Sure, I'd be happy to help you with these math problems step-by-step.
### 20. If [tex]\( p(x) = x^{51} + 51 \)[/tex], then the value of [tex]\( p(-1) \)[/tex] is:
We need to find [tex]\( p(-1) \)[/tex]:
[tex]\[ p(-1) = (-1)^{51} + 51 \][/tex]
Since any negative number raised to an odd power remains negative, [tex]\( (-1)^{51} = -1 \)[/tex]. Thus, we have:
[tex]\[ p(-1) = -1 + 51 \][/tex]
[tex]\[ p(-1) = 50 \][/tex]
So, the correct answer is (d) 50.
### 21. Degree of the polynomial [tex]\( p(x) = (x+2)(x-2) \)[/tex] is:
First, simplify the polynomial:
[tex]\[ p(x) = (x+2)(x-2) \][/tex]
[tex]\[ p(x) = x^2 - 4 \][/tex]
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the expression. Here, [tex]\( x^2 - 4 \)[/tex] has the highest power of [tex]\( 2 \)[/tex].
So, the correct answer is (a) 2.
### 22. Factorise: [tex]\( x^2 + (a+b+c) x + ab + bc \)[/tex]
Let’s check the option that correctly factorises this expression.
The correct factorisation is:
[tex]\[ (x + a)(x + b + c) \][/tex]
So, the correct answer is (a).
### 23. Factors of [tex]\( 42 - x - x^2 \)[/tex]
First, let's rewrite the polynomial in standard form:
[tex]\[ 42 - x - x^2 = -x^2 - x + 42 \][/tex]
To factorise, we look for two numbers that multiply to [tex]\(-42\)[/tex] (since [tex]\(-x^2\)[/tex] means leading coefficient is actually -1) and add up to [tex]\(-1\)[/tex]. These numbers are [tex]\(-7\)[/tex] and [tex]\(6\)[/tex].
So, the factors are:
[tex]\[ (x + 7), (x - 6) \][/tex]
So, the correct answer is (b) [tex]\( (x + 7), (x - 6) \)[/tex].
### 24. Find the value of [tex]\( x + y + z \)[/tex] if [tex]\( x^2 + y^2 + z^2 = 18 \)[/tex] and [tex]\( xy + yz + zx = 9 \)[/tex]
Using the identity:
[tex]\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx) \][/tex]
Substituting the given values:
[tex]\[ (x + y + z)^2 = 18 + 2 \times 9 \][/tex]
[tex]\[ (x + y + z)^2 = 18 + 18 \][/tex]
[tex]\[ (x + y + z)^2 = 36 \][/tex]
Taking the square root of both sides:
[tex]\[ x + y + z = \sqrt{36} \][/tex]
[tex]\[ x + y + z = 6 \][/tex]
So, the correct answer is (c) 6.
