QUADRATIC EQUATIONS

EXERCISE - 2.1

1. Which of the following are quadratic equations ?

(i) 11 = – 4x2 – x3          [Ans.]

(ii) -¾ y2 = 2y + 7          [Ans.]

(iii) (y – 2) (y + 2) = 0      [Ans.]

(iv) 3/y – 4 = y      [Ans.]

(v) m3 + m + 2 = 4m      [Ans.]

(vi) n – 3 = 4n      [Ans.]

(vii) y2 – 4 = 11y      [Ans.]

(viii) z – 7/z = 4z + 5    [Ans.]

(ix) 3y2 – 7 = √3 y      [Ans.]

(x) (q2 – 4)/q2 = - 3  [Ans.]

2. Write the following quadratic equations in standard form ax2 + bx + c = 0

(i) 7 – 4x –x2 = 0  [Ans]

(ii) 3y2 = 10y + 7     [Ans]

(iii) (m + 4) (m – 10) = 0    [Ans]

(iv) p(p – 6) = 0    [Ans]

(v) (x2/25)  – 4 = 0     [Ans]

(vi) n – (7/n) = 4     [Ans]

(vii) y2 – 9 = 13y     [Ans]

(viii) 2z – (5/z) = z – 6     [Ans]

(ix) x2 = –7 – √10 x     [Ans]

(x) (m2 +5)/m2  = –3     [Ans]

EXERCISE - 2.2

1. In each of the examples given below determine whether the values given against each of the quadratic equation are the roots of the equation or not.

(i) x2 + 3x – 4 = 0,  x = 1, –2, – 3 [Ans]

(ii) 4m2 – 9 = 0, m = 2, 2/3, 3/2   [Ans]

(iii) x2 + 5x – 14 = 0, x = √2 , –7, 3    [Ans]

(iv) 2p2 + 5p – 3 = 0, p = 1, ½,  –3    [Ans]

(v) n2 + 4n = 0, n = 0, – 2, – 4   [Ans]

2. If one root of the quadratic equation x2 – 7x + k = 0 is 4, then find the value of k.     [Ans]

3. If one root of the quadratic equation 3y2 – ky + 8 = 0 is 2/3,  then find the value of k.      [Ans]

4. State whether k is the root of the given equation y2 – (k – 4)y – 4k = 0.    [Ans]

5. If one root of the quadratic equation kx2 – 7x + 12 = 0 is 3, then find the value of k.     [Ans]

EXERCISE - 2.3

Solve the following quadratic equations by

factorization method..

(i) x2 – 5x + 6 = 0 [Ans.]

(ii) x2 + 10x + 24 = 0 [Ans.]

(iii) x2 – 13x – 30 = 0 [Ans.]

(iv) x2 – 17x + 60 = 0 [Ans.]

(v). m2 – 84 = 0 [Ans.]

(vi) x + 20/x – 12 = 0 [Ans.]

(vii) x2 = 2(11x – 48) [Ans.]

(viii) 21x = 196 – x2  [Ans.]

(ix) 2x - 10/x = 1 [Ans.]

(x) x2 – x – 132 = 0 [Ans.]

(xi) 5x2 – 22x – 15 = 0 [Ans.]

(xii) 3x2 – x – 10 = 0 [Ans.]

(xiii) 2x2 – 5x – 3 = 0 [Ans]

(xiv) x (2x + 3) = 35 [Ans.]

(xv) 7x2 + 4x – 20 = 0 [Ans]

(xvi) 10x2 + 3x – 4 = 0   [Ans.]

(xvii) 6x2 – 7x – 13 = 0  [Ans.]

(xviii) 3x2 + 34x + 11 = 0 [Ans.]

(xix) 3x2 – 11x + 6 = 0 [Ans.]

(xx) 3x2 – 10x + 8 = 0  [Ans.]

(xxi) 2m2 + 19m + 30 = 0 [Ans.]

(xxii) 7m2 – 84 = 0 [Ans.]

(xxiii)  x2 – 3√3 x + 6 = 0 [Ans.]

EXERCISE - 2.4

Solve the following quadratic equations by completing square.

(i) x2 + 8x + 9 = 0 [Ans]

(ii) z2 + 6z – 8 = 0 [Ans]

(iii) m2 – 3m – 1 = 0 [Ans]

(iv) y2 = 3 + 4y [Ans]

(v) p2 – 12p + 32 = 0 [Ans]

(vi) x (x – 1) = 1 [Ans]

(vii) 3y2 + 7y + 1 = 0 [Ans]

(viii) 4p2 + 7 = 12p [Ans]

(ix) 6m2 + m = 2  [Ans]

EXERCISE - 2.5

1. Solve the following quadratic equations by using formula.

(i) m2 – 3m – 10 = 0 [Ans]

(ii) x2 + 3x – 2 = 0 [Ans]

(iii) x2 + (x – 1)/3 = 0 [Ans]

(iv) 5m2 – 2m = 2 [Ans.]

(v) 7x + 1 = 6x2 [Ans.]

(vi) 2x2 – x – 4 = 0 [Ans.]

(vii) 3y2 + 7y + 4 = 0 [Ans.]

(viii) 2n2 + 5n + 2 = 0 [Ans.]

(ix) 7p2 – 5p – 2 = 0 [Ans.]

(x) 9s2 – 4 = – 6s [Ans.]

(xi) 3q2 = 2q + 8 [Ans.]

