Linear Equations in Two Variables

1. For what values of a and b does the following pair of equations have an infinite number of solutions.

        2x+3y=7
        a(x+y)-b(x-y)=3a+b-2

2. For what value of k will the following equations have infinitely many solutions?
        2x-3y=7
        (k+1)x+(1-2k)y=5k-4

3. The sum of denominator and numerator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is denominator is decreased by 1, the fraction becomes 1/2. Find the fraction.

4. A boat covers 14 kms in upstream and 20 kms downstream in 7 hours. Also it covers 22 kms upstream and 34 kms downstream in 10 hours. Find the speed of the boat in still water and of that the stream.

5. Draw the graph of 2x+y=6 and 2x-y+2=0. Shade the region bounded by these lines and x axis. Find the area of the shaded region.

6. When you add two numbers and the number obtained by reversing the order of its digits is 165. If the both numbers differ by three, find the number.

7. The sum of the digits of a two-digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number.

8. 4 men and 6 boys can finish a piece of work in 5 days while 3 men and 4 boys can finish it in 7 days. Find the time taken by 1 man alone and by 1 boy alone.

9. A man travels 600km apart by train and partly by car. It takes 8 hours and 40 minutes if he travels 320 km by train and rest by car. It would take 30 minutes more if he travels 200 km by train and the rest by the car/. Find the speed of the train and by car separately.

10. Solve the equations
        2x+y=2
        2y-x=4

11. Solve the following pair of equations.
        x+3y=6
        2x-3y=12

Polynomials

 

1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
        (i) 4u² + 8u
        (ii) t² – 15
        (iii) 3x² – x – 4
        (iv) 4s² – 4s + 1

2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
        (i) 1/4, -1
        (ii) 0, √5
        (iii) √2, 1/3 

3. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.

        (i) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x
        (ii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

4. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
        (i) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1
        (ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

5. Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are √(5/3) and –√(5/3)

6. On dividing (x³ – 3x² + x + 2) by a polynomial g(x), the quotient and remainder were (x – 2) and (–2x + 4), respectively. Find g(x).

7. If one zero of the quadratic polynomial x² + 3x + k is 2, then calculate the value of k.

8. Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficients of the polynomials:

        (i) 4x² – 3x – 1
        (ii) 3x² + 4x – 4