1) Add (2√3 +5√5 - 7√7) and (3√5 - √3 + √7)
2) Multiply: 7√6 by 5√24.
3) Simplify:
a) (√2+1)(√2+3).
b) (2√5 + 2√3)².
4) Rationalise the denominator:
a) √2/(√5+1).
b) 10/(7- 2√3)
c) 5/(4√3 - 3√2).
d) (√7 - √6)/(√7 + √6).
e) (7√3 - 5√2)/(√48 + √18)
f) 3/(√3 - √2 + √5).
5) Find a, b from the following:
a) (5+ 2√3)/(7+ 4√3)= a+ b√3.
b) (7+ √5)/(7- √5) - (7- √5)/(7+ √5) = a+ 7√5 b
6) Simplify:
a) 2√6/(√2 + √3) + 6√2/(√6 + √3).
b) 7√3/√10 + √3) - 2√5/(√6 + √5) - 3√2/(√15+ 3√2).
7) If a= 9- 4√5, find the value of √a - 1/√a.
8) If x= 3+ 2√2, find the value of x²+ 1/x²
9) If x= (√3+1)/(√3- 1) and y= (√3-1)/(√3+ 1), find the value of x²- xy + y²
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1) Find the product of
a) (2y -3)(2y -5)
b) (x -1)(x +1)(x²+1)(x⁴+1).
c) (x²+ x -1)(x²- x +1).
2) Evaluate with formula:
a) 108 x 108.
b) 105 x 98.
3) Simplify with formula:
a) 212 x 212 - 2 x 12 x 212 + 12 x 12..
b) {231 x 231 - (180 x 180)}/411
4) a) If x+ 1/x = 3, find x²+ 1/x².
b) If x² + 1/x² = 83, find x - 1/x.
c) If 4x²+ 9y²= 136 and xy= 10, find 2x + 3y.
d) If x¹ + 1/x² = 38, find (i) x + 1/x (ii) x - 1/x.
e) If 2x - √7 y =10 and xy= - √7, find 4x²+ 7y².
5) Write in expanded form:
a) (-2x + 5y - 3z)².
b) (3a/2 + b/4 - 2c)².
6) a) If x+ y+ z =6 and xy+ yz+ zx= 11. Find x²+ y²+ z².
b) If x² + y² + z² =24 and xy+ yz+ zx= 30. Find x + y + z.
7) Evaluate: 4x²+ y²+ 9z²- 4xy + 6yz - 12zx using the identity if x=3, y=1 and z= 2.
8) Write in expanded form:
a) (3x +1)³.
b) (x - 3y/5)³.
9) Evaluate:
a) (999)³.
b) (1.1)³.
10) a) If x + 1/x = 3, find x³+ 1/x³.
b) If x - 1/x = 4, find x³ - 1/x³.
c) If x² + 4/x² = 5, find x³+ 8/x³.
11) Find the following products:
a) (7y - 5z)(49y²+ 35yz + 25z²).
b) (0.4x + 0.5y)(0.16x² - 0.2xy + 0.25y²)
c) x(y + 9x)(y²- 9xy + 81x²).
12) Evaluate with formula:
a) (104)³ + (96)³.
b) (136x 136 x 136 + 64 x 64x 64)/(136x 136 - 136 x 64 + 64 x 64).
13) a) If x + y= 8 and xy= 6, find the value of x³+ y³.
b) If x - y= 10/9 and xy= 5/3, find the value of x³- y³.
c) If x²+ 1/x²= 18, find the value of x³ - 1/x³.
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Factorise the following:
1) 12x²+ 16xy.
2) 9a²(b - 2c)+ 6(b - 2c)².
3) 2xy - 3ab + 2bx - 3ay.
4) a²b² - (ab² - 5)b - 5a.
5) ax - bx + cy + by - cx - at.
6) x(2a + b)²+ 4ay + 2by + 8a + 4b
7) x²+ 1/x² - 2 - 3x + 3/x.
8) 81x²- 25y².
9) (2a + 3b)² - 9c²
10) 16x² - 1/x²
11) 3xy - 243xy⁵
12) x⁸ - y⁸.
13) 9 - 2xy - (x²+ y²)
14) (1- a²)(1- b²) + 4ab.
15) a⁴+ a²+ 1
16) 9a⁴+ 4b⁴ - 13a²b²
17) x²+ 14x + 48.
