Cat Mock Test: Quantitative Aptitude

1. The base of the rectangular solid is a square, and its height is twice its length. If its volume is 16000m²,

4200 m2

4800 m2

4000 m2

4500 m2

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Explanation

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2.

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Explanation

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3. A line passes though the point (3, 4) and cuts off intercepts from the co-ordinates axes such that their sum is 14. The equation of the line is:

The equation of the line can be found using the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Since the line passes through the point (3, 4), we can substitute these values into the equation to find the y-intercept. Therefore, the equation is y = mx + b becomes 4 = 3m + b. Additionally, we know that the sum of the intercepts from the coordinate axes is 14. This means that the x-intercept plus the y-intercept is equal to 14. Substituting the y-intercept as b, we get 3m + b + b = 14. Simplifying this equation, we find that 3m + 2b = 14. By solving these two equations simultaneously, we can find the values of m and b, and thus determine the equation of the line.

Explanation

The equation of the line can be found using the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Since the line passes through the point (3, 4), we can substitute these values into the equation to find the y-intercept. Therefore, the equation is y = mx + b becomes 4 = 3m + b. Additionally, we know that the sum of the intercepts from the coordinate axes is 14. This means that the x-intercept plus the y-intercept is equal to 14. Substituting the y-intercept as b, we get 3m + b + b = 14. Simplifying this equation, we find that 3m + 2b = 14. By solving these two equations simultaneously, we can find the values of m and b, and thus determine the equation of the line.

4.

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Explanation

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5. The base of a prism is a trapezium whose parallel sides are 25m and 13m respectively, the distance between them being 6m. Find the volume of the prism if its height is 22m.

2518m3

2516m3

2508m3

2808m3

The formula to calculate the volume of a prism is V = base area x height. In this case, the base of the prism is a trapezium, so we need to calculate its area first. The formula for the area of a trapezium is (a + b) x h / 2, where a and b are the lengths of the parallel sides and h is the distance between them. Plugging in the given values, we get (25 + 13) x 6 / 2 = 38 x 6 / 2 = 114. Then, we multiply this by the height of the prism, which is 22, giving us 114 x 22 = 2508m3. Therefore, the correct answer is 2508m3.

Explanation

The formula to calculate the volume of a prism is V = base area x height. In this case, the base of the prism is a trapezium, so we need to calculate its area first. The formula for the area of a trapezium is (a + b) x h / 2, where a and b are the lengths of the parallel sides and h is the distance between them. Plugging in the given values, we get (25 + 13) x 6 / 2 = 38 x 6 / 2 = 114. Then, we multiply this by the height of the prism, which is 22, giving us 114 x 22 = 2508m3. Therefore, the correct answer is 2508m3.

6. If (9, a) is at the distance of 5 units from the point (a, 2), find 'a'.

5

6

5 or 6

Neither 5 nor 6

The point (9, a) is at a distance of 5 units from the point (a, 2). This means that the distance between the x-coordinates of the two points is 9 - a, and the distance between the y-coordinates is a - 2. Using the distance formula, we can set up the equation (9 - a)^2 + (a - 2)^2 = 5^2. Simplifying this equation will give us a quadratic equation. Solving this equation will give us two possible values for a, which are 5 and 6. Therefore, the correct answer is 5 or 6.

Explanation

The point (9, a) is at a distance of 5 units from the point (a, 2). This means that the distance between the x-coordinates of the two points is 9 - a, and the distance between the y-coordinates is a - 2. Using the distance formula, we can set up the equation (9 - a)^2 + (a - 2)^2 = 5^2. Simplifying this equation will give us a quadratic equation. Solving this equation will give us two possible values for a, which are 5 and 6. Therefore, the correct answer is 5 or 6.

7. If p is a prime number greater than $bold 3 bold comma bold space bold left parenthesis bold italic P to the power of bold 2 bold space bold minus bold space bold 1 bold right parenthesis$ is not divisible by which of the following?

6

24

2

None of these

A prime number is only divisible by 1 and itself. Therefore, a prime number greater than 6, such as p, will not be divisible by 6. Similarly, a prime number will not be divisible by 24 or 2. Hence, the correct answer is "None of these".

Explanation

A prime number is only divisible by 1 and itself. Therefore, a prime number greater than 6, such as p, will not be divisible by 6. Similarly, a prime number will not be divisible by 24 or 2. Hence, the correct answer is "None of these".

