Past Questions

To find the missing digits in the unit's place (y) and hundred's place (x) respectively for the number 83x7y: 1. **Divisibility by 5:** A number is divisible by 5 if its unit's digit (y) is either 0 or 5. All options satisfy this condition. 2. **Divisibility by 3:** A number is divisible by 3 if the sum of its digits is a multiple of 3. The sum of the digits of 83x7y is 8 + 3 + x + 7 + y = 18 + x + y. Since 18 is already divisible by 3, the sum of the missing digits (x + y) must also be divisible by 3. Let's test the options (given as 'y, x' respectively): - **Option a (0,2):** y = 0, x = 2. Sum (x + y) = 2 (not divisible by 3). - **Option b (0,4):** y = 0, x = 4. Sum (x + y) = 4 (not divisible by 3). - **Option c (5,2):** y = 5, x = 2. Sum (x + y) = 7 (not divisible by 3). - **Option d (5,4):** y = 5, x = 4. Sum (x + y) = 9 (divisible by 3). Thus, 83475 is divisible by both 3 and 5.

To convert speed from meters per second (m/s) to kilometers per hour (km/h), we use the following conversion factors: 1. 1 kilometer (km) = 1,000 meters (m) 2. 1 hour (h) = 3,600 seconds (60 minutes × 60 seconds) Step-by-step calculation: Speed = 20 m/s To convert meters to kilometers, divide by 1,000: 20 / 1,000 km/s To convert seconds to hours, multiply by 3,600: (20 / 1,000) × 3,600 Simplified calculation: Speed = 20 × (3,600 / 1,000) Speed = 20 × 3.6 Speed = 72 km/h Alternatively, a quick shortcut for converting m/s to km/h is to multiply the value by 18/5: 20 × (18/5) = 4 × 18 = 72 km/h. Comparing this result to the options: a. 36 km/h (Incorrect: 10 m/s = 36 km/h) b. 45 km/h (Incorrect: 12.5 m/s = 45 km/h) c. 72 km/h (Correct) d. 84 km/h (Incorrect: 23.33 m/s = 84 km/h)

Step 1: Analyze the relationship in the original pair (DELTOID : MUSCLE). The deltoid is a specific type of muscle (specifically, the large triangular muscle covering the shoulder joint). Therefore, the relationship is 'Specific Example : General Category' or 'Type of'. Step 2: Evaluate the options based on this relationship: - Option (a) radius : bone: The radius is a specific bone in the forearm. This follows the 'Specific Example : General Category' relationship. - Option (b) brain : nerve: The brain is an organ composed of nerves, but it is not a 'type of' nerve. This is a 'Whole : Part' or 'Composition' relationship. - Option (c) tissue : organ: Tissues make up organs. This is a 'Part : Whole' relationship, and the order is reversed (General/Component : Specific/Complex). - Option (d) blood : vein: Blood flows through veins. This is a 'Content : Container' relationship. Step 3: Conclusion: Only option (a) mirrors the exact logical relationship of the original pair.

Step-by-step solution: 1. **Analyze the combined efficiency**: A and B together can complete the whole work in 5 days. Therefore, their combined rate of work is 1/5 of the work per day. 2. **Determine the remaining work**: The problem states that A left after 2/3 of the work was completed. The remaining work is: 1 - 2/3 = 1/3 of the total work. 3. **Calculate B's efficiency**: The remaining 1/3 of the work was completed by B alone in 3 days. Rate of B = (Work done) / (Time taken) = (1/3) / 3 = 1/9 of the total work per day. 4. **Calculate time for B to finish the whole work**: Since B completes 1/9 of the work in one day, the total time B would take to complete the entire work alone is: Total Time = 1 / (1/9) = 9 days. Verification of options: - Option (a) 9 days: Matches our calculation. - Option (b) 6 days: Incorrect. - Option (c) 5 days: This is the combined time for A and B, not B alone. - Option (d) 3 days: This is the time B took to finish only the remaining 1/3 of the work.

The relationship in the first pair (26 : 5) follows the pattern: (n² + 1). Step 1: Analyze the first pair. 5² = 25, and 25 + 1 = 26. So, the logic is (Second Number)² + 1 = First Number. Step 2: Apply the same logic to the second pair (65 : X). We need to find X such that X² + 1 = 65. Step 3: Solve for X: X² = 65 - 1 X² = 64 X = √64 X = 8. Therefore, the value of X is 8, which corresponds to option (a).

