BUNDAY TUTORIAL
Lagos State University
MR BUNDAY: 07011534711, 07053845576
MAT 101 PRACTICE QUESTIONS
THEORY OF QUADRATIC EQUATIONS
1. Find a if the equation (5a + 1)x2 − 8ax + 3a = 0 has equal roots
2. If the equation (7p + 1)x2 + (5p − 1)x + p = 1 has equal roots, find p
3. For what values of k does the equation x2 − (4 + k)x + 9 = 0 have real roots
4. Find the greatest value of λ for which the equation (λ − 1)x2 − 2x + (λ − 1) = 0 has real roots
5. The equation x2 + 2px + p2 + q 2 = r2 has equal roots. Show that r2 ≥ q 2
6. Find the values of a if the equation (a + 3)x2 − (11a + 1)x + a = 2(a − 5) has equal roots
7. Show that x2 + 4x + 13 > 0 for all values of x
8. Show that 16x2 − 24x + 10 > 0 for all values of x
9. If α and β are the roots of the equation 2x2 − 7x − 3 = 0, find the values of:
β β β
(a) α + β (b) αβ (c) αβ 2 + α2 β (d) α2 + β 2 (e) α3 + β 3 (f) α1 + β1 (g) β+1
α
+ α+1 (h) α
α+1 + β+1 (i) α
β + α (j) 3
β + 3
α
10. If α and β are the roots of the quadratic equation 4x2 − 5x − 1 = 0, find the quadratic equations whose roots are:
(a) α + 1 and β + 1 (b) 2 − α and 2 − β (c) α2 and β 2 (d) α1 and β1 (e) α22 and β22
11. If α and β are the roots of the quadratic equation px2 + qx + r = 0, express (2 + α2 )(2 + β 2 ) in terms of p,q and r
12. One of the roots of the quadratic equation ax2 + bx + c = 0 is twice the other. Show that 2b2 = 9ac
13. If α and β are the roots of the quadratic equation ax2 + bx + c = 0, express (1 − α3 )(1 − β 3 ) in terms of a,b and c
14. One of the roots of the quadratic equation x2 − (4 + p)x + 12 = 0 is three times the other. Find (a) the roots of the equation (b)
the possible values of p
1 1 α2 β2
15. If α and β are the roots of the equation 5x2 − 3x − 1 = 0, form the equations which have the roots (i) α2 and β2 (ii) β and α
16. One root of the equation x2 − px + q = 0 is the square of the other. Show that p3 − q(3p + 1) − q 2 = 0 provided q 6= 1
α β
17. If α and β are the roots of the equation ax2 + bx + c = 0, form the equation whose roots are β2 and α2
18. If one root of the equation px2 + qx + r = 0 is four times the other, show that 4q 2 − 25pr = 0
19. Form the quadratic equation for which the sum of the roots is 5 and the sum of the squares of the roots is 53
p
20. Find the relationship which must exist between a, b and c if the roots of the equation ax2 + bx + c = 0 are in the ratio q
21. γ and δ are the roots of the equation px2 + qx + r = 0. Find in terms of p, q and r. (i) γ + δ (ii) γ − δ (iii) γ 2 + δ 2 (iv) γ 2 − δ 2
(v) γ 3 + δ 3 (vi) γ 3 − δ 3 (vii) γ 4 + δ 4 (viii) γ 4 − δ 4
22. Find the set of values of p for which the equation x2 + px + 2p − 3 = 0 has no real roots (A) 2 < p < 6 (B) −2 < p < 6 (C)
−3 < p < 2 (D) 2 < p < 3
√ √
23. The roots of the equation x2 − 9x + 20 = 0 are α and β, where α > β. Find α − β (A) 3 (B) 1 (C) 3 (D) 3
24. If the roots of the equation x2 − 2(p − 2)x + 2p − 10 = 0 are real. Find the possible values of p when the roots of the equation
differ by 6 (A) -5,-1 (B) 5,1 (C) 5,-1 (D) -5,1
1 1
25. If α2 and β 2 are the roots of x2 − 21x + 4 = 0 and α and β are both positive. Find the equation with roots α2 and β2 (A)
x2 − 11x + 4 = 0 (B) 9x2 − 11x − 4 = 0 (C) x2 + 29 x + 32 = 0 (D) 9x2 + 11x + 4 = 0
26. The roots of the quadratic equation x2 + 2x + 3 = 0 are denoted by α, β. Without solving the equation. Find the quadratic
equation whose roots are α + β1 , β + α1 (A) x2 − 4x + 5 = 0 (B) x2 − 4x − 5 = 0 (C) 3x2 + 8x + 16 = 0 (D) 3x2 − 8x + 1 = 0
−1 −3 −7 1
27. Let α and β be the roots of the equation 3x2 − 7x − 1 = 0, find αβ (A) 3 (B) 7 (C) 3 (D) 3
28. Write the quadratic function f (x) = 3x2 −7x−1 in completed square form (A) 3(x− 76 )2 + 61 7 2 85 7 2 61
12 (B) 3(x− 6 ) + 36 (C) 3(x− 6 ) − 12
7 2 85
(D) 3(x − 6 ) − 36
29. If α and β are the roots of the equation x2 − 4x + 5 = 0. Find the value of α2 + β 2 (A) 5 (B) 6 (C) 4 (D) -6
Each question attracts #500. Come to the tutorial today, solve any question correctly and get #500
VENUE: Benson hall, opp. MBA, faculty of science.
