 1. Step-by-Step Solution: Step 1: Let the three consecutive natural numbers be n - 1, n and n + 1, where n is a positive integer. Step 2: Compute their sum: (n - 1) + n + (n + 1) = 3n. Step 3: The square of their sum is (3n)^2 = 9n^2. Step 4: Compute the sum of their squares: (n - 1)^2 + n^2 + (n + 1)^2. Step 5: Expand each square: (n - 1)^2 = n^2 - 2n + 1, n^2 = n^2 and (n + 1)^2 = n^2 + 2n + 1. Step 6: Add these results: (n^2 - 2n + 1) + n^2 + (n^2 + 2n + 1) = 3n^2 + 2. Step 7: The difference between the square of the sum and the sum of the squares is 9n^2 - (3n^2 + 2) = 6n^2 - 2. Step 8: We are told this difference equals 292, so set 6n^2 - 2 = 292. Step 9: Solve for n: 6n^2 = 292 + 2 = 294, so n^2 = 294 / 6 = 49. Step 10: Thus n = 7, since n is positive. Step 11: The three numbers are n - 1 = 6, n = 7 and n + 1 = 8. The largest is 8. Verification / Alternative check: Verify the condition using the numbers 6, 7 and 8. Their sum is 6 + 7 + 8 = 21. The square of the sum is 21^2 = 441. The sum of their squares is 6^2 + 7^2 + 8^2 = 36 + 49 + 64 = 149. The difference is 441 - 149 = 292, matching the value given in the question. This confirms that 6, 7 and 8 are the correct numbers and 8 is the largest. Why Other Options Are Wrong: If the largest number were 5, then the numbers would be 3, 4 and 5 or 4, 5 and 6, and substituting these into the statement would not produce a difference of 292. Similar checks show that sets with largest numbers 6 or 7 cannot satisfy the required condition. Only the triple 6, 7 and 8 works, corresponding to largest number 8. Common Pitfalls: Some learners choose to represent the consecutive numbers as n, n + 1 and n + 2. Although this also works, it leads to slightly more complicated expressions. Others may square the sum incorrectly or forget to fully expand and simplify the sum of squares. Using the symmetric representation n - 1, n and n + 1 and working carefully through each algebraic step helps avoid these mistakes. Final Answer: The largest of the three positive consecutive natural numbers is 8, which corresponds to option D.

Question 2

What is the value of 0.53 + 0.53?

Accepted Answer

Introduction / Context: This question is a simple test of decimal addition. You need to add 0.53 and 0.53 and express the result correctly in decimal form. Even though this looks straightforward, such questions are used to check basic accuracy with decimal place value and carrying over digits when the sum of decimal parts exceeds 9 in any column. Given Data / Assumptions: - The two numbers to be added are 0.53 and 0.53. - Both numbers have two decimal places. - We assume standard base ten arithmetic. Concept / Approach: When adding decimals, it is important to line up the decimal points and then add column by column, just as with whole numbers. The hundredths digits are added together, then the tenths, and finally the units digits, taking care to carry over if a sum in any column is 10 or more. Because the two numbers are equal, this is equivalent to doubling 0.53, which can also be computed as 2 * 0.53. Step-by-Step Solution: Step 1: Write the numbers vertically with decimal points aligned: 0.53 + 0.53 Step 2: Add the hundredths digits: 3 + 3 = 6. Place 6 in the hundredths position. Step 3: Add the tenths digits: 5 + 5 = 10. Write 0 in the tenths position and carry 1 to the units position. Step 4: Add the units digits: 0 + 0 + carried 1 = 1. Place 1 in the units position. Step 5: Write the decimal point directly below the decimal points in the numbers added. Step 6: The result is 1.06. Verification / Alternative check: An alternative way is to interpret 0.53 as 53/100. Then 0.53 + 0.53 = 53/100 + 53/100 = (53 + 53) / 100 = 106 / 100. Simplifying, 106 / 100 = 1.06. Both approaches produce the same answer, confirming that 1.06 is correct. Why Other Options Are Wrong: The values 1.053 and 1.063 come from incorrect placement of digits or confusion between hundredths and thousandths. The value 1.16 would result from incorrectly adding tenths as 5 + 5 = 11 and writing 1.11 then misplacing the decimal. None of these match the carefully computed sum of 1.06. Common Pitfalls: Learners sometimes ignore place value when adding decimals, treating 0.53 as 53 and doubling to get 106, then forgetting to divide by 100 again. Others may not line up decimal points and therefore add mismatched digits. Always align decimal points and remember that 0.53 has five in the tenths place and three in the hundredths place to avoid these errors. Final Answer: The value of 0.53 + 0.53 is 1.06, which corresponds to option A.

Question 3

What is the quotient when 1359 is divided by 48?

Accepted Answer

Introduction / Context: This question assesses basic division skills with three and two digit numbers. You are asked to find the quotient when 1359 is divided by 48. The focus is on performing integer division accurately, identifying how many full groups of 48 fit into 1359 and disregarding the remainder for the final answer. Given Data / Assumptions: - Dividend = 1359. - Divisor = 48. - We are asked for the quotient only, not the remainder. Concept / Approach: To find the quotient, we can estimate how many times 48 goes into 1359 by using nearby multiples. Multiplying 48 by convenient integers such as 20, 25 and 30 provides a range for the quotient. We then refine our guess by checking exact products until we find the largest multiple of 48 that does not exceed 1359. This multiple corresponds to the quotient, and the difference between 1359 and that multiple is the remainder (which is not required here but helps confirm the calculation). Step-by-Step Solution: Step 1: Start with an estimate. 50 * 20 = 1000, so 48 * 20 will be slightly less than 1000. This tells us that the quotient is greater than 20. Step 2: Compute 48 * 25 = 48 * (20 + 5) = 960 + 240 = 1200. Step 3: Since 1200 is still less than 1359, the quotient is greater than 25. Step 4: Compute 48 * 30 = 1440, which is larger than 1359, so the quotient must be less than 30. Step 5: Narrow the search between 25 and 29. Compute 48 * 28 = 48 * (20 + 8) = 960 + 384 = 1344. Step 6: Compute 48 * 29 = 48 * (30 - 1) = 1440 - 48 = 1392, which is greater than 1359. Step 7: Therefore the largest multiple of 48 that does not exceed 1359 is 1344, which is 48 * 28. Step 8: The quotient is 28 and the remainder is 1359 - 1344 = 15, though only the quotient is required. Verification / Alternative check: As a quick check, use approximate division. 1359 ÷ 48 is close to 1360 ÷ 50, which equals 27.2. This suggests the actual quotient should be slightly higher than 27 but less than 30, consistent with 28. The exact product calculations confirm that 28 is correct and 29 would overshoot the dividend. Why Other Options Are Wrong: Option A (29) corresponds to a product 1392, which is greater than 1359, so it cannot be the quotient. Option B (31) and option C (30) give even larger products and are clearly too big. Only 28 yields a product less than or equal to 1359 while 29 surpasses it, so 28 is the correct quotient. Common Pitfalls: Common mistakes include approximating too loosely and rounding to the nearest integer without checking the exact multiples. Another error is to ignore the fact that the quotient must give a product not exceeding the dividend. Always compute at least two nearby products to verify your chosen quotient in such questions. Final Answer: The quotient when 1359 is divided by 48 is 28, which corresponds to option D.

