Page No 99:
Answer:
We have:
Page No 99:
Question 2:
We have:
Answer:
We have:
Page No 99:
Question 3:
We have:
Answer:
We have:
Page No 99:
Question 4:
We have:
Answer:
We have:
Page No 99:
Question 5:
We have:
Answer:
We have:
Page No 99:
Question 6:
We have:
Answer:
We have:
Page No 99:
Question 7:
We have:
Answer:
We have:
Page No 99:
Question 8:
We have:
Answer:
We have:
Page No 99:
Question 9:
We have:
Answer:
We have:
Page No 99:
Question 10:
We have:
Answer:
We have:
Page No 99:
Question 11:
We have:
Answer:
We have:
Page No 99:
Question 12:
We have:
Answer:
We have:
Page No 99:
Question 13:
We have:
Answer:
We have:
Page No 99:
Question 14:
We have:
Answer:
We have:
Page No 99:
Question 15:
We have:
Answer:
We have:
Page No 99:
Question 16:
We have:
Answer:
We have:
Page No 99:
Question 17:
We have:
Answer:
We have:
Page No 99:
Question 18:
We have:
Answer:
We have:
Page No 99:
Question 19:
We have:
Answer:
We have:
Page No 99:
Question 20:
We have:
Answer:
We have:
Page No 99:
Question 21:
We have:
Answer:
We have:
Page No 99:
Question 22:
We have:
Answer:
We have:
Page No 100:
Question 23:
We have:
Answer:
We have:
Page No 100:
Question 24:
We have:
Answer:
We have:
Page No 100:
Question 25:
We have:
Answer:
We have:
Page No 100:
Question 26:
We have:
Answer:
We have:
Page No 100:
Question 27:
We have:
Answer:
We have:
Page No 100:
Question 28:
We have:
Answer:
We have:
Page No 100:
Question 29:
We have:
Answer:
We have:
Page No 100:
Question 30:
We have:
Answer:
We have:
Page No 100:
Question 31:
We have:
Answer:
We have:
Page No 100:
Question 32:
We have:
Answer:
We have:
Page No 100:
Question 33:
We have:
Answer:
We have:
Page No 100:
Question 34:
We have:
Answer:
Page No 105:
Question 1:
Answer:
Page No 105:
Question 2:
Answer:
Page No 105:
Question 3:
Answer:
Page No 105:
Question 4:
Answer:
Page No 105:
Question 5:
Answer:
Page No 105:
Question 6:
Answer:
Page No 105:
Question 7:
Answer:
Page No 105:
Question 8:
Answer:
Page No 105:
Question 9:
Answer:
Page No 105:
Question 10:
Answer:
Page No 105:
Question 11:
Answer:
Page No 105:
Question 12:
Answer:
Page No 105:
Question 13:
Answer:
Page No 105:
Question 14:
Answer:
Page No 105:
Question 15:
Answer:
Page No 105:
Question 16:
Answer:
Page No 105:
Question 17:
Answer:
Page No 105:
Question 18:
Answer:
Page No 105:
Question 19:
Answer:
Page No 105:
Question 20:
Answer:
Page No 105:
Question 21:
Answer:
Page No 105:
Question 22:
Answer:
Page No 105:
Question 23:
Answer:
Page No 105:
Question 24:
Answer:
Page No 105:
Question 25:
Answer:
Page No 105:
Question 26:
Answer:
Page No 105:
Question 27:
Answer:
Page No 105:
Question 28:
Answer:
Page No 105:
Question 29:
Answer:
Page No 105:
Question 30:
Answer:
Page No 105:
Question 31:
Answer:
Page No 105:
Question 32:
Answer:
Page No 105:
Question 33:
Answer:
Page No 105:
Question 34:
Answer:
Page No 105:
Question 35:
Answer:
Disclaimer: The expression of the question should be
. The same has been done before solving the question.
Page No 105:
Question 36:
Disclaimer: The expression of the question should be
. The same has been done before solving the question.
Answer:
Page No 105:
Question 37:
Answer:
Page No 105:
Question 38:
Answer:
Page No 105:
Question 39:
Answer:
Page No 105:
Question 40:
Answer:
Page No 114:
Question 1:
Answer:
We have:
We have to split 11 into two numbers such that their sum of is 11 and their product is 30.
Clearly, .
Page No 114:
Question 2:
We have:
We have to split 11 into two numbers such that their sum of is 11 and their product is 30.
Clearly, .