18) x²- 30x +216.
19) y²- 32y - 105.
20) 6x² + 5x - 6.
21) x²+ x/6 - 1/6.
22) 2x² - x + 1/8.
23) 12(x -2)² - 25(x -2)(y+1) + 12(y+ 1)²
24) 3(6x²+ 5x)² - 10(x²+ 5x) - 8.
25) x⁴+ 3x² - 28.
26) 4x⁴- 5x² +1.
27) a³+ 8.
28) 27y³+ 125z³.
29) x⁶/125 + 125/x⁶.
30) 2x⁸+ 54y⁸.
31) x³+ y³+ x + y
32) x⁶+ y⁶.
33) 8x³- 343y³
34) x³ - 27y³/8.
35) 5√5 a³ - 2√2 b³.
36) 3a⁷b - 24a⁴b⁴.
37) a³ - 8b³ + 2ax - 4bx.
38) x¹² - y¹².
39) x³- 1/x³ - 2x + 2/x.
_________________:::_______________________
) Evaluate:
a) (64)¹⁾³. 4
b) (8)⁵⁾³. 32
c) (64/25)⁻³⁾². 125/512
2) Simplify:
a) (81)³⁾⁴ - (32)⁻²⁾⁵ + (8)²⁾³ x (1/2)⁻¹ x 3⁰ - (1/81)⁻¹⁾². 22
b)(64/125)⁻²⁾³ + 4⁰ x 9⁵⁾² x 3⁻⁴ - √25/³√64 x (1/3)⁻¹. 13/16
3) Evaluate:
a) √36. 6
b) ³√125. 5
4) If √2= 1.414 and √3= 1.732, find the value of
a) √72. 8.484
b) √147 + √27. 17.32
c) 5 √125 + 2 √75 - 5 √108. 21.92
5) If 2160= 2ᵃ x 3ᵇ x 5ᶜ, find the values of a, b, c. Hence, find the value of 3ᵃ x 2⁻ᵇ x 5⁻ᶜ. 81/40
6) Simplify: (5ⁿ⁺³ - 6 x 5ⁿ⁺¹)/(9 x 5ⁿ - 2² x 5ⁿ). 19
7) Simplify:
a) (xᵇ/xᶜ)ᵃ . (xᶜ/xᵃ)ᵇ . (xᵃ/xᵇ)ᶜ. 1
b) (xᵃ/xᵇ)^(a²+ ab+ b²) . (xᵇ/xᶜ)^ (b²+ bc+ c²). (xᶜ/xᵃ)^(c²+ ca+ a²). 1
8) If aˣ = bʸ = cᶻ and b²= ac, show that y= 2xz/(z + x).
9) Solve:
a) 3³ˣ= 1/9 . -2/3
b) 2³(6⁰ + 3²ˣ)= 224/27. -3/2
c) √(3/5)¹⁻²ˣ = 125/27. 3.5
d) √(5⁰+ 2/3)= (0.6)²⁻³ˣ. 5/6
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1) x + 2y +1=0; 2x - 3y -12=0. 2) 3x + 2y +24= 0, 2x + y + 10= 0.
3) 2x + y = 35, 3x + 4y - 65= 0.
4) 2x - y = 6, x - y = 2.
5) (x + y)/xy =2; (x - y)/xy =6.
6) ax + by = a - b, bx - ay = a+ b.
7) x + ay = b, ax - by = c.
8) ax + by = a², bx - ay= b².
9) x/a + y/b = 2, ax - by = a² - b².
10) x/a + y/b = a+ b, x/a² + y/b² = 2.
11) x/a = y/b, ax + by = a² + b².
12) 5/(x + y) - 2/(x - y)= -1; 15/(x + y) + 7/(x - y)= 10.
13) 2/x + 3/y = 13; 5/x - 4/y = -2. Where x≠0 and y=0.
14) ax + by = (a + b)/2; 3x + 5y = 4.
15) 2ax + 3by = a + 2b, 3ax + 2by = 2a+ b.
16) 5ax + 6by = 28, 3ax+ 4by = 18.
17) (a + 2b)x + (2a - b)y = 2, (a - 2b)x + (2a+ b)y = 3.
18) x (a - b + ab/(a - b)) = y(a + b - ab/(a+ b); x + y= 2a².
