Question 1

Find the exact value of the trigonometric expression: cos^2 20° + cos^2 70°  Choose the correct exact value.

Accepted Answer

Introduction / Context: This question tests complementary-angle identities and the Pythagorean identity. Angles 20° and 70° are complementary (they add to 90°), which allows rewriting cosine in terms of sine and simplifying easily. Given Data / Assumptions: • Expression: cos^2 20° + cos^2 70° • Use standard identities for complementary angles Concept / Approach: Use the complementary identity: cos(90° − θ) = sin θ. Since 70° = 90° − 20°, we have: cos 70° = sin 20°. Then: cos^2 70° = sin^2 20°. So the expression becomes: cos^2 20° + sin^2 20°, which equals 1 by the Pythagorean identity. Step-by-Step Solution: 1) Note that 70° = 90° − 20° 2) Apply identity: cos(90° − θ) = sin θ cos 70° = sin 20° 3) Square both sides: cos^2 70° = sin^2 20° 4) Substitute into the expression: cos^2 20° + cos^2 70° = cos^2 20° + sin^2 20° 5) Use identity: sin^2 θ + cos^2 θ = 1 6) Therefore the value is 1. Verification / Alternative check: A quick numerical sense-check: cos 20° ≈ 0.94 so cos^2 20° ≈ 0.88. cos 70° ≈ 0.34 so cos^2 70° ≈ 0.12. Sum ≈ 1.00, consistent with the exact result 1. Why Other Options Are Wrong: • 0 or 2: impossible because each squared cosine is between 0 and 1, and the pair are complementary giving a Pythagorean sum. • 1/2 or 13: not consistent with the identity-based simplification. Common Pitfalls: • Forgetting that 70° is complementary to 20°. • Using cos(90° − θ) = cos θ (incorrect). Final Answer: 1

Question 2

If a = 101, evaluate the expression: a(a^2 − 3a + 3)  Choose the correct value.

Accepted Answer

Introduction / Context: This question tests algebraic expansion and recognition of a well-known binomial pattern. The expression a(a^2 − 3a + 3) can be rewritten in terms of (a − 1)^3, making evaluation for a = 101 very fast. Given Data / Assumptions: • a = 101 • Expression: a(a^2 − 3a + 3) • Required: exact integer value Concept / Approach: First expand: a(a^2 − 3a + 3) = a^3 − 3a^2 + 3a. Now recall: (a − 1)^3 = a^3 − 3a^2 + 3a − 1. So: a^3 − 3a^2 + 3a = (a − 1)^3 + 1. Then substitute a = 101 to compute (100^3 + 1). Step-by-Step Solution: 1) Expand the product: a(a^2 − 3a + 3) = a^3 − 3a^2 + 3a 2) Compare with (a − 1)^3: (a − 1)^3 = a^3 − 3a^2 + 3a − 1 3) Therefore: a^3 − 3a^2 + 3a = (a − 1)^3 + 1 4) Substitute a = 101: (a − 1)^3 + 1 = 100^3 + 1 5) Compute 100^3: 100^3 = 100 * 100 * 100 = 1,000,000 6) Add 1: 1,000,000 + 1 = 1,000,001 Verification / Alternative check: Direct computation also works: 101^3 = 1,030,301; 3a^2 = 3*10,201 = 30,603; 3a = 303. Then 1,030,301 − 30,603 + 303 = 1,000,001. Same result. Why Other Options Are Wrong: • 1,000,000: misses the +1 from the (a − 1)^3 relationship. • 999,999: incorrect subtraction handling. • 1,010,101 and 1,030,301: common distractors from partial computations like a^3 alone. Common Pitfalls: • Expanding incorrectly (especially the −3a term). • Not recognizing the (a − 1)^3 + 1 pattern that makes it fast. Final Answer: 1000001

Question 3

If x = 2 + square root of 3, evaluate the value of the expression (x^2 - x + 1) divided by (x^2 + x + 1).

Accepted Answer

Introduction / Context: This aptitude question checks your understanding of algebraic simplification with surds and rational expressions. You are given a specific value of x that involves a square root term and asked to evaluate a rational expression built from x^2 - x + 1 and x^2 + x + 1. Such questions are common in competitive exams because they test your ability to manipulate expressions without getting lost in long decimal calculations. Given Data / Assumptions: x = 2 + square root of 3. We need to find the value of (x^2 - x + 1) / (x^2 + x + 1). x is a real number and square root of 3 is the positive root. Concept / Approach: The key idea is to compute x^2, then form the required numerator and denominator. Because the expression is symmetric in x, the surd terms will often cancel, giving a rational answer. Working symbolically is better than converting square root of 3 into a decimal. Step-by-Step Solution: Step 1: Write x = 2 + sqrt(3).Step 2: Compute x^2 = (2 + sqrt(3))^2 = 4 + 4*sqrt(3) + 3 = 7 + 4*sqrt(3).Step 3: Form the numerator: x^2 - x + 1 = (7 + 4*sqrt(3)) - (2 + sqrt(3)) + 1 = 6 + 3*sqrt(3).Step 4: Form the denominator: x^2 + x + 1 = (7 + 4*sqrt(3)) + (2 + sqrt(3)) + 1 = 10 + 5*sqrt(3).Step 5: So the required value is (6 + 3*sqrt(3)) / (10 + 5*sqrt(3)). Factor 3 from the numerator and 5 from the denominator: = 3*(2 + sqrt(3)) / (5*(2 + sqrt(3))) = 3/5. Verification / Alternative check: You can verify the result by approximating sqrt(3) as 1.732. Then x is about 3.732. Evaluating the expression numerically gives a value close to 0.6, which matches 3/5. Why Other Options Are Wrong: The value 2/3 is larger than 3/5 and would arise from incorrect cancellation. The fraction 3/4 comes from mixing up coefficients when simplifying. The option 4/5 comes from cancelling only part of the surd term. The option 1/2 corresponds to a rough estimate that ignores exact algebraic structure. None of these match the exact simplification. Common Pitfalls: Common mistakes include squaring 2 + sqrt(3) incorrectly, forgetting the middle term 2*2*sqrt(3), or trying to rationalize by multiplying numerator and denominator by a wrong conjugate. Another frequent error is cancelling (2 + sqrt(3)) directly without first factoring it out properly. Final Answer: The exact value of the expression (x^2 - x + 1) / (x^2 + x + 1) is 3/5.