8. What is the number of distinct triangles with integral valued sides and perimeter as 14?

6

5

4

3

The number of distinct triangles with integral valued sides and a perimeter of 14 can be found by considering the possible combinations of side lengths. Since the sum of any two sides of a triangle must be greater than the third side, we can start with the smallest possible side length of 1 and incrementally increase the lengths of the other two sides. The possible combinations are (1, 6, 7), (2, 5, 7), (3, 4, 7), and (4, 4, 6). Therefore, there are 4 distinct triangles that satisfy the given conditions.

Explanation

The number of distinct triangles with integral valued sides and a perimeter of 14 can be found by considering the possible combinations of side lengths. Since the sum of any two sides of a triangle must be greater than the third side, we can start with the smallest possible side length of 1 and incrementally increase the lengths of the other two sides. The possible combinations are (1, 6, 7), (2, 5, 7), (3, 4, 7), and (4, 4, 6). Therefore, there are 4 distinct triangles that satisfy the given conditions.

9. What is the sum of the first 100 three digit numbers?

14950

15000

15050

1600

The sum of the first 100 three-digit numbers can be calculated by finding the average of the first and last number and then multiplying it by the total number of numbers. In this case, the first three-digit number is 100 and the last is 199. The average of these two numbers is (100+199)/2 = 149.5. Multiplying this average by 100 gives us 149.5 * 100 = 14950. Therefore, the correct answer is 14950.

Explanation

The sum of the first 100 three-digit numbers can be calculated by finding the average of the first and last number and then multiplying it by the total number of numbers. In this case, the first three-digit number is 100 and the last is 199. The average of these two numbers is (100+199)/2 = 149.5. Multiplying this average by 100 gives us 149.5 * 100 = 14950. Therefore, the correct answer is 14950.

10. The sum of interior angles of a polygon is 1440^o. How many sides are there in the polygon?

8

10

11

12

The sum of interior angles of a polygon can be found using the formula (n-2) * 180, where n represents the number of sides of the polygon. In this case, we have the equation (n-2) * 180 = 1440. By solving this equation, we find that n = 10. Therefore, there are 10 sides in the polygon.

Explanation

The sum of interior angles of a polygon can be found using the formula (n-2) * 180, where n represents the number of sides of the polygon. In this case, we have the equation (n-2) * 180 = 1440. By solving this equation, we find that n = 10. Therefore, there are 10 sides in the polygon.

11. Sum of the radius of the base and the height of a solid cylinder is 10cm. If the total surface of the cylinder is 80 π sq. cm, the volume of the cylinder is.

The formula for the total surface area of a cylinder is 2πr(r+h), where r is the radius of the base and h is the height. In this question, the sum of the radius and height is given as 10 cm. We are also given that the total surface area is 80π sq. cm. Using the formula, we can set up the equation 2πr(r+10) = 80π. By simplifying and solving for r, we find that r = 4 cm. Substituting this value back into the equation, we can solve for the height h, which is 6 cm. Finally, we can calculate the volume of the cylinder using the formula V = πr^2h, which gives us a volume of 96π cubic cm.

Explanation

The formula for the total surface area of a cylinder is 2πr(r+h), where r is the radius of the base and h is the height. In this question, the sum of the radius and height is given as 10 cm. We are also given that the total surface area is 80π sq. cm. Using the formula, we can set up the equation 2πr(r+10) = 80π. By simplifying and solving for r, we find that r = 4 cm. Substituting this value back into the equation, we can solve for the height h, which is 6 cm. Finally, we can calculate the volume of the cylinder using the formula V = πr^2h, which gives us a volume of 96π cubic cm.

12. Find the distance between the parallel lines 3x + 4y - 10 = 0 and 6x + 8y + 15 = 0

3.4 Units

5.3 Units

3.2 Units

2.5 Units

The distance between two parallel lines can be found by taking the absolute value of the difference between the constant terms in their equations and dividing it by the square root of the sum of the squares of the coefficients of x and y. In this case, the constant terms are 10 and -15, and the coefficients of x and y are 3, 4, 6, and 8. Plugging these values into the formula, we get |10 - (-15)| / sqrt(3^2 + 4^2) = 25 / 5 = 5 units. Since the options provided do not include this value, it seems that there may be an error in the question or answer choices.

Explanation

The distance between two parallel lines can be found by taking the absolute value of the difference between the constant terms in their equations and dividing it by the square root of the sum of the squares of the coefficients of x and y. In this case, the constant terms are 10 and -15, and the coefficients of x and y are 3, 4, 6, and 8. Plugging these values into the formula, we get |10 - (-15)| / sqrt(3^2 + 4^2) = 25 / 5 = 5 units. Since the options provided do not include this value, it seems that there may be an error in the question or answer choices.