(xii) 4x2 + 7x + 2 = 0 [Ans.]

EXERCISE - 2.6

1. Find the value of discriminant of each of the following equations :

(i) x2 + 4x + 1 = 0   [Ans]

(ii) 3x2 + 2x – 1 = 0  [Ans]

(iii) x2 + x + 1 = 0  [Ans]

(iv) √3 x2 + 2√2 x – 2√3 = 0 [Ans]

(v) 4x2 + kx + 2 = 0 [Ans]

(vi) x2 + 4x + k = 0  [Ans]

2. Determine the nature of the roots of the following equations from their  discriminants :

(i) y2 – 4y – 1 = 0 [Ans.]

(ii) y2 + 6y – 2 = 0  [Ans.]

(iii) y2 + 8y + 4 = 0  [Ans.]

(iv) 2y2 + 5y – 3 = 0  [Ans.]

(v) 3y2 + 9y + 4 = 0  [Ans.]

(vi) 2x2 + 5√3 x + 16 = 0   [Ans.]

3. Find the value of k for which given equation has real and equal roots :

(i) (k – 12)x2 + 2 (k – 12)x + 2 = 0  [Ans.]

(ii) k2x2 – 2 (k – 1)x + 4 = 0 [Ans.]

EXERCISE - 2.7

1. If one root of the quadratic equation kx2 – 5x + 2 = 0 is 4 times the other, find k.   [Ans.]

2. Find k, if the roots of the quadratic equation x2 + kx + 40 = 0 are in the ratio 2 : 5.   [Ans.]

3. Find k, if one of the roots of the quadratic equation kx2 – 7x + 12 = 0 is 3.   [Ans.]

4. If the roots of the equation x2 + px + q = 0 differ by 1, prove that p2= 1 + 4q.  [Ans.]

5. Find k, if the sum of the roots of the quadratic equation 4x2 + 8kx + k + 9 = 0 is equal to their product. [Ans.]

6. If  α  and β  are the roots of the equation x2 – 5x + 6 = 0, find [Ans.]

(i) α2+β2

(ii) α/β +β/α

7. If one root of the quadratic equation kx2 – 20x + 34 = 0 is 5 – 2√2 , find k. [Ans.]

EXERCISE - 2.8

1. Form the quadratic equation if its roots are

(i) 5 and – 7     [Ans.]

(ii) ½  and – ¾     [Ans.]

(iii) - 3 and –11    [Ans.]

(iv) -2 and  11/2    [Ans.]

(v) ½  and – ½      [Ans.]

(vi) 0 and – 4       [Ans.]

2. Form the quadratic equation if one of the root is

(i) 3 – 2√ 5 [Ans.]

(ii) 4 – 3√ 2  [Ans.]

(iii) √ 2 + √3 [Ans.]

(iv) 2√3 – 4 [Ans.]

(v) 2+√5  [Ans.]

(vi) √ 5 - √3 [Ans.]

3. If the sum of the roots of the quadratic is 3 and sum of their cubes is 63, find the quadratic equation. [Ans.]

4. If the difference of the roots of the quadratic equation is 5 and the difference  of their cubes is 215, find the quadratic equation. [Ans.]

EXERCISE - 2.9

Solve the following equations.

(i) x4 – 3x2 + 2 = 0 [Ans.]

(ii) (x2 + 2x) (x2 + 2x – 11) + 24 = 0 [Ans.]

(iii) 2(x2 + 1/x2 ) – 9(x+1/x) + 14 = 0 [Ans.]

(iv) 35y2 + 12/y2 = 44 [Ans.]

(v) x2 + 12/x2 = 7 [Ans.]

(vi) (x2 + x) (x2 + x – 7) + 10 = 0 [Ans.]

(vii) 3x4 – 13x2 + 10 = 0 [Ans.]

(viii) 2y2 + 15/y2 = 12 [Ans.]

EXERCISE - 2.10

1. The sum of the squares of two consecutive natural numbers is 113. Find the numbers. [Ans]

2. Tinu is younger than Pinky by three years. The product of their ages is 180. Find their ages.[Ans]

3. The length of the rectangle is greater than its breadth by 2 cm. The area of the rectangle is 24 sq.cm, find its length and breadth.[Ans]

4. The sum of the squares of two consecutive even natural numbers is 100. Find the numbers. [Ans]

5. A natural number is greater than twice its square root by 3. Find the number. [Ans]

6. The sum of a natural number and its reciprocal is 10/3 . Find the number. [Ans]

7. The sum of the ages of father and his son is 42 years. The product of their ages is 185, find their ages. [Ans]

8. Three times the square of a natural numbers is 363. Find the numbers. [Ans]

9. The length of one diagonal of a rhombus is less than the second diagonal by 4 cm. The area of the rhombus is 30 sq.cm. Find the length of the diagonals. [Ans]

10. A natural number is greater than the other  by 5. The sum of their squares is 73. Find those numbers. [Ans]

11. The sum ‘S’ of the first ‘n’ natural numbers is given by S = n (n + 1)/2 . Find ‘n’, if the sum (S) is 276. [Ans]

12. A rectangular playground is 420 sq.m. If its length is increases by 7 m and breadth is decreased by 5 metres, the area remains the same. Find the length and breadth of the playground ? [Ans]

13. The cost of bananas is increased by Re. 1 per dozen, one can get 2 dozen less for Rs. 840. Find the original cost of one dozen of banana.  [Ans]