NCTB Class 8 Math Chapter Four Exercise 4.1 Solution by Math Expert. Bangladesh Board Class 8 Math Solution Chapter 4 Algebraic formulae and applications Exercise 4.1 Solution.
Board |
NCTB |
Class |
8 |
Subject |
Mathematics |
Chapter |
4 |
Chapter Name |
Algebraic formulae and applications |
Exercise |
4.1 Solution |
Exercise 4.1
1> Find the Square of the following expression with the help of formulae:
(a) (5a+7b)
= (5a)2 + 2.5a.7b + (7b)2
= 25a2 + 70ab + 49b2
(b) (6x + 3 )2
= 36x2 + 2.6x.3 + 9
= 36x2 + 36x +9
(c) (7p – 2q)2
= (7p)2 -2.7p.2q + (2q)2
= 49p2 – 28pq + 4q2
(d) (ax – by )2
= (ax)2 – 2.ax.by + b2y2
= a2 x2 -2axby + b2 y2
(e) (X3+ xy)2
= (x3)2 + 2.x3 .xy + (xy)2
= X6 + 2x4y + x2y2
(f) (11a-12b)2
= (11a)2 – 2.11a.12b + (12b)2
= 121a2 – 243ab + 144b2
(g) 6x2 y – 5xy2
= 2xy ( 3x – 5y )2
= 2xy {(3x)2– 2.3x.5y+(5y)2}
= 2xy (9x2 – 30xy + 25y2)
(h) (-X-Y)2
= {-(X+Y)}2
= – (x2 + 2xy + y2)
= -x2 – 2xy – y2
(j) a2x3 – b2y4
= (a2x3 – b2y4)2
= (a2x3)2 – 2.(a2x3).(b2y4) + (b2y4)
= a4x6 – 2a2b2x3y4 + b4y6
(k) (108)2
= (100+8)2
= (100)2 + 2.100.8 + (8)2
= 10000 + 1600 + 64
= 11664
(L) (606)2
= (600+6)2
= (600)2 + 2.600.6 + (6)2
= 360000 + 7200 + 36
= 367236
(m) (597)2
= (600 – 3)2
= (600)2 – 2. 600. 3 + 32
= 360000 – 3600 + 9
= 356409
(n) (a-b+c)2
= (a+ (-b)+ c)2
= a2 + (-b)2 + c2 + 2a (-b) + 2 (b) c + 2ac
= a2 + b2 + c2 – 2ab – 2bc + 2ac
= a2 + b2 + c2 – 2ab – 2bc + 2ac
(o) (ax+b+2)2
= (ax)2 + (b)2 + (2)2 + 2.ax.b + 2b.2 + 2.ax.2
= a2x2 + b2 + 4 + 2abx + 4b + 4ax
(p) (xy + yz – zx)2
= (xy)2 + (yz)2 + (-zx)2 + 2.xy.yz + 2yz.xy + 2 (-zx).xy
= x2y2 + y2z2 + z2x2 + 2xy2z + 2xy2z – 2x2yz
(q) ( 3p+ 2q-5r)2
= (3p)2 + (2q)2 + (-5r)2 + 2.3q.2q + 2. 2q. (-5r) + 2. (-5r). 3p
= 9p2 + 4q2 + 25r2 + 12pq – 20qr – 30pr
(s) 7a2 + 8b2 + 5c2
= (7a2)2 + (8b2) + (-5c2)2 + 2.7a2.8b2 + 2. 8b2. (-5c2) + 2. (-5c2) . (7a)
= 49a4 + 64b4 + 25c4 + 112a2b2 – 80b2c2 – 70a2c2
2> Simplify :
(a) (X+Y)2 + 2(X+Y)(X-Y)+(X+Y)2
= (X+Y+X-Y)2
=(2X)2
=4X2
(b) (2a+3b)2 – 2 (2a + 3b)+ (3b-a)+(3b-a)2
= (2a+3b-3b+a)2
= (3a)2
= 9a2
(c) (3x2 + 7y2)2 + 2 (3x2 + 7y2) (3x2 – 7y2) + (3x2 + 7y2)2
= (3x2+7y2+3x2-7y2)
=(6x2)2
=36x4
(d) (8x + y)2 – (16x+2y) (5x+y) + (5x+y)2
= (8x + y)2 – 2 (8x+y) (5x + y ) + (5x + y)2
= (8x + y – 5x – y )2
= (3x)2
= 9x2
3> Find the product by applying formulae:
(a) ( x+7) (x-7)
= x2 – 72
= x2 – 49
(b) (5x + 13) (5x – 13)
= (5x)2 – (13)2
= 25x2 – 169
(c) (xy + yz) (xy – yz )
= (xy)2 – (yz)2
= x2y2 – y2z2
(d) (ax + b) (ax – b)
= (ax)2 – (b)2
= a2x2 – b2
(e) (a + 3) (a- 4)
= a2 – 4a + 3a – 12
= a2 – a – 12
(f) (ax + 3 ) (ax + 4)
