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Vidaara.orgClass 10 · Mathematics
CodeVID-M10-04-DIS-01
Discriminant & Nature of Roots — Assignment
Chapter: Quadratic Equations
Topic: Discriminant and Nature of Roots
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • All questions are compulsory.
  • Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
  • Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions 5 × 1 = 5 marks
1.
The discriminant is:
  • A.$b^2-4ac$
  • B.$b^2+4ac$
  • C.$4ac-b^2$
  • D.$2a$
2.
If $D>0$, roots are:
  • A.equal
  • B.real and distinct
  • C.not real
  • D.zero
3.
If $D=0$, roots are:
  • A.real and equal
  • B.distinct
  • C.not real
  • D.negative
4.
If $D<0$, roots are:
  • A.real and equal
  • B.distinct
  • C.not real
  • D.zero
5.
For $x^2-4x+4=0$, $D=$
  • A.$0$
  • B.$4$
  • C.$16$
  • D.$-4$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Find $D$ for $x^2-5x+6=0$.
7.
Find $D$ for $x^2+x+1=0$.
8.
State the nature of roots of $2x^2-4x+2=0$.
9.
State the nature of roots of $x^2+2x+5=0$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Find $k$ for which $x^2-4x+k=0$ has equal roots.
11.
For what $k$ does $kx^2-2x+1=0$ have real roots?
12.
Find the nature of the roots of $3x^2-2x+1=0$.
13.
Find $k$ if $2x^2+kx+3=0$ has equal roots.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Find the values of $k$ for which $2x^2+kx+8=0$ has equal roots, and find the roots for the positive $k$.
15.
If the roots of $(b-c)x^2+(c-a)x+(a-b)=0$ are equal, show that $2b=a+c$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $b^2-4ac$
  2. (B) real and distinct
  3. (A) real and equal
  4. (C) not real
  5. (A) $0$
Section B — Short Answer (2 marks)
  1. $1$.
  2. $-3$.
  3. Real and equal ($D=0$).
  4. Not real ($D<0$).
Section C — Short Answer (3 marks)
  1. $k=4$.
  2. $k\le1$.
  3. Not real ($D=-8$).
  4. $k=\pm2\sqrt6$.
Section D — Long Answer (5 marks)
  1. $k=\pm8$; for $k=8$, $x=-2$ (repeated).
  2. $2b=a+c$.
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