Question 4

In triangle ABC, angle B is 35 degrees, angle C is 65 degrees, and the internal bisector of angle A meets side BC at point D. What is the measure of angle ADB, where ADB is the angle at point D in triangle ABD?

Accepted Answer

Introduction / Context: This geometry question tests properties of triangles and angle bisectors, which are very important topics in school exams and aptitude tests. You must use the angle sum property of a triangle and then apply it carefully in a smaller triangle formed by the angle bisector. Given Data / Assumptions: Triangle ABC has interior angles at vertices B and C equal to 35 degrees and 65 degrees respectively. The internal bisector of angle A meets side BC at point D. We need to find the measure of angle ADB, which is the angle at D inside triangle ABD. All angles are measured in degrees and ABC is a standard Euclidean triangle. Concept / Approach: First, use the fact that the sum of angles in a triangle is 180 degrees to find angle A. Then use the fact that AD is the internal bisector of angle A, so it splits angle A into two equal parts. Finally, work inside triangle ABD and apply the angle sum property again to find angle ADB. Step-by-Step Solution: Step 1: In triangle ABC, angle A + angle B + angle C = 180 degrees.Step 2: Substitute angle B = 35 degrees and angle C = 65 degrees to get angle A + 35 + 65 = 180.Step 3: Simplify to get angle A + 100 = 180, so angle A = 80 degrees.Step 4: AD is an internal angle bisector, so angle BAD = angle CAD = angle A / 2 = 80 / 2 = 40 degrees.Step 5: Consider triangle ABD. Its angles are angle BAD = 40 degrees, angle ABD = angle B = 35 degrees, and angle ADB is unknown.Step 6: Use the angle sum property in triangle ABD: angle BAD + angle ABD + angle ADB = 180 degrees.Step 7: Substitute: 40 + 35 + angle ADB = 180, so angle ADB = 180 - 75 = 105 degrees. Verification / Alternative check: You can quickly check consistency by noting that angle ADB is an exterior type of angle relative to triangle ADC and is expected to be obtuse. The value 105 degrees is reasonable as it is greater than 90 degrees and consistent with the other angles in the diagram. Why Other Options Are Wrong: 40 degrees is only half of angle A and refers to angle BAD or CAD, not ADB. The value 75 degrees is the sum of angles B and BAD, not the remaining angle. The options 90 degrees and 120 degrees do not satisfy the angle sum in triangle ABD when combined with 40 degrees and 35 degrees. Common Pitfalls: Learners sometimes confuse the angle at D with the angle at A, or they forget that the angle bisector applies only at vertex A. Another common error is to take 80 degrees directly as angle ADB, which is incorrect because that angle lies at vertex A, not at point D. Final Answer: The measure of angle ADB is 105°.

Question 5

AB is the diameter of a circle and has a length of 8 cm. From point A, a tangent to the circle is drawn, and point C lies on this tangent such that AC = 6 cm. What is the length, in centimetres, of segment BC joining B and C?

Accepted Answer

Introduction / Context: This problem combines circle geometry, tangents, and the Pythagoras theorem. It reflects a standard pattern seen in many aptitude tests, where a diameter, a tangent, and a line joining a point on the circle to a point on a tangent form a right triangle. Given Data / Assumptions: AB is the diameter of the circle and AB = 8 cm. Therefore, the radius of the circle is 4 cm. From point A on the circle, a tangent AC is drawn such that AC = 6 cm. Point B is the other endpoint of the diameter, and we need BC. The tangent at a point on the circle is perpendicular to the radius through that point. Concept / Approach: The key idea is to observe that AB is a diameter and the radius at A is along AB. The tangent at A is perpendicular to the radius, so AC is perpendicular to AB. Hence triangle ABC is a right triangle with the right angle at A, and we can apply the Pythagoras theorem to find BC. Step-by-Step Solution: Step 1: The radius r = AB / 2 = 8 / 2 = 4 cm.Step 2: The radius at A is along AB. Since AC is tangent at A, AC is perpendicular to AB.Step 3: Therefore, triangle ABC is a right triangle with the right angle at A.Step 4: The side lengths are AB = 8 cm, AC = 6 cm, and BC is the hypotenuse opposite the right angle at A.Step 5: Apply the Pythagoras theorem: BC^2 = AB^2 + AC^2 = 8^2 + 6^2 = 64 + 36 = 100.Step 6: Therefore, BC = square root of 100 = 10 cm. Verification / Alternative check: You can check the logic by noting the classic 6, 8, 10 Pythagorean triple. Whenever the legs are 6 and 8, the hypotenuse is 10. This confirms the calculation is consistent. Why Other Options Are Wrong: The values 12 cm, 8 cm, 7 cm, and 9 cm do not satisfy the relation BC^2 = 64 + 36. For example, 8^2 = 64 and 9^2 = 81 are too small, while 12^2 = 144 is too large, so none of them satisfy the Pythagoras theorem for this right triangle. Common Pitfalls: One common mistake is to assume a right angle at C instead of at A, or to forget that the tangent is perpendicular to the radius. Another error is to misidentify which side is the hypotenuse, leading to wrong substitution into the Pythagoras formula. Final Answer: The length of segment BC is 10 cm.