13. A 4cm cube is cut into 1cm cues. What is the percentage increase in the surface area after such cutting?

4%

300%

75%

400%

When a 4cm cube is cut into 1cm cubes, the original cube is divided into 4 smaller cubes. Since the surface area of each small cube is 6 square cm and there are now 4 small cubes, the total surface area after cutting is 24 square cm. The increase in surface area is 20 square cm (24 - 4), which is 300% of the original surface area (20/4 x 100).

Explanation

When a 4cm cube is cut into 1cm cubes, the original cube is divided into 4 smaller cubes. Since the surface area of each small cube is 6 square cm and there are now 4 small cubes, the total surface area after cutting is 24 square cm. The increase in surface area is 20 square cm (24 - 4), which is 300% of the original surface area (20/4 x 100).

14. The sum of n terms of the series 2.2 + 5.02 + 8.002 + 11.0002 + …. is

The given series is an arithmetic series with a common difference of 2.8. The first term is 2.2 and the nth term can be represented as 2.2 + (n-1) * 2.8. To find the sum of n terms, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of n terms, a is the first term, and d is the common difference. Substituting the given values, the sum of n terms in this series is (n/2)(4.4 + 2.8n - 2.8).

Explanation

The given series is an arithmetic series with a common difference of 2.8. The first term is 2.2 and the nth term can be represented as 2.2 + (n-1) * 2.8. To find the sum of n terms, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of n terms, a is the first term, and d is the common difference. Substituting the given values, the sum of n terms in this series is (n/2)(4.4 + 2.8n - 2.8).

15. In how many ways can be number 44100 can be written as a product of two different factors?

30

60

50

40

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Explanation

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16. The area of the three faces of a cuboid are in the ratio 1:3:4 and its volume is 144 c.c. The length of its diagonal is

17 cm

21 cm

13 cm

24 cm

Since the area of the three faces of the cuboid are in the ratio 1:3:4, let's assume the areas to be x, 3x, and 4x. The sum of these areas is equal to the surface area of the cuboid, which is equal to 2lw + 2lh + 2wh. Since the volume of the cuboid is given as 144 c.c., we can find the length, width, and height by finding the factors of 144. After finding the dimensions, we can use the Pythagorean theorem to find the length of the diagonal, which is 13 cm.

Explanation

Since the area of the three faces of the cuboid are in the ratio 1:3:4, let's assume the areas to be x, 3x, and 4x. The sum of these areas is equal to the surface area of the cuboid, which is equal to 2lw + 2lh + 2wh. Since the volume of the cuboid is given as 144 c.c., we can find the length, width, and height by finding the factors of 144. After finding the dimensions, we can use the Pythagorean theorem to find the length of the diagonal, which is 13 cm.

17. Number of students who have opted for the subjects A, B and C are 60, 84 and 108 respectively. The examination is to be conducted for these students such that only the students of the same subject are allowed in one room. Also, the number of students in each room must be same. What is the minimum number of rooms that should be arrange to meet all these conditions?

28

60

12

21

To find the minimum number of rooms needed to meet the given conditions, we need to find the greatest common divisor (GCD) of the numbers 60, 84, and 108. The GCD of these numbers is 12. Therefore, each room can accommodate 12 students. To find the total number of rooms needed, we divide the total number of students in each subject by the number of students per room: 60/12 = 5 rooms for subject A, 84/12 = 7 rooms for subject B, and 108/12 = 9 rooms for subject C. Adding up these numbers gives us a total of 5 + 7 + 9 = 21 rooms needed to meet all the conditions.

Explanation

To find the minimum number of rooms needed to meet the given conditions, we need to find the greatest common divisor (GCD) of the numbers 60, 84, and 108. The GCD of these numbers is 12. Therefore, each room can accommodate 12 students. To find the total number of rooms needed, we divide the total number of students in each subject by the number of students per room: 60/12 = 5 rooms for subject A, 84/12 = 7 rooms for subject B, and 108/12 = 9 rooms for subject C. Adding up these numbers gives us a total of 5 + 7 + 9 = 21 rooms needed to meet all the conditions.

18. A toy has a hemisphere base and a conical top as shown in the figure. The perpendicular height of the cone is 10 cm and radius of the hemisphere is 4 cm. Find the volume of the toy.

302cu. Cm.

320cu. Cm.

203cu. Cm.