Question 4

If 49 is subtracted from the square of a number, the result is 576. What is that number?

Accepted Answer

Introduction / Context: This question involves a simple quadratic equation arising from a sentence about squares and subtraction. You are told that when 49 is subtracted from the square of a number, the result is 576. From this, you must determine the original number. The problem tests basic algebra skills and understanding of square roots. Given Data / Assumptions: - Let the unknown number be n. - The expression n^2 - 49 represents the square of the number minus 49. - We are told that n^2 - 49 = 576. - We are looking for a positive integer solution as suggested by the options. Concept / Approach: The key idea is to convert the verbal description into an algebraic equation and then isolate n. After adding 49 to both sides, we obtain n^2 on one side equal to a known value. Taking the positive square root of that value (because the context suggests a positive number) gives the answer. Recognizing perfect squares and their roots is very helpful here. Step-by-Step Solution: Step 1: Introduce a variable n for the number. The condition becomes n^2 - 49 = 576. Step 2: Add 49 to both sides of the equation to isolate the square term: n^2 = 576 + 49. Step 3: Compute 576 + 49 = 625. Step 4: Now we have n^2 = 625. Step 5: Take square roots of both sides: n = ±√625. Step 6: Compute √625. Since 25 * 25 = 625, √625 = 25. Step 7: Thus n = 25 or n = -25 formally, but the options list only positive numbers and the usual interpretation of such questions is the positive value. Step 8: Therefore the required number is 25. Verification / Alternative check: Verify by substituting n = 25 into the original statement. Compute the square: 25^2 = 625. Subtract 49: 625 - 49 = 576, which matches the given result. For completeness, note that n = -25 also satisfies the equation algebraically, since (-25)^2 is also 625, but this value is not among the listed options and problems of this type usually intend the positive root. Why Other Options Are Wrong: If n = 24, then n^2 - 49 = 576 - 49 = 527, not 576. If n = 23, n^2 is 529, and 529 - 49 = 480. If n = 27, n^2 is 729, and 729 - 49 = 680. None of these match the required result of 576. Only n = 25 produces the correct value. Common Pitfalls: Some learners mistakenly subtract 49 from 576 instead of adding it back to get n^2. Others may forget that squaring a negative number also produces a positive result, although in this context the positive root is the relevant one. Carefully translating the sentence into n^2 - 49 = 576 and then solving step by step avoids such mistakes. Final Answer: The number whose square minus 49 equals 576 is 25, which corresponds to option B.

Question 5

In triangle ABC, angle B is 90°. If cosec A = 13/12 and side AB = 10 cm, what is the length (in cm) of side AC?

Accepted Answer

Introduction / Context: This question involves a right angled triangle and basic trigonometric ratios. You are told that angle B is 90 degrees, cosec A is 13/12 and one of the sides, AB, is 10 cm. You are asked to find the length of side AC. This tests understanding of right triangle geometry, the definition of cosecant and the Pythagoras theorem. Given Data / Assumptions: - Triangle ABC is right angled at B, so angle B = 90°. - Side AB = 10 cm. - cosec A = 13 / 12. - AC is the hypotenuse (opposite the right angle at B). - All sides are positive real lengths. Concept / Approach: In a right triangle with angle B = 90°, side AC is the hypotenuse, side BC is opposite angle A and side AB is opposite angle C. The trigonometric ratio sine of angle A is defined as opposite over hypotenuse, that is sin A = BC / AC. Its reciprocal, cosec A, is AC / BC. We use the given value of cosec A to express AC in terms of BC, then apply the Pythagoras theorem AC^2 = AB^2 + BC^2 to find the actual lengths. Finally we solve the resulting equations to determine AC. Step-by-Step Solution: Step 1: In triangle ABC, since angle B is 90°, AC is the hypotenuse. Step 2: For angle A, the side opposite A is BC and the hypotenuse is AC. Step 3: By definition, sin A = BC / AC and cosec A = 1 / sin A = AC / BC. Step 4: We are given that cosec A = 13 / 12, so AC / BC = 13 / 12. Step 5: This implies AC : BC = 13 : 12. We can write AC = 13k and BC = 12k for some positive k. Step 6: We also know AB = 10 cm, and Pythagoras theorem gives AC^2 = AB^2 + BC^2. Step 7: Substitute AC = 13k and BC = 12k into the Pythagoras formula: (13k)^2 = 10^2 + (12k)^2. Step 8: Compute the squares: 169k^2 = 100 + 144k^2. Step 9: Subtract 144k^2 from both sides: 169k^2 - 144k^2 = 25k^2 = 100. Step 10: Solve for k^2: k^2 = 100 / 25 = 4, so k = 2 (taking the positive root since k represents a length factor). Step 11: Now AC = 13k = 13 * 2 = 26 cm. Verification / Alternative check: Verify by finding BC as well. BC = 12k = 12 * 2 = 24 cm. Check Pythagoras: AB^2 + BC^2 = 10^2 + 24^2 = 100 + 576 = 676. The square root of 676 is 26, so AC = 26 cm, consistent with our result. Also, check cosec A: AC / BC = 26 / 24 = 13 / 12, matching the given value. Both checks confirm the correctness of AC = 26 cm. Why Other Options Are Wrong: An answer of 24 cm would incorrectly treat BC as the hypotenuse or misapply the trigonometric ratio. Values of 12 cm or 14 cm are too small to serve as the hypotenuse in a right triangle with a leg of length 10 cm and do not satisfy either the Pythagoras theorem or the given cosec A value. Only 26 cm is consistent with all the given information. Common Pitfalls: Some learners mix up which side is opposite angle A and which side is the hypotenuse, leading to a wrong ratio for sine and cosecant. Others may forget that angle B is the right angle and misidentify the hypotenuse. Carefully labeling the triangle and writing down the definitions of the trigonometric ratios before substituting values helps avoid these errors. Final Answer: The length of side AC is 26 cm, which corresponds to option D.

Question 6

What least number must be added to 213 so that the resulting sum is exactly divisible by 9?

Accepted Answer

Introduction / Context: This question tests your understanding of the divisibility rule for 9 and your ability to adjust a number slightly so that it becomes a multiple of 9. You are asked to find the smallest number that must be added to 213 so that the sum is divisible by 9. Such problems are very common in aptitude tests and help reinforce patterns in digit sums and divisibility. Given Data / Assumptions: - The original number is 213. - We need to add a non negative integer k such that 213 + k is divisible by 9. - The smallest such k is required. Concept / Approach: The divisibility rule for 9 states that a number is divisible by 9 if and only if the sum of its digits is a multiple of 9. Therefore, we can find the sum of the digits of 213, determine the next higher multiple of 9 and then see how much needs to be added to the digit sum to reach that multiple. The same amount must be added to the original number to make it divisible by 9. Step-by-Step Solution: Step 1: Compute the sum of the digits of 213: 2 + 1 + 3. Step 2: 2 + 1 + 3 = 6. Step 3: The nearest multiple of 9 that is greater than or equal to 6 is 9. Step 4: To make the digit sum equal to 9, we must add 3 to the current digit sum of 6. Step 5: Adding 3 to the original number 213 gives 213 + 3 = 216. Step 6: Check the digit sum of 216: 2 + 1 + 6 = 9, which is a multiple of 9. Step 7: Therefore, 216 is divisible by 9, and 3 is the least number that must be added to 213. Verification / Alternative check: We can also verify by direct division. 216 ÷ 9 = 24, which is an integer, confirming that 216 is divisible by 9. If we tried adding 1 or 2 instead, we would get 214 or 215. The digit sums of 214 (2 + 1 + 4 = 7) and 215 (2 + 1 + 5 = 8) are not multiples of 9, so neither is divisible by 9. Thus 3 is indeed the smallest possible addition. Why Other Options Are Wrong: Adding 1 yields 214, which is not divisible by 9 since its digit sum is 7. Adding 2 gives 215, digit sum 8, also not divisible by 9. Adding 4 gives 217, with digit sum 10, again not a multiple of 9. Only adding 3 produces a digit sum that is a multiple of 9 and a number divisible by 9. Common Pitfalls: Some learners confuse the rules for divisibility by 3 and 9 or miscompute the digit sum. Others may attempt long division by 9 on several nearby numbers instead of using the quick digit sum method. Carefully applying the digit sum rule is faster and less error prone. Final Answer: The least number that must be added to 213 to make it divisible by 9 is 3, which corresponds to option A.

