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Question 1 of 5
1. Question
X3+X2 +16 is exactly divisible by x, where x is a positive integer. The number of all such possible values of x is
Correct
Ans :(c)
We have, x3 +x2+16 is exactly divisible by x.
∴(X3+X2+16)/X =X2+ X+(16/X) is positive integer
∴X is factor of 16.
i.e., 1,2, 4, 8, 16
∴Number of all possible value of x is 5.
Incorrect
Ans :(c)
We have, x3 +x2+16 is exactly divisible by x.
∴(X3+X2+16)/X =X2+ X+(16/X) is positive integer
∴X is factor of 16.
i.e., 1,2, 4, 8, 16
∴Number of all possible value of x is 5.
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Question 2 of 5
2. Question
The number of (a, b, c), where a,b, c are positive integers such that abc = 30, is
Correct
Ans: (b)
We have, abc = 30, where a, b,c are positive integer.
\30 = 1x 1×30 = 3 ways
30 = 1 x 2x 15 = 6 ways
30= 1x 3x 10 = 6 ways
30 = 1x 5 x 6 = 6 ways
30 = 2 x 3x 5 = 6 ways
Total number of (a, b, c) = 3+ 6+ 6+ 6+ 6 =27 ways
Incorrect
Ans: (b)
We have, abc = 30, where a, b,c are positive integer.
\30 = 1x 1×30 = 3 ways
30 = 1 x 2x 15 = 6 ways
30= 1x 3x 10 = 6 ways
30 = 1x 5 x 6 = 6 ways
30 = 2 x 3x 5 = 6 ways
Total number of (a, b, c) = 3+ 6+ 6+ 6+ 6 =27 ways
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Question 3 of 5
3. Question
What is the maximum value of m, if the number N = 90 x 42 x 324x 55 is divisible by 3m?
Correct
Ans: (b)
We have, N = 90 x 42 x 324 x 55
= 32 x 10x 3x 14x 34 x 4x 55
= 37 x 10x 14x 4x 55
Hence, N is divisible by 37
So, the maximum value of m is 7 when N is divisible by 37
Incorrect
Ans: (b)
We have, N = 90 x 42 x 324 x 55
= 32 x 10x 3x 14x 34 x 4x 55
= 37 x 10x 14x 4x 55
Hence, N is divisible by 37
So, the maximum value of m is 7 when N is divisible by 37
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Question 4 of 5
4. Question
Consider the following statements
- Every natural number is a real number.
- Every real number is an irrational number.
- Every integer is a real number.
- Every rational number is a real number
Which of the above statements are correct?
Correct
Ans:(b)
- Every natural number is a real number, which is correct.
- Every real number is an irrational number, which is incorrect as real number can be rational or irrational.
- Every integer is a real number, which is correct.
- Every rational number is a real number, which is correct.
Incorrect
Ans:(b)
- Every natural number is a real number, which is correct.
- Every real number is an irrational number, which is incorrect as real number can be rational or irrational.
- Every integer is a real number, which is correct.
- Every rational number is a real number, which is correct.
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Question 5 of 5
5. Question
The largest natural number which divides every natural number of the form (n3 – n) (n – 2), where n is a natural number greater than 2, is
Correct
Ans: (C)
Let x = (n3- n)(n-2), where n> 2
Take n= 3, we get
x=(33-3)(3 -2) (27 -3)(1)= 24
which is divisible by 6, 12 and 24
Take n= 4, we get
x = (43-4)(4-2)= (64- 4) x 2 =120
which is again divisible by 6, 12 and 24.
Now, take n = 5, we get
x= (53-5)(5-2)= (125- 5) x 3 = 120 x 3 =360
which is again divisible by 6, 12 and 24.
Hence, 24 is the largest natural number.
Incorrect
Ans: (C)
Let x = (n3- n)(n-2), where n> 2
Take n= 3, we get
x=(33-3)(3 -2) (27 -3)(1)= 24
which is divisible by 6, 12 and 24
Take n= 4, we get
x = (43-4)(4-2)= (64- 4) x 2 =120
which is again divisible by 6, 12 and 24.
Now, take n = 5, we get
x= (53-5)(5-2)= (125- 5) x 3 = 120 x 3 =360
which is again divisible by 6, 12 and 24.
Hence, 24 is the largest natural number.
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