Class 9 Mathematics Ganita Manjari Part 1 NCERT Solutions
Chapter-2: Introduction to Linear Polynomials

Exercise Set 2.2

1. Find the value of the linear polynomial \bm{5x-3} if:
Solution:
Given polynomial: p(x)=5x-3

(i) \bm{x=0}
     p(0)=5(0)-3
             =0-3
             =-3

(ii) \bm{x=-1}
      p(1)=5(-1)-3
             =-5-3
             =-8

(iii) \bm{x=2}
        p(2)=5(2)-3
               =10-3
               =7

2. Find the value of the quadratic polynomial \bm{7s^2-4s+6} if:
Solution:
Given polynomial: p(s)=7s^2-4s+6

(i) \bm{s=0}
     p(0)=7(0)^2-4(0)+6
            =0-0+6
            =6

(ii) \bm{s=-3}
      p(-3)=7(-3)^2-4(-3)+6
               =7(9)+12+6
               =63+12+6
               =81

(iii) \bm{s=4}
       p(4)=7(4)^2-4(4)+6
              =7(16)-16+6
              =112-16+6
              =102

3. The present age of Salil’s mother is three times Salil’s present age. After 5 years, their ages will add up to 70 years. Find their present ages.
Solution:
Let Salil’s present age =x
       Mother’s present age =3x
After 5 years,
               Salil’s age =x+5
              Mother’s age =3x+5
According to the question:
          (x+5)+(3x+5)=70
          x+5+3x+5=70
          4x+10=70
          4x=70-10
          4x=60
          x=\frac{60}{4}
          x=15

∴ Salil’s present age =15 years
   Mother’s present age =3(15)
                                          =45 years

4. The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.
Solution:
Ratio of the two integers = 2:5
Let the two integers be: 2x\text{ and }5x
Their difference is 63.
        5x-2x=63
        3x=63
        x=\frac{63}{3}
        x=21
Therefore, the first integer is: 2x=2(21)=42
The second integer is: 5x=5(21)=105

5. Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total ₹88, how many coins does she have of each type?
Solution:
Let number of five-rupee coins =x.
Number of two-rupee coins =3x
Total rupees = ₹88.
    ∴ 5x+2(3x)=88
       5x+6x=88
       11x=88
       x=\frac{88}{11}
       x=8
Therefore, number of five-rupee coins =8
Number of two-rupee coins =3x=3(8)=24

6. A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?
Solution:
Let length of shorter piece  =x
       Length of longer piece =4x
Total length is 300 feet.
        x+4x=300
        5x=300
        x=\frac{300}{5}
        x=60
∴ Length of shorter piece =60\text{ feet}
    Length of longer piece =4x=4(60)
                                             =240\text{ feet}

7. If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?
Solution:
Let the width of the rectangle =x
So the length of rectangle =2x+3

Perimeter of a rectangle =24 cm
        2(\text{Length}+\text{Width})=24
        2(2x+3+x)=24
        2(3x+3)=24
        3x+3=\frac{24}{2}
        3x+3=12
        3x=12-3
        3x=9
        x=3

Therefore, width =3\text{ cm}
                Length =2(3)+3
                             =6+3
                             =9\text{ cm}