To find the selling price after two successive discounts, we apply the discounts sequentially to the remaining balance. Step 1: Calculate the price after the first discount of 10%. First Discount = 10% of Rs. 500 = (10/100) * 500 = Rs. 50. Price after 1st discount = 500 - 50 = Rs. 450. Step 2: Calculate the price after the second discount of 10% (applied to the new price). Second Discount = 10% of Rs. 450 = (10/100) * 450 = Rs. 45. Final Selling Price = 450 - 45 = Rs. 405. Alternative Method (Formula): Selling Price = Marked Price * (1 - d1/100) * (1 - d2/100) Selling Price = 500 * (0.90) * (0.90) Selling Price = 500 * 0.81 = Rs. 405. Comparison with options: - Option (a) Rs. 400 is incorrect as it assumes a flat 20% discount (500 - 100). - Option (b) Rs. 405 is correct. - Option (c) Rs. 410 and (d) Rs. 415 are mathematically incorrect based on the given percentages.

Let's analyze the given premises step-by-step: 1. **Premise 1:** 60% of students play cricket. This implies 40% of students do not play cricket. 2. **Premise 2:** A student who does not play cricket plays football. This means all non-cricketers (40%) are football players. 3. **Premise 3:** Every football player has a two-wheeler. This means all 40% who play football have two-wheelers. Now let's evaluate the conclusions to see if they can be logically drawn: - **Conclusion i: '60% of the students do not have two-wheelers.'** We know that 40% (football players) definitely have two-wheelers. However, the data tells us nothing about the 60% who play cricket. They might have two-wheelers, or they might not. We cannot conclude that they definitely do not have them. Therefore, this cannot be drawn. - **Conclusion ii: 'No cricketer has a two-wheeler.'** Similar to conclusion (i), the premises state that football players have two-wheelers, but they do not state that *only* football players have them. A cricketer could also own a two-wheeler. There is no information to support the claim that no cricketer has one. Therefore, this cannot be drawn. - **Conclusion iii: 'Cricket players do not play football.'** The premise says 'A student who does not play cricket, plays football.' This establishes that Non-Cricketers $\rightarrow$ Football. It does NOT establish that Cricketers $\rightarrow$ Non-Football. It is possible for a student to play both cricket and football. Since we cannot prove they are mutually exclusive, this cannot be drawn. Since none of the three conclusions can be logically drawn from the given data, the correct option is 'i, ii and iii'.

In Microsoft Word, the shortcut key Ctrl + H is specifically assigned to open the 'Find and Replace' dialog box, focusing on the 'Replace' tab. This allows users to search for a specific word or phrase and replace it with another throughout the document. Analysis of other options: - Option (a) Copy: The shortcut key for Copy is Ctrl + C. - Option (b) Find: The shortcut key for Find is Ctrl + F. - Option (d) Highlight: There is no standard single-combination Ctrl shortcut for the Highlight tool; it is typically accessed via the Ribbon menu.

To find the radius of the circle, we use the formula for the area of a sector: Area = (θ / 360°) * π * r² Given: - Area = 17.6 cm² - θ = 56° - π ≈ 22/7 Substituting these values into the formula: 17.6 = (56 / 360) * (22 / 7) * r² Simplify the expression: 17.6 = (8 / 360) * 22 * r² 17.6 = (1 / 45) * 22 * r² 17.6 = (22 / 45) * r² Solving for r²: r² = (17.6 * 45) / 22 r² = 0.8 * 45 r² = 36 Taking the square root of both sides: r = 6 cm Therefore, the radius of the circle is 6 cm (Option a). Options b, c, and d are incorrect because substituting their values into the sector area formula does not yield the given area of 17.6 cm².

Step 1: Analyze the relationship between the first pair of words. 'Alleviate' means to make a problem or pain less severe, while 'Aggravate' means to make a problem or injury worse. These two words are antonyms (opposites). Step 2: Apply the same relationship to the second pair. The word 'Elastic' refers to a material or object that is flexible and can return to its original shape after being stretched or compressed. Step 3: Evaluate the options to find the antonym of 'Elastic': - (a) Flexible: This is a synonym of elastic, not an antonym. - (b) Rigid: This means unable to bend or be forced out of shape; not flexible. This is the direct opposite of elastic. - (c) Strong: This refers to power or durability, which is not the opposite of elasticity. - (d) Malleable: This refers to a material that can be hammered or pressed permanently out of shape without breaking, which is a form of flexibility, not the opposite. Conclusion: Since the first pair consists of antonyms, the second pair must also consist of antonyms. Therefore, 'Rigid' is the correct counterpart to 'Elastic'.

