Maths Chapter-12: Algebraic Expressions
Exercise 12.1
Q1: Using arithmetic operations, constants and variables find the algebraic expressions of the cases given below:
(i) Numbers a and b both squared and added.
(ii) Number 5 added to three times the product of s and t.
(iii) One-fourth of the product of numbers m and n.
(iv) One-half of the sum of numbers a and b.
(v) Product of numbers e and f subtracted from 10.
(vi) Subtraction of v from u.
(vii) Sum of numbers s and t subtracted from their product
(viii) The number x multiplied by itself.
Sol:
(i) \(a^{2}+b^{2}\)
(ii) \(3st+5\)
(iii) \(\frac{mn}{4}\)
(iv) \(\frac{a+b}{2}\)
(v) \(10- ef\)
(vi) u-v
(vii) \(st-(s+t)\)
(viii) \(x^{2}\)
Q2:
(a) Figure out the terms and their factors in the expression given below and show them by the help of tree diagram
(i) \(a-3\)
(ii) \(1+a+a^{2}\)
(iii) \(y-y^{3}\)
(iv)\(5ab^{2}+7x^{2}y\)
(v) \(-xy+2y^{2}-3x^{2}\)
(b) Figure out the terms and factors in the expressions below:
(i) \(-4a+5\)
(ii) \(-4a+5b\)
(iii) \(5a+3a^{2}\)
(iv) \(ab+2a^{2}b^{2}\)
(v) \(ab+b\)
(vi) \(1.2xy-2.4y+3.6x\)
(vii) \(\frac{3}{4}x+\frac{1}{4}\)
(viii) \(0.1a^{2}+0.2b^{2}\)
Also show the terms and factors by tree diagram.
Sol:
(a)
(i) \(a-3\)
(ii) \(1+a+a^{2}\)
(iii) \(y-y^{3}\)
(iv) \(5ab^{2}+7x^{2}y\)
(v) \(-xy+2y^{2}-3x^{2}\)
(b)-
(i) \(-4a+5\)
Terms: \(-4a,5\)
Factors: \( -4,\; a;\; 5\)
(ii) \(-4a+5b\)
Terms: \(-4a,5b\)
Factors: \(-4,\;a;\;5,\;b\)
(iii) \(5a+3a^{2}\)
Terms: \(5a,3a^{2}\)
Factors: \(5,\;a;\;3,\;a\;a\)
(iv) \(ab+2a^{2}b^{2}\)
Terms: \(ab,2a^{2}b^{2}\)
Factors: \(a,\;b;\;2,\;a,\;a;\;b,\;b\)
(v) \(ab+b\)
Terms: \(ab,b\)
Factors: \(a,\;b;\;b\)
(vi) \(1.2xy-2.4y+3.6x\)
Terms: \(1.2xy,-2.4y,3.6x\)
Factors: \(1.2,\;x,\;y;\;-2.4,\;y;\;3.6\; x\)
(vii) \(\frac{3}{4}x+\frac{1}{4}\)
Terms: \(\frac{3}{4}x,\frac{1}{4}\)
Factors: \(\frac{3}{4},\;x;\; \frac{1}{4}\)
(viii) \(0.1a^{2}+0.2b^{2}\)
Terms: \(0.1a^{2},0.2b^{2}\)
Factors: \(0.1,\;a,\;a;\;0.2,\;b\;b\)
Q3: Other than the constants figure out the numerical coefficients of the given expressions:
(i) \(5-3a^{2}\)
(ii) \(1=a+a^{2}+a^{3}\)
(iii) \(a+2ab+3b\)
(iv) \(100x+100y\)
(v) \(-x^{2}y^{2}+7xy\)
(vi) \(1.2x+0.8y\)
(vii) \(3.14x^{2}\)
(viii) \(2(a+b)\)
(ix) \(0.1x+0.01x^{2}\)
| S.no | Expression | Terms | Numerical Coefficient | ||||||
| (i) | \(5-3a^{2}\) | \(-3a^{2}\) | |||||||
| (ii) | \(1=a+a^{2}+a^{3}\) |
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| (iii) | \(a+2ab+3b\) |
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| (iv) | \(100x+100y\) |
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| (v) | \(-x^{2}y^{2}+7xy\) |
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| (vi) | \(1.2x+0.8y\) |
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| (vii) | \(3.14x^{2}\) | \(3.14x^{2}\) | 3.14 | ||||||
| (viii) | \(2(a+b)\) |
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| (ix) | \(0.1x+0.01x^{2}\) |
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Q4:
(a) Identify the terms which contain ‘a’ and give the coefficient of a.