Question 7

The sum of a non-zero number and twenty times its reciprocal is 9. What is the larger of the possible values of this number?

Accepted Answer

Introduction / Context: This problem explores the relationship between a number and its reciprocal. You are told that the sum of a non zero number and twenty times its reciprocal equals 9. From this, you must find the larger of the possible values of the number. The question leads naturally to a quadratic equation and tests your ability to handle reciprocals and solve quadratics. Given Data / Assumptions: - Let the non zero number be x. - Its reciprocal is 1 / x. - The equation is x + 20 * (1 / x) = 9. - x is non zero, so division by x is valid. - We are specifically asked for the larger possible value of x. Concept / Approach: The main idea is to convert the given expression into a standard quadratic equation. Multiplying through by x eliminates the denominator and yields a polynomial equation. Factoring this quadratic gives two possible values for x. Because the problem asks for the larger value, we pick the bigger of the two real roots. This process combines algebraic manipulation with an understanding of reciprocals. Step-by-Step Solution: Step 1: Start from the equation x + 20 * (1 / x) = 9. Step 2: Write it as x + 20 / x = 9. Step 3: Multiply both sides of the equation by x to clear the denominator: x^2 + 20 = 9x. Step 4: Rearrange the equation into standard quadratic form: x^2 - 9x + 20 = 0. Step 5: Factor the quadratic: x^2 - 9x + 20 = (x - 4)(x - 5) = 0. Step 6: Set each factor to zero to find the roots: x - 4 = 0 or x - 5 = 0. Step 7: Thus, x = 4 or x = 5. Step 8: Both solutions are non zero and satisfy the original equation, as can be checked by substitution. Step 9: The larger of these two values is x = 5. Verification / Alternative check: Verify each root. For x = 4, compute x + 20 / x = 4 + 20 / 4 = 4 + 5 = 9. For x = 5, compute x + 20 / x = 5 + 20 / 5 = 5 + 4 = 9. Both satisfy the original equation. Since the question asks for the larger value, we select 5. This confirms that 5 is the correct answer. Why Other Options Are Wrong: Option A (-5) and option C (-3) give negative values that do not satisfy x + 20 / x = 9 when substituted. Option B (4) is a valid solution of the equation but it is the smaller of the two possible values, while the question explicitly asks for the larger one. Therefore, only 5 correctly answers what is asked. Common Pitfalls: Some learners try to guess x by mental arithmetic without forming the quadratic, which can be unreliable. Others may forget to multiply the entire equation by x, leading to an incorrect quadratic. A few may stop after finding only one root, ignoring the possibility of a second solution. Writing the equation in standard form, factoring carefully and then comparing the roots ensures a correct and complete solution. Final Answer: The larger possible value of the number is 5, which corresponds to option D.

Question 8

In an arithmetic progression, the 3rd term is 13 and the 5th term is 21. What is the 13th term of this progression?

Accepted Answer

Introduction / Context: This problem deals with arithmetic progressions (A.P.), which are sequences where each term after the first is obtained by adding a constant difference. You are given the 3rd and 5th terms of an A.P. and asked to find the 13th term. This tests your ability to use the general term formula of an arithmetic progression and to solve for the first term and common difference from partial information. Given Data / Assumptions: - Let the first term of the A.P. be a. - Let the common difference be d. - The 3rd term is given as 13. - The 5th term is given as 21. - We need to find the 13th term. Concept / Approach: In an arithmetic progression, the n-th term T_n is given by T_n = a + (n - 1)d. Using this formula, we can express the 3rd and 5th terms in terms of a and d, giving two equations. Solving these equations yields the values of a and d. Then we substitute these into the formula for the 13th term to find its value. This is a standard method in A.P. questions. Step-by-Step Solution: Step 1: Write the formula for the n-th term: T_n = a + (n - 1)d. Step 2: For the 3rd term, n = 3, so T_3 = a + 2d. We are given T_3 = 13, so a + 2d = 13. Step 3: For the 5th term, n = 5, so T_5 = a + 4d. We are given T_5 = 21, so a + 4d = 21. Step 4: Subtract the first equation from the second to eliminate a: (a + 4d) - (a + 2d) = 21 - 13. Step 5: This gives 2d = 8, so d = 4. Step 6: Substitute d = 4 back into a + 2d = 13: a + 2 * 4 = 13. Step 7: So a + 8 = 13, which gives a = 5. Step 8: Now find the 13th term using T_13 = a + 12d. Step 9: Substitute a = 5 and d = 4: T_13 = 5 + 12 * 4. Step 10: Compute 12 * 4 = 48, so T_13 = 5 + 48 = 53. Verification / Alternative check: We can list some terms of the progression to verify. With a = 5 and d = 4, the sequence starts as 5, 9, 13, 17, 21, ... The 3rd term is 13 and the 5th term is 21, matching the given information. Continuing, the 6th term is 25, and so on. The 13th term is obtained by adding 4 twelve times to 5, which is 5 + 48 = 53, consistent with our formula based calculation. Why Other Options Are Wrong: Values like 49, 57 or 61 would arise from incorrect values of the common difference or arithmetic mistakes when computing T_13. For example, using d = 3 or miscounting the number of increments would lead to such errors. Only 53 is consistent with both given terms and the arithmetic progression formula. Common Pitfalls: Common mistakes include miswriting the general term formula as a + nd instead of a + (n - 1)d, or incorrectly subtracting equations and getting d wrong. Another pitfall is miscounting the steps to the 13th term and using 11d instead of 12d. Carefully applying the formula and double checking each algebraic step prevents these issues. Final Answer: The 13th term of the arithmetic progression is 53, which corresponds to option A.

Question 9

In what ratio does the x-axis divide the line segment joining the points (-1, -12) and (3, 4)?