To find the height of the electric pole, we can use trigonometric ratios: 1. Let the height of the tower be $H = 21\text{ m}$, the height of the electric pole be $h$, and the horizontal distance between them be $d$. 2. From the bottom of the pole, the angle of elevation to the top of the tower is $60^\circ$: $$\tan(60^\circ) = \frac{H}{d} \implies \sqrt{3} = \frac{21}{d} \implies d = \frac{21}{\sqrt{3}} = 7\sqrt{3}\text{ m}$$ 3. From the top of the pole, the angle of elevation to the top of the tower is $30^\circ$. The vertical height difference is $H - h$: $$\tan(30^\circ) = \frac{H - h}{d} \implies \frac{1}{\sqrt{3}} = \frac{21 - h}{7\sqrt{3}}$$ 4. Multiplying both sides by $\sqrt{3}$ gives: $$1 = \frac{21 - h}{7} \implies 7 = 21 - h \implies h = 14\text{ m}$$ Thus, the height of the electric pole is $14\text{ m}$ (Option a). Other options (b, c, and d) are incorrect because they do not satisfy these geometric and trigonometric relationships. For example, a height of $7\text{ m}$ (Option c) would result in a different angle of elevation from the top of the pole.

Let the original number be represented by 'x'. We can translate the word problem into a mathematical equation step-by-step: 1. 'When 5 is added to a certain number': (x + 5) 2. 'the sum is multiplied by 3': 3 * (x + 5) 3. 'the product is divided by 10': [3 * (x + 5)] / 10 4. '1 is added to the quotient': ([3 * (x + 5)] / 10) + 1 5. 'the result is 16': ([3 * (x + 5)] / 10) + 1 = 16 Now, we solve for x: - Subtract 1 from both sides: [3 * (x + 5)] / 10 = 15 - Multiply both sides by 10: 3 * (x + 5) = 150 - Divide both sides by 3: x + 5 = 50 - Subtract 5 from both sides: x = 45 Verification: (45 + 5) = 50 50 * 3 = 150 150 / 10 = 15 15 + 1 = 16 The result matches the problem statement. Therefore, the original number is 45.

Step-by-step solution: 1. **Identify the given values**: - Diameter of the spherical ball (d) = 21 cm - Radius of the spherical ball (r) = d / 2 = 21 / 2 = 10.5 cm - Side of each cube (a) = 1 cm 2. **Calculate the volume of the spherical ball**: The formula for the volume of a sphere is V = (4/3)πr³. Using π ≈ 22/7: V_sphere = (4/3) * (22/7) * (21/2) * (21/2) * (21/2) V_sphere = (4/3) * (22/7) * (9261 / 8) V_sphere = (4 * 22 * 9261) / (3 * 7 * 8) V_sphere = 813048 / 168 V_sphere = 4845.5 (approx) or more precisely: V_sphere = (4/3) * (22/7) * (10.5)³ = 4851 cm³ Calculation check: (4/3) * (22/7) * (21/2)^3 = (4/3) * (22/7) * (9261/8) = (22 * 9261) / (3 * 7 * 2) = 203742 / 42 = 4851 cm³. 3. **Calculate the volume of one cube**: The formula for the volume of a cube is V = a³. V_cube = 1³ = 1 cm³. 4. **Find the number of cubes**: Number of cubes = Total Volume of Sphere / Volume of one Cube Number of cubes = 4851 / 1 = 4851. Therefore, the number of cubes formed is 4851, which corresponds to option (d).