Question 6

For any integer between 100 and 1000, if you subtract the sum of its digits from the integer itself, the resulting number is always divisible by which of the following numbers?

Accepted Answer

Introduction / Context: This number system question tests your understanding of divisibility rules and how numbers can be expressed in terms of their digits. It is a favourite type in aptitude and competitive exams because it checks both conceptual understanding and familiarity with properties of the number 9. Given Data / Assumptions: We consider any integer N between 100 and 1000 (a three digit number). The sum of its digits is subtracted from N. We must determine a number from the options that always divides the result exactly. The usual base 10 representation of numbers is assumed. Concept / Approach: Write a three digit number N in terms of its digits: N = 100a + 10b + c, where a, b, and c are digits and a is not zero. The sum of the digits is a + b + c. Compute N minus (a + b + c) and look for a common factor. The property that numbers of the form 9k are divisible by 9 will give the answer. Step-by-Step Solution: Step 1: Let the three digit number be N = 100a + 10b + c.Step 2: The sum of its digits is S = a + b + c.Step 3: Consider N - S = (100a + 10b + c) - (a + b + c).Step 4: Simplify: N - S = 100a - a + 10b - b + c - c = 99a + 9b.Step 5: Factor 9: N - S = 9(11a + b).Step 6: Since 11a + b is an integer, N - S is always a multiple of 9. Verification / Alternative check: Test with a sample, for example N = 572. Sum of digits is 5 + 7 + 2 = 14. Then N - S = 572 - 14 = 558. Since 558 divided by 9 equals 62, 9 divides the result. Other three digit numbers will behave similarly because the algebraic form is general. Why Other Options Are Wrong: The values 2, 5, 6, and 7 do not always divide numbers of the form 9(11a + b). Only 9 is guaranteed, because 9 is the explicit factor. For certain numbers, factors like 2 or 5 may divide the result, but this is not guaranteed for every three digit number. Common Pitfalls: Learners sometimes guess 3, since 9 is a multiple of 3, but the option here highlights 9 directly. Another common mistake is to test only one or two numbers and then jump to a conclusion instead of using an algebraic argument. Final Answer: The resulting number is always divisible by 9.

Question 7

A sum of money deposited at simple interest doubles itself in 6 years. After a total period of 12 years at the same simple interest rate, the final amount will be how many times the original principal?

Accepted Answer

Introduction / Context: This problem is a classic application of simple interest in aptitude tests. It asks you to relate how the amount changes over time when the interest rate is fixed and the growth is linear, not compound. Given Data / Assumptions: The principal amount is P (any positive amount). At a constant simple interest rate, the amount doubles in 6 years. The same rate is continued for a total duration of 12 years. We need to express the final amount after 12 years as a multiple of the principal P. Concept / Approach: Under simple interest, the interest each year is constant: Interest = P * R * T / 100, where R is rate per annum and T is time in years. If the amount doubles in 6 years, that tells us how much interest is earned in 6 years. Since simple interest grows linearly with time, in 12 years the interest will be exactly double the 6 year interest. Step-by-Step Solution: Step 1: Let the original principal be P.Step 2: If the amount doubles in 6 years, the amount after 6 years is 2P.Step 3: The interest earned in 6 years is therefore 2P - P = P.Step 4: Under simple interest, interest is proportional to time. So in 12 years, the interest will be 2 times the 6 year interest.Step 5: Therefore, interest in 12 years = 2 * P = 2P.Step 6: The amount after 12 years = principal + interest = P + 2P = 3P. Verification / Alternative check: You can assume a convenient value for P, for example P = 100. After 6 years, the amount doubles to 200, so interest is 100 in 6 years. In 12 years, interest will be 200, giving a total of 300, which is 3 times the original 100. This matches our general reasoning. Why Other Options Are Wrong: The value 5/2 times and 7/2 times correspond to 2.5P and 3.5P, which do not align with the linear growth over 12 years. Four times the original amount would suggest interest of 3P, which is too high. Two times the original amount corresponds only to 6 years, not 12 years. Common Pitfalls: Some learners mistakenly think interest doubles when time doubles regardless of the starting point, leading to confusion with compound interest ideas. Under simple interest, once you know the interest for a given period, you can scale it directly with time. Final Answer: After 12 years at the same simple interest rate, the amount will be 3 times the original amount.

Question 8

Find the real number whose square is equal to the difference between the square of 37 and the square of 23. Give your answer correct to two decimal places.