(i) \(b^{2}a+b\)
(ii) \(13b^{2}-8ab\)
(iii) \(a+b+15\)
(iv) \(5+m+ma\)
(v) \(1+a+ab\)
(vi) \(12ab^{2}+10\)
(vii) \(7a+am^{2}\)
(b) Figure out the terms which contain \(b^{2}\) and also give the coefficient of the same term.
(i) \(8-ab^{2}\)
(ii) \(5b^{2}+10a\)
(iii) \(2a^{2}b-5ab^{2}+15b^{2}\)
Sol:
| S.no | Expression | Terms with factor a | Coefficient of a | ||||
| (i) | \(b^{2}a+b\) | \(b^{2}a\) | \(b^{2}\) | ||||
| (ii) | \(13b^{2}-8ab\) | \(-8ab\) | \(-8b\) | ||||
| (iii) | \(a+b+15\) | a | 1 | ||||
| (iv) | \(5+m+ma\) | ma | m | ||||
| (v) | \(1+a+ab\) |
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| (vi) | \(12ab^{2}+10\) | \(12ab^{2}\) | \(12b^{2}\) | ||||
| (vii) | \(7a+am^{2}\) |
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(b)
| S.no | Expression | Terms containing \(b^{2}\) | Coefficient of \(b^{2}\) | ||||
| (i) | \(8-ab^{2}\) | \(-ab^{2}\) | \(-a\) | ||||
| (ii) | \(5b^{2}+10a\) | \(5b^{2}\) | 5 | ||||
| (iii) | \(2a^{2}b-5ab^{2}+15b^{2}\) |
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Q5: Classify into monomials, binomials and trinomials:
(i) \(4b-7a\)
(ii) \(b^{2}\)
(iii) \(a+b-ab\)
(iv) \(50\)
(v) \(ab+b+a\)
(vi) \(5+10x\)
(vii) \(15a^{2}b-10ab^{2}\)
(viii) \(10yz\)
(ix) \(x^{2}+10x-5\)
(x) \(x^{2}+y^{2}\)
(xi) \(x^{2}+y\)
(xii) \(a^{2}+a+50\)
Sol:
| S.no | Expression | Type of Polynomial |
| (i) | \(4b-7a\) | Binomial |
| (ii) | \(b^{2}\) | Monomial |
| (iii) | \(a+b-ab\) | Trinomial |
| (iv) | \(50\) | Monomial |
| (v) | \(ab+b+a\) | Trinomial |
| (vi) | \(5+10x\) | Binomial |
| (vii) | \(15a^{2}b-10ab^{2}\) | Binomial |
| (viii) | \(10yz\) | Monomial |
| (ix) | \(x^{2}+10x-5\) | Trinomial |
| (x) | \(x^{2}+y^{2}\) | Binomial |
| (xi) | \(x^{2}+y\) | Binomial |
| (xii) | \(a^{2}+a+50\) | Trinomial |
Q6: State whether a given pair of term is of like or unlike terms:
(i) 1,100
(ii) \(-20x, \frac{1}{2}x\)
(iii) \(-10x, -10 y\)
(iv) \(50ab,30ba\)
(v) \(2 a^{2}b,8ab^{2}\)
(vi) \(10ab, 20 a^{2}b\)
Sol:
| S.no | Pair of terms | Like/Unlike terms |
| (i) | 1,100 | Like terms |