Step-by-step analysis of the seating arrangement: 1. **Analyze the layout**: There are two rows of three people each. Let's call them Row 1 (facing South) and Row 2 (facing North). 2. **Position of P and R**: 'P is second to the left of R'. In a row of three, for someone to be second to the left of another, they must be at the extreme ends. - If R is at the right end (facing North), P is at the left end. - If R is at the left end (facing South), P is at the right end. Let's assume R and P are in Row 2 (facing North). So, P is at the left end and R is at the right end. 3. **Position of N**: 'N is the neighbour of R'. Since R is at the right end, N must be in the middle of Row 2. Row 2 (North facing): P | N | R 4. **Position of O**: 'O is sitting diagonally opposite to P'. Since P is at the left end of Row 2, O must be at the right end of Row 1 (facing South). 5. **Position of Q**: 'O is the neighbour of Q' and 'Q is not at the end of any row'. Since O is at the right end of Row 1, Q must be in the middle of Row 1. Row 1 (South facing): _ | Q | O 6. **Position of M**: Only one person (M) and one spot (left end of Row 1) remain. Row 1 (South facing): M | Q | O Final Arrangement: Row 1 (Facing South): M | Q | O Row 2 (Facing North): P | N | R Comparing the positions: - M is facing P - Q is facing N - O is facing R Wait, let's re-verify the 'diagonally opposite' and 'left' constraints. If Row 1 faces South: Left is to the viewer's right. If P is second to the left of R (Row 2, North): P(Left) - N(Mid) - R(Right). O is diagonally opposite P: O must be at the right end of Row 1 (from viewer's perspective, but for someone facing South, the right end is the viewer's left). Let's re-map: Row 1 (South): [Right End] [Mid] [Left End] Row 2 (North): [Left End] [Mid] [Right End] P is 2nd left of R: P is at Left End (Row 2), R is at Right End (Row 2). N is Mid (Row 2). O is diagonally opposite P: O must be at the Right End of Row 1 (viewer's perspective: Right end of Row 1 is the 'Left' side for someone facing South). Q is neighbor of O and not at end: Q is Mid (Row 1). M is the remaining person: M is at the Left End of Row 1 (viewer's perspective). Facing pairs: - M (Left end Row 1) faces R (Right end Row 2) - Q (Mid Row 1) faces N (Mid Row 2) - O (Right end Row 1) faces P (Left end Row 2) Let's check the 'diagonally opposite' again. P is at (Row 2, Col 1). Diagonally opposite is (Row 1, Col 3). O is at (Row 1, Col 3). This matches. Now, who is facing M? M is at (Row 1, Col 1). He faces (Row 2, Col 1), which is P. Let's re-read: 'O is the neighbour of Q and is sitting diagonally opposite to P'. If P is at Row 2 Left, O is at Row 1 Right. Q is Mid Row 1. M is Row 1 Left. Row 1: M(L) Q(M) O(R) Row 2: P(L) N(M) R(R) Facing: M-P, Q-N, O-R. Let's try the other orientation: If P and R are in Row 1 (Facing South). P is 2nd left of R. For South facing, Left is viewer's right. So P is at viewer's right, R is at viewer's left. Row 1: R(L) N(M) P(R) O is diagonally opposite P: O is at Row 2 Left. Q is neighbor of O and not at end: Q is Row 2 Mid. M is Row 2 Right. Row 2: O(L) Q(M) M(R) Facing: R-O, N-Q, P-M. In both valid configurations, M faces P. However, looking at the options provided (a. Q, b. R, c. N, d. P), P is an option. Let me re-verify the 'diagonally opposite' logic. If P is at one corner, O is at the opposite corner. Row 1: M Q O Row 2: P N R Facing: M-P, Q-N, O-R. Wait, let's re-read the 'P is second to the left of R' carefully. If they face each other, Row 1 faces South, Row 2 faces North. If R is in Row 1 (South), his left is viewer's right. P is 2nd to his left -> P is at viewer's right. Row 1: R(L) N(M) P(R) O is diagonally opposite P -> O is at Row 2 Left. Q is neighbor of O and not at end -> Q is Row 2 Mid. M is Row 2 Right. Row 2: O(L) Q(M) M(R) Facing: R-O, N-Q, P-M. In both cases, M faces P. Let me double check the question source or common variations. Often in these puzzles, 'facing' refers to the person directly opposite. Wait, if Row 1: O Q M and Row 2: R N P. O is diagonally opposite P. Q is neighbor of O. Q is mid. M is end. P is 2nd left of R (North facing: P is left, R is right). Row 2: P N R. Row 1: O Q M. Facing: O-P, Q-N, M-R. In this case, M faces R. Let's re-evaluate 'O is diagonally opposite to P'. If P is Row 2 Left, O must be Row 1 Right. Row 2: P N R Row 1: M Q O Facing: M-P, Q-N, O-R. If P is Row 2 Right, O must be Row 1 Left. But P is 2nd left of R, so P MUST be the left-most person in a North-facing row. So P is Row 2 Left. Then O is Row 1 Right. Row 2: P N R Row 1: M Q O Facing: M-P, Q-N, O-R. Is there any other way? What if Row 1 faces North and Row 2 faces South? Row 1 (North): P N R Row 2 (South): M Q O Facing: P-M, N-Q, R-O. In all standard interpretations, M faces P. Let me check the options again. If the answer is Q, then M must be in the middle. But Q is in the middle. M cannot face Q because Q is facing N. Let's