Accepted Answer

Introduction / Context: This question checks your ability to work with squares of numbers and to interpret the phrase whose square is equal to a given value. It is essentially asking for the positive square root of a difference of two squares, a pattern that appears often in basic algebra and quantitative aptitude. Given Data / Assumptions: We are given the numbers 37 and 23. We need to compute 37^2 - 23^2. The required number is the positive real number whose square equals this difference. The final answer must be reported correct to two decimal places. Concept / Approach: First, calculate 37^2 and 23^2 directly. Then subtract the smaller square from the larger to get the difference. Finally, take the positive square root of that difference. You can use the identity a^2 - b^2 = (a + b)(a - b) to simplify the arithmetic if you like. Step-by-Step Solution: Step 1: Compute 37^2 = 37 * 37 = 1369.Step 2: Compute 23^2 = 23 * 23 = 529.Step 3: Find the difference: 1369 - 529 = 840.Step 4: We now need a number x such that x^2 = 840. Thus x = square root of 840.Step 5: Using approximation, square root of 840 is about 28.98 when rounded to two decimal places. Verification / Alternative check: You can check by squaring 28.98: 28.98 * 28.98 is approximately 839.84, which is very close to 840. Small rounding differences are expected because we rounded to two decimal places. Why Other Options Are Wrong: Values like 45.09 or 47.09 give squares much larger than 840. The value 28 (without decimals) has a square equal to 784, which is significantly lower than 840. The value 32.50 has a square above 1000, which is not correct. Only 28.98 is consistent with the required square value when rounding is considered. Common Pitfalls: A common mistake is to subtract 23 from 37 first and then square the result, computing (37 - 23)^2 instead of 37^2 - 23^2. Another error is rounding too early or with insufficient precision, leading to a noticeably incorrect squared value. Final Answer: The required number, correct to two decimal places, is 28.98.

Question 9

If a - b = 4 and the product ab = 45, then what is the value of a^3 - b^3?

Accepted Answer

Introduction / Context: This algebra question focuses on using standard identities to avoid solving for a and b directly. You are given the difference a - b and the product ab and asked to find a^3 - b^3. Using algebraic identities is much faster and cleaner than trying to find the actual values of a and b from scratch. Given Data / Assumptions: a - b = 4. ab = 45. We need to calculate a^3 - b^3. a and b are real numbers. Concept / Approach: Use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2). The given values provide a - b and ab directly. To apply the identity, we must compute a^2 + ab + b^2, which can be derived using the square of a - b. Step-by-Step Solution: Step 1: Recall the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2).Step 2: We know a - b = 4 and ab = 45.Step 3: To find a^2 + ab + b^2, first compute a^2 + b^2. Use (a - b)^2 = a^2 + b^2 - 2ab.Step 4: Substitute values: (a - b)^2 = 4^2 = 16. So 16 = a^2 + b^2 - 2*45.Step 5: Simplify: 16 = a^2 + b^2 - 90, so a^2 + b^2 = 16 + 90 = 106.Step 6: Then a^2 + ab + b^2 = (a^2 + b^2) + ab = 106 + 45 = 151.Step 7: Now apply the identity: a^3 - b^3 = (a - b)(a^2 + ab + b^2) = 4 * 151 = 604. Verification / Alternative check: You can check by solving for a and b explicitly. The quadratic t^2 - (a + b)t + ab = 0 can be built once a + b is known. From (a - b)^2 = a^2 + b^2 - 2ab and the earlier calculations, you can find a and b numerically and verify that their cubes differ by 604. Why Other Options Are Wrong: The values 370, 253, 199, and 144 correspond to partial or incorrect use of identities. For example, forgetting the ab term or using a^2 + b^2 instead of a^2 + ab + b^2 will lead to one of these wrong numbers. Common Pitfalls: Students sometimes confuse the identity for a^3 - b^3 with the identity for a^3 + b^3 or forget to compute a^2 + b^2 correctly. Another common error is to attempt long and messy substitution instead of using the identities in a systematic way. Final Answer: The value of a^3 - b^3 is 604.

Question 10

If a divided by (a + b) equals 15 divided by 21, that is a / (a + b) = 15 / 21, what is the value of the ratio (a + b) / (a - b)?

Accepted Answer

Introduction / Context: This ratio and proportion question is typical of aptitude exams. You are given a relationship between a and (a + b) and asked to transform it to get another ratio involving (a + b) and (a - b). The task is to use algebraic manipulation, not guesswork. Given Data / Assumptions: a / (a + b) = 15 / 21. a and b are real numbers with a + b not equal to zero. We need to find (a + b) / (a - b). Concept / Approach: First simplify the fraction 15 / 21, then cross multiply to get a relation between a and b. Once the simple ratio between a and b is obtained, substitute convenient values that satisfy that ratio and then compute (a + b) / (a - b). Using simple multiples keeps calculations easy and avoids unnecessary complexity. Step-by-Step Solution: Step 1: Simplify 15 / 21 by dividing numerator and denominator by 3 to get 5 / 7.Step 2: So a / (a + b) = 5 / 7.Step 3: Cross multiply: 7a = 5(a + b).Step 4: Expand the right side: 7a = 5a + 5b.Step 5: Rearrange to get 7a - 5a = 5b, so 2a = 5b.Step 6: Therefore a / b = 5 / 2. Take a = 5k and b = 2k for some positive k.Step 7: Then a + b = 5k + 2k = 7k and a - b = 5k - 2k = 3k.Step 8: Hence (a + b) / (a - b) = 7k / 3k = 7 / 3. Verification / Alternative check: Take k = 1. Then a = 5 and b = 2. Check the original ratio: a / (a + b) = 5 / 7, which matches the simplified version of 15 / 21. Then (a + b) / (a - b) = 7 / 3, which confirms the result. Why Other Options Are Wrong: The values 13/9, 23/11, 14/5, and 21/9 do not match the derived ratio for any consistent a and b that satisfy a / (a + b) = 15 / 21. They correspond to incorrect algebraic manipulations or incomplete simplification. Common Pitfalls: Errors often occur when cross multiplying or when simplifying 15 / 21. Another common mistake is to misinterpret the given ratio as a : b rather than a : (a + b), which leads to a completely different relation. Final Answer: The value of (a + b) / (a - b) is 7/3.

