1. Home
  2. |  
  3. CBSE Board |  

NCERT Exemplar Solution for CBSE Class 10 Mathematics: Quadratic Equations (Part-II)

Jun 28, 2017 13:23 IST

    Class 10 Maths NCERT Exemplar, Quadratic Equations NCERT ExemplarHere you get the CBSE Class 10 Mathematics chapter 4, Quadratic Equations: NCERT Exemplar Problems and Solutions (Part-II). This part of the chapter includes solutions for Exercise 4.2 of NCERT Exemplar Problems for Class 10 Mathematics Chapter: Quadratic Equations. This exercise comprises of only the Short Answer Type Questions framed from various important topics in the chapter. Each question is provided with a detailed solution.

    NCERT Exemplar problems are a very good resource for preparing the critical questions like Higher Order Thinking Skill (HOTS) questions. All these questions are very important to prepare for CBSE Class 10 Mathematics Board Examination 2017-2018 as well as other competitive exams.

    Find below the NCERT Exemplar problems and their solutions for Class 10 Mathematics Chapter, Quadratic Equations:

    Exercise 4.2

    Very Short Answer Type Questions

    Question11. State whether the following quadratic equations have two distinct real roots. Justify your answer.

    Equations with distinct real roots

    Solution:

    A quadratic equation, ax2 + bx + c = 0; a ≠ 0 will have two distinct real roots if its discriminant, D = b2 - 4ac > 0.

    (i) Given equation is x2 – 3x + 4 = 0

     

    On comparing with standard equation we have,

    a = 1, b = –3 and c = 4

    Now, D = b2 – 4ac = (–3)2 –4(1) (4) = 9 – 16 = – 7< 0

    Hence, the equation x2 –3x + 4 = 0 has no real roots.

    (ii) Given equation is, 2x2 + x – 1 = 0

    On comparing with standard equation we have,

    a = 2, b = 1 and c = –1

    Now, D = b2 – 4ac = (1)2 – 4(2) (–1) = 1 + 8 = 9 > 0

    Hence, the equation 2x2 + x – 1 = 0 has two distinct real roots.

    Quadratic Equations NCERT Exemplar

    (iv) Given, 3x2 –4x + 1 = 0

    On comparing with standard equation we have,

    a = 3, b = –4 and c = 1

    Now, D = b2 –4ac = (–4)2 – 4(3) (1) = 16 – 12 = 4 > 0

    Hence, the equation 3x2 – 4x + 1 = 0 has two distinct real roots.

    (v) Given, (x + 4)2 –8x = 0

    Simplifying the above equation, we have:

                x2+ 16 + 8x – 8x = 0        [∵(a + b)2 = a2+ 2ab + b2]

    ⟹        x2 + 16 = 0

    ⟹       x2 + 0. x + 16 = 0

    On comparing with standard equation we have:

    a = 1, b = 0 and c = 16

    Now, D = b2 –4ac = (0)2 – 4 (1) (16) = – 64 < 0

    Hence, the equation (x + 4)2 – 8x = 0 has imaginary roots.

    Class 10 Maths NCERT Exemplar

    (viii) Given, x (1 - x) -2 = 0

    Simplifying above equation we have:

    x - x2 − 2 = 0

    x2x + 2 = 0

    On comparing with standard equation we have,

    a = 1, b = –1 and c = 2

    Now, D = b2 – 4ac = (-1)2 –4(1) (2)=1 – 8 = – 7 < 0

    Hence, the equation x (1 – x) –2 = 0 has imaginary roots.

    (ix) Given, (x – 1) (x + 2) + 2 = 0

    Simplifying above equation we have:

    x2 + x – 2 + 2 = 0

    x2 + x + 0 = 0

    On comparing with standard equation we have,

    a = 1, b = 1 and c = 0

    Now, D = b2 – 4ac = 1 – 4(1) (0) = 1 > 0

    Hence, equation (x – 1) (x + 2) + 2 = 0 has two distinct real roots.

