10th Class Test Linear Equation , Real Numbers , Polynomials

10th Class 

Test 

 Linear Equation , Real Numbers , Polynomials 

1. If α and β are the zeroes of the quadratic polynomial f(x) = x² – 5x + k such that α - β =1, then find the  value of k
2. If α and β are the zeroes of the quadratic polynomial f(t)=t²−4t+3, find the value of α⁴β³+ α³β⁴
3. If one zero of the polynomial (a2 + 9)x² + 13x + 6a is reciprocal of the other, find the value of a.
4. Find the zeroes of the quadratic polynomial p(x) = x² – (Ö3+1)x+Ö3 and verify the relationship between the zeroes and its coefficients.
5. A baker has 444 sweet biscuits and 276 salty biscuits. He wants to stack them in such a way that each stack has the same number and same type of biscuits and they take up the least area of the tray. What is the number of biscuits that can be placed in each stack for this purpose?
6. Show that any positive even integer is of the form 6m, 6m + 2 or 6m + 4. where m is some integer.
7. Find the value of k for which the following system of equations have infinitely many solutions.                      2x – 3y = 7, (k + 2)x – (2k + 1)y = 3(2k – 1)
8. Determine a and b for which the following system of linear equations has infinite number of solutions                             2x – (a-4)y = 2b + 1 , 4x – (a – 1)y = 5b – 1
9. Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
10. Solve for u and v by changing into linear equations 2(3u – v) = 5uv; 2(u + 3v) = 5uv.
11. It takes 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately?
12. There are two classrooms A and B containing students. If 5 students are shifted from room A to room B, the resulting number of students in the two rooms become equal. If 5 students are shifted from room B to room A, the resulting number of students in room A becomes double the number of students left in room B. Find the original number of students in the two rooms.