Section-A
1575
1575
103
(d)
1575
102
104
10
(c) 9
(d) V8
(6) V121
(d) None of these
1. Express 15.75 in the form
9
1575
(a)
(C)
(6)
2. Find the irrational number from the following.
(a) 136
3. If HCF of 26 and 91 is 13, then find its LCM.
4. The zeros of quadratic equation x² + 7x + 10 are
5. Zeros of polynomial x2 – 11 are:
(a) 711, V11
(b) 111, -V11
(c) 11, -11
6. If P(x) = x² – 3x – 4, then find the value of P(2).
a 떠
7. If in equations aqx + bay +41 = 0 and a2x + b2y + C2 = 0,
following is true?
(a) Intersecting lines (6) Coincident lines (c) Parallel lines
8. The product of two roots of the quadratic equation 3x + 2x – 5 = 0, will be
-5
(a)
(b)
(c)
3
3
5
9. The value of k for quadratic equation 3x? – kx + 5 = 0 has two equal roots is
(16)
bi
b2
C1, then which
2.
C2
(d) None of these
3
5
(d) None of these1
Answers
Answer:
)
(c)
3
3
5
9. The value of k for quadratic equation 3x? – kx + 5 = 0 has two equal roots is
(16)
bi
b2
C1, then which
2.
C2
(d) None of these
3
5
(d) None of these1
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.