Algebra - Quadratic Equations in one variable Practice Question for ICSE Class 10 2021
Syllabus for ICSE 2021
(a) Nature of roots
- Two distinct real roots if b2 – 4ac > 0
- Two equal real roots if b2 – 4ac = 0
- No real roots I b2 – 4ac < 0
(b) Solving Quadratic equations by:
- Factorisation
- Using Formula.
(c) Solving simple quadratic equation problems
Some Points to Look into Quadratic Roots
The nature of the roots depends on the value of b2 – 4ac which is called the discriminant of the quadratic equation ax2 + bx + c = 0 and is generally denoted by D.
∴ D = b2 – 4ac
a) If D > 0, which means if b2 – 4ac > 0 , so the term is positive; the roots are real and unequal. Also,
- If b2 – 4ac is a perfect square, the roots are rational and unequal.
- If b2 – 4ac is positive but not perfect square, the roots are irrational and unequal.
b) If D = 0, which means if b2 – 4ac = 0; the roots are real and equal.
c) If D < 0, which means b2 – 4ac < 0; so the term is negative; the roots are not real,infact the roots are imaginary and it is termed as complex numbers which you will study in future class.
Solving the Factorization
I am very clear to my students that this is the most important section in algebra where student loses marks. There are certain approach to deal with it:
1) Finding Common Terms
2) Mid Term Separation
3) Completing the square
4) Checking if the equation follows any identity trait.
Important Questions for ICSE 2021 class 10 Exams boards
1. Check whether it is quaratic equation - (2x + 1) (3x – 2) = 6(x + 1) (x – 2)
2. Determine whether the given numbers are roots of the given equations or not
(i) x² – x + 1 = 0; 1, – 1
(ii) x² – 5x + 6 = 0; 2, – 3
3. If 2/3 is a solution of the equation 7x² + kx – 3 = 0, find the value of k.
4. Find the root using formula
(i)256x² – 32x + 1 = 0
(ii) 25x² + 30x + 7 = 0
5. Find the discriminant of the following equations and hence find the nature of roots:
(i) 3x² – 5x – 2 = 0
(ii) 2x² – 3x + 5 = 0
(iii) 7x² + 8x + 2 = 0
(iv) 16x² – 40x + 25 = 0
6. Find the value(s) of m for which each of the following quadratic equation has real and equal roots:
(i) (3m + 1)x² + 2(m + 1)x + m = 0
(ii) x² + 2(m – 1) x + (m + 5) = 0
7. Find the value(s) of p for which the quadratic equation (2p + 1)x² – (7p + 2)x + (7p – 3) = 0 has equal roots. Also find these roots.
8. Solve the following equation by using quadratic equations for x and give your
(i) x² – 5x – 10 = 0
(ii) 5x(x + 2) = 3
9. The sum of two numbers is 9 and the sum of their squares is 41. Taking one number as x, form ail equation in x and solve it to find the numbers.
10. There are three consecutive positive integers such that the sum of the square of the first and the product of other two is 154. What are the integers?
11. A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.
12. A rectangular garden 10 m by 16 m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is 120 square metres, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x.
13. The perimeter of a rectangular plot is 180 m and its area is 1800 m². Take the length of the plot as x m. Use the perimeter 180 m to write the value of the breadth in terms of x. Use the values of length, breadth and the area to write an equation in x. Solve the equation to calculate the length and breadth of the plot.
14. in a P.T. display, 480 students are arranged in rows and columns. If there are 4 more students in each row than the number of rows, find the number of students in each row.
15. A bus covers a distance of 240 km at a uniform speed. Due to heavy rain, its speed gets reduced by 10 km/h and as such it takes two hours longer to cover the total distance. Assuming the uniform speed to be ‘x’ km/h, form an equation and solve it to evaluate x.
(x−3)3+5=x3+7x2−1
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