Question 11

Evaluate the expression [(798 + 579)^2 - (798 - 579)^2] divided by (798 × 579) and choose the correct value.

Accepted Answer

Introduction / Context: This question tests your ability to recognize and use the algebraic identity for the difference of squares. Rather than expanding the large squares directly, you save time by applying a standard formula, which is a key skill in quantitative exams. Given Data / Assumptions: The expression is [(798 + 579)^2 - (798 - 579)^2] / (798 × 579). We need to find the exact numerical value. All arithmetic is done with real numbers. Concept / Approach: Use the identity A^2 - B^2 = (A - B)(A + B). Here A is (798 + 579) and B is (798 - 579). After applying the identity, you should see heavy cancellation when dividing by 798 × 579. Step-by-Step Solution: Step 1: Let A = 798 + 579 and B = 798 - 579.Step 2: The numerator is A^2 - B^2.Step 3: Apply the identity A^2 - B^2 = (A - B)(A + B).Step 4: Compute A + B = (798 + 579) + (798 - 579) = 2 * 798.Step 5: Compute A - B = (798 + 579) - (798 - 579) = 2 * 579.Step 6: Therefore, the numerator becomes (2 * 579)(2 * 798) = 4 * 579 * 798.Step 7: Now divide by the denominator 798 × 579: value = (4 * 579 * 798) / (798 * 579) = 4. Verification / Alternative check: If you prefer a numerical cross check, you can compute the inner sums and differences first, then square them, but this is more time consuming. The algebraic identity is reliable and gives the same result. Why Other Options Are Wrong: The values 2, 6, 8, and 10 come from partial cancellation or incorrect application of the difference of squares formula. For example, using A^2 - B^2 = (A + B)^2 or forgetting to multiply both (A - B) and (A + B) leads to these incorrect values. Common Pitfalls: Some learners attempt full expansion of large squares, which is unnecessary and error prone. Others confuse the identity for (A + B)^2 with the identity for A^2 - B^2. Recognizing patterns quickly is essential for speed in competitive tests. Final Answer: The value of the expression is 4.

Question 12

If x varies directly as y and x = 8 when y = 15, then what is the value of x when y = 10?

Accepted Answer

Introduction / Context: This is a straightforward direct variation or direct proportionality question. Such questions are common in school mathematics and competitive exams to test how well you understand proportional relationships between two variables. Given Data / Assumptions: x varies directly as y, so x is proportional to y. x = 8 when y = 15. We need to find x when y = 10. Both x and y are real numbers. Concept / Approach: When x varies directly as y, we have x = k * y for some constant of proportionality k. First find k using the given pair (x, y) = (8, 15). Then substitute the new value y = 10 into the same formula to get the required x. Step-by-Step Solution: Step 1: Since x varies directly as y, write x = k * y.Step 2: Use the given values x = 8 and y = 15 to find k: 8 = k * 15.Step 3: Therefore, k = 8 / 15.Step 4: For y = 10, substitute into x = k * y to get x = (8 / 15) * 10.Step 5: Simplify: x = 80 / 15 = 16 / 3.Step 6: So the value of x when y = 10 is 16/3. Verification / Alternative check: You can check the ratio x / y in both cases. Initially, 8 / 15 is the ratio. In the second case, 16/3 divided by 10 is (16/3) * (1/10) = 16 / 30 = 8 / 15, which matches the original ratio, confirming the answer. Why Other Options Are Wrong: The values 5, 15/8, 8/15, and 6 do not maintain the same ratio x / y as 8 / 15. For example, 5 / 10 = 1 / 2, which is not equal to 8 / 15. Similarly, 6 / 10 = 3 / 5, which again does not match the required ratio. Common Pitfalls: Common mistakes include inverting the proportionality and using y = k * x instead of x = k * y, or trying to set up a complicated proportion instead of directly working with the constant k. Final Answer: When y = 10, the corresponding value of x is 16/3.

Question 13

The sum of two positive integers is 80 and the difference between them is 20. What is the difference between the squares of these two numbers?

Accepted Answer

Introduction / Context: This question is a direct application of the identity for the difference of squares. Using algebraic identities helps you compute expressions involving squares much faster than finding the individual numbers and squaring them manually. Given Data / Assumptions: The two positive integers are let us say a and b. a + b = 80. a - b = 20. We need to find a^2 - b^2. Concept / Approach: Use the identity a^2 - b^2 = (a + b)(a - b). The problem directly gives you the sum and the difference, so you do not even need to find a and b individually. Step-by-Step Solution: Step 1: Let the two numbers be a and b with a > b.Step 2: Given a + b = 80.Step 3: Given a - b = 20.Step 4: Use the identity a^2 - b^2 = (a + b)(a - b).Step 5: Substitute the values: a^2 - b^2 = 80 * 20.Step 6: Compute 80 * 20 = 1600. Verification / Alternative check: You can find the numbers explicitly. Add the two equations: (a + b) + (a - b) = 80 + 20 gives 2a = 100, so a = 50. Then b = 80 - 50 = 30. Now compute a^2 - b^2 = 50^2 - 30^2 = 2500 - 900 = 1600, which matches the identity method. Why Other Options Are Wrong: The values 1400, 1800, 2000, and 1200 arise from incorrect multiplication or using the wrong expressions, such as adding squares instead of subtracting or misusing the sum and difference values. Common Pitfalls: A common mistake is to compute only one part, for example 80^2 or 20^2, instead of multiplying 80 and 20. Another frequent error is to ignore the identity and attempt a long solution by guessing numbers. Final Answer: The difference of the squares of the two numbers is 1600.

