Table of Contents
CBSE Board Half Yearly Exam Question Paper Class-X Quadratic Equation 2022
ABC School
SUBJECT: MATHEMATICS
CLASS : X
CBSE Board Half Yearly Exam Sample Paper- 2022
NO. OF BLOCKS: 3
TOPIC: QUADRATIC EQUATIONS
Guidelines:-
Dear students, kindly refer to the following notes/video links for the Chapter- “Quadratic Equations” and thereafter do the questions in your math notebook.
SUBTOPICS:-
(i)Quadratic Equation – Introduction
(ii) Solution of Quadratic equation by factorization
(iii) Forming a quadratic equation in a given situation and solving it
(iv) Solution of Quadratic equation using quadratic formula
(v) Forming a quadratic equation in a given situation and solving it
(vi) Nature of roots of the given quadratic equation
INSTRUCTIONAL AIDS:- Presentation by screen sharing, offline whiteboard, online whiteboard, You tube links, E-lesson
LEARNING OUTCOMES:- Each student will be able to
- Recall the concept of quadratic polynomials and correlate with linear equation and quadratic equation
- Represent the equation n general form as ax2 + bx + c = 0 where a, b, c are real numbers a ≠ 0
- Solve the quadratic equation by using different methods
- Recall factorization method and
- apply the same to quadratic equation
- Apply the method of completing the squares to solve a quadratic equation and show the same geometrically
- Calculate discriminant to find nature of roots and apply the same to solve a quadratic equation.
Block 1
Lesson Development
The word “quadratic” comes from “quadratum”, the Latin word for square. Hence, we define a quadratic equation as an equation where the variable is of the second degree. Therefore, a quadratic equation is also called an “Equation of Degree 2”.
Quadratic Equation
ax2 + bx + c = 0, a ≠ 0, is called the standard form of quadratic equation For example :
(i) 2×2+5x+3=0; Here, a=2, b=3 and c=5
(ii) (ii) x 2-3x=0; Here, a=1 , b=-3 and c=0
But sometimes, the quadratic equation does not come in the standard form. These are the hidden quadratic equations which we may have to reduce to the standard form. Here are some examples:
ASSIGNMENT : Q1 (v) to (viii), Q2 (ii)
Solution of a Quadratic Equation by Factorisation
A real number α is called a root of the quadratic equation
ax2 + bx + c = 0, a ≠ 0 if a α 2+ bα + c = 0.
We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation.
Note that the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same
Relation with other subjects
Sports In athletic events that involve throwing objects like the shot put, balls or javelin, quadratic equations become highly useful. Distance Formula in physics If you know how fast and how long something was traveling, you can solve for the distance traveled.
EXAMPLAR QUESTIONS
Question 1:
Which of the following is a quadratic equation?
(a) x2 + 2x + 1 = (4 – x)2 + 3
(b) – 2x2 = (5 – x) (2x – 2/5
(c) (k + 1)x2 + – 3/2x = 7, where k = -1
(d) x3 – x2 =(x -1)3
Question 2:
Which of the following is not a quadratic equation?
(a) 2 (x -1)2 = 4x2 – 2x +1
(b) 2x – x2 = x2 + 5
(c) (-√2X +√3)2 = 3x2 – 5x
(d) (x2 + 2x)2 = x4 + 3 + 4x2
Question 3:
Which of the following equations has 2 as a root?
(a) x2-4x + 5=0
(b) x2 + 3x-12 =0
(c) 2x2 – 7x + 6 = 0
(d) 3x2 – 6x – 2 = 0
Question 4:
If 1/2 is a root of the equation x2 + kx –5/4 = 0, then the value of k is
(a)2
(b)-2
(c) ¼
(d)½
Life skill: Group Project Work
Real Life applications of Quadratic equations
Students will make a project work in a pair to explore the real life applications of Quadratic polynomials. Their projects can be presented in the class to enhance their understanding related to the topic.
Block 2
Lesson Development
Completion of square is deleted so do all problems by quadratic formula
Solving Quadratic equations using Quadratic Formula
Block 3
Lesson Development
Examplar Questions:
Question 1:
(x2 +1)2 – x2 = 0 has
(a) four real roots (b) two real roots (c) no real roots (d) one real root
Question 2:
Which of the following equations has no real roots?
(a) x2 – 4x + 3√2 =0
(b)x2+4x-3√2=0
(c) x2 – 4x – 3√2 = 0
(d) 3x2 + 4√3x + 4=0
Question 3:
The quadratic equation 2×2 – √5x + 1 = 0 has
(a) two distinct real roots
(b) two equal real roots
(c) no real roots
(d) more than 2 real roots
Question 4:
Which of the following equations has two distinct real roots?
(a) 2x2-3√2x +9/4=0
(b) x2 + x – 5 =0
(c) x2 + 3x + 2√2 =0
(d) 5x2-3x + 1=0
Question 5:
Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.
Question 6:
A train, travelling at a uniform speed for 360 km, would have taken 48 min less to travel the same distance, if its speed were 5 km/h more. Find the original speed of the train.
Question 7:
In the centre of a rectangular lawn of dimensions 50m x 40m, a rectangular pond has to be constructed, so that the area of the grass surrounding the pond would be 1184 m2[see
figure]. Find the length and breadth of the pond.
CASE STUDY 1:
Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet.
Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.
2. The shape of the path traced shown is
a) Spiral
b) Ellipse
c) Linear
d) Parabola
3. The graph of parabola opens upwards, if
a) a = 0
b) a < 0
c) a > 0
d) a ≥ 0
4. Observe the following graph and answer
In the above graph, how many zeroes are there for the polynomial?
a) 0
b) 1
c) 2
d) 3
5. The three zeroes in the above shown graph are
a) 2, 3,-1
b) -2, 3, 1
c) -3, -1, 2
d) -2, -3, -1
6. What will be the expression of the polynomial?
a) x3 + 2x2 – 5x – 6
b) x3 + 2x2 – 5x + 6
c) x3 + 2x2 + 5x – 6
d) x3 + 2x2 + 5x + 6
Maths fun activity :
Fun with Blocks:
Reema was playing with blocks of equal sizes.During the play she placed the blocks in different positions over each other. As shown below
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