JOIN BUNDAY TUTORIAL TODAY...............your distinctions are our success
Lagos State University
MR BUNDAY: 07011534711, 07053845576
MAT 101 PRACTICE QUESTIONS
THEORY OF QUADRATIC EQUATIONS
1. Find a if the equation (5a + 1)x2 − 8ax + 3a = 0 has equal roots
2. If the equation (7p + 1)x2 + (5p − 1)x + p = 1 has equal roots, find p
3. For what values of k does the equation x2 − (4 + k)x + 9 = 0 have real roots
4. Find the greatest value of λ for which the equation (λ − 1)x2 − 2x + (λ − 1) = 0 has real roots
5. The equation x2 + 2px + p2 + q 2 = r2 has equal roots. Show that r2 ≥ q 2
6. Find the values of a if the equation (a + 3)x2 − (11a + 1)x + a = 2(a − 5) has equal roots
7. Show that x2 + 4x + 13 > 0 for all values of x
8. Show that 16x2 − 24x + 10 > 0 for all values of x
9. If α and β are the roots of the equation 2x2 − 7x − 3 = 0, find the values of:
β β β
(a) α + β (b) αβ (c) αβ 2 + α2 β (d) α2 + β 2 (e) α3 + β 3 (f) α1 + β1 (g) β+1
α
+ α+1 (h) α
α+1 + β+1 (i) α
β + α (j) 3
β + 3
α
10. If α and β are the roots of the quadratic equation 4x2 − 5x − 1 = 0, find the quadratic equations whose roots are:
(a) α + 1 and β + 1 (b) 2 − α and 2 − β (c) α2 and β 2 (d) α1 and β1 (e) α22 and β22
11. If α and β are the roots of the quadratic equation px2 + qx + r = 0, express (2 + α2 )(2 + β 2 ) in terms of p,q and r
12. One of the roots of the quadratic equation ax2 + bx + c = 0 is twice the other. Show that 2b2 = 9ac
13. If α and β are the roots of the quadratic equation ax2 + bx + c = 0, express (1 − α3 )(1 − β 3 ) in terms of a,b and c
14. One of the roots of the quadratic equation x2 − (4 + p)x + 12 = 0 is three times the other. Find (a) the roots of the equation (b)
the possible values of p
1 1 α2 β2
15. If α and β are the roots of the equation 5x2 − 3x − 1 = 0, form the equations which have the roots (i) α2 and β2 (ii) β and α
16. One root of the equation x2 − px + q = 0 is the square of the other. Show that p3 − q(3p + 1) − q 2 = 0 provided q 6= 1
α β
17. If α and β are the roots of the equation ax2 + bx + c = 0, form the equation whose roots are β2 and α2
18. If one root of the equation px2 + qx + r = 0 is four times the other, show that 4q 2 − 25pr = 0
19. Form the quadratic equation for which the sum of the roots is 5 and the sum of the squares of the roots is 53
p
20. Find the relationship which must exist between a, b and c if the roots of the equation ax2 + bx + c = 0 are in the ratio q
21. γ and δ are the roots of the equation px2 + qx + r = 0. Find in terms of p, q and r. (i) γ + δ (ii) γ − δ (iii) γ 2 + δ 2 (iv) γ 2 − δ 2
(v) γ 3 + δ 3 (vi) γ 3 − δ 3 (vii) γ 4 + δ 4 (viii) γ 4 − δ 4
22. Find the set of values of p for which the equation x2 + px + 2p − 3 = 0 has no real roots (A) 2 < p < 6 (B) −2 < p < 6 (C)
−3 < p < 2 (D) 2 < p < 3
√ √
23. The roots of the equation x2 − 9x + 20 = 0 are α and β, where α > β. Find α − β (A) 3 (B) 1 (C) 3 (D) 3
24. If the roots of the equation x2 − 2(p − 2)x + 2p − 10 = 0 are real. Find the possible values of p when the roots of the equation
differ by 6 (A) -5,-1 (B) 5,1 (C) 5,-1 (D) -5,1
1 1
25. If α2 and β 2 are the roots of x2 − 21x + 4 = 0 and α and β are both positive. Find the equation with roots α2 and β2 (A)
x2 − 11x + 4 = 0 (B) 9x2 − 11x − 4 = 0 (C) x2 + 29 x + 32 = 0 (D) 9x2 + 11x + 4 = 0
26. The roots of the quadratic equation x2 + 2x + 3 = 0 are denoted by α, β. Without solving the equation. Find the quadratic
equation whose roots are α + β1 , β + α1 (A) x2 − 4x + 5 = 0 (B) x2 − 4x − 5 = 0 (C) 3x2 + 8x + 16 = 0 (D) 3x2 − 8x + 1 = 0
−1 −3 −7 1
27. Let α and β be the roots of the equation 3x2 − 7x − 1 = 0, find αβ (A) 3 (B) 7 (C) 3 (D) 3
28. Write the quadratic function f (x) = 3x2 −7x−1 in completed square form (A) 3(x− 76 )2 + 61 7 2 85 7 2 61
12 (B) 3(x− 6 ) + 36 (C) 3(x− 6 ) − 12
7 2 85
(D) 3(x − 6 ) − 36
29. If α and β are the roots of the equation x2 − 4x + 5 = 0. Find the value of α2 + β 2 (A) 5 (B) 6 (C) 4 (D) -6
Each question attracts #500. Come to the tutorial today, solve any question correctly and get #500
VENUE: Benson hall, opp. MBA, faculty of science.
JOIN BUNDAY TUTORIAL TODAY...............your distinctions are our success
