Q: IF (14) 3 is added to the square of a number, the answer to obtained is 4425. What is the number?

Let the number be x. According to the question, x2 + (14)3 = 4425 ⇒ x2 = 4425 - (14)3 ⇒ x2 = 4425 - 2744 ⇒ x2 = 1681 ∴ x = √1681 = 41

Q: Which among ∜10 and ∛8 is greater?

∜10 and ∛8 = 101/4 and 81/3 Since, LCM of 4 and 3 is 12. ∴ 101/4 = 103/12 = (103)1/12 = (1000)1/12 = 12√1000 and 81/3 = 84/12 = (84)1/12 (4096)1/2 = 12√4096 ∴ 4096 > 1000 Hence, 81/3 >101/4 ∴ ∛8 > ∜10

Question 1

If x = √3 + √2, then the value of (x + 1)/(x - 1) - (x - 1)/(x + 1) is?

Accepted Answer

(x + 1)/(x - 1) - (x - 1)/(x + 1) = {(x + 1)2 - (x - 1)2} / {(x - 1) (x + 1)} = (x2 + 1 + 2x) - (x2 + 1 - 2x) / (x2 - 1) = 4x / (x2 - 1) = 4(√3 + √2) / {(√3 + √2)2 - 1} = 4(√3 + √2) / (3 + 2 + 2√6 - 1) = 4(√3 + √2) / (4 + 2 √6) = 4(√3 + √2) / 2 √2 (√3 + √2) = √2

Question 2

Which among ∛8 and ∛1000 is greater?

Accepted Answer

∛8 and ∛1000 = 81/3 and 10001/3 Since 1000 > 8 ∴ ∛1000 > ∛8

Question 3

Which among √81 and √1 is smaller?

Accepted Answer

√81 and √1 = 811/2 and 11/2 Since, 1 < 81 ∴ √1 < √81

Question 4

IF (14) 3 is added to the square of a number, the answer to obtained is 4425. What is the number?

Accepted Answer

Let the number be x. According to the question, x2 + (14)3 = 4425 ⇒ x2 = 4425 - (14)3 ⇒ x2 = 4425 - 2744 ⇒ x2 = 1681 ∴ x = √1681 = 41

Question 5

Which among ∜10 and ∛8 is greater?

Accepted Answer

∜10 and ∛8 = 101/4 and 81/3 Since, LCM of 4 and 3 is 12. ∴ 101/4 = 103/12 = (103)1/12 = (1000)1/12 = 12√1000 and 81/3 = 84/12 = (84)1/12 (4096)1/2 = 12√4096 ∴ 4096 > 1000 Hence, 81/3 >101/4 ∴ ∛8 > ∜10

Question 6

Find the least number which multiplied with 74088 will make it a perfect square?

Accepted Answer

74088 = 2 x 2 x 2 x 3 x 3 x 3 x 7 x 7 x 7 = (2 x 2) x (3 x 3) x (7 x 7) x ( 2 x 3 x 7) ∴ Least number which should be multiplied to 74088 to make it a perfect suqare = 2 x 3 x 7 = 42

Question 7

Find the number whose square is equal to the difference between the square of 75.15 and 60.12.?

Accepted Answer

Let the number be x. Then, accordingto the question, x2 = (75.15)2 - (60.12)2 ⇒ x2 = 5647.52 - 3614.41 ⇒ x2 = 2033.11 ∴ x = 45.09

Question 8

A general wishing to draw up his 16160 soldiers in the form as a solid square, found that he had 31 soldiers left over. Find the number of soldiers in the front.?

Accepted Answer

Let the total number of soliders in front be x. Then, total number of soliders in form of solid square = x2 According to the question, x2 + 31 = 16160 ⇒ x2 = 16129 ⇒ x = √16129 ∴ x = 127

Question 9

A certain number of people agree to subscribe as many rupees each as there are subscribers. The whole subscription is Rs. 2582449. Find the number of subscriber.

Accepted Answer

Let the total number of subbscribers be x. Then, x2 = 2582449 ⇒ x = √2582449 = 1607

Question 10

Each student of class IX contributed some money for a picnic. The money contributed by each student was equal to the cube of the total number of student. If the total collected amount was Rs. 29791, then find the total number of students.

Accepted Answer

Total collected amount = Rs. 29791 ∴ Total number of student = ∛29791 = 31

Question 11

If x = (√2 + 1)/(√2 - 1) and y = (√2 - 1)/(√2 + 1), then 3 (xy + 1) is euqal to?

Accepted Answer

xy = [(√2 + 1)/(√2 - 1)] x [(√2 - 1)/(√2 + 1)] = 1 ∴ 3(xy + 1) = 3(1 + 1) = 6 Now, x + y = [(√2 + 1)/(√2 - 1)] + [(√2 - 1)/(√2 + 1)] = (√2 + 1)2 + (√2 - 1)2 / (√2 - 1) (√2 + 1) = (2 + 1 +2√2) + (2 + 1 - 2√2) / (2 - 1) = 6 ∴ 3(xy + 1) = x + y

Question 12

The least perfect square which is divisible by each of 21, 36 and 66 is equal to?

Accepted Answer

Ans. LCM of 21, 36 and 66 = 2772 Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11 To make it a perfect square, it must be multiplied by 77 [for (7 x 11)]. ∴ Required number = 22 x 32 x 72 x 112 = 213444

Question 13

In an opera house, there are 7777 chairs to be placed but the organizer of the event arranged all the chair in such a way that these were as many columns as there were rows, so he had to remove minimum n chairs from the total 7777 chairs. The minimu…

Accepted Answer

7777 - N must be a perfect square. Now, using the option, we find that N = 33 7777 - 33 = 7744 is perfect square.

Question 14

In a bangel shop, the shopkeeper displays the bangels in the form of a square, then he is left with 38 bangles with him. If he wants to increse the size of square by one unit each side of the square, he found that 25 bangles fall short of in complet…

Accepted Answer

Let the number of bangles be x. Then, x - 38 is a perfect square and x + 25 is a perfect square. Now, using the option, we find that x = 999 999 - 38 = 961 is a perfect square. 999 + 25 = 1024 is a perfect square

Question 15

The sum of two number, when multiplied with each of the number separately, then the results of these multiplication are 2418 and 3666, respectively. Find the difference between the numbers.?

Accepted Answer

Let the number be x and y. Now, according to the question, x(x + y) = 2418 .....(i) and y(x + y) = 3666 .....(ii) On adding Eqs. (i) and (ii), we get x2 + xy + yx + y2 = 6084 ⇒ x2 + 2xy + y2 = 6084 ⇒ (x + y)2 = 6084 ∴ x + y = √6084 = 78 On subtracting Eq. (ii) from Eq. (i), we get x2 + xy - yx - y2 = -1248 ⇒ x2 - y2 = -1248 ⇒ (x + y) (x - y) = -1248 ⇒ 78(x - y) = -1248 ⇒ x - y = - 1248 / 78 = -16 ⇒ (y - x) = 16

Question 16

One-fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

Accepted Answer

Let the total number of camels be x. Then, acording to the question, x/4 + 2√x + 15 = x ⇒ x - x/4 - 2 √x = 15 ⇒ 3x - 8√x - 60 = 0 ⇒ √x = [8 ± √64 + 720] / 6 = (8 ± 28) / 6 ∴ x = 36

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