= a2x2 + 4ax + 3ax + 12
= a2x2 + 7ax + 12
(g) (6x + 17 ) (6x – 13)
= 36x2 – 78x + 102x – 221
= 36x2 + 24x -221
(h) (a2 + b2) (a2 – b2) (a4 + b4)
= (a2)2 – (b2)2 (a4+ b4)
= (a4 – b4) (a4– b4)
= (a4)2 – (b4)2
= a8 – b8
(i) (ax + by + cz) (ax + by – cz)
= (ax +by)2 – (cz)2
= a2x2 + 2.ax.by + b2y2 – c2z2
= a2x2 + 2abxy + b2y2 – c2z2
(j) (3a – 10) (3a – 5)
= 9a2 – 15 – 30a + 50
= 9a2 – 30a + 35
(k) (5a + 2b – 3c) (5a + 2b + 3c)
= (5a+ 2b)2 – (3c)2
= (5a)2 + 2.5a.2b + (2b)2 – 9c2
= 25a2 +20ab + 4b2 – 9c2
(l) (ax + by + 5) (ax + by + 3)
= a2x2 +axby+ 3ax+ abxy + b2y2 + 3by+ 5ax+5by + 15
= a2x2 + 2axby+ 8ax+b2y2 +8by+ 15
4> If a=4, b=6 and c=3, find the 4a2b2 – 16ab2c + 16b2c2
= (2ab)2 – 2.2ab.4bc + (4bc)2
= (2ab + 4bc)2
= (2x4x6+4x6x3)2
= (48 + 72)2
= (120)2
= 14400
5> x- 1/x = 3, find the value of x2+1/x2
= (x-1/x)2 + 2.x.1/x
= (3)2 + 2
= 9+2
=11
6> If a+1/a=4, what is the value of a4 + 1/a4 ?
Or, (a+1/a)2 = 42
Or, a2 + 2.a.1/a + (1/a)2 = 16
Or, a2 + 1/a2 = 16-2
= 14
Therefore, a4 + 1/a4
= (a2)2 + (1/a2)2
= (a2 + 1/a2) – 2.a2.1/a2
= (14)2 – 2
= 196-2
= 194
8> If a-1/a =m, show that a4+ 1/a4 =m4+4m2 +2
Or (a-1/a)2 = m2
Or, a2-2.a.1/a2 = m2
Or, a2 + 1/a2 = m2+2
Or, (a2+1/a2)2 = (m2 + 2)2
Or, (a2)2 + 2.a2.1/a2 + 1/a4 = m4 + 2.m2.2 + 4
Or, a4 + 2+ 1/a4 = m4 + 4m2 + 4
Or, a4 + 2 + 1/a4 = m4 + 4m2+4
Or, a4+ 1/a4 = m4 + 4m2 + 4-2
= m4+4m2 +2 (Proved)
9> If x-1/x=4, Prove that x2+(1/x)2 =18
x- 1/x =4 (given)
or, (x – 1/x)2 = (4)2
or, x2 – 2.x.1/x + 1/x2 =16
or, x2 + 1/x2 = 16+2
= 18 (proved)
10> If m+1/m = 2, prove that m4 + 1/m4 =2
M+1/m=2 (Given)
Or, (m+1/m)2 =4
Or, m2+2.m.1/m+1/m2 =4
Or, m2 + 1/m2 = 4-2
=2
Or, ( m2+1/m2)2 =4
Or, m4 + 2.m2.1/m2+1/m4 =4
Or, m4 + 1/m4 = 4-2
=2 (proved)
11> If x+y=12 and xy=27, find the value of (x-y)2 and x2 + y2
= x2+y2 – 2.x.y
= 90 – 2.27
= 90 – 54
= 36
X2 + y2
= (x+y)2 + 2xy
= (12)2 – 2.27
= 144- 54
=90
12> If a+b=13 and a-b=3, find the Value of 2a2 +2b2 and ab.
2a2 + 2b2
= 2(a2 + b2)
= 2{(a+b)2 – 2.ab}
= 2 {(13)2 – 2.ab
a-b=3
or, (a-b)2 =9
or, a2 – 2ab + b2 =9
or, (a+b)2 + 2ab =9
13> Express the difference between of the square of two expression
(a) (5p – 3q)2 – (p+7q)2
= (5q-3q+p+7q) (5p-3q-p-7q)
= (6p+4q) (4p-10q)
= 24p2+60pq + 16pq – 40q2
= 24p2 – 44pq – 40q2
e
7 no math koi
Nai
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Write, and where’s 12 er baki ongsho
13 er baki gula koi
7No. Math na dile kmne bujhbo?
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14 no question er ans dily kub upakar hoto
4 number ta wrong .. (a-b)^2 er rule hbe but ase (a-b)^2 er