4 Mark Question

Question 14 Marks

If $P=a^2-b^2+2 a b, Q=a^2+4 b^2-6 a b, R=b^2+b, S=a^2-4 a b$ and $T=-2 a^2+b^2-a b+a$. Find $P+Q+R+S-T$.

Answer

We have,
$P+Q+R+S-T=\left[\left(a^2-b^2+2 a b\right)+\left(a^2+4 b^2-6 a b\right)+\left(b^2+b+\left(a^2-4 a b\right)\right]-\left(-2 a^2+b^2-a b+a\right)\right.$
$=\left[a^2-b^2+2 a b+a^2+4 b^2-6 a b+b^2+b+a^2-4 a b\right]-\left(-2 a^2+b^2-a b+a\right)$
$=\left[3 a^2+4 b^2-8 a b+b\right]-\left(-2 a^2+b^2-a b+a\right)$
$=3 a^2+4 b^2-8 a b+b+2 a^2-b^2+a b-a$
Collecting positive and negative like terms together, we get
$3 a^2+2 a^2+4 b^2-b^2-8 a b+a b-a+b$
$=5 a^2+3 b^2-7 a b-a+b$

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Question 24 Marks

How much does $a^2-3 a b+2 b^2$ exceed $2 a^2-7 a b+9 b^2$ ?

Answer

Required expression:
$=\left(a^2-3 a b+2 b^2\right)-\left(2 a^2-7 a b+9 b^2\right)$
$=a^2-3 a b+2 b^2-2 a^2+7 a b-9 b^2$
Collecting positive and negative like terms together, we get
$=a^2-2 a^2-3 a b+7 a b+2 b^2-9 b^2$
$=-a^2+4 a b-7 b^2$

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Question 34 Marks

From the sum of $3 x^2-5 x+2$ and $-5 x^2-8 x+9$ subtract $4 x^2-7 x+9$

Answer

Required expression:
$=\left[\left(3 x^2-5 x+2\right)+\left(-5 x^2-8 x+9\right)\right]-\left(4 x^2-7 x+9\right)$
$=\left[3 x^2-5 x+2-5 x^2-8 x+9\right]-\left(4 x^2-7 x+9\right)$
$=\left[3 x^2-5 x^2-5 x-8 x+2+9\right]-\left(4 x^2-7 x+9\right)$
$=\left[-2 x^2-13 x+11\right]-\left(4 x^2-7 x+9\right)$
$=-2 x^2-13 x+11-4 x^2+7 x-9$
$=-2 x^2-4 x^2-13 x+7 x+11-9$
$=-6 x^2-6 x+2$

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Question 44 Marks

What must be added to $12 x^3-4 x^2+3 x-7$ to make the sum $x^3+2 x^2-3 x+2$ ?

Answer

Let ' $M$ ' be the required expression:
Thus, we have
$12 x^3-4 x^2+3 x-7+M$
$=x^3+2 x^2-3 x+2$
Therefore,
$M=\left(x^3+2 x^2-3 x+2\right)-\left(12 x^3-4 x^2+3 x-7\right)$
$=x^3+2 x^2-3 x+2-12 x^3+4 x^2-3 x+7$
Collecting positive and negative like terms together, we get
$x^3-12 x^3+2 x^2+4 x^2-3 x-3 x+2+7$
$=-11 x^3+6 x^2-6 x+9$

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Question 54 Marks

If $P=7 x^2+5 x y-9 y^2, Q=4 y^2-3 x^2-6 x y$ and $R=-4 x^2+x y+5 y^2$, show that $P+Q+R=0$.

Answer

We have,
$P+Q+R=\left(7 x^2+5 x y-9 y^2\right)+\left(4 y^2-3 x^2-6 x y\right)+\left(-4 x^2+x y+5 y^2\right)$
$=7 x^2+5 x y-9 y^2+4 y^2-3 x^2-6 x y-4 x^2+x y+5 y^2$
Collecting positive and negative like terms together, we get
$7 x^2-3 x^2-4 x^2+5 x y-6 x y+x y-9 y^2+4 y^2+5 y^2$
$=7 x^2-7 x^2+6 x y-6 x y-9 y^2+9 y^2$
$=0$

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Question 64 Marks

Subtract the sum of 13x - 4y + 7z and -6z + 6x + 3y from the sum of 6x - 4y - 4z and 2x + 4y - 7.

Answer

Sum of (13x - 4y + 7z) and (-6z + 6x + 3y)
= (13x - 4y + 7z) + (-6z + 6x + 3y)
= (13x - 4y + 7z - 6z + 6x + 3y)
= (13x + 6x - 4y + 3y + 7z - 6z)
= (19x - y + z)
Sum of (6x - 4y - 4z) and (2x + 4y - 7)
= (6x - 4y - 4z) + (2x + 4y - 7)
= (6x - 4y - 4z + 2x + 4y - 7)
= (6x + 2x - 4z - 7)
= (8x - 4z - 7)
Now, required expression:
= (8x - 4z - 7) - (19x - y + z)
= 8x - 4z - 7 - 19x + y - z
= 8x - 19x + y - 4z - z - 7
= -11x + y - 5z - 7

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Question 74 Marks

From the sum of $x^2+3 y^2-6 x y, 2 x^2-y^2+8 x y, y^2+8$ and $x^2-3 x y$ subtract $-3 x^2+4 y^2-x y+x-y+3$

Answer

Sum of $\left(x^2+3 y^2-6 x y\right),\left(2 x^2-y^2+8 x y\right),\left(y^2+8\right) \text { and }\left(x^2-3 x y\right)$
$=\left[\left(x^2+3 y^2-6 x y\right)+\left(2 x^2-y^2+8 x y\right)+\left(y^2+8\right)+\left(x^2-3 x y\right)\right]$
$=\left[x^2+3 y^2-6 x y+2 x^2-y^2+8 x y+y^2+8+x^2-3 x y\right]$
$=\left[x^2+2 x^2+x^2+3 y^2-y^2+y^2-6 x y+8 x y-3 x y+8\right]$
$=4 x^2+3 y^2-x y+8$
Now, required expression:
$=\left(4 x^2+3 y^2-x y+8\right)-\left(-3 x^2+4 y^2-x y+x-y+3\right) \\
=4 x^2+3 y^2-x y+8+3 x^2-4 y^2+x y-x+y-3 \\
=4 x^2+3 x^2+3 y^2-4 y^2-x+y-3+8 \\
=7 x^2-y^2-x+y+5$

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Maths 4 Mark Question

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