Accepted Answer

Introduction / Context: This coordinate geometry question asks you to find the ratio in which the x-axis divides the line segment joining two points, one below and one above the x-axis. The x-axis corresponds to all points with y = 0, so the point of intersection lies where the y coordinate of the segment becomes zero. This problem tests your understanding of section formula, linear interpolation and the idea that the line segment is divided in proportion to distances from the intercept point. Given Data / Assumptions: - Point A is (-1, -12). - Point B is (3, 4). - The x-axis is the line y = 0. - The x-axis intersects the segment AB at some point P where y = 0. - We need the ratio in which P divides AB, typically written as AP : PB. Concept / Approach: The line segment between A and B is divided by the x-axis at a point with y coordinate zero. Because the y coordinates change linearly along the segment, the distances from P to A and from P to B are proportional to the magnitudes of the y coordinates at A and B. Alternatively, we can use the section formula for internal division: if a point P divides the segment joining A(x1, y1) and B(x2, y2) in the ratio m : n, then its coordinates are ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)). We use the condition that the y coordinate of P is zero to find the ratio m : n. Step-by-Step Solution: Step 1: Let A(-1, -12) and B(3, 4). Step 2: Suppose the x-axis (y = 0) cuts AB at point P, dividing AB in the ratio m : n, where AP : PB = m : n. Step 3: By the section formula, the y coordinate of P is (m * 4 + n * (-12)) / (m + n). Step 4: Since P lies on the x-axis, its y coordinate is 0. Therefore, (4m - 12n) / (m + n) = 0. Step 5: The numerator must be zero, so 4m - 12n = 0. Step 6: Divide both sides by 4 to get m - 3n = 0, so m = 3n. Step 7: Hence the ratio m : n is 3 : 1, meaning AP : PB = 3 : 1. Verification / Alternative check: You can also reason using vertical distances from the x-axis. Point A is 12 units below the x-axis (y = -12), and point B is 4 units above it (y = 4). The x-axis will divide the segment in the inverse ratio of these distances, giving AP : PB = 3 : 1 because 12 : 4 simplifies to 3 : 1. This matches the algebraic result from the section formula. Why Other Options Are Wrong: Ratios such as 1 : 3, 3 : 2 and 2 : 3 do not match the derived relationship m = 3n. For example, 1 : 3 would imply that AP is shorter than PB, which contradicts the fact that the point is closer in y value to B (4 units) than to A (12 units). Only 3 : 1 correctly reflects the distances to the x-axis and satisfies the section formula equation. Common Pitfalls: Some learners mistakenly invert the ratio, writing 1 : 3 instead of 3 : 1, or confuse AP : PB with BP : PA. Others may misapply the section formula by swapping x and y coordinates or by setting the wrong coordinate to zero. Being clear about which segment corresponds to which part of the ratio and carefully using the y coordinate to enforce y = 0 helps avoid these errors. Final Answer: The x-axis divides the line segment joining (-1, -12) and (3, 4) in the ratio 3 : 1, which corresponds to option C.

Question 10

Triangle ABC is right angled at B. BD is the altitude drawn from B to hypotenuse AC. If DC = 9 cm and AC = 25 cm, what is the length of side BC (in cm)?

Accepted Answer

Introduction / Context: This question uses a special property of right angled triangles and the altitude from the right angle to the hypotenuse. In triangle ABC, angle B is 90 degrees and BD is the altitude from B to the hypotenuse AC. You are given the length of the hypotenuse AC and the segment DC on it, and asked to find the length of side BC. This tests your knowledge of geometric mean relationships in right triangles and the Pythagoras theorem. Given Data / Assumptions: - Triangle ABC is right angled at B. - BD is drawn from B perpendicular to AC, so BD is an altitude. - AC is the hypotenuse and has length 25 cm. - DC, a segment of the hypotenuse, has length 9 cm. - We need to find the length of side BC. Concept / Approach: In a right triangle, when an altitude is drawn from the right angle to the hypotenuse, several useful relationships hold. One of them states that the square of each leg equals the product of the hypotenuse and the projection of that leg onto the hypotenuse. Specifically, BC^2 = AC * DC and AB^2 = AC * AD, where AD and DC are the segments into which the altitude divides the hypotenuse. We use this geometrical mean relation directly since AC and DC are given. Step-by-Step Solution: Step 1: Note that AC is the hypotenuse of the right triangle ABC and AC = 25 cm. Step 2: BD is the altitude from the right angle B to hypotenuse AC, dividing AC into two segments AD and DC. Step 3: We are given DC = 9 cm. Then the other segment AD is AC - DC = 25 - 9 = 16 cm (this may be useful for cross checking). Step 4: Use the right triangle property: The square of the leg adjacent to DC, which is BC, satisfies BC^2 = AC * DC. Step 5: Substitute the known values: BC^2 = 25 * 9. Step 6: Compute 25 * 9 = 225. Step 7: So BC^2 = 225. Taking the positive square root, BC = √225 = 15 cm. Verification / Alternative check: We can also verify by finding AB using the related formula AB^2 = AC * AD = 25 * 16 = 400, so AB = 20 cm. Check Pythagoras theorem in triangle ABC: AB^2 + BC^2 = 20^2 + 15^2 = 400 + 225 = 625. The square root of 625 is 25, equal to AC. This confirms that the side lengths 15, 20 and 25 form a right triangle and that BC = 15 cm is correct. Why Other Options Are Wrong: Values like 12, 18 or 16 do not satisfy the relationship BC^2 = 25 * 9. For example, if BC were 12, BC^2 would be 144, which is not equal to 225. Similarly, 16^2 = 256 and 18^2 = 324 both fail to match 225. Thus these options contradict the geometric mean property and Pythagoras theorem for the given triangle. Common Pitfalls: Students sometimes attempt to use Pythagoras theorem directly without recognizing the role of the altitude and the segments on the hypotenuse, which leads to missing information. Others may incorrectly apply the formula BC^2 = AD * DC instead of AC * DC. Remember that the leg squared equals hypotenuse times the adjacent hypotenuse segment, while the altitude squared equals the product of the two hypotenuse segments. Keeping these relationships straight is essential. Final Answer: The length of side BC is 15 cm, which corresponds to option D.

Question 11

Find the average of all prime numbers between 20 and 50.

Accepted Answer

Introduction / Context: This question asks you to work with prime numbers in a given interval and compute their average. You need to identify all primes between 20 and 50, sum them and divide by how many there are. It tests your ability to recognize prime numbers, perform accurate addition and correctly apply the formula for the arithmetic mean. Given Data / Assumptions: - We are interested in all prime numbers strictly between 20 and 50. - A prime number has exactly two distinct positive divisors: 1 and the number itself. - We must compute their arithmetic mean, that is, sum divided by count. Concept / Approach: The first step is to list all primes between 20 and 50 by checking each integer for divisibility by smaller primes such as 2, 3, 5 and 7. After listing the primes, we add them together and then divide the sum by the number of primes. Using careful arithmetic avoids errors in both the list and the calculations. Since the options are given with decimal values, we may express the average as a fraction first and then convert to decimal form. Step-by-Step Solution: Step 1: List integers between 20 and 50 and identify which are prime: 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49. Step 2: Check each candidate. - 23 is prime (no divisors other than 1 and 23). - 29 is prime. - 31 is prime. - 37 is prime. - 41 is prime. - 43 is prime. - 47 is prime. Step 3: Composite numbers like 21, 22, 24, 25, 26, 27, 28, 30, 32 and so on have obvious divisors and are not primes. Step 4: The full list of primes between 20 and 50 is therefore: 23, 29, 31, 37, 41, 43 and 47. Step 5: Count them: there are 7 prime numbers. Step 6: Compute their sum: 23 + 29 = 52; 52 + 31 = 83; 83 + 37 = 120; 120 + 41 = 161; 161 + 43 = 204; 204 + 47 = 251. Step 7: The average is sum divided by count: 251 / 7. Step 8: Divide 251 by 7. Since 7 * 35 = 245, the remainder is 6, so 251 / 7 = 35 + 6/7. Step 9: 6/7 is approximately 0.857 (because 6 ÷ 7 ≈ 0.857). So the average is about 35.857. Verification / Alternative check: You can verify the decimal more precisely. 6/7 = 0.857142..., which rounds to 0.857 to three decimal places. Adding this to 35 gives 35.857. Comparing with the options shows that 35.857 exactly matches option A. This cross check ensures both the prime list and arithmetic are correct. Why Other Options Are Wrong: Values 35.657, 34.857 and 36.657 correspond to averages computed from incorrect sums or wrong counts of primes. For example, missing one prime or including a composite number would change both the sum and the count, producing a different average. None of these alternatives simplify to 251 / 7. Common Pitfalls: Students sometimes forget to include 23 or 47 or mistakenly classify a composite like 33 or 39 as prime. Others may misadd the primes, especially if done quickly. Carefully checking divisibility and summing step by step helps ensure accuracy. Using the fraction 251 / 7 before converting to decimal also reduces rounding errors. Final Answer: The average of all prime numbers between 20 and 50 is 35.857, which corresponds to option A.

