III secuon A questions from Sl.Nos.1 to 10 are of 'Very short answer type'.
Each question carries to marks.
In Section 'B', question from Sl.Nos 11 to 17 are of 'Short Answer type'.
Each question carries four marks.
In section 'C', questions from Sl.Nos 18 to 24 are of 'Long answer type'. Each question carries seven mar
SECTION-I
Answer all the following questions:
10 x2 =20M
Form the quadratic equation whose roots are 7+2/5.
Find the maximum (or) minimum value of the quadratic expression 3x +2x+11
If - 1, 2 and a are the roots of 2r? + - 7x-6= 0, then find a.
14
4.
Find the 7th term in the expansion of
2
Find the equation of the circle passing through the point (1,1) and concentric wi
x2 + y2 - 6x-4y - 12 = 0
Find the pole of the line 3x + 4y - 12 =0 with respect to the circle ? + y2 = 24
es of the point on the parabola y2 = 8x, Whose focal distanc
Answers
Answer:
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Answer:
The roots of quadratic equation are the values of the variable that satisfy the equation. They are also known as the "solutions" or "zeros" of the quadratic equation. For example, the roots of the quadratic equation x2 - 7x + 10 = 0 are x = 2 and x = 5 because they satisfy the equation. i.e.,
when x = 2, 22 - 7(2) + 10 = 4 - 14 + 10 = 0.
when x = 5, 52 - 7(5) + 10 = 25 - 35 + 10 = 0.
But how to find the roots of a general quadratic equation ax2 + bx + c = 0? Let us try to solve it for x by completing the square.
ax2 + bx = - c
Dividing both sides by 'a',
x2 + (b/a) x = - c/a
Here, the coefficient of x is b/a. Half of it is b/(2a). Its square is b2/4a2. Adding b2/4a2 on both sides,
x2 + (b/a) x + b2/4a2 = (b2/4a2) - (c/a)
[ x + (b/2a) ]2 = (b2 - 4ac) / 4a2 (using (a + b)² formula)
Taking square root on both sides,
x + (b/2a) = ±√ (b² - 4ac) / 4a²
x + (b/2a) = ±√ (b² - 4ac) / 2a
Subtracting b/2a from both sides,
x = (-b/2a) ±√ (b² - 4ac) / 2a (or)
x = (-b ± √ (b² - 4ac) )/2a
This is known as the quadratic formula and it can be used to find any type of roots of a quadratic equation.