Answer:
We have:
We have to split 18 into two numbers such that their sum is 18 and their product is 32.
Clearly, .
Page No 114:
Question 3:
We have:
We have to split 18 into two numbers such that their sum is 18 and their product is 32.
Clearly, .
Answer:
Page No 114:
Question 4:
Answer:
Page No 114:
Question 5:
Answer:
Page No 114:
Question 6:
Answer:
Page No 114:
Question 7:
Answer:
Page No 114:
Question 8:
Answer:
Page No 114:
Question 9:
Answer:
Page No 114:
Question 10:
Answer:
Page No 114:
Question 11:
Answer:
Page No 114:
Question 12:
Answer:
Page No 114:
Question 13:
Answer:
Page No 114:
Question 14:
Answer:
Page No 114:
Question 15:
Answer:
Page No 114:
Question 16:
Answer:
Page No 114:
Question 17:
Answer:
Page No 114:
Question 18:
Answer:
Page No 114:
Question 19:
Answer:
Page No 114:
Question 20:
Answer:
Page No 114:
Question 21:
Answer:
Page No 114:
Question 22:
Answer:
Page No 114:
Question 23:
Answer:
Page No 114:
Question 24:
Answer:
Page No 114:
Question 25:
Answer:
We have:
We have to split (1) into two numbers such that their sum is (1) and their product is (156).
Clearly, .
Page No 114:
Question 26:
We have:
We have to split (1) into two numbers such that their sum is (1) and their product is (156).
Clearly, .
Answer:
Page No 114:
Question 27:
Answer:
Page No 114:
Question 28:
Answer:
Page No 114:
Question 29:
Answer:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is (180), i.e., .
Clearly, .
â
Page No 114:
Question 30:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is (180), i.e., .
Clearly, .
â
Answer:
We have:
We have to split 11 into two numbers such that their sum is 11 and their product is (42), i.e., .
Clearly, .
Page No 114:
Question 31:
We have:
We have to split 11 into two numbers such that their sum is 11 and their product is (42), i.e., .
Clearly, .
Answer:
We have:
We have to split 2 into two numbers such that their sum is 2 and their product is (120), i.e., .
Clearly, .
Page No 114:
Question 32:
We have:
We have to split 2 into two numbers such that their sum is 2 and their product is (120), i.e., .
Clearly, .
Answer:
Page No 114:
Question 33:
Answer:
We have:
We have to split (41) into two numbers such that their sum is (41) and their product is 288, i.e., .
Clearly, .
Page No 114:
Question 34:
We have:
We have to split (41) into two numbers such that their sum is (41) and their product is 288, i.e., .
Clearly, .
Answer:
Hence, factorisation of 3
x2 – 14
x + 8 is
.
Page No 114:
Question 35:
Hence, factorisation of 3
x2 – 14
x + 8 is
.
Answer:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is (180), i.e., .
Clearly, .
Page No 114:
Question 36:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is (180), i.e., .
Clearly, .
Answer:
We have:
We have to split 2 into two numbers such that their sum is 2 and product is (15), i.e.,.
Clearly, .
Page No 114:
Question 37:
We have:
We have to split 2 into two numbers such that their sum is 2 and product is (15), i.e.,.
Clearly, .
Answer:
We have:
We have to split 1 into two numbers such that their sum is 1 and product is 30, i.e.,.
Clearly, .
Page No 114:
Question 38:
We have:
We have to split 1 into two numbers such that their sum is 1 and product is 30, i.e.,.
Clearly, .
Answer:
We have:
We have to split into two numbers such that their sum is and product is 14.
Clearly, .
Page No 114:
Question 39:
We have:
We have to split into two numbers such that their sum is and product is 14.
Clearly, .
Answer:
We have:
Now, we have to split (47) into two numbers such that their sum is (47) and their product is 90.
Clearly, .
Page No 114:
Question 40:
We have:
Now, we have to split (47) into two numbers such that their sum is (47) and their product is 90.
Clearly, .
Answer:
We have:
We have to split 20 into two numbers such that their sum is 20 and their product is 75.
Clearly,
Page No 114:
Question 41:
We have:
We have to split 20 into two numbers such that their sum is 20 and their product is 75.
Clearly,
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 42:
Hence, factorisation of
is
.
Answer:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is 2, i.e., .
Clearly, .
Page No 114:
Question 43:
We have:
We have to split 3 into two numbers such that their sum is 3 and their product is 2, i.e., .
