JIPMAT 2023 (QA) - <p>The expression 2x<sup>3</sup> + ax<sup>2</sup> + 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) =</p> | PYQs + Solutions | AfterBoards
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JIPMAT 2023

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Easy

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) =

Correct Option: 2
1) The expression: 2x3+ax2+bx+32x^3 + ax^2 + bx + 3
2) Since x−1x-1 is a factor, the expression should be equal to 00 when x=1x=1\newline - When x=1x=1: \newline2(1)3+a(1)2+b(1)+3=02(1)^3 + a(1)^2 + b(1) + 3 = 0\newline - Simplifies to: 2+a+b+3=02 + a + b + 3 = 0\newline - Therefore: a+b=−5a + b = -5 ...(1)
3) When divided by x+2x+2, remainder is 1515\newline - When x=−2x=-2: \newline2(−2)3+a(−2)2+b(−2)+3=152(-2)^3 + a(-2)^2 + b(-2) + 3 = 15\newline - −16+4a−2b+3=15-16 + 4a - 2b + 3 = 15\newline - 4a−2b=284a - 2b = 28 ...(2)\newline - 2a−b=142a - b = 14
4) We now have two equations:\newline - a+b=−5a + b = -5 ...(1)\newline - 2a−b=142a - b = 14 ...(2)
5) From (1): b=−5−ab = -5 - a\newline Substitute in (2):\newline - 2a−(−5−a)=142a - (-5-a) = 14\newline - 2a+5+a=142a + 5 + a = 14\newline - 3a=93a = 9\newline - a=3a = 3
6) Put back to find bb:\newline - b=−5−3=−8b = -5 - 3 = -8
Therefore, (a,b)=(3,−8)(a,b) = (3,-8)

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JIPMAT 2022

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 1323×5\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:

JIPMAT 2023

Amicable numbers are a pair of distinct natural numbers (a, b) such that the sum of the proper divisors of a equals b and the sum of the proper divisors of b equals a. Given that (220, y) is a pair of amicable numbers, y equals:

JIPMAT 2023

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.

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