re-read: 'O is the neighbour of Q and is sitting diagonally opposite to P'. If O is Row 1 Left, P is Row 2 Right. If P is Row 2 Right, then R must be to his right? No, 'P is second to the left of R'. If Row 2 faces North, P(Left) ... R(Right). If Row 2 faces South, P(Right) ... R(Left). Case 1: Row 2 North. P=Left, N=Mid, R=Right. O is diag opposite P -> O=Row 1 Right. Q is neighbor of O and not end -> Q=Row 1 Mid. M=Row 1 Left. Facing: M-P, Q-N, O-R. Case 2: Row 2 South. P=Right, N=Mid, R=Left. O is diag opposite P -> O=Row 1 Left. Q is neighbor of O and not end -> Q=Row 1 Mid. M=Row 1 Right. Facing: M-P, Q-N, O-R. In both cases, M faces P. Wait, let me re-read the prompt's options. If the official answer is 'a. Q', I must have missed something. 'Q is not at the end of any row.' (Confirmed: Q is mid). 'P is second to the left of R.' (Confirmed: P and R are ends). 'O is the neighbour of Q and is sitting diagonally opposite to P.' (Confirmed: O and P are opposite corners). 'N is the neighbour of R.' (Confirmed: N is mid). If M faces Q, then M must be in the middle. But Q is in the middle. Two people cannot both be in the middle of their respective rows and face each other unless they are the only ones in the middle. If M is in the middle, then Q cannot be in the middle. But the problem says Q is NOT at the end, meaning Q MUST be in the middle. Therefore, M cannot face Q. Let me re-read: 'Who is facing M?' If the arrangement is: Row 1: O Q M Row 2: R N P Then M faces P. Wait, let's try: Row 1: P N R Row 2: O Q M Facing: P-O, N-Q, R-M. Here M faces R. Check constraints: 1. Q not at end? Yes (Mid). 2. P second to left of R? (Row 1 North: P is left, R is right). Yes. 3. O neighbor of Q? Yes. 4. O diagonally opposite P? P is Row 1 Left, O is Row 2 Left. That is NOT diagonally opposite. Diagonally opposite to P(1,1) is (2,3). So O must be at Row 2 Right. If O is Row 2 Right, then Q (neighbor) is Row 2 Mid. Then M is Row 2 Left. Row 1: P N R Row 2: M Q O Facing: P-M, N-Q, R-O. Again, M faces P. Let me double check the 'diagonally opposite' for P and O. If P is (1,1), O is (2,3). If P is (1,3), O is (2,1). If P is (1,1) [Row 1 Left], then R is (1,3) [Row 1 Right] because P is 2nd left of R. Then N is (1,2). O is (2,3) [Row 2 Right]. Q is (2,2) [Row 2 Mid]. M is (2,1) [Row 2 Left]. Facing: (1,1)- (2,1) => P faces M. Is it possible that 'second to the left' means something else? No. Is it possible that M is facing Q? For M to face Q, M must be (1,2) or (2,2). But Q is already (1,2) or (2,2). If Q is (1,2), then M must be (2,2). But if M is (2,2), then Q's neighbor O must be (2,1) or (2,3). If O is (2,1), then P (diagonally opposite O) must be (1,3). If P is (1,3), then R must be... (P is 2nd left of R). If Row 1 faces North, P(1,3) is the right end. He cannot be to the left of anyone. If Row 1 faces South, P(1,3) is the left end. R would be (1,1). Then N is (1,2). But we said Q is (1,2)! Contradiction. Therefore, M must face P. Wait, let me check the options again. a. Q, b. R, c. N, d. P. If the answer is 'a', there is a fundamental misunderstanding of the text or a typo in the question/options. However, based on the logic, M faces P. Let me re-read one more time. 'O is the neighbour of Q and is sitting diagonally opposite to P'. Could 'diagonally opposite' mean something else? No. Could 'second to the left' mean something else? No. Let's try: Row 1: M N R Row 2: P Q O Check: Q not at end? Yes. P 2nd left of R? P(2,1) is 2nd left of R(1,3)? No, they are in different rows. Let's try: Row 1: P N R Row 2: O Q M Check: Q not at end? Yes. P 2nd left of R? Yes (P is left, R is right). O neighbor of Q? Yes. O diag opposite P? P(1,1), O(2,1). No. Let's try: Row 1: P N R Row 2: M Q O Check: Q not at end? Yes. P 2nd left of R? Yes. O neighbor of Q? Yes. O diag opposite P? P(1,1), O(2,3). Yes. Facing: P-M, N-Q, R-O. Result: M faces P. Wait, I just noticed something. If the question is 'Who is facing Q?', the answer would be N. If 'Who is facing N?', the answer would be Q. If the answer is 'a. Q', then the question must be 'Who is facing N?'. But the question is 'Who is facing M?'. Let me re-verify the 'diagonally opposite' one more time. If P is (1,1), O is (2,3). If P is (1,3), O is (2,1). If P is (1,3), then for P to be 2nd left of R, Row 1 must face South. Row 1 (South): R(1,1) N(1,2) P(1,3). O is (2,1). Q is (2,2). M is (2,3). Facing: R-O, N-Q, P-M. In every single scenario, M faces P. Let me check if I misread 'O is the neighbour of Q'. If O is (2,1) and Q is (2,2), they are neighbours. Is it possible that M is facing Q? If M is (2,2), then Q is (1,2). O is neighbor of Q -> O is (1,1) or (1,3). If O is (1,1), P is (2,3). If P is (2,3), is P 2nd left of R? If Row 2 faces North, P(2,3) is the right end. No. If Row 2 faces South, P(2,3) is the left end. R would be (2,1). Then N is (2,2). But we said M is (2,2)! Contradiction. Conclusion: M faces P. Option d.