| (ii) | \(-20x, \frac{1}{2}x\) | Like terms |
| (iii) | \(-10x, -10 y\) | Unlike terms |
| (iv) | \(50ab,30ba\) | Like terms |
| (v) | \(2 a^{2}b,8ab^{2}\) | Unlike terms |
| (vi) | \(10ab, 20 a^{2}b\) | Unlike terms |
Q7: Identify the like terms in the following:
(a) \(-a^{2}b,-4ab^{2},9a^{2},2ab^{2},10a,-20a^{2},-30a, -5a^{2}b,-2ab,35a\)
(b) \(10pq,10p,5q,2p^{2}q^{2},-5pq,-50q,-30,18p^{2}q^{2},55,100p,-30pq, 105p^{2}q,-200\)
Sol:
(a) Like terms are:
(i) \(-a^{2}b, -5a^{2}b\)
(ii) \(-4ab^{2}, 2ab^{2}\)
(iii) \(9a^{2},-20a^{2} \)
(iv) \(10a,-30a,35a\)
(v) \(-2ab \)
(b) Like terms are:
(i) \(10pq,-5pq,-30pq \)
(ii) \( 10p,100p,\)
(iii) \(5q, -50q \)
(iv) \(2p^{2}q^{2}, 18p^{2}q^{2}\)
(v) \(-30, 55,-200\)
(vi) \(105p^{2}q \)
Exercise 12.2
Q1: Simplify the terms:
(i) \(21a-32+7a-20a\)
(ii) \(-x^{2}+13x^{2}-5x+7x^{3}-15x\)
(iii) \(a-(a-b)-b-(b-a)\)
(iv) \(3x-2y-xy-(x-y+xy)+3xy+y-x\)
(v) \(5a^{2}b-5a^{2}+3a^{2}b-3b^{2}+a^{2}-b^{2}+8ab^{2}-3b^{2}\)
(vi) \((3b^{2}+5b-4)-(8b-b^{2}-4)\)
Sol:
(i) \(21a-32+7a-20a=21a+7a-20b-32\)
\(\Rightarrow 8b-32\)
(ii) \(-x^{2}+13x^{2}-5x+7x^{3}-15x=7x^{3}+13x^{2}-x^{2}-5x-15x\)
\(=7x^{3}+12x^{2}-20x\)
(iii) \(a-(a-b)-b-(b-a)=a-a+b-b-b+a\)
\(=a-b\)
(iv) \(3x-2y-xy-(x-y+xy)+3xy+y-x=3x-2y-xy-x+y-xy+3xy+y-x\)
\(=3x-x-x+y+y-2y-xy-xy+3xy\) \(=x-2xy+3xy\)
(v) \(5a^{2}b-5a^{2}+3a^{2}b-3b^{2}+a^{2}-b^{2}+8ab^{2}-3b^{2}\)
\(5a^{2}b+3a^{2}b+8ab^{2}-5a^{2}+a^{2}-3b^{2}-b^{2}-3b^{2}=8a^{2}b+8ab^{2}-4a^{2}-7b^{2}\)
(vi) \((3b^{2}+5b-4)-(8b-b^{2}-4)\)
\(3b^{2}+5b-4-8b+b^{2}+4=3b^{2}+b^{2}+5b-8b+4-4\) \(=4b^{2}-3b\)
Q2: Add:
(i) \(3mn,-5mn,8mn,-4mn\)
(ii) \(a-8ab,3ab-b,b-a\)
(iii) \(-7mn+5,12mn+2, 9mn-8, 2mn-3\)
(iv) \(a+b-3,b-a+3,a-b+3\)
(v) \(14x+10y-12xy-13,18-7x-10y+8xy,4xy\)
(vi) \(5m-7n,3n-4m+2,2m-3mn-5\)
(vii) \(4x^{2}y,-3xy^{2},-5xy^{2},5x^{2}y\)
(viii) \(3p^{2}q^{2}-4pq+5,-10p^{2}q^{2},15+9pq+7p^{2}q^{2}\)
(ix) \(ab-4a,4b-ab,4a-4b\)
(x) \(x^{2}-y^{2}-1,y^{2}-1-x^{2},1-x^{2}-y^{2}\)
Sol:
(i) \(3mn,-5mn,8mn,-4mn\)
\(3mn+(-5mn)+8mn(-4mn)=(3-5+8-4)mn\) \(=(2)mn\)
(ii) \(a-8ab,3ab-b,b-a\)