19) bx + cy = a+ b; ax{1/(a - b) - 1/(a - b)} + cy{1/(b - a) - 1/(b + a)}= 2a/(a + b).
20) (a - b)x + (a + b)y = 2a² - 2b²,
(a +b)(x+ y) = 4ab.
21) a²x + b²y = c² , b²x + a²y = d².
22) 57/(x + y) + 6/(x - y)= 5
38/(x + y) + 21/(x - y)= 9.
23) 2(ax - by) + a + 4b =0;
2(bx + ay) + b - 4a = 0.
24) 6(ax + by) = 3a + 2b; + 6(bx - ay)= 3b - 2a.
EXERCISE - B
1) In each of the following systems equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is unique solution, find it.
a) 5x + 2y = 16, 7x - 4y = 2. 2,3
b) 5x + 2y = 16, 3x +6y/5 = 2. No solution
c) 4x + 7y = 10, 10x +35y/2 = 25. Infinitely many solutions
d) 5x + 2y = 16, 15x/2 +3y = 24. Infinitely many solutions
e) 3x - 5y = 20, 7x +2y = 17. 125/41,-89/41
f) -3x +4y = 5, 9x/2 - 6y = 15/2. No solution
g) x - 3y = 3, 3x - 9y = 2. No solution
h) 2x + y = 5, 4x + 2y = 10. Infinitely many solutions
i) 3x - 5y = 20, 6x - 10y = 40. Infinitely many solutions
j) x - 2y = 8, 5x - 10y = 10. No solution
2) Find the value of k for which the following system of equations has a unique solution:
a) kx + 2y = 5, 3x + y = 1. k≠ 6
b) x + 2y = 5, 3x + y = 1. k≠ -6
c) 4x - 5y = k, 2x - 3y = 12. k is any real number
d) x + 2y = 3, 5x + ky +7 = 0. k≠ 10
3) Find the value of k for which each of the following system of equations have infinitely many solution:
a) 2x + 3y = 5, 6x + ky - 15= 0. 9
b) 4x + 5y = 3, kx + 15y = 9. 12
c) kx - 2y +6= 0, 4x -3y+9 = 0.; 8/3
d) 8x + 5y = 9, kx + 10y = 18. 16
e) 2x - 3y = 7, (k +2)x - (2k +1) y = 3(2k -1). 4
f) 2x + 3y = 2, (k +2)x + (2k +1) y = 2(2k -1). 4
g) x + (k+ 1)y = 4, (k +1)x + 9y = 5k +2. 2
h) kx + 3y = 2k +1, 2(k +1)x + 9y = 7k + 1. 2
i) 2x + (k -2)y = k, 6x + (2k -1) y = 2k + 5. 5
j) 2x + 3y = 7, (k +1)x - (2k -1) y = 4k +1. 5
k) 2x + 3y = k, (k -)x + (k +2) y = 3k. 7
4) Find the value of k for which the following system of equations has no solution:
a) kx - 5y = 2, 6x + 2y = 7. -15
b) x + 2y = 0, 1x + ky = 5. 4
c) 3x - 4y +7= 0, kx + 3y - 5= 0. -9/4
d) 2x - ky + 3 = 0, 3x + 2y -1= 0. -4/3
e) 2x + ky = 11, 5x - 7y = 5. -14/5
f) cx + 3y = 3, 12x + cy = 6. -6
5) For what values of k the following system of equations will be inconsistent?
4x + 6y = 11, 2x + ky = 7. 3
6) For what values of k the system of equations will have no solution?
kx + 3y = k - 3, 12x + ky = k. -6
7) Find the value of k for which the system has
a) unique solution .
b) no solution.
kx + 2y = 5, 3x + y = 1. k≠ 6, k=6
8) Prove that there is a value of c(≠ 0) for which the system 6x + 3y = c -3; 12x + cy = c has infinitely many solutions . Find the value . 6
9) Find the values of k for which the system
2x + ky = 1, 3x - 5y = 7 will have
a) a unique solution
b) no solution
Is there a value of k for which the system has infinitely many solutions? -10/3, no
10) For what values of k the following system of equations will represent the coincident lines?
x + ²y +7= 0 , 2x + ky +14 = 0 4
11) Obtain the condition for the following system of linear equations to have a unique solution.
ax + by = c, lx + my = n. am ≠ bl
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