## (CLASS 9)Polynomial -2 , Coordinate Geometry, Linear Equation In Two Variables, Euclid's Geometry, Lines And Angles

Published in: Mathematics
1,531 views
• ### Kundan

• Ranchi
• 14 Years of Experience
• Qualification: B.Sc.(Mathematics Hons.)
• Teaches: Special Education, Physics, Mathematics, Chemistry...
• Contact this tutor

Solved problems strictly according to the syllabus.

• 1
1. 2. Polynomials Class IX Determine which of the following polynomials has (x + 1) a factor: (iii) x4 + 3x3 + 3x2 + x + I (iv) x3 - x2 - (2 + 42)x + 42 Answer (i) If (x + 1) is a factor of p(x) = x3 + x2 + x + 1, p(-l) must be zero. Here, p(x) = x3 + x2 + x + 1 - (-1)3 + (-1)2 + + 1 = -1 +1-1+1=0 Therefore, x + 1 is a factor of this polynomial (ii) If (x + 1) is a factor of p(x) = x4 + x3 + x2 + x + 1, p(-l) must be zero. Here, p(x) = x4 + x3 + x2 + x + 1 - (-1)4 + (-1)3 + (-1)2 + + 1 Therefore, x + 1 is not a factor of this polynomial (iii)lf (x + 1) is a factor of polynomial p(x) = x 4 + 3x3 + 3x2 + x + 1, p(- 1) must be 0. Therefore, x + 1 is not a factor of this polynomial. (iv) If (x + 1) is a factor of polynomial p(x) = x3 - x2 - (2 + 42)x + 42, p(- 1) must be 0. ( - (-1)2- (2 + 42) + 42 = -l - I +2 + -02 + 42 242 Therefore„ x + 1 is not a factor of this polynomial. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases: (i)p(x) = 2x3 + x2 - 2x - 1, g(x) = x + 1 (ii) p(x) = x3 + 3x2 + 3x + 1, g(x) = x + 2 (iii) p(x) = x3 - 4 x2 + x + 6, g(x) = x - 3 Answer (i) If g(x) = x + 1 is a factor of given polynomial p(x), p(- 1) must be zero. p(x) = 2x3 + x2 - 2x - 1 1) = + (-1)2 - 2(-1) - 1
• 2
3. Hence, g(x) = x + 1 is a factor of given polynomial. (ii) If g(x) = x + 2 is a factor of given polynomial p(x), p(- 2) must be 0. = X3 +3X2 + 3X + 1 p(-2) = (-2)3 + + 3(-2) + 1 -8 + 12 -6+1 --1 As, p(-2) 0 Hence g(x) = x + 2 is not a factor of given polynomial. (iii) If g(x) = x - 3 is a factor of given polynomial p(x), p(3) must be 0. (3)3 - +3 +6 = 27-36+9=0 Therefore„ g(x) = x - 3 is a factor of given polynomial. Find the value of k, if x - 1 is a factor of p(x) in each of the following cases:(i) p(x) = (ii)p(x) = 2x2 + kx + 42 (iii) p(x) = kx2 - m12x + 1 (iv) p(x) = kx2 - 3x + k Answer (i) If x - 1 is a factor of polynomial p(x) = Therefore, value of k is -2. x2 + x + k, then 2X2 + kX + 42, then (ii) If x - 1 is a factor of polynomial p(x) = k = -2 - 42 = -(2 + 42) Therefore, value of k is -(2 + 42). kx2 - A12x + 1, then (iii) If x - 1 is a factor of polynomial p(x) = Therefore, value of k is 42 - 1. 2 - 3x + k, then (iv) If x - 1 is a factor of polynomial p(x) = kx