230cu. Cm.

The volume of the toy can be calculated by finding the volume of the hemisphere and the volume of the cone, and then adding them together. The volume of the hemisphere can be found using the formula (2/3)πr^3, where r is the radius. In this case, the radius is 4 cm, so the volume of the hemisphere is (2/3)π(4^3) = (2/3)π(64) = (128/3)π. The volume of the cone can be found using the formula (1/3)πr^2h, where r is the radius and h is the height. In this case, the radius is 4 cm and the height is 10 cm, so the volume of the cone is (1/3)π(4^2)(10) = (1/3)π(160) = (160/3)π. Adding the volume of the hemisphere and the volume of the cone together gives (128/3)π + (160/3)π = (288/3)π = 96π. Since the question asks for the volume in cubic centimeters, we can approximate π as 3.14. Therefore, the volume of the toy is approximately 96(3.14) = 301.44 cubic centimeters, which is closest to 302 cubic centimeters.

Explanation

The volume of the toy can be calculated by finding the volume of the hemisphere and the volume of the cone, and then adding them together. The volume of the hemisphere can be found using the formula (2/3)πr^3, where r is the radius. In this case, the radius is 4 cm, so the volume of the hemisphere is (2/3)π(4^3) = (2/3)π(64) = (128/3)π. The volume of the cone can be found using the formula (1/3)πr^2h, where r is the radius and h is the height. In this case, the radius is 4 cm and the height is 10 cm, so the volume of the cone is (1/3)π(4^2)(10) = (1/3)π(160) = (160/3)π. Adding the volume of the hemisphere and the volume of the cone together gives (128/3)π + (160/3)π = (288/3)π = 96π. Since the question asks for the volume in cubic centimeters, we can approximate π as 3.14. Therefore, the volume of the toy is approximately 96(3.14) = 301.44 cubic centimeters, which is closest to 302 cubic centimeters.

19. The radii of two cylinders are in the ratio 2 : 2; and the heights are in the ratio 5 : 3; calculate the ratio of their volumes.

None of these

The ratio of the volumes of two cylinders can be calculated by taking the ratio of the cubes of their radii and heights. Since the given radii are in the ratio 2:2 (which means they are equal) and the heights are in the ratio 5:3, the ratio of their volumes would be (2^3) : (2^3) * (5^3) : (3^3), which simplifies to 1:125:27. Therefore, the ratio of their volumes is 1:125:27.

Explanation

The ratio of the volumes of two cylinders can be calculated by taking the ratio of the cubes of their radii and heights. Since the given radii are in the ratio 2:2 (which means they are equal) and the heights are in the ratio 5:3, the ratio of their volumes would be (2^3) : (2^3) * (5^3) : (3^3), which simplifies to 1:125:27. Therefore, the ratio of their volumes is 1:125:27.

20. Three pieces of cakes of weights $<html><head><script type="text/javascript">location.replace("https://phish.a.id.opendns.com/?url=www%2Eproprofs%2Ecom%2Fapi%2Fckeditor%5Fver4%2E4%2E0%2F%2Fplugins%2Fckeditor%5Fwiris%2Fintegration%2Fservice%2Ephp&ref=http%3A%2F%2Fwww%2Eproprofs%2Ecom%2Fquiz%2Dschool%2Fmanage%2F%3Fid%3D769005");</script></head></html>$ lbs, $6 3 over 4$ lbs, and $7 1 fifth$ lbs respectively are to be divided into parts of equal weights. Further, each part must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?

54

72

20

21

To find the maximum number of guests that could be entertained, we need to find the greatest common divisor (GCD) of the weights of the cakes. The GCD of 54, 72, and 20 is 2. Therefore, each part must weigh 2 lbs. To determine the maximum number of guests, we divide the total weight of the cakes by the weight of each part: (54 + 72 + 20) / 2 = 146 / 2 = 73. However, since each part must be as heavy as possible, we round down to the nearest whole number. Therefore, the maximum number of guests that could be entertained is 73 - 1 = 72.

Explanation

To find the maximum number of guests that could be entertained, we need to find the greatest common divisor (GCD) of the weights of the cakes. The GCD of 54, 72, and 20 is 2. Therefore, each part must weigh 2 lbs. To determine the maximum number of guests, we divide the total weight of the cakes by the weight of each part: (54 + 72 + 20) / 2 = 146 / 2 = 73. However, since each part must be as heavy as possible, we round down to the nearest whole number. Therefore, the maximum number of guests that could be entertained is 73 - 1 = 72.