IMO Practice Test: Polynomials | Class 10 Mathematics

Question 1 of 6 medium

If α and β are zeros of $x^{2} - 3x - 2$, find the value of 1/α + 1/β.

3/2

−3/2

2/3

−2/3

Explanation: 1/α+1/β = (α+β)/αβ = (3)/(−2) = −3/2

Question 2 of 6 medium

The zeros of p(x) = $x^{2} - 2x - 8$ differ by:

2

4

6

8

Explanation: Zeros: x=4,−2; difference=4−(−2)=6

Question 3 of 6 medium

Find k such that the sum of zeros of $x^{2}$ − (k+1)x + (2k−3) is equal to their product.

2

3

4

5

Explanation: For $x^{2} - (k+1)x + (2k-3)$, the sum of zeros is $k+1$ and the product is $2k-3$. Setting sum equal to product: $k+1 = 2k-3 \Rightarrow k = 4$. Check: at $k=4$, both the sum and the product equal $5$.

Question 4 of 6 medium

If the zeros of $x^{2} + 5x + k$ are reciprocals of each other, find k.

1

−1

5

−5

Explanation: Product αβ = c/a = k/1 = k. For reciprocals, αβ = 1 → k=1

Question 5 of 6 medium

Find the quadratic whose zeros are 1/α and 1/β, where α,β are zeros of $2x^{2} - 5x + 2$.

$2x^{2} - 5x + 2$

$x^{2} - 5x + 4$

$2x^{2} + 5x + 2$

$2x^{2} - 5x - 2$

Explanation: From $2x^{2}-5x+2$, α+β=5/2, αβ=1. New sum=1/α+1/β = (α+β)/αβ = (5/2)/1=5/2. New product = 1/(αβ)=1. Polynomial: $x^{2}$−(5/2)x+1 multiply by 2 → $2x^{2}-5x+2$ (same as original!)

Question 6 of 6 medium

If α,β are zeros of $x^{2} - px + q$, find the value of α$^{2}$ + β$^{2}$.

$p^{2} - 2q$

$p^{2} + 2q$

$q^{2} - 2p$

$q^{2} + 2p$

Explanation: α$^{2}$+β$^{2}$ = (α+β)$^{2} - 2$αβ = $p^{2} - 2q$

IMO Practice Test — Polynomials