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QUADRATIC EQUATIONS
OMTEX CLASSES www.omtexclasses.com
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.]
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2. Write the following quadratic equations in standard form ax
2 + 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]
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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]
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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]
QUADRATIC EQUATIONS
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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.]
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(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.]
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(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.]
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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]
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(viii) 4p2 + 7 = 12p [Ans]
(ix) 6m2 + m = 2 [Ans]
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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.]
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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.]
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(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.]
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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.]
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(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.]
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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.]
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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.]
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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.]
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EXERCISE - 2.9
Solve the following equations.
(i) x4 – 3x
2 + 2 = 0 [Ans.]
(ii) (x2 + 2x) (x
2 + 2x – 11) + 24 = 0 [Ans.]
(iii) 2(x2 +
1/x
2 ) – 9(x+
1/x) + 14 = 0 [Ans.]
(iv) 35y2 +
12/y
2 = 44 [Ans.]
(v) x2 +
12/x
2 = 7 [Ans.]
(vi) (x2 + x) (x
2 + x – 7) + 10 = 0 [Ans.]
(vii) 3x4 – 13x
2 + 10 = 0 [Ans.]
(viii) 2y2 +
15/y
2 = 12 [Ans.]
QUADRATIC EQUATIONS
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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]
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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]