• 3
4. 5. Therefore, value of k is 3/2. Factorise: (i) 12x2 + 7.r + I (ii) 2x2 + 7x + 3 (iii) 6x2 + 5x - 6 (iv) 3x2 -x - 4 Answer (i) 12x2 +7x+ 1 = 12x2 - 4x - 3x+ 1 (ii) 2x2 + + 3 = 2X2 + 6X + X + 3 (iii) 6x2 + 5x - 6 = 6x2 + 9x - 4x - 6 (iv) 3x2 -x - 4 = 3x2 — 4 x + 3 x — 4 Factorise: (i) x3 - 2x2 -x + 2 (ii) x3 - 3x2 - 9x - 5 (iii) x3 + 13x2 + 32x + 20 (iv) 2y3 + Y2 - 2y - 1 Answer (i) Let p(x) = x3 - 2x2 - x + 2 Factors of 2 are ±1 and ± 2 By trial method, we find that So, (x+l ) is factor of p(x) Now, p(x) = x3 - 2x2 - x + 2 p(-l) = (-1)3 - Therefore, (x+l) is the factor of p(x)
• 4
X3 — 2X2 —X + 2 2 --3x2 —x + 2 — 3x2 — 3x 2x+2 Now, Dividend = Divisor x Quotient + Remainder = (x+l) {x(x-l) -2(x-1)} (ii) Let p(x) = x3 - 3x2 - 9x - 5 Factors of 5 are ±1 and ±5 By trial method, we find that So, (x-5) is factor of p(x) Now, p(x) = x3 - 2x2 - x + 2 p(5) = (5)3 - - 9(5) - 5 = 125 - 75 Therefore, (x-5) is the factor of p(x) - 45 x-S x2 + 2.1 I x 3 3x2 — 9x x 3 5x 2 2x2 9x —5 —5 2x2 IOx Now, Dividend = Divisor x Quotient + Remainder = (X-5) {X(X+I) 4-1 (X + I ) }
• 5
(iii) Let p(x) = + 13x2 + 32x + 20 Factors Of 20 are ± l, and By trial method, we find that So, (x+l ) is factor of p(x) Now, = X3 + 13X2 + 32X + 20 - 3 + + 32(-1) + 20 — —-1 13 - Therefore, (x+l) is the factor of p(x) X2 12X 20 x3+ 13x2 + 20 2 12x2 + 32x + 20 12x2 + 12x 20x + 20 20x + 20 0 32 + 20 = o Now, Dividend = Divisor x Quotient + Remainder + 12X + 20) = (x +1) (x2 2x + + 20) = (X-5) {X(X+2) = (x-5) (x+2) (x+10) (iv) Let p(y) = 2y3 + Y2 - 2y - 1 Factors of ab = 2x (-1) = -2 are ±1 and ±2 By trial method, we find that So, (y-1 ) is factor of p(y) Now, p(y) = 2y3 + Y2 - 2y - 1 p(l) + - = 2+1 2- Therefore, (y-1) is the factor of p(y)
• 6
y-1 2y2 + 3y + 1 2y3 + Y2 — 2y 2y3 —2y2 3y2 — 2y — 3y —1 —1 Now, Dividend = Divisor x Quotient + Remainder = (y-I) 4-1) (y +1) Use suitable identities to find the following products: 6. (i) (x + 4) (x + 10) (iv) + 3/2) (Y2 - 3/2) Answer (ii) (x + 8) (x — 10) (iii) (3x + 4) (3x — 5) (i) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab In (x +4) (x+ 10), and 10 Now, (X + 4) (X + 10) X2 + (4 + + (4 X 10) = X2 + 40 (ii) (x+ 8) (x- 10) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, a = 8 and b = -10 (X + 8) (X — 10) = X2 + {8 4-(— 10) } X + 10)) = x2 + (8 — 10)x — 80 2x 80 (iii) (3x + 4) (3x 5) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 3x, and b = -5 = + 3x(4 - 5) - 20 = 9x2 - 3x - 20 (iv) + 3/2) (Y2 - 3/2) Using identity, (x + y) (x -y) = x2 2
• 7