    (x) Given, (x + 1) (x – 2) + x = 0

    Simplifying above equation we have:

    x2 + x – 2x – 2 + x = 0

    x2 – 2 = 0

    x2 + 0 x – 2 = 0

    On comparing with ax2 + bx + c = 0, we have

    a = 1, b = 0 and c = –2

    Now, D = b2 –4ac = (0)2 – 4(1) (–2) = 0 + 8 = 8 > 0

    Hence, the equation (x + 1) (x – 2) + x = 0 has two distinct real roots.

    Question 2. Write whether the following statements are true or false. Justify your answers.

    (i) Every quadratic equation has exactly one root.

    (ii) Every quadratic equation has atleast one real root.

    (iii) Every quadratic equation has atleast two roots.

    (iv) Every quadratic equation has almost two roots.

    (v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.

    (vi) If the coefficient of x2and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.

    Solution:

    (i) False. For example, a quadratic equation x2 – 9 = 0 has two distinct roots – 3 and 3.

    (ii) False. For example, equation x2 + 4 = 0 has no real root.

    (iii) False. For example, a quadratic equation x2 – 4x + 4 = 0 has only one root which is 2.

    (iv) True, because every quadratic polynomial has almost two roots.

    (v) True, because in this case discriminant is always positive. For example, in ax2+ bx + c = 0, as a and c have opposite sign so, ac < 0 ⟹ Discriminant = b2 – 4ac > 0.

    (vi) True, because in this case discriminant is always negative. For example, in ax2+ bx + c = 0, as b = 0, and a and c have same sign then ac > 0 ⟹ discriminant = b2 – 4ac = – 4 a c < 0

    Question3. A quadratic equation with integral coefficient has integral roots. Justify your answer.

    Solution:

    No, a quadratic equcation with integral coefficient (0, ± 1, ± 2, ± 3…) can have fractional or non integral roots.

    For example, the quadratic equation 8x2 – 2x – 1 = 0 have integral coefficients (8, −2 and −1)

    quadratic equation with integral coefficient and non integral roots

    Question4. Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

    Solution:

    Yes, a quadratic equation having coefficients as rational number, has irrational roots.

    For example, the quadratic equation x2 – 4x – 3 = 0 has rational coefficients.

    Now, b2 – 4ac = 16 + 12 = 28.

    Class 10 maths NCERT exemplar

    Question5. Does there exist a quadratic equation whose coefficient are all distinct irrationals but both the roots are rationals? Why?

    Solution:

    Yes, there may be a quadratic equation whose coefficients are all distinct irrationals, but both the roots are rational.

    Class 10 Maths NCERT Exemplar

    Hence roots are rational.

    Question6. Is 0.2 a root of the equation x2 –0.4 = 0? Justify your answer.

    Solution:

    No, because 0.2 does not satisfy the given quadratic equation, as,

    (0.2)2 – 0.4 = 0.04 – 0.4 = − 0.36 ≠ 0.

    Question7. If b = 0, c < 0, is it true that the roots of x2 + bx + c = 0 are numerically equal and opposite in sign? Justify your answer.

    Solution:

    Given quadratic equation is, x2 + bx + c = 0   ...(i)

    Also given that b = 0 and c < 0

    Putting b = 0 in equation (i), we get:

     

    Hence, roots of x2 + bx +c = 0 are numerically equal and opposite in sign.Quadratic Equations NCERT Exemplar

    You may also like to read:

    CBSE Class 10 Mathematics Syllabus 2017-2018

    CBSE Class 10 NCERT Textbooks & NCERT Solutions

    NCERT Solutions for CBSE Class 10 Maths

    NCERT Exemplar Problems and Solutions Class 10 Science: All Chapters

    Latest Videos

    Register to get FREE updates

      All Fields Mandatory
    • (Ex:9123456789)
    • Please Select Your Interest
    • Please specify

    • ajax-loader
    • A verifcation code has been sent to
      your mobile number

      Please enter the verification code below

    This website uses cookie or similar technologies, to enhance your browsing experience and provide personalised recommendations. By continuing to use our website, you agree to our Privacy Policy and Cookie Policy. OK
    X

    Register to view Complete PDF