Question 12

Divide 20 into two parts such that the sum of the squares of the two parts is 232. What are the two parts?

Accepted Answer

Introduction / Context: This algebra question asks you to split the number 20 into two parts such that the sum of their squares equals 232. You must find the specific pair of numbers that satisfies both the sum condition and the sum of squares condition. This tests your ability to form and solve quadratic equations using basic identities. Given Data / Assumptions: - Let the two parts be x and y. - Their sum is x + y = 20. - The sum of their squares is x^2 + y^2 = 232. - We seek real number solutions, and the answer options suggest positive integers. Concept / Approach: We have two equations involving x and y: one for their sum and one for the sum of their squares. Using the identity (x + y)^2 = x^2 + y^2 + 2xy, we can express xy in terms of known quantities. Once we know x + y and xy, we can form a quadratic equation with roots x and y. Solving this quadratic yields the required two numbers. This method avoids guessing and ensures that both conditions are satisfied. Step-by-Step Solution: Step 1: Let x and y be the two parts with x + y = 20. Step 2: We are given x^2 + y^2 = 232. Step 3: Use the identity (x + y)^2 = x^2 + y^2 + 2xy. Step 4: Substitute the known values: (20)^2 = 232 + 2xy. Step 5: Compute 20^2 = 400. Step 6: The equation becomes 400 = 232 + 2xy. Step 7: Subtract 232 from both sides: 400 - 232 = 2xy, so 168 = 2xy. Step 8: Divide by 2 to find xy: xy = 84. Step 9: Now x and y are roots of the quadratic equation t^2 - (sum)t + (product) = 0, that is t^2 - 20t + 84 = 0. Step 10: Solve t^2 - 20t + 84 = 0 by factoring. We need two numbers whose product is 84 and whose sum is 20. Step 11: The pair 6 and 14 satisfies 6 * 14 = 84 and 6 + 14 = 20. Step 12: Therefore the two parts are 6 and 14. Verification / Alternative check: Verify both conditions with x = 6 and y = 14. The sum is 6 + 14 = 20, which matches the requirement. The sum of squares is 6^2 + 14^2 = 36 + 196 = 232, exactly as given in the problem. Therefore this pair satisfies both conditions perfectly and is the correct answer. Why Other Options Are Wrong: For 8 and 12, the sum is 20 but the sum of squares is 64 + 144 = 208, not 232. For 4 and 16, the sum is 20 but 4^2 + 16^2 = 16 + 256 = 272. For 10 and 10, the sum is 20 but 10^2 + 10^2 = 200. None of these match the required sum of squares of 232, so they are incorrect. Common Pitfalls: Some learners may try to guess pairs that add to 20 and check their squares one by one, which is inefficient and prone to error. Others may misuse the identity or forget the factor of 2 in 2xy, leading to wrong values for xy. Using the identity systematically and forming the quadratic equation ensures a clear and reliable solution path. Final Answer: The two parts into which 20 must be divided are 6 and 14, which corresponds to option A.

Question 13

Of three numbers whose average is 64, the first number is equal to one-third of the sum of the other two numbers. What is the value of this first number?

Accepted Answer

Introduction / Context: This problem tests the concept of averages and how to convert a verbal condition about three numbers into algebraic equations. Questions of this type are common in aptitude exams because they check whether the learner can translate sentences into equations and then solve them logically. Understanding how the average relates to the total sum is particularly important for many quantitative reasoning topics. Given Data / Assumptions: There are three numbers, let them be a, b, and c. The average of the three numbers is 64. The first number a is equal to one-third of the sum of the other two numbers, so a = (b + c) / 3. All numbers are assumed to be real numbers, and we are asked to find the value of the first number a. Concept / Approach: The key ideas are: average = (sum of numbers) / (number of numbers), and the ability to express the verbal condition as an equation. Once we know the average, we can compute the total sum of the three numbers. Then we use the relationship between a, b, and c to form a system of equations and solve for a. Because there are two equations involving a, b, and c, we can eliminate b and c to get a single equation in a, which is easy to solve. Step-by-Step Solution: Step 1: Let the three numbers be a, b, and c.Step 2: The average is 64, so (a + b + c) / 3 = 64.Step 3: Multiply both sides by 3 to get the total sum: a + b + c = 64 * 3 = 192.Step 4: The condition says the first number is one-third of the sum of the other two, so a = (b + c) / 3.Step 5: Rearrange this to get b + c = 3a.Step 6: Substitute b + c = 3a into the sum equation a + b + c = 192, giving a + 3a = 192.Step 7: Simplify to get 4a = 192.Step 8: Divide both sides by 4 to obtain a = 192 / 4 = 48. Verification / Alternative check: We can check the result by constructing one possible set of numbers. Suppose a = 48. Then b + c = 3a = 144. One simple example is b = 70 and c = 74, whose sum is 144. Then a + b + c = 48 + 70 + 74 = 192. The average is 192 / 3 = 64, which matches the given condition. This confirms that a = 48 is consistent with the problem statement. Why Other Options Are Wrong: Option 32: If a = 32, then b + c = 3a = 96 and a + b + c = 128, so the average would be 128 / 3, which is not 64. Option 72: If a = 72, then b + c = 216 and a + b + c = 288, giving an average of 96, not 64. Option 96: If a = 96, then b + c = 288 and a + b + c = 384, giving an average of 128, again not 64. Option 56: If a = 56, then b + c = 168 and a + b + c = 224, giving an average of 224 / 3, which is not 64. Common Pitfalls: A common mistake is to treat the phrase one-third of the sum of the other two numbers as a = (b + c) / 2 or a = b / 3 + c, which is incorrect. Another error is to confuse average with sum and forget to multiply the average by the number of terms. Learners might also incorrectly assume symmetry between the three numbers and think they are all equal, which violates the given condition about the first number. Careful translation of the language into equations is essential. Final Answer: The value of the first number is 48.

Question 14

In an arithmetic progression, the 7th term is -15 and the 12th term is 5. What is the value of the 16th term of this arithmetic progression?