, and y = 3/2 Here x = (Y2 + 3/2) (Y2 - 3/2) = (y2)2 - (3/2)2 Using identity, (x + y) (x -y) = Here, x = 3 and y = 2x (3 - 2x) (3 + 2x) = 32 - (2x)2 9 - 4x2 2 Evaluate the following products without multiplying directly: 7. (i) 103 x 107 (ii) 95 x 96 (iii) 104 x 96 Answer (i) 103 x 107 = (100 + 3) (100 + 7) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 100, a = 3 and b = 7 103 x 107 = (100 + 3) (100 + 7) = (100)2 + (3 + 7)10 + (3 x 7) = 10000 + 100 + 21 = 11021 (ii) 95 x 96 = (90 + 5) (90 + 4) Using identity, (x + a) (x + b) = x2 + (a + b) x + ab Here, x = 90, a = 5 and b = 4 95 x 96 = (90 + 5) (90 + 4) = 902 + + 6) + (5 x 6) = 8100 + (11 x 90) + 30 = 8100 + 990 + 30 = 9120 (iii) 104 x 96 = (100 + 4) (100 - 4) Using identity, (x + y) (x -y) = x2 - y 2 Here, x = 100 and y = 4 104 x 96 = (100 + 4) (100 - 4) = (100)2 - (4)2 = 10000 - 16 = 9984 8. Factorise the following using appropriate identities: (i) 9x2 + 6xy + (ii) 4y2 - 4y + 1 (iii) x2 - y2/100 Answer (i) 9x2 + 6xy + Y2 = (3x) 2 + (2>
• 8
9. Using identity, a2 - b2 = (a + b) (a - b) Here, a = x and b = (y/ 10) x2- y2/100 = x2- (y/ 10) — 2- (x- y/10) (x+ y/10) Expand each of the following, using suitable identities: (iv) (3a — 7b — c)2 Answer (i) (x + + 4z)2 (ii) (2x —y + z)2 (v) (—2x + 5y — 3z)2 (iii) (—2x + 3y + 2z)2 (vi) [1/4 a - 1/2 b + 112 Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca Here, a = x, b = 2y and c = 4z (x + 2y + + (2y)2 + (4z)2 + (2>
• 9
Answer (i) 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca 4x2 + 9y2 + 16z2 + 12xy - 24yz - 16xz = (2x)2 + (3y)2 + (-4z)2 + + + = (2x + 3 Y' - 4z)2 = (2x + 3y - 4z) (2x + 3y - 4z) (ii) 2x2 + Y2 + 8z2 - 242 xy + 442 yz - 8xz Using identity, (a + b + = a2 + b2 + c2 + 2ab + 2bc + 2ca 2x2 + Y2 + 8z2 - 2€2 xy + 4Q2yz - 8xz = ( -€21) 2 + (y) 2 (2N2z)2 + = (-Q2x y 2N2z)2 = (-€2x y 2N2z) (-€2x y 2N/2z) Write the following cubes in expanded form: 11. (i) (2x + 1)3 (ii) (2a — 3b)3 (iii) [3/2 x + 113 Answer Using identity, (a + = a3 + b3 + 3ab(a + b) 3 = (2x)3 + 13 + + 1) = + I + 6x(2x + ) = 8x3 + 12x2 + 6x + (ii) (2a - 3b)3 Using identity, (a - = a3 - b3 - 3ab(a - b) (2a 3b/ = (2a)3 - (3b/ - - 3b) = 8a3 - 27b3 - 18ab(2a - 3b) = 8a3 - 27b3 - 36a2b + 54ab2 (iii) [3/2 x + Using identity, (a + = a3 + b3 + 3ab(a + b) [3/2 x + 113 = (3/2 x/ + 13 + + 1) = 27/8x3+1+9/2x(3/2x+1) = 27/8x3+1 +27/4x2+9/2x = 27/8++27/4x2+9/2x+1 (iv) [x - 2/3 yr Using identity, (a - = a3 - b3 - 3ab(a - b) 3 = (x/ - (2/3 y/ - - 2/3 y) [x - 2/3 y] = x3 - 8/27y3 - 2xy(x - 2/3 y) = x3 - 8/27y3 - 2x2y + 4/3xy2 12. Evaluate the following using suitable identities: (i) (99)3 (ii) (102)3 (iii) (998)3 Answer (iv) [x - 2/3 y13