Clearly, .
Answer:
We have:
We have to split into two numbers such that their sum is and their product is 6, i.e.,.
Clearly, .
Page No 114:
Question 44:
We have:
We have to split into two numbers such that their sum is and their product is 6, i.e.,.
Clearly, .
Answer:
We have:
We have to split (1) into two numbers such that their sum is (1) and their product is (420), i.e., .
Clearly, .
Page No 114:
Question 45:
We have:
We have to split (1) into two numbers such that their sum is (1) and their product is (420), i.e., .
Clearly, .
Answer:
We have:
We have to split (5) into two numbers such that their sum is (5) and their product is (126), i.e., .
Clearly, .
Page No 114:
Question 46:
We have:
We have to split (5) into two numbers such that their sum is (5) and their product is (126), i.e., .
Clearly, .
Answer:
We have:
We have to split (7) into two numbers such that their sum is (7) and their product is (30), i.e., .
Clearly, .
Page No 114:
Question 47:
We have:
We have to split (7) into two numbers such that their sum is (7) and their product is (30), i.e., .
Clearly, .
Answer:
We have:
We have to split (16) into two numbers such that their sum is (16) and their product is (105), i.e., .
Clearly, .
Page No 114:
Question 48:
We have:
We have to split (16) into two numbers such that their sum is (16) and their product is (105), i.e., .
Clearly, .
Answer:
Hence, factorisation of 6
x2 – 11
x – 35 is
.
Page No 114:
Question 49:
Hence, factorisation of 6
x2 – 11
x – 35 is
.
Answer:
Hence, factorisation of 9
x2 – 3
x – 20 is
.
Page No 114:
Question 50:
Hence, factorisation of 9
x2 – 3
x – 20 is
.
Answer:
We have:
We have to split (9) into two numbers such that their sum is (9) and their product is (70), i.e., .
Clearly, .
Page No 114:
Question 51:
We have:
We have to split (9) into two numbers such that their sum is (9) and their product is (70), i.e., .
Clearly, .
Answer:
Now, we have to split (
32) into two numbers such that their sum is (
32) and their product is 112, i.e.,
.
Clearly,
.
Page No 114:
Question 52:
Now, we have to split (
32) into two numbers such that their sum is (
32) and their product is 112, i.e.,
.
Clearly,
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 53:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 54:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 55:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 56:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 57:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 58:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 114:
Question 59:
Hence, factorisation of
is
.
Answer:
We have:
Thus, the given expression becomes
Now, we have to split (9) into two numbers such that their sum is (9) and their product is (10).
Clearly, .
Putting , we get:
Page No 114:
Question 60:
We have:
Thus, the given expression becomes
Now, we have to split (9) into two numbers such that their sum is (9) and their product is (10).
Clearly, .
Putting , we get:
Answer:
We have:
Thus, the given expression becomes
Now, we must split (4) into two numbers such that their sum is (4) and their product is (117).
Clearly, .
Putting , we get:
Page No 115:
Question 61:
We have:
Thus, the given expression becomes
Now, we must split (4) into two numbers such that their sum is (4) and their product is (117).
Clearly, .
Putting , we get:
Answer:
Hence, factorisation of
is
.
Page No 115:
Question 62:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 115:
Question 63:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 115:
Question 64:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 115:
Question 65:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of 4
x4 + 7
x2 – 2 is
.
Page No 115:
Question 66:
Hence, factorisation of 4
x4 + 7
x2 – 2 is
.
Answer:
Hence, {(999)
2 – 1} = 998000.
Page No 119:
Question 1:
Hence, {(999)
2 – 1} = 998000.
Answer:
Page No 119:
Question 2:
Answer:
Page No 119:
Question 3:
Answer:
Page No 119:
Question 4:
Answer:
Page No 119:
Question 5:
Answer:
Page No 119:
Question 6:
Answer:
Hence, 16
x2 + 4
y2 + 9
z2 – 16
xy – 12
yz + 24
xz =
.
Page No 119:
Question 7:
Hence, 16
x2 + 4
y2 + 9
z2 – 16
xy – 12
yz + 24
xz =
.
Answer:
Page No 123:
Question 1:
Answer:
Page No 123:
Question 2:
Answer:
Page No 123:
Question 3:
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 4:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 5:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 6:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 7:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 8:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
is
.
Page No 123:
Question 9:
Hence, factorisation of
is
.
Answer:
Hence, factorisation of
a3 – 12
a(
a – 4) – 64 is
.
