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4. Quadratic Equations

Quadratic Equations

Chapter 4. Quadratic Equations 

Class 10 Maths Chapter 4 Quadratic Equations : Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :

 Class 10 Quadratic Equations Multiple Choice Questions and Answers

SECTION = A

 Q1. The roots of the quadratic equation  are :

(a)  3 , –                    

(b)  – 3  ,                 

(c)  3  ,                      

(d) – 3 , –  

Solution:  (c)  3  ,                     

[ We have ,  

 

 

So,    or     

Q2. The nature of quadratic equation  are :

(a) two distinct real roots      

(b) two equal roots         

(c) no real roots      

(d) none of these . 

Solution: (b) two equal roots  

[ We have,  

Here,   ,      ,   

The discriminant  

Hence, the given quadratic equation has two equal real roots .       ]

Q3. A quadratic equation  has two distinct real roots , then

(a)      

(b)       

(c)     

(d)

Solution:       (a)       .

Q4.  A quadratic equation  has coincident roots (two equal roots) , then  [SEBA 2014]

(a)      

(b)     

(c)      

(d)  

Solution:  (d)    .

Q5. The quadratic equation whose roots are 1 and  , then the equation is :

(a)     

(b)        

(c)       

(d)

Solution:  (d)  

[ The quadratic equation,

 

 

     ]

Q6. Which constant should be added and subtracted to solve the quadratic equation  by the method of completing the square ?

(a)                       

(b)                           

(c)                         

(d)   

Solution:  (a)        

[ We have ,

    ]

Q7.  Which of the following equations has 2 as a root ?  

(a)          

(b)           

(c)         

(d)   

Solution: (c)    

 [  Given ,  

         ]

Q8.  A quadratic equation  has no  real roots , then

(a)     

(b)     

(c)      

(d)  

Solution: (b)       .

Q9. Which of the following is not a quadratic equation ?

(a)                                  

(b)   

(c)                                

(d) 

Solution:     (d)   .

Q10. If   is a root of the equation   , then the value of  is :

(a)  2                       

(b)   – 2                     

(c)                           

(d)   

Solution:  (a)  2   

[  Given ,      

 

 

 

        ]

Q11. If the roots of  are reciprocal of each other, then

(a)               

(b)               

(c)              

(d) 

Solution: (d)      

[  let  and  are two roots .

A/Q,         ]

Q12. If the roots of the equation  are in the ratio 3 : 2 , then  is :

 (a)  – 5                         

(b)  + 5                      

(c)                        

(d)  6

Solution: (c)    

[  let  and  are two roots . 

Given ,   

  

    

and  

 

 

   ]

Q13. Which of the following equations has the sum of its roots as 3 ?

 (a)                                         

(b) 

(c)                                       

(d)   

Solution:  (b)   .   

[ We have,  the sum of roots   ]

Q14. The discriminant of the quadratic equation  is    [CBSE2020 basic]

 (a)  12                        

(b) 84                           

(c)                                 

(d)  – 12 

Solution:  (d)  – 12 

[ We have ,   ;

Here ,   ,    ,  

The discriminant   ]

Q15. The value of  for which the quadratic equation  has equal roots , is

(a)  4                             

(b)                        

(c) – 4                   

(d) 0

Solution:  (b)                         

[ We have, 

 A/Q,    

     ] 

Q16. Which one of the following is not a quadratic equation ?  [SEBA 2017]

(a)                             

(b)    

(c)                                  

(d)  

Solution:  (b)      .

Q17. One root of a quadratic equation is 2 and sum of the two roots is 0 , the equation is : [SEBA 2016]

 (a)              

(b)               

(c)              

(d)   

Solution: (b)               

[  Here,    

and   

The quadratic equation, 

 

  

        ]

Q18. The roots of a quadratic equation are and  , then the equation is :    [SEBA 2013]

 (a)      

(b)        

(c)      

(d)    

Solution:  (b)    

 [ The quadratic equation,   

     ]

Q19. Under what condition the roots of the quadratic equation  will be real and unequal ?   [ SEBA 2015]

(a)             

(b)            

(c)             

(d)  

Solution:  (c)    .   