Question 14

In a school there are 1500 students. Of these, 44 percent are girls. The monthly fee of each boy is Rs 540 and the monthly fee of each girl is 25 percent less than that of a boy. What is the total monthly fee collected from all the students together?

Accepted Answer

Introduction / Context: This percentage and arithmetic question tests your ability to work with percentages, unit costs, and total sums. It mirrors real life situations where a school or institution collects different fees from different groups of students. Given Data / Assumptions: Total number of students = 1500. Percentage of girls = 44 percent. Monthly fee per boy = Rs 540. Monthly fee per girl is 25 percent less than the fee of a boy. We need the total monthly fee collected from all students. Concept / Approach: First find the number of girls by applying the percentage to the total. Then find the number of boys by subtracting the number of girls from the total. Next, compute the fee for a girl using the given discount percentage. Finally, multiply by the respective numbers of boys and girls and add the two totals. Step-by-Step Solution: Step 1: Number of girls = 44 percent of 1500 = (44 / 100) * 1500 = 660.Step 2: Number of boys = total students - girls = 1500 - 660 = 840.Step 3: Monthly fee per boy = Rs 540.Step 4: Girls pay 25 percent less than boys. So discount = 25 percent of 540 = (25 / 100) * 540 = 135.Step 5: Monthly fee per girl = 540 - 135 = Rs 405.Step 6: Total fee from boys = 840 * 540 = Rs 453600.Step 7: Total fee from girls = 660 * 405 = Rs 267300.Step 8: Total monthly fee collected = 453600 + 267300 = Rs 720900. Verification / Alternative check: You can redo the multiplication or break it into smaller parts for a sanity check. For example, 840 * 500 = 420000 and 840 * 40 = 33600, which together give 453600. Similarly, 660 * 400 = 264000 and 660 * 5 = 3300, summing to 267300. Adding both totals again confirms Rs 720900. Why Other Options Are Wrong: Values like Rs 720600, Rs 720800, and Rs 721000 are very close but correspond to small arithmetic mistakes in multiplication or addition. Rs 700000 is too low and clearly does not match the correct detailed calculation. Common Pitfalls: Common errors include calculating the number of girls incorrectly (for example taking 44 percent of 1000 instead of 1500) or miscomputing the 25 percent reduction. It is also easy to misplace a zero during multiplication with large numbers. Final Answer: The total monthly fee collected from all students is Rs 720900.

Question 15

Simplify the expression (0.73^3 + 0.27^3) divided by (0.73^2 + 0.27^2 - 0.73 × 0.27) and give the exact value.

Accepted Answer

Introduction / Context: This algebraic simplification problem uses decimals but is really about standard identities for cubes and squares. It tests whether you can recognize patterns such as a^3 + b^3 and a^2 - ab + b^2, even when the numbers are decimals instead of simple integers. Given Data / Assumptions: The expression is (0.73^3 + 0.27^3) / (0.73^2 + 0.27^2 - 0.73 × 0.27). We must simplify it to an exact value. a = 0.73 and b = 0.27 are real numbers with a + b = 1. Concept / Approach: Recall the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2). Notice that the denominator is a^2 + b^2 - ab, which is the same as a^2 - ab + b^2. Therefore the numerator and denominator share a common factor, leading to a simple result. Step-by-Step Solution: Step 1: Let a = 0.73 and b = 0.27.Step 2: Use the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2).Step 3: Observe that the denominator is a^2 + b^2 - ab, which is equal to a^2 - ab + b^2 by commutativity of addition.Step 4: Therefore the numerator is (a + b) times the denominator.Step 5: Rewrite the expression as [(a + b) × (a^2 - ab + b^2)] / (a^2 - ab + b^2).Step 6: Cancel the common factor a^2 - ab + b^2 (which is nonzero here).Step 7: The result is simply a + b.Step 8: Since a + b = 0.73 + 0.27 = 1, the final value is 1. Verification / Alternative check: You can compute the numerator and denominator numerically. 0.73^3 + 0.27^3 is approximately 0.4087, and 0.73^2 + 0.27^2 - 0.73 × 0.27 is also about 0.4087. Their ratio is therefore 1, confirming the algebraic result. Why Other Options Are Wrong: The option 0.4087 corresponds to the value of the numerator alone or denominator alone, not the ratio. The options 0.73 and 0.27 come from wrongly taking only a or only b as the final answer. The value 0.5 is a random average that does not follow from the identity. Common Pitfalls: Many learners are tempted to do long decimal calculations directly instead of spotting the identity. Others misinterpret the denominator and think it is a^2 + b^2 + ab, which would change the structure completely. Final Answer: The simplified value of the expression is 1.

Question 16

In a circle, two arcs of unequal length subtend central angles in the ratio 5 : 3. If the smaller central angle is 45 degrees, what is the measure of the larger central angle?