\(a-8ab+3ab-b+b-a=a-a+b-b-8ab+3ab\) \(=-5ab\)
(iii) \(-7mn+5,12mn+2, 9mn-8, 2mn-3\)
\(-7mn+5+12mn+2+9mn-8+2mn-3=-7mn+12mn+9mn+5+2-8-3\) \(=(-7+12+9)mn+(5+2-8-3)=14mn+2\)
(iv) \(a+b-3,b-a+3,a-b+3\)
\(a+b-3+b-a+3+a-b+3=a+a-a+b+b-b+3+3-3\) \(=(1+1-1)a+(1+1-1)b+(3+3-3)=a+b+3\)
(v) \(14x+10y-12xy-13,18-7x-10y+8xy,4xy\)
\(=14x-7x+10y-10y+8xy+4xy-12xy+18-13=7x+18\) \(=7x+18\)
(vi) \(5m-7n,3n-4m+2,2m-3mn-5\)
\(5m-4m+2m-7n+3n+2-5-3mn=3m-4n-3mn-3\)
(vii) \(4x^{2}y,-3xy^{2},-5xy^{2},5x^{2}y\)
\(4x^{2}y+(-3xy^{2})+(-5xy^{2})+5x^{2}y=4x^{2}y+5x^{2}y-3xy^{2}-5xy^{2}\) \(=9x^{2}y-8xy^{2}\)
(viii) \(3p^{2}q^{2}-4pq+5,-10p^{2}q^{2},15+9pq+7p^{2}q^{2}\)
\(3p^{2}q^{2}-4pq+5+(-10p^{2}q^{2})+15+9pq+7p^{2}q^{2}=3p^{2}q^{2}+7p^{2}q^{2}-10p^{2}q^{2}+9pq-4pq+15-5\) \(=5pq+10\)
(ix) \(ab-4a,4b-ab,4a-4b\)
\(ab-4a+4b-ab+4a-4b=4a-4a+4b-4b+ab-ab\) \(=0\)
(x) \(x^{2}-y^{2}-1,y^{2}-1-x^{2},1-x^{2}-y^{2}\)
\(x^{2}-y^{2}-1+y^{2}-1-x^{2}+1-x^{2}-y^{2}=x^{2}-x^{2}-x^{2}+y^{2}-y^{2}-y^{2}+1-1-1\) \(=-x^{2}-y^{2}-1\)
Q3: Subtract:
(i) \(-5y^{2}\) from \(y^{2}\)
(ii) \(6xy\) from \(-12xy\)
(iii) \((a-b)\) from \((a+b)\)
(iv) \(a(b-5)\) from \(b(5-a)\)
(v) \(-m^{2}+5mn\) from \(4m^{2}-3mn+8\)
(vi) \(-x^{2}+10x-5\) from \(5x-10\)
(vii) \(5a^{2}-7ab+5b^{2}\) from \(3ab-2a^{2}-2b^{2}\)
(viii) \(4pq-5q^{2}-3p^{2}\) from \(5p^{2}+3q^{2}-pq\)
Sol:
(i) \(y^{2}-(-5y^{2})\)
\(=y^{2}+5y^{2}\)
\(=6y^{2}\)
(ii) \(-12xy-6xy\)
\(=-18xy\)
(iii) \((a+b)-(a-b)\)
\(=a+b-a+b\) \(=2b\)
(iv) \(b(5-a)-a(b-5)\)
\(=5b-ab-ab+5a\) \(=5a+5b-2ab\)
(v) \(4m^{2}-3mn+8-(-m^{2}+5mn)\)
\(=4m^{2}-3mn+8+m^{2}-5mn\) \(=5m^{2}-8mn+8\)
(vi) \(5x-10-(-x^{2}+10x-5)\)
\(=5x-10+x^{2}-10x+5\) \(=x^{2}-5x-5\)
(vii) \(3ab-2a^{2}-2b^{2}-(5a^{2}-7ab+5b^{2})\)
\(=3ab-2a^{2}-2b^{2}-5a^{2}+7ab-5b^{2}\) \(=3ab+7ab-2a^{2}-5a^{2}-2b^{2}-5b^{2}\) \(=10ab-7a^{2}-7b^{2}\)
(viii) \(5p^{2}+3q^{2}-pq-(4pq-5q^{2}-3p^{2})\)
\(=5p^{2}+3q^{2}-pq-4pq+5q^{2}+3p^{2}\) \(=5p^{2}+3p^{2}+3q^{2}+5q^{2}-pq-4pq\) \(=8p^{2}+8q^{2}-5pq\)
Q4: (a) What should be added to \(x^{2}+xy+y^{2}\) to obtain \(2x^{2}+3xy\) ?