Q20. The roots of the quadratic equation   is :

(a)  – 5 , – 1                

(b)  2 , 3                   

(c)  6 ,  – 1                  

(d)  – 2 , – 3   

Solution:  (a)  – 5 , – 1  

[  We have ,   

 

  ] 

 Class 10 Quadratic Equations Fill in the blanks

Q1. If  – 5 is a root of the quadratic equation  , then the value of  is  .

Solution:   7           

[     Given,  

 

  

  

  

    ]    

Q2. If the quadratic equation has equal roots , then the value of  is    .

Solution:   

[  We have,   and let  be the root of quadratic equation.

A/Q, 

 

and  

 

 

  

      ]

Q3. If  and  are the roots of equation  and  , then is equal to  .

Solution:   – 24    

[ We have,   

A/Q,   

and      

 

  

From  , we get 

      ]

Q4. Given  and  are roots of quadratic equation, if  and , then the equation is  .

Solution:      .

[ Given ,  and  

The quadratic equation,  

 

     ]

Q5. If the quadratic equation  has two equal real roots , then  is  .

Solution:             

[ We have,    

  

 

     ]

Q6. The discriminant of the quadratic equation  is  .

Solution:  0        

[ We have,   

Here,  ,  ,

The discriminant    ]  

 Class 10 Quadratic Equations Answers following the question

Q1. If  is one root of the quadratic equation, then find the value of    .  [CBSE2018]

Solution:   Given ,  

 

 

Q2. Find the nature of roots of quadratic equation   [CBSE 2019]

Solution:  We have ,   

 Here,   ,      , 

 The discriminant 

  

Therefore, the given equation has no real roots .

Q3. Find the roots of the quadratic equation  

Solution:  We have , 

 

Q4. Check whether the equation is a quadratic equation : 

Solution:  We have,  

 

  is a quadratic equation.

Q5. Find the roots of the quadratic equation :  

Solution: We have,

 

  

  

    or  

Q6. If  is a solution of the quadratic equation  , find the value of  . [DELHI 2015]                                                                                                      

Solution:  Given,  

    We have ,  

    

  

    

Q7. If  and  are the roots of  , then find the value of  .  

Solution:  Since ,  and  are the roots of   .

    

and     

 Now,        

   CASE  STUDY Section = II

Case study based questions are compulsory . Attempt any four sub parts of each question . Each subpart carries 1 marks.

Q1.  A quadratic equation in the variable  is of the form  , where  are real numbers and  .

(a)  Which of the following is a quadratic equation ?

 (i)                                    

(ii)   

(iii)                           

(iv)   

(b)  The product of roots of quadratic equation  is :

(i)           (ii)         (iii)       (iv) 

(c) If  is a example of quadratic equation ,then the roots of  given equation is :

(i)       (ii)       (iii)        (iv)  

(d)  The nature of the quadratic equation  is :

(i)  coincident roots    (ii)  no real roots      

(iii)  two distinct roots   (iv)  two equal roots

Solution:  (a)  (iv)    

 [ We have , 

  ]

(b)  (iii)                 

[ The product of  roots  .  ]

(c)  (i)                    

 [ We have ,

 

 

 

    ]

  (d) (ii)  no real roots .

  [  Here,   ,    ,  

 The discriminant  

   

 So, there are no real roots for the given equation .  ] 

Q3. Suppose a charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth . [ We assume  be the breadth of the prayer hall ]

            

  Answer the questions based upon this situation.

  (a) What is the length of the prayer hall  (in metres)  ?

   (i)                         

(ii)                       

(iii)                      

(iv)                

(b)  What is the quadratic  equation of the above statement ?

(i)                                                     

(ii)                            

(iii)                                                    

(iv)                            

(c) What should be the length and breadth of the hall  ( in metres) ?