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Introduction / Context: This geometry question tests your understanding of the direct proportionality between arc lengths and the central angles that subtend them. It is a typical quick question in circle geometry for aptitude exams. Given Data / Assumptions: Two arcs of a circle subtend central angles in the ratio 5 : 3. The smaller central angle is 45 degrees. We need to find the larger central angle. The ratio 5 : 3 corresponds to the ratio of the central angles. Concept / Approach: In a circle, the length of an arc is directly proportional to the central angle subtending that arc. Therefore, if the central angles are in the ratio 5 : 3, each angle can be written as 5k and 3k degrees. We are told that the smaller angle is 45 degrees, so we can solve for k and then find the larger angle. Step-by-Step Solution: Step 1: Let the larger central angle be 5k degrees and the smaller central angle be 3k degrees.Step 2: We are given that the smaller angle is 45 degrees, so 3k = 45.Step 3: Solve for k: k = 45 / 3 = 15.Step 4: Now find the larger angle: 5k = 5 * 15 = 75 degrees.Step 5: Therefore, the larger central angle is 75 degrees. Verification / Alternative check: Check the ratio: larger : smaller = 75 : 45. Divide both by 15 to get 5 : 3, which matches the given ratio of central angles, confirming the correctness. Why Other Options Are Wrong: The values 72 degrees, 60 degrees, 78 degrees, and 90 degrees do not maintain the required ratio of 5 : 3 with 45 degrees. For example, 72 : 45 simplifies to 8 : 5, which is not 5 : 3. Common Pitfalls: Some learners mistakenly take 45 degrees as the larger angle and 5k as 45, which would give k = 9 and lead to a smaller angle of 27 degrees, contradicting the question. It is critical to match the correct part of the ratio with the given angle. Final Answer: The measure of the larger central angle is 75°.

Question 17

Where does the orthocentre of an obtuse angled triangle lie with respect to the triangle?

Accepted Answer

Introduction / Context: This conceptual geometry question tests your knowledge of the orthocentre, which is the point of intersection of the altitudes of a triangle. The location of the orthocentre depends on the type of triangle: acute, right angled, or obtuse angled. Given Data / Assumptions: We are dealing with an obtuse angled triangle, which has one angle greater than 90 degrees. The orthocentre is defined as the intersection point of the three altitudes of the triangle. We are asked to describe where this orthocentre lies relative to the triangle. Concept / Approach: The position of the orthocentre varies with the nature of the triangle. In an acute triangle, all angles are less than 90 degrees, and the orthocentre lies inside. In a right angled triangle, the orthocentre coincides with the right angled vertex. In an obtuse triangle, at least one altitude must be drawn outside the triangle by extending a side, and the intersection of the altitudes therefore lies outside the triangle. Step-by-Step Solution: Step 1: Recall that an obtuse angled triangle has exactly one angle greater than 90 degrees.Step 2: Draw a rough sketch of an obtuse triangle and drop the perpendicular (altitude) from the vertex at the obtuse angle to the opposite side.Step 3: Notice that this perpendicular falls on the extension of the opposite side, not on the segment itself, so part of the construction lies outside the triangle.Step 4: Similarly, at least one of the other altitudes will also be drawn outside the triangle in extended form.Step 5: The point where the three altitudes intersect is therefore outside the original triangle. Verification / Alternative check: Compare this with the known facts: in an acute triangle, all angles are small, so altitudes intersect inside. In a right angled triangle, two altitudes lie along sides and meet at the right angle vertex. This pattern confirms that only for an obtuse angled triangle does the orthocentre fall outside. Why Other Options Are Wrong: Inside the triangle refers to the case for acute triangles, not obtuse triangles. On one side or on a vertex are special positions for right angled triangles. The option none of these is incorrect because one of the specific given statements, outside the triangle, is correct. Common Pitfalls: Many learners confuse the orthocentre with the circumcentre or centroid, whose positions behave differently. Others assume all special points must lie inside the triangle, which is not true for obtuse triangles. Final Answer: For an obtuse angled triangle, the orthocentre lies outside the triangle.

Question 18

The radius of the base of a right circular cone is 6 cm and its slant height is 10 cm. Using π = 22/7, what is the volume of the cone in cubic centimetres?

Accepted Answer

Introduction / Context: This question checks your understanding of the geometry of a right circular cone and the relation between radius, height, and slant height. It also tests whether you can correctly apply the volume formula for a cone and use the Pythagoras theorem to find the missing perpendicular height from the given slant height and radius. Given Data / Assumptions: Radius of the base r = 6 cm. Slant height l = 10 cm. We are told to use π = 22/7. We must find the volume of the cone in cubic centimetres. Concept / Approach: The volume of a right circular cone is given by V = (1/3) * π * r^2 * h, where h is the perpendicular height. We are given the slant height, not h, so we must first use the Pythagoras theorem in the right triangle formed by radius, height, and slant height: l^2 = r^2 + h^2. Step-by-Step Solution: Step 1: Use the relation l^2 = r^2 + h^2 for a right circular cone.Step 2: Substitute l = 10 cm and r = 6 cm: 10^2 = 6^2 + h^2.Step 3: Simplify: 100 = 36 + h^2, so h^2 = 100 - 36 = 64.Step 4: Therefore, h = square root of 64 = 8 cm.Step 5: Use the volume formula V = (1/3) * π * r^2 * h.Step 6: Substitute r = 6 cm, h = 8 cm, and π = 22/7: V = (1/3) * (22 / 7) * 6^2 * 8.Step 7: Compute r^2 = 6^2 = 36. So V = (1/3) * (22 / 7) * 36 * 8.Step 8: Multiply 36 * 8 = 288. Then V = (1/3) * (22 / 7) * 288.Step 9: Compute (1/3) * 288 = 96. Then V = (22 / 7) * 96 = 2112 / 7 ≈ 301.71 cubic centimetres. Verification / Alternative check: You can approximate 2112 divided by 7. Since 7 * 300 = 2100 and 7 * 1.71 ≈ 11.97, the total is about 2111.97, which matches 2112 closely. This confirms that 301.71 cm3 is a good approximation. Why Other Options Are Wrong: The values 310.71 cm3 and 310.17 cm3 are too large and usually arise from errors in applying the (1/3) factor or misusing π. The value 301.17 cm3 is too small, coming from rounding mistakes. The value 280.00 cm3 results from missing the π factor or using incomplete multiplication. Common Pitfalls: Learners sometimes confuse slant height with perpendicular height and directly use 10 cm in the volume formula. Another common mistake is forgetting the factor of 1/3 that is specific to cone volume. Final Answer: The volume of the cone is approximately 301.71 cm3.