(b) What should be subtracted from \(2a+8b+10\) to get \(-3a+7b+16\)?
Sol:
(a) Let a should be added
Then according to the question
\(x^{2}+xy+y^{2}+a=2x^{2}+3xy\)
\(\Rightarrow a=2x^{2}+3xy-(x^{2}+xy+y^{2})\)
\(\Rightarrow a=2x^{2}+3xy-x^{2}-xy-y^{2}\)
\(\Rightarrow a=2x^{2}-x^{2}-y^{2}+3xy-xy\)
\(\Rightarrow a=x^{2}-y^{2}+2xy\)
Hence the value of a comes out to be \(x^{2}-y^{2}+2xy\).
Hence \(x^{2}-y^{2}+2xy\) should be added.
(b) Let b should be subtracted
Then according to the question,
\(2a+8b+10-q= -3a+7b+16\)
\(2a+8b+10-q= -3a+7b+16\)
\( q = 2a+8b+10-( -3a+7b+16)\)
\( q = 2a+8b+10+3a-7b-16\)
\( q = 2a+3a+8b-7b+10-16\)
\( q = 5a+b-6\)
Q5: What should be taken from 3x2-4y2+5xy+20 to obtain –x2-y2+6xy+20 ?
Sol:
Let a be subtracted
Then according to the question,
3x2 -4y2+5xy+20 – q= –x2-y2+6xy+20
q= 3x2 -4y2+5xy+20 -(–x2-y2+6xy+20)
q= 3x2-4y2+5xy+20+x2+y2-6xy-20
q=3x2+x2-4y2+y2+5xy-6xy +20 -20
q=4x2-3y2-xy
Hence, 4x2-3y2-xy should be subtracted in the given equation.
Q6:
(a) From the sum of 3x – y + 11 and – y – 11, subtract 3x – y – 11.
(b) From the sum of 4 + 3x and 5 – 4x + 2x2 , subtract the sum of 3x2 – 5x and –x2 + 2x + 5.
Sol:
(a)According to the question
(3x – y + 11)+( – y – 11)-( 3x – y – 11)= 3x – y + 11 – y – 11- 3x + y + 11
= 3x-3x+y-y+11+11-11
=11
(b) According to question,
(4 + 3x)+( 5 – 4x + 2x2)-( 3x2 – 5x)-( –x2 + 2x + 5)
= 4 + 3x+ 5 – 4x + 2x2– 3x2 + 5x +x2 – 2x – 5
= 3x-4x+5x-2x +2x2– 3x2+x2+4+5-5
= 2x+4
Exercise 12.3
Q1: If a=2, find the values of:
(i) a-2
(ii) 3a-5
(iii) 9-5a
(iv) \(3a^{2}-2a-7\)
(v) \(\frac{5m}{2}-4\)
Sol:
(i) a-2 =2-2 (Putting a=2)
=0
(ii) 3a-5= \(3\times 2-5\) (Putting a=2)
=1
(iii) 9-5a=\(9-5\times 2\) (Putting a=2)
= -1
(iv) \(3a^{2}-2a-7=3\times 2^{2}-2\times 2-7\) (Putting a=2)
=12-4-7
=1
(v) \(\frac{5m}{2}-4 =\frac{5\times 2}{2}-4 =5-4\) (Putting a=2)
=1
Q2: If x=-2, find
(i) 4x+7
(ii) \(-3x^{2}+4x+7\)
(iii) \(-2x^{3}-3x^{2}+4x+7\)
Sol:
(i) 4x+7=4(-2)+7 (Putting x= -2)
= -8+7=-1
(ii) \(-3x^{2}+4x+7=-3(-2)^{2}+4(-2)+7\)
= -3(4)-8+7=-12-8+7
= -13
(iii) \(-2x^{3}-3x^{2}+4x+7=-2(-2)^{3}-3(-2)^{2}+4(-2)+7\) (Putting x= -2)