(i)  24  , 12                      

(ii)  26 , 12                    

(iii) 25 , 12                

(iv) 25 , 13

(d)  In the given figure, if 13m decrease of length of the prayer hall and  13m increase of the breadth , then the shape of prayer hall is :

 (i)  Rectangle                     

(ii)  Square                    

(iii)  parallelogram            

(iv)  Rhombus

Solution:  (a)   (iii)              

 [ Given   be the breadth of prayer hall , then the length of prayer hall    ]

 (b)  (iv)          

 [ Given  be the breadth of prayer hall , then the length of prayer hall is  

  A/Q ,  

          ]

(c)   (iii)  25 , 12

[  Given   be the breadth of prayer hall , then the length of prayer hall is  

A/Q ,       

 

  

  

          

or    

 Thus,  the breadth of prayer hall is 12 m and  the length of prayer hall is  m      ]

(d)   (ii)   Rectangle  .   

Class 10 Quadratic Equations 2 Marks Questions and Answers

SECTION = B

Q1. Write the nature of roots of quadratic equation :   .

Solution:  We have ,  

 

  

    

 Here,   ,     ,

   The discriminant

Thus , the given equation has two distinct real roots .

Q2. Find the roots of the quadratic equation  using the quadratic formula .

Solution:  We have ,

Here ,    ,         ,   

Using the quadratic formula ,  

 

Thus, the roots are     and    .

Q3. Solve for   :    [CBSE 2014F , 2016]

Solution:  We have , 

 

  

 

      or  

Q4.  Check whether the equation is a quadratic equation : 

 Solution:   We have,  

 

 

It is of the form of   . So, the given equation is a quadratic equation .        

Q5. Solve the quadratic equation  for  .  [Delhi 2014]

Solution:   We have,

 

 

 

or   

Q6. If   and  are roots of the quadratic equation  , find the values of  and  .  [CBSE 2016]

Solution:  Since, and  are roots of  

   The sum of roots 

 

and

Product of roots  

   

Q7. Find the roots of the quadratic equation  .

Solution: We have,

 

 

    or  

 Therefore , the roots are   and    .

Q8. Find the roots of the quadratic equation

Solution:  We have ,

            or  

Thus, the roots of the quadratic equation are    and     .

Q9. Solve for    : 

Solution:  We have , 

Here ,    ,      ,

Using the quadratic formula , 

 

    or  

  or     

Q10. Find the roots the equation:        ;   

Solution:  We have,    

  

  

  

       ; 

Here,     ,      , 

Using the quadratic formula , 

 

 

Thus , the roots are   and     .

Q11. Find the value of  for the following quadratic equation , so that it has two equal roots :   .[SEBA 2020]

 Solution:  We have ,  

  

  Here ,     ,    ,    

      

 

 

 

 

   (impossible)    or   

Therefore, the value of  is   . 

Class 10 Quadratic Equations 3 Marks Questions and Answers     

SECTION = C

Q1. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is  . Find his present age .

Solution : Let  (in years)be present age of Rehman .

3 years ago , the age of Rehman was  years 

and 5 years from now , Rehman age will be  years .

A/Q ,   

   

 

 

 

 

 

   or  

 Therefore, the roots of the equation is 7 and – 3 .

Q2. In a class test, the sum of Shefali’s marks in Mathematics and English is 30 . Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210 . Find her marks in the two subjects .

Solution : Let  be the marks in Mathematics of Shefali and her English marks will be  .

A/Q ,

 

 

 

 

 

 

 

 

    or  

Therefore, the marks obtained by Shefali is 12 or 18 and 13 or 17 respectively .

Q3. Solve the equation    , for   .  [Delhi 2014 , CBSE 2013]

Solution: We have,   

 

  

  

  

 

 

 

  

  

  

          or     

Thus , the value of  are  1 or  – 2  .

Q4. Solve for    :     

Solution:   We have ,   

 

     

or   

Therefore , the value of  are   and    .

Q6. Find that non-zero value of  , for which the quadratic equation  has equal roots. Hence find the roots of the equation. [Delhi2015 , CBSE2002C]

Solution:  We have,  

  Here ,    ,    ;    

     

  

  

  

  

  

  

      (impossible)   or    

   

 

  

 

 

            

or         

Thus, the equation of the roots are    and     . 

