Number System

Miscellaneous

Medium

Given below are two statements:

Statement (I): The difference, the sum, and the product of two numbers are in the ratio 1: 5: 12. The product of the two numbers is 18.
Statement (II): The last digit in the decimal representation of 2⁹¹ is 6.

In light of the above statements, choose the most appropriate answer from the options given below.

Both Statement (I) and Statement (II) are true.

Both Statement (I) and Statement (II) are false.

Statement (I) is true but Statement (II) is false.

Statement (I) is false but Statement (II) is true.

✅ Correct Option: 2

Let me solve this step by step:

Statement I:

Let two numbers be x and y

Given their difference:sum:product $= 1:5:12$

And product $= 18$

So, $|x-y| : (x+y) : xy = 1:5:12$

Since $xy = 18$ , and using ratios:

$|x-y| = \frac{18}{12} = \frac{3}{2}$

$x+y = \frac{18×5}{12} = \frac{15}{2}$

From these equations:

When $x > y$ :

$x - y = \frac{3}{2}$ and $x + y = \frac{15}{2}$

Solving gives $x = 4.5, y = 3$

Numbers are not integers, Statement I is false.

Statement II:

To find last digit of $2^{91}$ :

Pattern of last digit of powers of $2: 2,4,8,6,2,4,8,6...$

$91$ mod $4$ = $3$

So last digit is $8$ , not $6$

Therefore Statement II is also false.

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The expression 2x³ + ax² + bx + 3, where a and b are constants, has a factor of x-1 and leaves a remainder of 15 when divided by x + 2. Then, (a, b) =

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Given below are two statements:

Statement I : The sum of exponents of prime factors in the prime factorization of 392 is 5.

Statement II : The decimal representation $\frac{13}{2^3 \times 5}$ will terminate after 2 decimal places.

In the light of the above statements, choose the correct answer from the options given below:

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(B) $\frac{7}{11}$

(D) $\frac{11}{17}$

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After adding 5 to both numerator and denominator of the following fractions, arrange them in ascending order of the magnitude of change in their values.

A. $\frac{2}{3}$

B. $\frac{3}{4}$

C. $\frac{4}{5}$

D. $\frac{5}{6}$

Solution

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