Number System
- The difference between two positive numbers is 3. If the sum of their squares is 369, then the sum of the numbers is :
-
View Hint View Answer Discuss in Forum
Let the number be x and y and x > y.
x – y = 3 ...(i)
x2 + y2 = 369 ....(ii)
From equation (i)
x – y = 3
⇒ (x – y)2 = 32
⇒ x2 + y2 – 2xy = 9
⇒ = (x2 + y2) – 9
= 369 – 9 = 360
From equation (ii)
Now,(x2 + y2) = x2 + y2 + 2xy
= 369 + 360 = 729
∴ x + y = √729 = 27
∴ Required sum = 27Correct Option: C
Let the number be x and y and x > y.
x – y = 3 ...(i)
x2 + y2 = 369 ....(ii)
From equation (i)
x – y = 3
⇒ (x – y)2 = 32
⇒ x2 + y2 – 2xy = 9
⇒ = (x2 + y2) – 9
= 369 – 9 = 360
From equation (ii)
Now,(x2 + y2) = x2 + y2 + 2xy
= 369 + 360 = 729
∴ x + y = √729 = 27
∴ Required sum = 27
- Thrice the square of a natural number decreased by four times the number is equal to 50 more than the number. The number is:
-
View Hint View Answer Discuss in Forum
Let the number be x.
According to the question,
3 × x2 – 4 × x = x + 50
⇒ 3x2 – 5x – 50 = 0
⇒ 3x2 – 15x + 10x – 50 = 0
⇒ 3x (x – 5) + 10 (x – 5) = 0
⇒ (x – 5) (3x – 10) = 0⇒ x = 5 or −10 3
But the number is natural.∴ x ≠ −10 3
Hence, the required number = 5.Correct Option: B
Let the number be x.
According to the question,
3 × x2 – 4 × x = x + 50
⇒ 3x2 – 5x – 50 = 0
⇒ 3x2 – 15x + 10x – 50 = 0
⇒ 3x (x – 5) + 10 (x – 5) = 0
⇒ (x – 5) (3x – 10) = 0⇒ x = 5 or −10 3
But the number is natural.∴ x ≠ −10 3
Hence, the required number = 5.
- A number of boys raised ₹ 400 for a famine relief fund, each boy giving as many 25 paise coins as there were boys. The number of boys was :
-
View Hint View Answer Discuss in Forum
Let the number of boys = x
∴ Number of 25 paise coins= x2
According to question,25 × x2 = 400 100 ⇒ x2 = 400 ⇒ x2 = 1600 4
⇒ x = √1600 = 40
Correct Option: A
Let the number of boys = x
∴ Number of 25 paise coins= x2
According to question,25 × x2 = 400 100 ⇒ x2 = 400 ⇒ x2 = 1600 4
⇒ x = √1600 = 40
- In a test, 1 mark is awarded for each correct answer and one mark is deducted for each wrong answer. If a boy answers all 20 items of the test and gets 8 marks, the number of questions answered correct by him was
-
View Hint View Answer Discuss in Forum
Let the number of correct answers be x
∴ The no. of incorrect answers = 20 – x
According to the question,
x – (20 – x) = 8
⇒ x – 20 + x = 8
⇒ 2x = 28 ⇒ x = 14Correct Option: B
Let the number of correct answers be x
∴ The no. of incorrect answers = 20 – x
According to the question,
x – (20 – x) = 8
⇒ x – 20 + x = 8
⇒ 2x = 28 ⇒ x = 14
- In a two digit number if it is known that its units digit exceeds its tens digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is
-
View Hint View Answer Discuss in Forum
Let the ten’s digit be x
∴ Unit’s digit = x + 2
Therefore, the two digit number
= 10x + x + 2
= 11x + 2 ...(i)
Again,
(11x + 2) (x + x + 2)
= 144
⇒ (11x + 2) (2x + 2) = 144
(11x + 2) (x + 1) = 72
⇒ 11x2 + 2x + 11x + 2 = 72
⇒ 11x2 + 13x – 70 = 0
⇒ 11x2 – 22x + 35x – 70 = 0
⇒ 11x (x – 2) + 35 (x – 2) = 0
⇒ (x – 2) (11x + 35) = 0⇒ x= 2 , − 35 11 ⇒ x= 2 , − 35 11
is not admissible.
∴ The number = 11x + 2
= 11 × 2 + 2 = 24Correct Option: D
Let the ten’s digit be x
∴ Unit’s digit = x + 2
Therefore, the two digit number
= 10x + x + 2
= 11x + 2 ...(i)
Again,
(11x + 2) (x + x + 2)
= 144
⇒ (11x + 2) (2x + 2) = 144
(11x + 2) (x + 1) = 72
⇒ 11x2 + 2x + 11x + 2 = 72
⇒ 11x2 + 13x – 70 = 0
⇒ 11x2 – 22x + 35x – 70 = 0
⇒ 11x (x – 2) + 35 (x – 2) = 0
⇒ (x – 2) (11x + 35) = 0⇒ x= 2 , − 35 11 ⇒ x= 2 , − 35 11
is not admissible.
∴ The number = 11x + 2
= 11 × 2 + 2 = 24