Q7. Find the roots of the equation  by the method of completing the square .

Solution:  We have,

  

  

  

  

 

    or 

Q8. Solve the following quadratic equation by applying the quadratic formula : 

Solution:  Here,      ,  ,  

Applying the quadratic formula,    

  

  

 

 

 

    or  

   or  

   or  

Q9. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them :

Solution:  Here ,   ,      ,    

  

 Hence, the given quadratic equation has two equal real roots  and the roots are exist .

 Applying quadratic formula,

  or

Q10. Find two consecutive odd positive integers, sum of whose squares is 290 .

Solution:  let the two consecutive odd positive integers are  and  respectively .

A/Q ,       

         or  

Thus , the two consecutive odd integers are 11 and 13    . 

Q11. Find two consecutive positive integers, sum of whose squares is 365 .

Solution:  Let  and are two consecutive positive integers respectively .

  A/Q ,  

 

 

 

 

 

 

      or 

So, the two consecutive positive integers are 13 and 14 (= 13+1) .

Q12. The altitude of a right triangle is 7 cm less than its base . If the hypotenuse is 13 cm , find the other two sides .

Solution : Let  (in cm) be the altitude of a right triangle and the base is  cm

  A/Q,  

 

 

 

 

 

  

  or  (impossible)

So, the two sides of the triangle are 12 cm and 5 cm (12 – 7 = 5) .

Class 10 Quadratic Equations 4 Marks Questions and Answers SECTION = D

Q1. Sum of the areas of two squares is 468   . If the difference of their perimeters is 24 m , find the sides of the two squares .  [SEBA 2016]

Solution:  let  and  are the  side of two square respectively.

 A/Q ,     

 

 

  

 and    

 

  

  

  

  

  

  

  

  

     (impossible)    

or   

Putting   in  , we get 

Therefore,  m and m are the  side of two square respectively.

Q2. Sum of the areas of two squares is 544   . If the difference of their perimeters is 32 m , find the sides of the two squares . [CBSE 2020]

Solution:  let   and   are the  side of two square respectively.

A/Q ,     

 

and      

 

 

 

 

 

 

 

 

         (impossible)   or  

Putting  in  , we get   

Therefore,  m and m are the  side of two square respectively.

Q3.  Find in terms of  ,  and      ;   [CBSE 2016]

Solution:  We have ,    

 

 

  

 

 

 

 

 

  

         or   

         or     

Q4.If the roots of the equation  are equal, then prove that that 

Solution:  Given, the equation is  

 Here ,   ,    ;    

A/Q ,    

 

 

 

 

 

      Proved.

Q5. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed . If it takes 3 hours to complete the total journey , what is its first speed ? .

Solution:  let,  (in km/h) be the speed of the first train.

A/Q,     3  

 3

 

 

 

 

 

 

 

 

         or  

             

 Therefore , the speed of the first train is 36 km/h.

Q6. If the equation  has equal roots , show that  . [CBSE 2018]

 Solution:  Given, the equation is  

   ;       and    

         

  

  

 

 

      

       proved.

 Q7. Find the roots of the equations :    

Solution:  We have ,   

      

= 0

    

 

   or

    

Therefore , the roots of the equations are 1 and 2 .

Q8. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Solution:   let,  be the shorter side of a rectangular field and the longer side will be (  m

 Therefore, the diagonal of a rectangular field is  m .

    A/Q ,    

 

 

 

        

 

      

 

          

and     (Impossible)

 Thus, the shorter side of a rectangular field is 90 m and the longer side is 

Q9. If the roots of the equation  are equal, prove that   

 Solution:  We have ,  

 Here,    ;     ;    

 A/Q ,       

 

 

 

    

  

  

       Proved.

Q10. A motor boat whose sped is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot . Find the speed of the stream .  [CBSE 2018]

Solution:  let  (in km/h) be the speed of the stream .

So, the speed of the boat upstream km/h and  the speed of the boat downstream  km/h

 The time taken to go upstream  

and the time taken go downstream  

A/Q,

     (impossible)

or     

Thus, the speed of the stream is 6 km/h .