= -2(-8)-3(4)+4(-2)+7
= 16-12-8+7
=3
Q3: Find the value of the following expressions, when x= -1:
(i) 5x-35
(ii) -2x+4
(iii) \(3x^{2}+6x+3\)
(iv) \(6x^{2}-3x-6\)
Sol:
(i) 5x-35 = 5(-1)-35 =-5-35 [Putting x= -1 ]
= -40
(ii) -2x+4 = -2(-1)+4 [Putting x= -1 ]
= 2 + 4 = 6
(iii) \(3x^{2}+6x+3\) = \(3(-1)^{2}+6(-1)+3\) [Putting x= -1 ]
= 3-6+3 =0
(iv) \(6x^{2}-3x-6\) = \(6(-1)^{2}-3(-1)-6\) [Putting x= -1 ]
= 6+1-6 =1
Q 4: If x=2, y= -2, find the value of:
(i) \(x^{2}+y^{2}\)
(ii) \(x^{2}+xy+y^{2}\)
(iii) \(x^{2}-y^{2}\)
Sol:
(i) \(x^{2}+y^{2}\) = \(2^{2}+(-2)^{2}\) [Putting a=2, b= -2 ]
= 4 + 4 = 8
(ii) \(x^{2}+xy+y^{2}\) = \(2^{2}+2(-2)+(-2)^{2}\) [Putting a=2,b= -2 ]
= 4 – 4 + 4 = 4
(iii) \(x^{2}-y^{2}\) = \((2)^{2}-(-2)^{2}\) [Putting a=2, b= -2]
= 4 – 4 = 0
Q5: When x=0,y= -1, find the value of the given expressions:
(i) 2x+2y
(ii) \(2x^{2}+y^{2}+1\)
(iii) \(2x^{2}y+2xy^{2}+xy\)
(iv) \(x^{2}+xy+2\)
Sol:
(i) 2x+2y = 2(0)+2(-1) [Putting x=0,y= -1 ]
= 0 – 2 = -2
(ii) \(2x^{2}+y^{2}+1\) = \(2(0)^{2}+(-1)^{2}+1\) [Putting x=0, y=-1 ]
= 0 + 1 + 1 = 2
(iii) \(2x^{2}y+2xy^{2}+xy\) = \(2(0)^{2}(-1)+2(0)(-1)^{2}+0(-1)\) [Putting x=0, y= -1]
= 0 + 0 + 0 = 0
(iv) \(x^{2}+xy+2\) = \((0)^{2}+(0)(-1)+2\) [Putting x=0, y= -1 ]
= 0 + 0 + 2 = 2
Q6: Simplify the following expressions and find the value at a= 2:
(i) a+7+4(a-5)
(ii) 3(a+2)+5a-7
(iii) 10a+4(a-2)
(iv) 5(3a-2)+4a+8
Sol:
(i) a+7+4(a-5) = a+7+4a-20
=4a+a+7-20 =5a-13
= 5(2)-13 =10-13 [Putting a=2 ]
= -3
(ii) 3(a+2)+5a-7 = 3a+6+5a-7
= 3a+5a+6-7 = 8a-1
= 8( 2) – 1 [Putting a=2 ]
= 16 – 1 = 15
(iii) 10a+4(a-2) = 10a+4a-8
= 14a-8
= 14( 2) – 8 [Putting a= 2 ]
= 28 – 8 = 20
(iv) 5(3a-2)+4a+8 = 15a-10+4a+8
=15a+4a-10+8 = 19a-2 [Putting =2 ]
= 19(2)-2 = 38-2
= 36
Q7: Simplify the expression given below and find the value at x=3, y= -1, z= -2 :
(i) 8x-10-3x+5
(ii) 10-5x+3x+6
(iii) 5y+3-2y+6
(iv) 5-8z-12-4z
(v) 3y-5z-6x+15
Sol:
(i) 8x-10-3x+5 = 8x-3x-10+5
=5x-5 = 5(3)-5 [Putting x=3 ]
= 15-5 = 0
(ii) 10-5x+3x+6 = 10+6-5x+3x
= 16-2x = 16-2(3) [Putting x= 3 ]
= 16-6 =10
