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Math Olympiad Test: Polynomials - 3 - Class 10 MCQ


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10 Questions MCQ Test Olympiad Preparation for Class 10 - Math Olympiad Test: Polynomials - 3

Math Olympiad Test: Polynomials - 3 for Class 10 2024 is part of Olympiad Preparation for Class 10 preparation. The Math Olympiad Test: Polynomials - 3 questions and answers have been prepared according to the Class 10 exam syllabus.The Math Olympiad Test: Polynomials - 3 MCQs are made for Class 10 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Math Olympiad Test: Polynomials - 3 below.
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Math Olympiad Test: Polynomials - 3 - Question 1

If sum of all zeros of the polynomial 5x2 – (3 + k)x + 7 is zero, then zeroes of the polynomial 2x2 – 2(k + 11)x + 30 are

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 1

Sum of zeroes of polynomial
5x– (3 + k)x + 7 is 
According to the question,
((3 + k)/5) = 0 ⇒ k = –3
Now, 2x2 – 2(k + 11)x + 30 becomes 2x2 – 16x + 30.
i.e., 2x2 – 16x + 30 = 0 or x2 – 8x + 15 = 0
⇒ x = 3, 5
Hence, zeroes of polynomial 2x2 – 16x + 30 are 3, 5.

Math Olympiad Test: Polynomials - 3 - Question 2

px3 + qx2 + rx + s = 0 is said to be cubic polynomial, if _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 2

For a cubic polynomial to exist, coefficient of term x3 must not be equal to zero.

Math Olympiad Test: Polynomials - 3 - Question 3

If a and b are the roots of the quadratic equation x2 + px + 12 = 0 with the condition a – b = 1, then the value of ‘p’ is _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 3

Given equation is x2 + px + 12 = 0
Now, if a and b are its roots, then sum of roots, a + b = –p and product of roots, a × b = 12
Also, a – b = 1 (Given)
We know that, (a – b)2 = (a + b)2 – 4ab
⇒ 1 = p2 – 4 × 12 ⇒ 1 = p2 – 48 ⇒ p2 = 49 ⇒ p = ±7

Math Olympiad Test: Polynomials - 3 - Question 4

If the sum of the product of the zeroes taken two at a time of the polynomial f(x) = 2x3 – 3x2 + 4tx – 5 is –8, then the value of t is _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 4

Given polynomial is 2x3 – 3x2 + 4tx – 5
Sum of product of roots taken two at a time is 4t/2.
∴ 4t/2 = -8 ⇒ t = –4

Math Olympiad Test: Polynomials - 3 - Question 5

A cubic polynomial with sum of its zeroes, sum of the product of its zeroes taken two at a time and the product of its zeroes as –3, 8, 4 respectively, is _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 5

For a cubic polynomial, ax3 + bx2 + cx + d
Sum of zeroes = -(b/a)
Sum of the product of zeroes taken two at a time = (c/a).
Product of zeroes = -(d/a)
We have, -(b/a) = -3, (c/a) = 8 and (-d)/a = 4
∴ x3 + 3x2 + 8x – 4 is the required polynomial.

Math Olympiad Test: Polynomials - 3 - Question 6

What will be the value of p(3), if 3 is one of zeroes of polynomial p(x) = x3 + bx + D?

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 6

Since, 3 is one of the zeroes of polynomial p(x).
So, p(3) = 0

Math Olympiad Test: Polynomials - 3 - Question 7

When x3 – 3x2 + 3x + 5 is divided by x2 – x + 1, the quotient and remainder are _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 7

This can be done either by long division as shown in the image
We can see the division gives us a quotient (x − 2) and a remainder 3

Math Olympiad Test: Polynomials - 3 - Question 8

If p, q are the zeroes of the polynomial f(x) = x2 + k(x – 1) – c, then (p – 1)(q – 1) is equal to _______.

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 8

Given equation is x2 + k(x – 1) – c
= x2 + kx – (k + c)
Since, p and q are the zeroes,
∴ = – k and pq = – (k + c)
Now, (p – 1) (q – 1) = pq – q – p + 1
= pq – (p + q) + 1 = – (k + c) – (–k) + 1
= – k – c + k + 1 = 1 – c

Math Olympiad Test: Polynomials - 3 - Question 9

Which of the following graph has more than three distinct real roots?

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 9

For more than three distinct real roots the graph must cut x-axis at least four times. So, graph in option (C) has more than three distinct real roots.

Math Olympiad Test: Polynomials - 3 - Question 10

What should be subtracted from f (x) = 6x3 + 11x2 – 39x – 65 so that f(x) is exactly divisible by x2 + x – 1?

Detailed Solution for Math Olympiad Test: Polynomials - 3 - Question 10

By long division method, we have

We must subtract the remainder so that f(x) is exactly divisible by x2 + x – 1
Hence, –38x – 60 is to be subtracted.

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