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20. If [tex][tex]$p(x)=x^{51}+51$[/tex][/tex], then the value of [tex]$p(-1)$[/tex] is:
a) 0
b) 51
c) 49
d) 50

21. The degree of the polynomial [tex]$p(x)=(x+2)(x-2)$[/tex] is:
a) 2
b) 1
c) 0
d) 3

22. Factorize: [tex][tex]$x^2+(a+b+c)x+ab+bc$[/tex][/tex]
a) [tex]$(x+a)(x+b+c)$[/tex]
b) [tex]$(x+a)(x+a+c)$[/tex]
c) [tex][tex]$(x+b)(x+a+c)$[/tex][/tex]
d) [tex]$(x+b)(x+b+c)$[/tex]

23. Factors of [tex]$(42-x-x^2)$[/tex] are:
a) [tex][tex]$(x-7),(x-6)$[/tex][/tex]
b) [tex]$(x+7),(x-6)$[/tex]
c) [tex]$(x+7),(6-x)$[/tex]
d) [tex][tex]$(x-7),(x+6)$[/tex][/tex]

24. Find the value of [tex]$x+y+z$[/tex], if [tex]$x^2+y^2+z^2=18$[/tex] and [tex][tex]$xy+yz+zx=9$[/tex][/tex].
a) 9
b) 3
c) 6
d) 8



Answer :

GinnyAnswer
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.

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