(iii) 5y+3-2y+6 = 5y-2y+3+6
= 3y+9 = 3(-1)+9 [Putting y= -1 ]
= -3 + 9 = 6
(iv) 5-8z-12-4z = 5-12-8z-4z
= -7-12 z [Putting z= -2 ]
= -7 -12(-2) = -7+24
= 17
(v) 3y-5z-6x+15
= 3(-1)-5(-2)-6(3)+15 [Putting x=3, y=-1, z=-2]
= -3+10-18+15
= 25-21
= 4
Q8:
(i) If x= 10, find the value of \(x^{3}-3x^{2}-5x+15\) .
(ii) If y= -10, find the value of \(2y^{2}-3y+50\)
Sol:
(i) \(x^{3}-3x^{2}-5x+15\)
= \(10^{3}-3(10)^{2}-5(10)+15\) [Putting x=10 ]
=1000-300-50+15
= 665
(ii) \(2y^{2}-3y+50\)
=\(2(-10)^{2}-3(-10)+50\) [Putting y= -10 ]
=200+30+50
=280
Q9: What should be the value of p if the value of \(2a^{2}+a-p=5\) equals to 5, when a=0 ?
Sol:
\(2a^{2}+a-p=5\)
\(2(0)^{2}+0-p=5\) [ Putting x= 0 ]
\(-p=5\)
Hence, the value of p is -5.
Q10: Simplify the expression and find its value when x= 5 and y= -3:
\(2(x^{2}+xy)+3-xy\).
Sol:
Given:
\(2(x^{2}+xy)+3-xy\)
\(\Rightarrow 2x^{2}+2xy+3-xy\)
\(\Rightarrow 2x^{2}+2xy-xy+3\)
\(\Rightarrow 2x^{2}+xy+3\)
\(\Rightarrow 2(5)^{2}+(5)(-3)+3\) [Putting x=5, y= -3 ]
\(\Rightarrow 2(25)+(-15)+3\)
\(\Rightarrow 50-15+3\) \(\Rightarrow 38\)
Exercise 12.4
Q1: Observe the pattern made from the line segment, which are of equal length which are found in display of calculators and digital speedometer: If n is the number of digits, and the number of required segments to form the digit n is given by the algebraic expression on the right of the digit. So how many segments are required to form 5,10,100 digits of the kind .
Sol:
| S.no | Symbols | Digit’s number | Pattern Formulae | No. of segments | ||||||
| (i) |
|
\(5n+1\) |
|
|||||||
| (ii) |
|
\(3n+1\) |
|
|||||||
| (iii) |
|
\(5n+2\) |
|
(i) \(5n+1\)
Putting n=5, \(5\times 5+1=25+1=26\)
Putting n=10, \(5\times 10+1=50+1=51\)
Putting n=100, \(5\times 100+1=500+1=501\)
(ii) \(3n+1\)
Putting n=5, \(3\times 5+1=15+1=16\)
Putting n=10, \(3\times 10+1=30+1=31\)
Putting n=100, \(3\times 100+1=300+1=301\)
(iii) \(5n+2\)
Putting n=5, \(5\times 5+2=25+2=27\)
Putting n=10, \(5\times 10+2=50+1=52\)
Putting n=100, \(5\times 100+2=500+1=502\)
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