Page No 123:
Question 10:
Hence, factorisation of
a3 – 12
a(
a – 4) – 64 is
.
Answer:
Page No 129:
Question 1:
Answer:
Page No 129:
Question 2:
Answer:
We know that
Given: 27a3 + 64b3
x = 3a, y = 4b
Page No 129:
Question 3:
We know that
Given: 27a3 + 64b3
x = 3a, y = 4b
Answer:
Page No 129:
Question 4:
Answer:
Page No 129:
Question 5:
Answer:
Page No 129:
Question 6:
Answer:
Page No 129:
Question 7:
Answer:
Page No 129:
Question 8:
Answer:
Page No 129:
Question 9:
Answer:
Page No 129:
Question 10:
Answer:
Page No 129:
Question 11:
Answer:
Page No 129:
Question 12:
Answer:
Page No 129:
Question 13:
Answer:
Page No 129:
Question 14:
Answer:
We know
We have,
So,
Page No 129:
Question 15:
We know
We have,
So,
Answer:
Page No 129:
Question 16:
Answer:
Page No 129:
Question 17:
Answer:
Page No 129:
Question 18:
Answer:
Using the identity
Page No 129:
Question 19:
Using the identity
Answer:
Page No 129:
Question 20:
Answer:
Page No 129:
Question 21:
Answer:
Page No 129:
Question 22:
Answer:
Page No 129:
Question 23:
Answer:
Page No 129:
Question 24:
Answer:
Page No 129:
Question 25:
Answer:
Page No 129:
Question 26:
Answer:
Page No 129:
Question 27:
Answer:
Page No 129:
Question 28:
Answer:
Page No 129:
Question 29:
Answer:
a12 – b12
Page No 129:
Question 30:
a12 – b12
Answer:
Let
So, the equation becomes
Page No 129:
Question 31:
Let
So, the equation becomes
Answer:
x3 – 3
x2 + 3
x + 7
Page No 129:
Question 32:
x3 – 3
x2 + 3
x + 7
Answer:
(x +1)
3 + (
x – 1)
3
Page No 129:
Question 33:
(x +1)
3 + (
x – 1)
3
Answer:
(2a +1)3 + (a – 1)3
Page No 129:
Question 34:
(2a +1)3 + (a – 1)3
Answer:
8(x +y)3 – 27(x – y)3
Page No 129:
Question 35:
8(x +y)3 – 27(x – y)3
Answer:
(x +2)
3 + (
x – 2)
3
Page No 129:
Question 36:
(x +2)
3 + (
x – 2)
3
Answer:
(x + 2)
3 – (
x – 2)
3
Page No 129:
Question 37:
(x + 2)
3 – (
x – 2)
3
Answer:
Thus, LHS = RHS
Page No 129:
Question 38:
Thus, LHS = RHS
Answer:
Thus, LHS=RHS
Page No 136:
Question 1:
Thus, LHS=RHS
Answer:
Page No 136:
Question 2:
Answer:
(x – y − z) (x2 + y2 + z2 + xy – yz + xz)
Page No 136:
Question 3:
(x – y − z) (x2 + y2 + z2 + xy – yz + xz)
Answer:
Page No 136:
Question 4:
Answer:
Page No 136:
Question 5:
Answer:
Page No 136:
Question 6:
Answer:
Page No 136:
Question 7:
Answer:
Page No 136:
Question 8:
Answer:
Page No 136:
Question 9:
Answer:
Page No 136:
Question 10:
Answer:
Page No 136:
Question 11:
Answer:
Page No 136:
Question 12:
Answer:
Page No 137:
Question 13:
Answer:
Page No 137:
Question 14:
Answer:
Page No 137:
Question 15:
Answer:
Page No 137:
Question 16:
Answer:
Page No 137:
Question 17:
Answer:
Page No 137:
Question 18:
Answer:
Page No 137:
Question 19:
Answer:
Page No 137:
Question 20:
Answer:
Page No 137:
Question 21:
Answer:
Page No 137:
Question 22:
Answer:
(i) (–12)3 + 73 + 53
(ii) (28)3 + (–15)3 + (–13)3
Page No 137:
Question 23:
(i) (–12)3 + 73 + 53
(ii) (28)3 + (–15)3 + (–13)3
Answer:
Page No 137:
Question 24:
Answer:
Thus, we have:
Page No 137:
Question 25:
Thus, we have:
Answer:
a + b + c = 9
We know,
(
a3 + b3 + c3 – 3
abc) =
Page No 138:
Question 1:
a + b + c = 9
We know,
(
a3 + b3 + c3 – 3
abc) =
Answer:
(c) 2
Page No 138:
Question 2:
(c) 2
Answer:
(249)2 – (248)2
We know
Hence, the correct answer is option (d).