Let's break down the relationships step-by-step: 1. **Lura and Liani**: Lura is the only son of Liani. (Liani is the mother, Lura is the son). 2. **Liani and Lawma**: Liani is the only daughter of Lawma. (Lawma is the grandparent of Lura). 3. **Sawmi and Liani**: Sawmi is the daughter-in-law of Liani. Since Lura is Liani's only son, Sawmi must be married to Lura. 4. **Sawmi and Lala**: Sawmi is the mother of Lala. Since Sawmi is married to Lura, Lala is the child of Lura and Sawmi. 5. **Lala and Liani**: Liani is the mother of Lura, and Lura is the father of Lala. Therefore, Lala is the grandchild of Liani. Evaluating the options: - (a) Son: Lura is the son, not Lala. - (b) Grandson: Lala is the child of Liani's son, making Lala the grandson (or granddaughter, but 'Grandson' is the only fitting option provided). - (c) Husband: Lala is a child, not Liani's husband. - (d) None of these: Incorrect, as 'Grandson' fits the relationship.

Step-by-step solution: 1. **Understand the Ratio**: The total sum of money is divided between the group of boys and the group of girls in the ratio 8:9. This means for every 8 units of money the boys receive as a group, the girls receive 9 units. 2. **Individual Share**: The problem states that each individual boy and each individual girl receives an equal amount of money. Let this individual amount be 'x'. 3. **Total Amount for Boys**: There are 48 boys. Since each gets 'x', the total amount for boys = 48 * x. 4. **Total Amount for Girls**: Let the number of girls be 'G'. Since each gets 'x', the total amount for girls = G * x. 5. **Set up the Proportion**: According to the given ratio (Boys:Girls = 8:9): (48 * x) / (G * x) = 8 / 9 6. **Simplify and Solve**: The 'x' cancels out from both the numerator and denominator: 48 / G = 8 / 9 Cross-multiplying gives: 8 * G = 48 * 9 8G = 432 G = 432 / 8 G = 54 Therefore, the number of girls is 54. Option (c) is correct. Options (a), (b), and (d) are incorrect as they do not satisfy the ratio 48:G = 8:9.