Accepted Answer

Introduction / Context: This question checks understanding of arithmetic progressions, especially how to use information about two non-consecutive terms to find any other term. In aptitude exams and competitive tests, candidates are often required to manipulate the general term formula of an arithmetic progression to obtain unknown values such as the first term, common difference, or any specific nth term. Given Data / Assumptions: The sequence is an arithmetic progression (A.P.). The 7th term of the A.P. is -15. The 12th term of the A.P. is 5. We are asked to find the 16th term of the same A.P. Let a be the first term and d be the common difference. Concept / Approach: For an arithmetic progression, the nth term is given by the formula Tn = a + (n - 1) * d. When we know two terms, say Tm and Tn, we can form two equations in a and d. Solving these equations simultaneously gives the values of a and d. Once a and d are known, we can compute any required term such as T16 using the same nth term formula. The problem is a straightforward example of forming and solving linear equations. Step-by-Step Solution: Step 1: Let a be the first term and d be the common difference.Step 2: The 7th term is T7 = a + (7 - 1) * d = a + 6d. We are given T7 = -15, so a + 6d = -15.Step 3: The 12th term is T12 = a + (12 - 1) * d = a + 11d. We are given T12 = 5, so a + 11d = 5.Step 4: Subtract the first equation from the second: (a + 11d) - (a + 6d) = 5 - (-15).Step 5: This simplifies to 5d = 20, so d = 4.Step 6: Substitute d = 4 into a + 6d = -15 to find a: a + 6 * 4 = -15, so a + 24 = -15 and a = -39.Step 7: The 16th term is T16 = a + (16 - 1) * d = a + 15d.Step 8: Substitute a = -39 and d = 4: T16 = -39 + 15 * 4 = -39 + 60 = 21. Verification / Alternative check: We can list a few terms around the 7th and 12th positions to verify. Starting with a = -39 and d = 4, the terms are: T1 = -39, T2 = -35, T3 = -31, T4 = -27, T5 = -23, T6 = -19, T7 = -15, T8 = -11, T9 = -7, T10 = -3, T11 = 1, T12 = 5. These match the given values. Continuing forward, T13 = 9, T14 = 13, T15 = 17, T16 = 21, which confirms our answer. Why Other Options Are Wrong: Option 25: This would correspond to a different common difference, inconsistent with T7 and T12. Option 29: This value would require the common difference to be higher than 4, which would break the given term values. Option 33: This is larger still and would make T7 and T12 incorrect if we recalculated them. Option 17: This is smaller than the correct value and would not satisfy the pair of equations derived from the 7th and 12th terms. Common Pitfalls: Candidates sometimes use the wrong index in the nth term formula, writing Tn = a + nd instead of a + (n - 1) * d. Another frequent mistake is to directly subtract the term values without forming proper equations, or to mix up which equation to subtract from which, leading to sign errors. Careful algebra and clear identification of T7 and T12 are important to avoid mistakes. Final Answer: The 16th term of the arithmetic progression is 21.

Question 15

What is the least number by which 37044 must be divided so that the resulting quotient is a perfect cube?

Accepted Answer

Introduction / Context: This question is about perfect cubes and prime factorization. It asks for the smallest divisor that, when used to divide a given number, produces a perfect cube. Problems like this are common in number theory sections of aptitude tests and help learners practice expressing numbers in terms of their prime factors and analysing exponents to achieve required properties such as being a perfect square or perfect cube. Given Data / Assumptions: The given number is 37044. We are looking for the least positive integer k such that 37044 / k is a perfect cube. All calculations are done with exact integer arithmetic. A perfect cube is a number of the form n^3 where n is an integer. Concept / Approach: The main idea is to factorize 37044 into its prime factors. For a number to be a perfect cube, the exponent of every prime in its factorization must be a multiple of 3. We need to find the smallest divisor k that we can remove from 37044 so that the remaining factorization has all exponents divisible by 3. This turns into an exponent adjustment problem, where we choose exponents for the divisor in such a way that the remaining exponents are multiples of 3. Step-by-Step Solution: Step 1: Factorize 37044 into prime factors.Step 2: One finds that 37044 = 2^2 * 3^3 * 7^3.Step 3: For a perfect cube, each prime exponent in the quotient must be a multiple of 3.Step 4: Suppose we divide 37044 by k = 2^a * 3^b * 7^c, where 0 ≤ a ≤ 2, 0 ≤ b ≤ 3, 0 ≤ c ≤ 3.Step 5: The quotient will then be 2^(2 - a) * 3^(3 - b) * 7^(3 - c).Step 6: For this quotient to be a perfect cube, 2 - a, 3 - b, and 3 - c must each be multiples of 3.Step 7: For the prime 2, exponents available are 2, 1, 0. Only 0 is a multiple of 3, so 2 - a = 0 which gives a = 2.Step 8: For the prime 3, we have 3 - b. Values 3 and 0 are multiples of 3, so b can be 0 or 3. To keep k minimal, choose b = 0.Step 9: For the prime 7, similarly 3 - c must be 3 or 0, so c = 0 or 3. To keep k minimal, choose c = 0.Step 10: Therefore the least such k is 2^2 = 4. Verification / Alternative check: If we divide 37044 by 4, we get 37044 / 4 = 9261. Now check if 9261 is a perfect cube. Note that 21^3 = 21 * 21 * 21 = 441 * 21 = 9261. Therefore 9261 is indeed a perfect cube, confirming that 4 is valid. If we tried k = 2, the quotient would be 2 * 3^3 * 7^3, which has exponent 1 for prime 2, not a multiple of 3, so it is not a perfect cube. Thus 4 is the least valid divisor. Why Other Options Are Wrong: Option 2: Leaves a factor of 2^1 in the quotient, so the exponent of 2 is not a multiple of 3. Option 14: Includes unnecessary factors of 7, making k larger than needed without improving the cube condition beyond what 4 already achieves. Option 21: Similar to 14, it includes extra factors which are not required to adjust exponents to multiples of 3. Option 6: Equal to 2 * 3 and again leaves the exponent of 2 as 1 in the quotient, which fails the cube requirement. Common Pitfalls: Some learners try to guess divisors instead of systematically using prime factorization, which often leads to incorrect answers. Another mistake is to confuse the condition for a perfect square (exponents must be even) with that for a perfect cube (exponents must be multiples of 3). It is also easy to forget that we want the least divisor, so including unnecessary prime factors in k leads to a larger number than required. Final Answer: The least number by which 37044 must be divided to obtain a perfect cube is 4.

Question 16

What is the greatest four-digit number that is exactly divisible by 93 without leaving any remainder?

