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Mathematics, a subject not of everyone’s taste, some may find it interesting some may find it difficult and boring. However, there is no escaping from it as it is one of the compulsory subjects till 10^th standard. So, here is an article that contains solutions that will guide you in solving problems without wasting your time in figuring how to solve.

Exercise 2.1

• Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.

i.        4x2–3x+7	ii.        y2+√2
iii.        3√t+t√2	iv.        y+2/y
v.        x10+y3+t50

• Write the coefficients of x2 in each of the following:

 

i.        2+x2+x	ii.        2–x2+x3
iii.            x2+x	iv.        √2x-1

• Give one example each of a binomial of degree 35, and of a monomial of degree 100.

• Write the degree of each of the following polynomials:

i.        5x3+4x2+7x	ii.        4–y2
iii.        5t–√7	iv.        3

• Classify the following as linear, quadratic and cubic polynomials:

i.        x2+x	ii.        x–x3	iii.        y+y2+4	iv.        1+x
v.        3t	vi.        r2	vii.        7x3

Solutions

Exercise 2.2

• Find the value of the polynomial (x)=5x−4x²+3 

i.        x = 0

ii.        x = – 1

iii.        x = 2

• Find p(0), p(1) and p(2) for each of the following polynomials:

i.        p(y)=y2−y+1	ii.        p(t)=2+t+2t2−t3
iii.        p(x)=x3	iv.        P(x) = (x−1)(x+1)

• Verify whether the following are zeroes of the polynomial, indicated against them:

i.  p(x)=3x+1, x=−1/3	ii.        p(x)=5x– Π, x = 4/5
iii.  p(x)=x2−1, x=1, −1	iv.        p(x) = (x+1)(x–2), x =−1, 2
v.  p(x) = x2, x = 0	vi.        p(x) = lx+m, x = −m/l
vii.  p(x) = 3x2−1, x = -1/√3 , 2/√3	viii.        p(x) =2x+1, x = 1/2

Solutions

Exercise 2.3

• Find the remainder when x³+3x²+3x+1 is divided by:

i.             x+1	ii.            x−1/2
iii.           x	iv.           x+Π
v.            5+2x

• Find the remainder when x³−ax²+6x−a is divided by x-a.
• Check whether 7+3x is a factor of 3x³+7x.

Solutions

Exercise 2.4

• Determine which of the following polynomials has (x + 1) a factor:

i.             x3+x2+x+1	ii.            x4+x3+x2+x+1
iii.           x4+3x3+3x2+x+1	iv.           x3 – x2– (2+√2)x +√2

•  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)=x3–4x2+x+6, g(x) = x–3

Find the value of k, if x–1 is a factor of p(x) in each of the following cases:

i.             p(x) = x2+x+k	ii.            p(x) = 2x2+kx+√2
iii.           p(x) = kx2–√2x+1	iv.           p(x)=kx2–3x+k

•  Factorize:

i.             12x2–7x+1	ii.            2x2+7x+3
iii.           6x2+5x-6	iv.           3x2–x–4

Factorize: 

i.             x3–2x2–x+2	ii.            x3–3x2–9x–5
iii.           x3+13x2+32x+20	iv.           2y3+y2–2y–1

Solutions

Exercise 2.5

i.             (x+4)(x +10)	ii.            (x+8)(x –10)
iii.           (3x+4)(3x–5)	iv.           (y2+3/2)(y2-3/2)

• Use suitable identities to find the following products:
• Evaluate the following products without multiplying directly:

i.             103×107	ii.            95×96
iii.           104×96

•  Factorize the following using appropriate identities:

i.             9x2+6xy+y2	ii.            4y2−4y+1
iii.           x2–y2/100

• Expand each of the following, using suitable identities:

i.             (x+2y+4z)2	ii.            (2x−y+z)2
iii.           (−2x+3y+2z)2	iv.           (3a –7b–c)2
v.            (–2x+5y–3z)2	vi.           ((1/4)a-(1/2)b+1)2

• Factorize:

i.             4x2+9y2+16z2+12xy–24yz–16xz

ii.            2x2+y2+8z2–2√2xy+4√2yz–8xz

 Write the following cubes in expanded form:

i.             (2x+1)3	ii.            (2a−3b)3
iii.           ((3/2)x+1)3	iv.           (x−(2/3)y)3

Evaluate the following using suitable identities: 

i.             (99)3	ii.            (102)3
iii.           (998)3

Factorise each of the following:

i.             8a3+b3+12a2b+6ab2

ii.            8a3–b3–12a2b+6ab2

iii.           27–125a3–135a +225a2

iv.           64a3–27b3–144a2b+108ab2

v.            27p3–(1/216)−(9/2) p2+(1/4)p

• Verify:

i.             x3+y3 = (x+y)(x2–xy+y2)

ii.            x3–y3 = (x–y)(x2+xy+y2)

• Factorize each of the following:

i.             27y3+125z3

ii.            64m3–343n3

• Factorise: 27x³+y³+z³–9xyz
• Verify that: x³+y³+z³–3xyz = (1/2) (x+y+z)[(x–y)²+(y–z)²+(z–x)²]
• If  x+y+z = 0, show that x³+y³+z³= 3xyz.
• 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

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:

• Area : 25a²–35a+12
• Area : 35y²+13y–12

16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

1. Volume : 3x²–12x
2. Volume : 12ky²+8ky–20k

Solutions

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