Page No 138:
Question 3:
(249)2 – (248)2
We know
Hence, the correct answer is option (d).
Answer:
(c) 0
2 + y2 = xy
⇒ x2+ y2 + xy = 0
Thus, we have:
Page No 138:
Question 4:
(c) 0
2 + y2 = xy
⇒ x2+ y2 + xy = 0
Thus, we have:
Answer:
(d) 3abc
Page No 139:
Question 5:
(d) 3abc
Answer:
Hence, the correct answer is option (c).
Page No 139:
Question 6:
Hence, the correct answer is option (c).
Answer:
(x + 3)3
So, the coefficient of x in (x + 3)3 is 27.
Hence, the correct answer is option (d).
Page No 139:
Question 7:
(x + 3)3
So, the coefficient of x in (x + 3)3 is 27.
Hence, the correct answer is option (d).
Answer:
(x + y)3 – (x3 + y3)
Thus, the factors of (x + y)3 – (x3 + y3) are 3xy and (x + y).
Hence, the correct answer is option (d).
Page No 139:
Question 8:
(x + y)3 – (x3 + y3)
Thus, the factors of (x + y)3 – (x3 + y3) are 3xy and (x + y).
Hence, the correct answer is option (d).
Answer:
So, the factors of
are (5
x + 1) and 10
x
Hence, the correct answer is option (d).
Page No 139:
Question 9:
So, the factors of
are (5
x + 1) and 10
x
Hence, the correct answer is option (d).
Answer:
(b) 5
Page No 139:
Question 10:
(b) 5
Answer:
(b) m = 7, n = −18
Let:
Now,
(x + 2) is a factor of p(x).
So, we have p(2)=0
Now,
Also,
(x 1) is a factor of p(x).
We have:
p(1) = 0
By substituting the value of m in (i), we get n = −18.
∴ m = 7 and n = −18
Page No 139:
Question 11:
(b) m = 7, n = −18
Let:
Now,
(x + 2) is a factor of p(x).
So, we have p(2)=0
Now,
Also,
(x 1) is a factor of p(x).
We have:
p(1) = 0
By substituting the value of m in (i), we get n = −18.
∴ m = 7 and n = −18
Answer:
(b) 9984
Page No 139:
Question 12:
(b) 9984
Answer:
(c) 93940
Page No 139:
Question 13:
(c) 93940
Answer:
(b) 39951
Page No 139:
Question 14:
(b) 39951
Answer:
(a) (2a + b + 2)2
Page No 139:
Question 15:
(a) (2a + b + 2)2
Answer:
(c) (x − 7)(x + 3)
Page No 139:
Question 16:
(c) (x − 7)(x + 3)
Answer:
(c) (2x + 3) (2x − 1)
Page No 139:
Question 17:
(c) (2x + 3) (2x − 1)
Answer:
(b) (2x + 5)(3x + 1)
Page No 139:
Question 18:
(b) (2x + 5)(3x + 1)
Answer:
(c) x3 − 2x2 − x − 2
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (−1) = 0.
We have:
Hence, (x + 1) is not a factor of .
Now,
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (1) = 0.
We have:
Hence, (x + 1) is not a factor of .
Now,
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (1) = 0.
We have:
Hence, (x + 1) is a factor of .
Page No 140:
Question 19:
(c) x3 − 2x2 − x − 2
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (−1) = 0.
We have:
Hence, (x + 1) is not a factor of .
Now,
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (1) = 0.
We have:
Hence, (x + 1) is not a factor of .
Now,
Let:
By the factor theorem, (x + 1) will be a factor of f (x) if f (1) = 0.
We have:
Hence, (x + 1) is a factor of .
Answer:
(d) (3x + 2)(x2 + 1)
Page No 140:
Question 20:
(d) (3x + 2)(x2 + 1)
Answer:
(d) 3
Thus, we have:
Page No 140:
Question 21:
(d) 3
Thus, we have:
Answer:
(a) 108
Page No 140:
Question 22:
(a) 108
Answer:
2 + b2 = ab
2 + b2 + ab = 0
Thus, we have:
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