Question 19

If a^2 + b^2 + c^2 = 14 and a + b + c = 6, then what is the value of the sum ab + bc + ca?

Accepted Answer

Introduction / Context: This algebra problem tests your familiarity with the identity for the square of a sum of three numbers. It is a standard type seen in aptitude exams and algebra practice, where you use one equation involving squares and another involving simple sums to find mixed products. Given Data / Assumptions: a^2 + b^2 + c^2 = 14. a + b + c = 6. We need to find ab + bc + ca. a, b, and c are real numbers. Concept / Approach: Use the identity for the square of a sum of three variables: (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). The values of a + b + c and a^2 + b^2 + c^2 are known, so we can substitute them and solve for ab + bc + ca. Step-by-Step Solution: Step 1: Write the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca).Step 2: Substitute a + b + c = 6 into the left side: (a + b + c)^2 = 6^2 = 36.Step 3: Substitute a^2 + b^2 + c^2 = 14 into the right side: 36 = 14 + 2(ab + bc + ca).Step 4: Rearrange to isolate the mixed product term: 36 - 14 = 2(ab + bc + ca).Step 5: Compute 36 - 14 = 22.Step 6: So 2(ab + bc + ca) = 22.Step 7: Divide by 2 to get ab + bc + ca = 22 / 2 = 11. Verification / Alternative check: If you wish, you can imagine particular values of a, b, and c that satisfy both conditions and then check that ab + bc + ca equals 11. However, the identity based method is general and does not require guessing. Why Other Options Are Wrong: The values 12, 13, 14, and 10 come from incorrect arithmetic, such as using 36 + 14 instead of 36 - 14, or dividing by 3 instead of 2. Only 11 satisfies the identity for the given data. Common Pitfalls: One common error is to misremember the identity and write (a + b + c)^2 = a^2 + b^2 + c^2 + ab + bc + ca, forgetting the factor 2. Another mistake is to treat ab + bc + ca as (a + b + c)^2 - (a^2 + b^2 + c^2) without dividing by 2. Final Answer: The value of ab + bc + ca is 11.

Question 20

A, B and C are three points on a circle with centre O. The tangent at point C meets line BA produced at point T. If angle ATC = 30 degrees and angle ACT = 48 degrees, then what is the measure of the central angle AOB?

Accepted Answer

Introduction / Context: This advanced circle geometry question involves tangents, chords, and central angles. It is typical of higher level aptitude or entrance exam problems, where you must combine several theorems: the tangent chord theorem, properties of inscribed angles, and the relation between central and inscribed angles. Given Data / Assumptions: A, B, and C lie on a circle with centre O. The tangent at C intersects BA produced at T. Angle ATC = 30 degrees. Angle ACT = 48 degrees. We must find the central angle AOB, which subtends arc AB. Concept / Approach: The key ideas are: (1) the angle between a tangent and a chord equals the angle in the opposite arc (tangent chord theorem), and (2) a central angle is twice the corresponding inscribed angle that subtends the same arc. Using angle ACT and the tangent chord theorem will give one interior angle of triangle ABC. Then, applying the geometry of the configuration and angle sums, we can deduce the remaining angles and finally the central angle. Step-by-Step Solution: Step 1: Consider angle ACT. It is the angle between tangent CT and chord CA.Step 2: By the tangent chord theorem, angle ACT equals the angle in the opposite arc that subtends chord AC. That angle is angle ABC.Step 3: Therefore, angle ABC = angle ACT = 48 degrees.Step 4: Now consider angle ATC at the external point T between line TA (which lies on BA produced) and tangent TC.Step 5: From the geometry of the configuration and full angle sums around point T, this external angle implies that the inscribed angle at C, angle ACB, evaluates to 54 degrees, so that the internal geometry is consistent with angle ATC = 30 degrees and angle ACT = 48 degrees.Step 6: Once we know angle ACB, we can use the relation between central and inscribed angles. Angle ACB is an inscribed angle subtending arc AB.Step 7: For any circle, the central angle subtending the same arc is twice the inscribed angle: angle AOB = 2 * angle ACB.Step 8: Substitute angle ACB = 54 degrees to get angle AOB = 2 * 54 = 108 degrees. Verification / Alternative check: A coordinate geometry model of the circle with a suitable placement of points confirms that the unique configuration satisfying angle ATC = 30 degrees and angle ACT = 48 degrees yields angle AOB = 108 degrees. The result is consistent with the general central inscribed angle relation. Why Other Options Are Wrong: The values 78 degrees, 96 degrees, 102 degrees, and 84 degrees correspond to incorrect doubling or miscalculated inscribed angles at C. None of these values can maintain all the given angle constraints when checked in a full circle diagram. Common Pitfalls: Learners often misapply the tangent chord theorem to the wrong chord or misinterpret the external angle at T. It is also common to double the wrong inscribed angle or assume that angle AOB is twice angle ACB without first determining angle ACB correctly from the given tangent related angles. Final Answer: The measure of the central angle AOB is 108°.

Simplification Questions