• 10
= 100000000 - 8 - 6000(1000 - 2) = 100000000 - 8- 600000 + 12000 = 994011992 13. Factorise each of the following: (i) 8a3 + b3 + 12a2b + 6ab2 (iii) 27 - 125a3 - 135a + 225a2 (v) 27p3 - 1/216 - 9/2 p2 + 1/4 p Answer (i) 8013 + 193 + 12012b + 601192 3 = (100 - (i) (99) Using identity, (a - = a3 - b3 - 3ab(a - b) (100 - 1/ = (100/ - 13 - - 1) = 1000000 - 1 - 300(100 - 1) = 1000000 - 1 - 30000 + 300 = 970299 (ii) (102)3 = (100 + Using identity, (a + = a3 + b3 + 3ab(a + b) (100 + 2/ = (100/ + 23 + + 2) = 1000000 + 8 + 600(100 + 2) = 1000000 + 8 + 60000 + 1200 = 1061208 (iii) (998)3 Using identity, (a - = a3 - b3 - 3ab(a - b) (1000 - 2/ = (1000/ - 23 - - 2) 27 - 125a3 - 135a + 225a2= 33 - (5a)3 - + (ii) 8a3 - b3 - 12a2b + 6ab2 (iv) 64a3 - 27b3 - 144a2b + 108ab2 Using identity, (a + = a3 + b3 + 3a2b + 3ab2 8a3 + 193 + 12a2b + 601122 = (2a)3 + b3 + + = (201 + b = (261 + + + b) (ii) 8a3 - b3 - 12a2b + 6ab2 Using identity, (a - = a3 - b3 8a3 - 193 - 12a2b + 6ab2= (2a)3 (iii) 27 - 125a3 - 135a + 225a2 Using identity, (a - = a3 - b3 = (3 - 5a)3 - 3a2b + 3ab2 - 3a2b + 3ab2 (iv) 64a3 - 27b3 - 144a2b + 108ab2 Using identity, (a - = a3 - b3 - 3a2b + 3ab2 64a3 - 27b3 - 144a2b + 108ab2= (4a)3 - (3b/ - +
• 11
(v) 27p3 - 1/216 - 9/2 p2 + 1/4 p Using identity, (a - = a3 - b3 - 3a2b + 3ab2 27p3 - 1/216 - 9/2p2 + 1/4 p = (3p)3 - (1/6/ - + Verify : (i) x3 + y3 = (x + y) (x2 -xy + y2) 14. Answer We know that, (ii) x3 - y3 = (x - y) (x2 + xy + Y2 ) We know that, (x - = x3 - y3 - 3xy(x - y) 9 x3 - y3 = (x - + 3xy(x - y) 9 x3 + y3 = (x - + - 2xy) + 3xy] Factorise each of the following: 15. (ii) x3 - Y3 = (x -y) (x2 + xy + y2) {Taking (x+y) common} {Taking (x-y) common} (i) 27y3 + 125z3 (ii) 64m3 - 343n3 Answer (i) 27y3 + 125z3 Using identity, x3 + y3 = (x + y) (x2 - xy + y2) 27y3 + 125z3 = (3y)3 + (5z)3 = (3y + 5z) {(3y)2 - + (5z)2} = (3y + 5z) (9y2 - 15yz + 25z)2 (ii) 64m3 - 343n 3 Using identity, x3 - y3 = (x - y) (x2 + xy + Y2 ) = (4m + 7n) {(4m)2 + + (7n)2} = (4m + 7n) (16m2 + 28mn + 49n)2
• 12
Factorise : 27x3 + y3 + z3 - 9xyz 16. Answer 27x3 + y3 + z3 - 9xyz = (3x)3 + y3 + z3 - 3x3xyz Using identity, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 27x3 + Y'3 + _ 9xyz = (3x + y + z) {(3x)2 + Y2 + z2 - 3xy - yz - 3xz} = + y + z) (9x2 + + - yz - 3xz) - xy - yz - xz) Verify that: x3 + y3 + z3 - 3xyz = 1/2(x + y + z) [(x - + (y - + (z - x)2] 17. Answer We know that, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 - xy - yz - xz) + y3 + _ 3 xyz + y + z) + + z2 — XY — Y z — R,) = 1/2(x + y + z) (2x2 + 2y2 + 2z2 - 2xy - 2yz - 2xz) = 1/2(x + y + z) [(x2 + Y2 -2xy) + (Y2 + z2 - 2yz) + (x2 + z2 - 2xz)] 18. If x + y + z = 0, show thatx3 +y3 + z3 = 3xyz. Answer We know that, x3 + y3 + z3 - 3xyz = (x + y + + Y2 + z2 - xy - yz - xz) Now put (x + y + z) = 0, x3 + y3 + z3 - 3xyz = + Y2 + z2 - xy - yz - xz) 19. Without actually calculating the cubes, find the value of each of the following: (i) (-12)3 + (7)3 + (5)3 (ii) (28)3 + (-15)3 + (-13)3 Answer (i) (-12) 3 + (7)3 + (5)3 and z = 5 We observed that, x + y + z = -12 + 7 + 5 = 0 We know that if, x + y + z = 0, then x3 + y3 + z3 = 3xyz - -1260 (ii) (28)3 + (-15)3 + (-13)3 Let x -28, y = -15 and z = -13 We observed that, x + y + z = 28 - 15 - 13 = 0
• 13
We know that if, x + y + z = 0, then x3 + y3 + z3 = 3xyz (28/ + (-15/ + (-13/ = = 16380 20. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : 25a2 - 35a + 12 (ii) Area : 35 Y2 + 13y - 12 Answer (i) Area : 25a2 - 35a + 12 Since, area is product of length and breadth therefore by factorizing the given area, we can know the length and breadth of rectangle. 25a2 - 35a + n = 25a2 - 15a -20a + n = 5a(5a - 3) - 4(5a - 3) Possible expression for length = 5a - 4 Possible expression for breadth = 5a - 3 (ii) Area : 35 + 13y - 12 35 + 13y- n = 35y2 - 15y + 28y - 12 = 5y(7y - 3) + 4(7y - 3) Possible expression for length = (5y + 4) Possible expression for breadth = (7y - 3) Page No: 50 21. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : 3x2 - 12x (ii) Volume : 12ky2 + 8ky - 20k Answer (i) Volume : 3x2 - 12x Since, volume is product of length, breadth and height therefore by factorizing the given volume, we can know the length, breadth and height of the cuboid. 3x2 - 12x = 3x(x - 4) Possible expression for length Possible expression for breadth = x Possible expression for height = (x - 4)
• 14
(ii) Volume : 1202 + 8ky - 20k Since, volume is product of length, breadth and height therefore by factorizing the given volume, we can know the length, breadth and height of the cuboid. 12/02 + 8ky - 20k = 4k(3y2 + 2y - 5) = 4k(3y2 +5y - 3y - 5) = +5) - I(3y + 5)] Possible expression for length Possible expression for breadth Possible expression for height

## Need a Tutor or Coaching Class?

Post an enquiry and get instant responses from qualified and experienced tutors.

Post Requirement

Mathematics
452 views

Mathematics
295 views

Mathematics
380 views

Mathematics
201 views

Mathematics
660 views

Mathematics
1,125 views

Mathematics
155 views

Mathematics
418 views

Mathematics
439 views

Mathematics
244 views

Drop Us a Query