To find the area of the path, we first determine the dimensions of the square park and the outer square (park + path): 1. **Side of the square park**: Since the area of the park is $169\text{ m}^2$, its side length is $\sqrt{169} = 13\text{ m}$. 2. **Side of the outer square**: The path of width $2\text{ m}$ runs all around the park, so we add $2\text{ m}$ to all four sides. The side length of the outer square is $13 + 2 + 2 = 17\text{ m}$. 3. **Area of the outer square**: The total area including the path is $17^2 = 289\text{ m}^2$. 4. **Area of the path**: Subtract the area of the park from the total area: $289\text{ m}^2 - 169\text{ m}^2 = 120\text{ m}^2$. * **Option b** is correct as it matches this calculation. * **Option a (56m²)** is incorrect; it represents the incorrect calculation where the path is added to only one side of the length ($15^2 - 169 = 56\text{ m}^2$). * **Options c (192m²)** and **d (116m²)** are incorrect due to arithmetic errors in calculating the outer area or the path width.

The formula for permutations is nPr = n! / (n - r)!. Given nP3 = 504, we can write this as: n * (n - 1) * (n - 2) = 504. We are looking for three consecutive integers whose product is 504. Let's test the options provided: - Option a (n=10): 10 * 9 * 8 = 720 (Too high) - Option b (n=6): 6 * 5 * 4 = 120 (Too low) - Option c (n=12): 12 * 11 * 10 = 1320 (Too high) - Option d (n=9): 9 * 8 * 7 = 504 (Correct) Since 9 * 8 * 7 = 504, the value of n is 9.

In Microsoft Excel, cell referencing determines how a cell address behaves when a formula is copied or dragged to another cell: 1. **Relative References**: These are the default references (e.g., A1). When you copy a formula containing a relative reference, Excel automatically adjusts the reference based on the relative position of the rows and columns. For example, if you copy '=A1' one row down, it becomes '=A2'. 2. **Absolute References**: These are locked references denoted by dollar signs (e.g., $A$1). The dollar signs fix both the column and the row. No matter where you copy the formula, the reference always points to the exact same cell. 3. **Mixed References**: These lock only one part of the reference—either the row or the column (e.g., $A1 or A$1). In $A1, the column is fixed but the row changes; in A$1, the row is fixed but the column changes. Evaluating the options: - Option (a) is incorrect because it swaps the definitions of relative and absolute. - Option (b) is incorrect because it also swaps the definitions. - Option (d) is incorrect because mixed and relative references behave differently. - Option (c) accurately describes all three types of referencing.

To find the time (n), we use the Compound Interest formula: A = P(1 + R/100)^n Given: Principal (P) = Rs. 2,000 Amount (A) = Rs. 2,315.25 Rate (R) = 5% per annum Step 1: Substitute the values into the formula: 2,315.25 = 2,000(1 + 5/100)^n Step 2: Simplify the expression inside the bracket: 1 + 5/100 = 1 + 1/20 = 21/20 Step 3: Isolate the term with the exponent: 2,315.25 / 2,000 = (21/20)^n Step 4: Simplify the fraction on the left side: 2,315.25 / 2,000 = 231,525 / 200,000 Dividing both by 25: 231,525 ÷ 25 = 9,261 200,000 ÷ 25 = 8,000 So, 9,261 / 8,000 = (21/20)^n Step 5: Recognize the powers: We know that 21^3 = 9,261 and 20^3 = 8,000. Therefore, (21/20)^3 = 9,261 / 8,000. Comparing (21/20)^n = (21/20)^3, we get n = 3. Verification of options: - If n=1: A = 2000 * 1.05 = 2100 - If n=2: A = 2100 * 1.05 = 2205 - If n=3: A = 2205 * 1.05 = 2315.25 (Matches the given amount) - If n=4: A = 2315.25 * 1.05 = 2431.01 Thus, the correct time is 3 years.

To find the number of ways to arrange the letters of the word 'APPLE', we use the formula for permutations of a multiset: n! / (p1! * p2! ... pk!), where 'n' is the total number of letters and 'p' represents the frequency of each repeating letter. 1. Count the total number of letters in 'APPLE': n = 5. 2. Identify the repeating letters: The letter 'P' appears 2 times. 3. Apply the formula: Total arrangements = 5! / 2! 4. Calculate the factorials: 5! = 5 × 4 × 3 × 2 × 1 = 120; 2! = 2 × 1 = 2. 5. Final calculation: 120 / 2 = 60. Therefore, there are 60 distinct ways to arrange the letters. Option (a) 30 is incorrect as it divides by 4; Option (c) 90 is mathematically unrelated to the factorial of 5; Option (d) 120 is incorrect because it fails to account for the identical 'P's, which makes some arrangements indistinguishable.

