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**1.** A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.**Hint: **Let the natural number be x

∴ x + 12 = 160/x

⇒ x(x + 12) = 160

⇒ x^{2} + 12x – 160 = 0**2. **By increasing the list price of a book by ₹ 10, a person can buy 10 books less for ₹ 1200. Find the original list price of the book.**3.** The hypotenuse of a right-angled triangle is 1 cm more than twice the shortest side. If the third side is 2 cm less than the hypotenuse, find the sides of the triangle.**4. **A passenger train takes 2 hours less for a journey of 300 km, if its speed is increased by 5 km/hr from its usual speed. Find its usual speed.**5.** The numerator of a fraction is one less than its denominator. If three is added to each of the numerator and denominator, the fraction is increased by 3/28. Find the fraction.**6.** The difference of squares of two natural numbers is 45. The square of the smaller number is four times the larger number. Find the numbers**7.** Solve: **8.** A train travels a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km an hour, the journey would have taken two hours less. Find the original speed of the train.**9.** The speed of a boat in still water is 11 km/ hr. It can go 12 km upstream and returns downstream to the original point in 2 hours 45 minutes. Find the speed of the stream.**10.** Determine the value of k for which the quadratic equation 4x^{2} - 3kx + 1 = 0 has equal roots.**11.** Using quadratic formula, solve the following equation for ‘x’:

ab x^{2} + (b^{2} - ac) x - bc = 0**12. **The sum of the numerator and the denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.**13**. Rewrite the following as a quadratic equation in x and then solve for x:**14. **A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.**15.** A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/ hr more, it would have taken 30 minutes less for the journey. Find the original speed of the train.**16.** Solve for x: **17.** Using quadratic formula, solve the following for x: 9x^{2} − 3 (a^{2} + b^{2}) x + a^{2} b^{2} = 0

**18.** Find the equation whose roots are reciprocal of the roots of 3x^{2} − 5x + 7 = 0**19.** A number consists of two digits whose product is 18. When 27 is subtracted from the number, the digits change their places.

Find the number.**20. **A number consisting of two digits is seven times the sum of its digits. When 27 is subtracted from the number, the digits are reversed. Find the number.**21.** The sum of the squares of two consecutive odd numbers is 394. Find the nimbers.**Hint:**

Let the two consecutive odd numbers be x and x + 2.

∴ x^{2} + (x + 2)^{2} = 394

⇒ x^{2} + x^{2} + 4x + 4 = 394

⇒ 2x^{2} + 4x – 390 = 0

⇒ x^{2} + 2x – 195 = 0

⇒ x^{2} + 15x – 13x – 195 = 0

or

x(x + 15) – 13(x + 15) = 0

⇒ x= 13 or x = – 15

∴ For x = 13, x + 2 = 13 + 2 = 15

Thus, the required numbers are 13 and 15.

**22.** An aeroplane left 30 minutes later than its scheduled time and in order to reach its destination 1500 km away in time, it has to increase its speed by 250 km/hr from its usual speed. Determine its usual speed.

**23.** Using quadratic formula, solve for x:

**24.** Find the number which exceeds its positive square root by 20.

**25. **The sum of two numbers is 15 and the sum of their reciprocals is 1/3. Find the numbers.**26.** A two digit number is such that the product of its digits is 14. If 45 is added to the number, the digits interchange their places.

Find the number.**27. **A two digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.**28.** A two digit number is such that the product of its digits is 15. If 8 is added to the number, the digits interchange their places. Find the number.

**29. **Solve for x :

**30.** Solve for x: ab x2 + (b2 − ac) x − bc = 0

**31.** The sum of two numbers is 18. The sum of their reciprocals is 1/4. Find the numbers.

**33.** The sum of two numbers ‘a’ and ‘b’ is 15, and sum of their reciprocals 1/a and 1/b is 3/10. Find the numbers ‘a’ and ‘b’.

**34.** Solve for x: **35.** Find the roots of the following quadratic equation:**Hint:**

⇒ 2x^{2} − 5x − 3 = 0

⇒ (2x + 1) (x − 3) = 0**36.** Find the roots of the equation:**37. **A natural number when subtracted from 28, becomes equal to 160 times its reciprocal.

Find the number.

38. Find two consecutive odd positive integers, sum of whose squares is 290.**39. **Find the values of k for which the quadratic equation

(k + 4) x^{2} + (k + 1) x + 1 = 0 has equal roots.

Also find these roots.**Hint:** From (k + 4)x^{2} + (k + 1)x + 1 = 0, we get

a = (k + 4),b = (k + 1) and c = 1

∴ b^{2} – 4ac = (k + 1)^{2} – 4 (k + 4) (1)

= k^{2} – 2k – 15

For equal roots, b^{2} – 4ac = 0

∴ k^{2} – 2k – 15 = 0

⇒ k = 5 or k = – 3

For k = 5, we have (k + 4)x^{2} + (k + 1) x + 1 = 0

≡ 9x^{2} + 6x + 1 = 0

Solving for x, we get x = -1/3, -1/3

For k = – 3, we have

x^{2} – 2x + 1 = 0

∴ solving it, we get x = 1, x = 1**40. **Solve for x :**Hint:**

or

16(x + 1) – 15(x) = x(x + 1) ⇒ x^{2} = 1 6

x = ± 4**41. **Solve for x : **Hint:** L.H.S.

⇒

(By cross multiplication)

⇒

or

3[2x^{2} – 14x + 22]

= 10 [x^{2} – 8x + 15]

On simplification, we get

2x^{2} – 19x + 42 = 0 ⇒ x = 7/2**42.** Find the value of k, for which one root of quadratic equation kx^{2} – 14x + 8 = 0 is six times the other.**43.** If x = 2/3 and x = –3 are roots of the quadratic equation ax^{2} + 7x + b = 0, find the value of a and b.**44.** If –5 is a root of the quadratic equation 2x^{2} + px – 15 = 0 and the quadratic equation p (x^{2} + x) + k = 0 has equal roots, find the value of k.**45.** Find x in terms of a, b and c :**Hint :**

⇒ **ANSWERS****1.** 8

**2.** ₹ 30

**3.** 8 cm, 15 cm, 17 cm

**4. **25 km/hr

**5.** 3/4

**6. **9 and 6

**7. **

**8.** 25 km/hr

**9.** 5 km/hr

**10.**

**11.**

**12.** 5/7

**13.** –2, 1

**14.** 92

**15.** 45 km/hr

**16.** x = –5, 2

**17.****18.** 7x^{2} – 5x + 3 = 0

**19.** 63

**20.** 63

**21.** 14

**22.** 750 km/h

**23.** x = a/3 , b/3

**24.** x = 16

**25.** 5, 10

**26.** 27

**27.** 45

**28.** 35

**29.**

**30.** x = –b

**31.** 6, 12

**32. **12, 14

**33.** a = 5 or 10, b = 10 or 5

**34.** x = –a

**35.**

**36.****37.** 8

**38.** 11 and 13

**42**. k = 3

**43.** a = 3, b = –6

**44.** k = 7/4

**45.**

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