Accepted Answer

Introduction / Context: This problem is a classic example of divisibility and working with the largest or smallest numbers satisfying a condition. Here, we look for the largest four-digit number that is divisible by 93. Such questions are common in number system and arithmetic sections and help develop fluency in handling division, remainders, and bounds like four-digit limits. Given Data / Assumptions: We are interested in four-digit numbers, that is, numbers from 1000 to 9999. The divisor is 93. We need the largest four-digit number that is exactly divisible by 93, which means the remainder must be zero. All calculations are done with integer arithmetic. Concept / Approach: To find the largest four-digit multiple of 93, we start from the largest four-digit number, 9999, and see how it behaves when divided by 93. If 9999 is not divisible by 93, the remainder tells us how much we must subtract to reach the nearest lower multiple. This uses the simple relation: number - remainder = nearest smaller multiple of the divisor. This is an efficient method compared to listing multiples one by one. Step-by-Step Solution: Step 1: The largest four-digit number is 9999.Step 2: Divide 9999 by 93 to find the quotient and remainder.Step 3: When 9999 is divided by 93, the quotient is 107 and the remainder is 48. So 9999 = 93 * 107 + 48.Step 4: To get the largest multiple of 93 that is not greater than 9999, subtract the remainder from 9999.Step 5: Compute 9999 - 48 = 9951.Step 6: Therefore, 9951 is a multiple of 93 and is less than or equal to 9999.Step 7: To confirm, divide 9951 by 93: 9951 / 93 = 107 exactly, so there is no remainder. Verification / Alternative check: An alternative viewpoint is to directly compute 93 * 108, which gives 10044. This is a five-digit number and therefore larger than any allowed four-digit number. So the previous multiple, 93 * 107, must be the largest four-digit multiple. As already computed, 93 * 107 = 9951, which matches our earlier result and confirms that 9951 is correct. Why Other Options Are Wrong: Option 9981: When divided by 93, it does not give an integer quotient; the remainder is not zero. Option 9971: Also fails divisibility by 93 and produces a non-zero remainder. Option 9961: Not a multiple of 93 and thus cannot be the correct answer. Option 9924: Although closer, it is smaller than 9951 and still not the largest possible multiple if 9951 is a valid multiple. Common Pitfalls: One common mistake is to divide 1000 or 9999 roughly and then round incorrectly, leading to an off-by-one error in the multiple. Another pitfall is attempting to list many multiples manually, which is time-consuming and prone to arithmetic mistakes. A structured approach using division with remainder is faster and more reliable for exam conditions. Final Answer: The largest four-digit number divisible exactly by 93 is 9951.

Question 17

In an arithmetic progression, the 4th term is 11 and the 7th term is -4. What is the value of the 15th term of this arithmetic progression?

Accepted Answer

Introduction / Context: This question examines understanding of arithmetic progressions and how to derive unknown terms when two non-consecutive terms are given. It is a standard application of the nth term formula and solving simultaneous linear equations, which makes it a typical aptitude problem on sequences and series. Given Data / Assumptions: The sequence is an arithmetic progression. The 4th term is 11. The 7th term is -4. We must find the 15th term. Let a denote the first term and d denote the common difference of the arithmetic progression. Concept / Approach: For an arithmetic progression, the nth term is Tn = a + (n - 1) * d. Given T4 and T7, we can form two equations in a and d. Subtracting these equations lets us find d, and then substituting back gives us a. After we know a and d, we use the same formula to calculate T15. This illustrates how relationships between different terms in an arithmetic progression can be exploited to find any term in the sequence. Step-by-Step Solution: Step 1: Let a be the first term and d be the common difference.Step 2: The 4th term is T4 = a + (4 - 1) * d = a + 3d. We are given T4 = 11, so a + 3d = 11.Step 3: The 7th term is T7 = a + (7 - 1) * d = a + 6d. We are given T7 = -4, so a + 6d = -4.Step 4: Subtract the first equation from the second to eliminate a: (a + 6d) - (a + 3d) = -4 - 11.Step 5: This simplifies to 3d = -15, so d = -5.Step 6: Substitute d = -5 into a + 3d = 11. Then a + 3 * (-5) = 11, so a - 15 = 11 and a = 26.Step 7: The 15th term is T15 = a + (15 - 1) * d = a + 14d.Step 8: Substitute a = 26 and d = -5: T15 = 26 + 14 * (-5) = 26 - 70 = -44. Verification / Alternative check: We can verify by listing some terms. Starting from a = 26 and d = -5, the terms are: T1 = 26, T2 = 21, T3 = 16, T4 = 11, T5 = 6, T6 = 1, T7 = -4. These match the given 4th and 7th terms. Continuing, T8 = -9, T9 = -14, T10 = -19, T11 = -24, T12 = -29, T13 = -34, T14 = -39, T15 = -44, which confirms the answer. Why Other Options Are Wrong: Option -49: This would correspond to a different common difference and would not keep T4 and T7 equal to 11 and -4. Option -39: This is actually T14 in this progression, not T15. Option -34: This is T13, so it is not the requested term either. Option -29: This equals T12, again not matching the index n = 15. Common Pitfalls: Students sometimes write Tn = a + nd instead of a + (n - 1) * d, which shifts all indices and leads to incorrect results. Others may incorrectly subtract equations, introducing sign errors. It is also easy to miscount when moving from one term to another without relying on the general term formula. Using the formula systematically avoids most counting mistakes. Final Answer: The 15th term of the arithmetic progression is -44.

Question 18

The sum of the first 10 terms of an arithmetic series is 390, and the third term of the series is 19. What is the value of the first term of this series?

Accepted Answer

Introduction / Context: This question combines the formula for the sum of an arithmetic series with the formula for the nth term. It tests whether you can handle multiple relationships at once: the total sum over a number of terms and the value of a particular term. Such questions help build algebraic manipulation skills and are frequently used in aptitude tests involving sequences. Given Data / Assumptions: The series is arithmetic. The sum of the first 10 terms, S10, is 390. The third term, T3, is 19. We are required to find the first term a. Let d be the common difference of the series. Concept / Approach: For an arithmetic progression, the nth term is Tn = a + (n - 1) * d, and the sum of the first n terms is Sn = n / 2 * (2a + (n - 1) * d). We are given T3 and S10, so we can write one equation using T3 and another using S10. Solving these two equations simultaneously will give us a and d. Once a is determined, we have answered the question. The important skill is substituting correctly into the formulas and keeping track of the algebraic manipulations. Step-by-Step Solution: Step 1: Let a be the first term and d be the common difference.Step 2: The third term is T3 = a + (3 - 1) * d = a + 2d. We are given T3 = 19, so a + 2d = 19.Step 3: The sum of the first 10 terms is S10 = 10 / 2 * (2a + (10 - 1) * d) = 5 * (2a + 9d).Step 4: We are given S10 = 390, so 5 * (2a + 9d) = 390.Step 5: Divide both sides by 5 to get 2a + 9d = 78.Step 6: Now we have a system of equations: (1) a + 2d = 19 and (2) 2a + 9d = 78.Step 7: From equation (1), express a in terms of d: a = 19 - 2d.Step 8: Substitute a = 19 - 2d into equation (2): 2(19 - 2d) + 9d = 78.Step 9: Simplify: 38 - 4d + 9d = 78, so 38 + 5d = 78.Step 10: Subtract 38 from both sides: 5d = 40, so d = 8.Step 11: Substitute d = 8 back into a = 19 - 2d: a = 19 - 16 = 3. Verification / Alternative check: We can quickly check by constructing the series. With a = 3 and d = 8, the first few terms are: 3, 11, 19, 27, 35, 43, 51, 59, 67, 75. The third term is 19, as required. Now sum these 10 terms: (3 + 11 + 19 + 27 + 35) + (43 + 51 + 59 + 67 + 75). The first group sums to 95, the second group also sums to 295, giving a total of 390. This confirms the correctness of a = 3. Why Other Options Are Wrong: Option 5: If a = 5, we would not be able to get T3 = 19 and S10 = 390 with a single common difference. Option 7: Leads to inconsistent values for the sum when combined with T3 = 19. Option 8: Would make T3 too large relative to 19 for any reasonable d. Option 1: Would force d to be unrealistically large to reach T3 = 19, and the sum would not match 390. Common Pitfalls: A common error is misusing the sum formula or writing S10 as 10 * (a + d) instead of 10 / 2 * (2a + 9d). Another pitfall is trying to guess the first term and common difference without forming equations, which is inefficient and error-prone. Careful substitution and stepwise algebra are the best way to handle such problems. Final Answer: The first term of the arithmetic series is 3.