Step-by-step reasoning: 1. **Understand the Original Series**: The standard English alphabet is A(1), B(2), C(3), D(4), E(5), F(6), G(7), H(8), I(9), J(10), K(11), L(12), M(13), N(14), O(15), P(16), Q(17), R(18), S(19), T(20), U(21), V(22), W(23), X(24), Y(25), Z(26). 2. **Analyze the Switching Pattern**: The problem states that letters are switched in pairs: - 1st and 2nd (A ↔ B) - 3rd and 4th (C ↔ D) - 5th and 6th (E ↔ F) - ... and so on. 3. **Identify the Rule**: In this pattern, every odd-positioned letter (n) moves to the next even position (n+1), and every even-positioned letter (n) moves to the previous odd position (n-1). 4. **Determine the 20th Letter**: We are looking for the letter at the 20th position in the new series. Since 20 is an even number, it will be occupied by the letter that was originally at the 19th position (20 - 1 = 19). 5. **Find the 19th Letter**: In the standard English alphabet, the 19th letter is 'S'. 6. **Verification**: - 19th letter (S) moves to 20th position. - 20th letter (T) moves to 19th position. Therefore, the 20th letter from the left in the new series is S. Comparing with options: - a. T (This is the 19th letter in the new series) - b. S (Correct) - c. N (14th letter) - d. M (13th letter)

Step-by-step solution: 1. Let the three numbers be represented as 3x, 2x, and 5x, where 'x' is a common multiplier based on the given ratio 3:2:5. 2. According to the problem, the sum of their squares is 1862. Therefore, we set up the equation: (3x)² + (2x)² + (5x)² = 1862 3. Calculate the squares: 9x² + 4x² + 25x² = 1862 4. Combine the like terms: 38x² = 1862 5. Solve for x²: x² = 1862 / 38 x² = 49 6. Find the value of x: x = √49 = 7 7. Now, find the three numbers: First number = 3x = 3 * 7 = 21 Second number = 2x = 2 * 7 = 14 Third number = 5x = 5 * 7 = 35 8. The question asks for the smallest number. Comparing 21, 14, and 35, the smallest number is 14. Verification of options: - Option (a) 14: Correct (Smallest number). - Option (b) 21: Incorrect (This is the middle number). - Option (c) 35: Incorrect (This is the largest number). - Option (d) 7: Incorrect (This is the value of the multiplier 'x', not one of the numbers).

Step-by-step solution: 1. **Identify given values**: - Area of the square ($A_{square}$) = $2772\text{ cm}^2$. - Relationship: Area of the square is twice the area of the circle ($A_{square} = 2 \times A_{circle}$). - Value of $\pi = 22/7$. 2. **Find the area of the circle**: Since $A_{square} = 2 \times A_{circle}$, then: $A_{circle} = A_{square} / 2$ $A_{circle} = 2772 / 2 = 1386\text{ cm}^2$. 3. **Use the circle area formula to find the radius ($r$)**: The formula for the area of a circle is $A = \pi r^2$. $1386 = (22/7) \times r^2$ 4. **Solve for $r^2$**: $r^2 = (1386 \times 7) / 22$ $r^2 = 63 \times 7$ (since $1386 / 22 = 63$) $r^2 = 441$ 5. **Calculate $r$**: $r = \sqrt{441} = 21\text{ cm}$. Comparing this result to the options: - Option (a) 21 cm matches our calculation. - Options (b), (c), and (d) are incorrect as their squares multiplied by $\pi$ do not equal 1386.

Let's break down the relationships step-by-step: 1. A is the brother of B: This means A is male and A and B are siblings. 2. B is the sister of C: This means B is female and B, A, and C are all siblings. 3. C is the father of D: This means C is male and D is the child of C. Now, we determine the relationship between D and A: - Since A is the brother of C, and C is the father of D, A is the paternal uncle of D. - Conversely, D is the child of A's brother (C). - In blood relation terminology, the child of one's sibling is either a 'Nephew' (if male) or a 'Niece' (if female). Crucially, the problem does not specify the gender of D. Without knowing if D is male or female, we cannot determine if D is A's nephew or A's niece. Therefore, the exact relationship cannot be established from the given information.