Please finish these sums I want to submit tomorrow
Answers
Step-by-step explanation:
1. Find the remainder when 2x² - 5x - 3 is divided by
x - 3
Solution : p(x) = 2x² - 5x - 3 & g(x) = x - 3
g(x) = x - 3
x - 3 = 0
x = 3
putting the value of x in p(x)
= 2x² - 5x - 3
= 2(3)² - 5(3) - 3
= 18 - 15 - 3
= 18 - 18
= 0
remainder = 0
2. Find the value of k, if (x - 1) is a factor of
p(x) = 4x² + 3x² - 4x + k
Solution : p(x) = (4x² + 3x² - 4x + k) & g(x) = (x - 1)
g(x) = (x - 1)
x - 1 = 0
x = 1
p(x) is a factor of g(x) [given]
p(x) = 0
(4x² + 3x² - 4x + k) = 0
putting the value of x in p(x)
4(1)² + 3(1)² - 4(1) + k = 0
4 + 3 - 4 + k = 0
3 + k = 0
k = -3
3. If x² + Kx + 6 = (x + 2)(x + 3)
x² + Kx + 6 = x² + 3x + 2x + 6
x² + Kx + 6 = x² + 5x + 6
k = 5
4. p(t) = t² - t + 2 , find p(1/3)
= (1/3)² - 1/3 + 2
= 1/9 - 1/3 + 2
= 1/9 - 3/9 + 18/9
= (1 - 3 + 18)/9
= 16/9
5. determine whether indicated numbers are zeroes of the polynomial f(x) = x³ - 6x² + 11x - 6 ; x = 1, 3.
Solution :
putting value of x = 1 in f(x)
x³ - 6x² + 11x - 6
= (1)³ - 6(1)² + 11(1) - 6
= 1 - 6 + 11 - 6
= 0
1 is the zeroes of the polynomial
putting value of x = 3 in f(x)
x³ - 6x² + 11x - 6
= (3)³ - 6(3)² + 11(3) - 6
= 27 - 54 + 33 - 6
= 60 - 60
= 0
3 is also the zeroes of the polynomial
6. factorise x² - (y/100)²
Identities a² - b² = (a + b)(a - b)
x² - (y/100)²
= (x + y/10)(x - y/10)
7. (0.75 × 0.75 × 0.75 + 0.25 × 0.25 × 0.25)/(0.75 × 0.75 - 0.75 × 0.25 + 0.25 × 0.25)
= {(0.75)³ + (0.25)³}/(0.75)² - 0.75 × 0.25 + (0.25)²}
[(a + b)(a² – ab + b²) = a³ + b³]
= [(0.75 + 0.25){(0.75)² - 0.75 × 0.25 + (0.25)²]/(0.75)² - 0.75 × 0.25 + (0.25)²
= (0.75 + 0.25)
= 1.00
= 1
L.H.S = R.H.S.
hence proved
8. If x + 1/x = 3, find x² + 1/x²
Solution :
Squaring both sides
(x + 1/x)² = (3)²
x² + 1/x² + 2(x)(1/x) = 9
x² + 1/x² + 2 = 9
x² + 1/x² = 9 - 2
x² + 1/x² = 7
9. If -1 is a zero of the polynomial p(x) = ax³ - x² + x + 4,
find the value of a.
ax³ - x² + x + 4 = 0
putting value of x = (-1)
a(-1)³ - (-1)² + (-1) + 4 = 0
-1a - 1 - 1 + 4 = 0
-1a + 2 = 0
-1a = -2
a = (-2)/(-1)
a = 2
10. factorise 7√2x² - 10x - 4√2
= 7√2x² - 14x + 4x - 4√2
= 7√2x(x - √2) + 4(x - √2)
= (7√2x + 4)(x - √2)
11. x² + 3√3x + 6
= x² + 2√3x + √3x + 6
= x(x + 2√3) + √3(x + 2√3)
= (x + √3)(x + 2√3)
12. expand (x - 2y - 3z)²
[(a - b - c)² = a² + b² + c² - 2ab + 2bc - 2ac]
= (x - 2y - 3z)²
= x² + 4y² + 9z² -2(x)(2y) + 2(2y)(3z) - 2(x)(3z)
= x² + 4y² + 9z² - 4xy + 12yz - 6xz
13. Simplify (x + 1)³ - (x - 1)³
(x + 1)³ - (x - 1)³
= {x³ + (1)³ + 3(x)(1)(x + 1)} - {x³ - (1)³ - 3(x)(1)(x - 1)}
= (x³ + 1 + 3x² + 3x) - (x³ - 1 - 3x² + 3x)
= x³ + 1 + 3x² + 3x - x³ + 1 + 3x² - 3x
= 2 + 6x²
14. Evaluate (99)³
(99)³
= (100 - 1)³
[(a + b)³ = a³ + b³ + 3ab(a + b)]
= [(100)³ + (1)³ + 3(100)(1)(100 + 1)]
= 1000000 + 1 + 30000 + 300
= 1030301
15. factorise a² + b² - 2(ab - ac + bc)
= a² + b² - 2(ab - ac + bc) + c² - c²
= (a - b + c)² - (c)²
= (a - b + c + c)(a - b + c - c)
= (a - b + 2c)(a - b)
16. if x + 1/x = 7 , find the value of x³ + 1/x³
(x + 1/x)³ = (7)³
x³ + 1/x³ + 3(x)(1/x)(x + 1/x) = 343
x³ + 1/x³ + 3(7) = 343
x³ + 1/x³ + 21 = 343
x³ + 1/x³ = 343 - 21
x³ + 1/x³ = 322