Question 19

In an arithmetic progression, the sum of the first 11 terms is required, given that the first term is -31 and the last (11th) term is 29. What is the sum of these first 11 terms?

Accepted Answer

Introduction / Context: This problem uses the sum formula for an arithmetic progression when the first and last terms are known. It is a very standard type of series question in aptitude tests and helps learners become comfortable with quickly computing sums without listing all individual terms. Given Data / Assumptions: The sequence is an arithmetic progression. The first term, T1, is -31. The last term (which is T11) is 29. The number of terms n is 11. We need to compute the sum of the first 11 terms, S11. Concept / Approach: For an arithmetic progression, the sum of the first n terms is given by Sn = n / 2 * (first term + last term), when the first and nth terms are known. This formula comes from pairing terms from the beginning and end of the sequence, each pair having the same sum. Because both the first and the 11th term are given, we can apply the formula directly without first finding the common difference. Step-by-Step Solution: Step 1: Identify the given values: n = 11, first term a = -31, and last term l = 29.Step 2: Use the sum formula for an arithmetic progression: Sn = n / 2 * (a + l).Step 3: Substitute the values: S11 = 11 / 2 * (-31 + 29).Step 4: Compute the sum inside the parentheses: -31 + 29 = -2.Step 5: So S11 = 11 / 2 * (-2).Step 6: Simplify: 11 / 2 * (-2) = 11 * (-1) = -11. Verification / Alternative check: We can verify by noting that the average of the first and last term is (a + l) / 2 = (-31 + 29) / 2 = -2 / 2 = -1. In an arithmetic progression, the average of the first and last term equals the average of all terms when they are evenly spaced. Therefore, the average of all 11 terms is -1. Then the total sum is average * number of terms = -1 * 11 = -11. This matches the result from the formula, confirming that the answer is correct. Why Other Options Are Wrong: Option 42: This would correspond to a positive average value, which contradicts the fact that the first term is strongly negative and the last term is only slightly positive. Option 28: Also suggests a positive average, inconsistent with the endpoints. Option 12: This does not match the required average when consistent with the first and last terms. Option 0: Would require the average of the sequence to be 0, which would need symmetric terms around 0, but -31 and 29 are not symmetric around 0. Common Pitfalls: Some learners forget that the sum formula uses (a + l) and mistakenly use only the first term or only the last term. Others try to compute all 11 terms explicitly, which is unnecessary and time-consuming. It is also easy to miscalculate -31 + 29 and accidentally write +2 instead of -2, which would completely change the answer. Careful handling of negative numbers is important. Final Answer: The sum of the first 11 terms of the arithmetic progression is -11.

Question 20

Evaluate the sum 10.2 + 10.02 + 10.002 + 10.0002 + 10.00002 + 10.000002 + 10.0000002 + 10.00000002 + 10.000000002, that is, 10.2 + 10.02 + 10.002 + ... up to 9 terms.

Accepted Answer

Introduction / Context: This question involves a decimal series where each term has an increasing number of zeros after the decimal point before the digit 2. It is a nice example of applying geometric series ideas to decimal numbers. Instead of trying to add all terms directly, which can be messy, we can recognise a pattern and rewrite the sum in a more convenient algebraic form. Such problems test the ability to work with place values and geometric progressions in decimal form. Given Data / Assumptions: The series is 10.2 + 10.02 + 10.002 + 10.0002 + 10.00002 + 10.000002 + 10.0000002 + 10.00000002 + 10.000000002. There are exactly 9 terms. Each term consists of 10 plus a small decimal part. We are asked to find the total sum of all 9 terms. Concept / Approach: Each term can be viewed as 10 plus a fractional part. If we separate the constant 10 from each term, we can sum the integer contributions and the decimal contributions independently. The decimal parts form a geometric progression with first term 0.2 and common ratio 0.1. Using the sum formula for a geometric progression, we can compute the total of the decimal parts exactly. Then, adding the result to the sum of the integer parts gives the final answer. Step-by-Step Solution: Step 1: Observe that each term is 10 plus a decimal part: 10.2 = 10 + 0.2, 10.02 = 10 + 0.02, 10.002 = 10 + 0.002, and so on.Step 2: There are 9 terms, so the integer part contributes 9 * 10 = 90.Step 3: The decimal parts form the series 0.2 + 0.02 + 0.002 + 0.0002 + 0.00002 + 0.000002 + 0.0000002 + 0.00000002 + 0.000000002.Step 4: This decimal series is a geometric progression with first term a = 0.2 and common ratio r = 0.1, since each term is obtained by multiplying the previous term by 0.1.Step 5: The sum of the first n terms of a geometric progression is S = a * (1 - r^n) / (1 - r).Step 6: Here, n = 9, a = 0.2, and r = 0.1, so S = 0.2 * (1 - 0.1^9) / (1 - 0.1).Step 7: Compute the denominator: 1 - 0.1 = 0.9.Step 8: The numerator is 0.2 * (1 - 0.1^9). Since 0.1^9 is very small (0.000000001), we keep it symbolically to stay exact: S = 0.2 * (1 - 0.000000001) / 0.9.Step 9: So S = 0.2 / 0.9 * (1 - 0.000000001) = (2 / 9) * (0.999999999).Step 10: The exact value is 0.222222222, because (2 / 9) * (0.999999999) equals 0.222222222.Step 11: Therefore, the total sum of the original series is 90 (integer part) plus 0.222222222 (decimal part) = 90.222222222. Verification / Alternative check: We can approximate by ignoring the extremely tiny term 0.000000002 temporarily. Add the first few decimal parts: 0.2 + 0.02 + 0.002 + 0.0002 + 0.00002 ≈ 0.22222. Adding the rest brings it close to 0.222222222. When this is added to 90, the result is about 90.2222, which is consistent with the exact calculation of 90.222222222. Thus the chosen option matches both the exact and approximate reasoning. Why Other Options Are Wrong: Option 90.022222222: This is too small, as it effectively reduces the contribution from the decimal series. Option 92.222222222: This assumes the integer part sums to something larger than 90, which is impossible as we only have nine tens. Option 89.222222222: This would result from using 9 * 9 instead of 9 * 10 for the integer part. Option 90.202222222: This slightly underestimates the geometric sum of the decimal series and does not match the exact calculation. Common Pitfalls: A frequent mistake is to try to add all decimals manually without recognising the geometric pattern, leading to rounding errors. Another error is to forget that each term includes 10, and instead only add the decimal parts. Some learners also misapply the geometric series formula or mix up the values of a and r. Keeping the structure clear, splitting the sum into integer and decimal parts, and then methodically applying the geometric sum formula avoids these issues. Final Answer: The value of 10.2 + 10.02 + 10.002 + ... up to 9 terms is 90.222222222.

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