A small luminous object is placed at a distance of 48 cm from a convex lens of focal length 30 cm on its principal axis. A concave length of focal length 60 cm is place coaxially on the other side of the convex lens at a distance of x cm from it. If a real image is formed by the combination at a point 1.6 m from the convex lens, find the value of x. 1. 80cm 2. 20 cm 3. 40 cm no bs please ans.
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To find the sum of all possible non-negative
integral values of k for which the roots of the
equation 4x² - 3kx + 2k = 0 are real and Ikl
≤ 13, we need to use the discriminant of the
quadratic equation.
The quadratic equation in standard form is
ax² + bx + c = 0, where a = 4, b = -3k, and c =
2k.
For the roots to be real, the discriminant (D)
should be greater than or equal to zero:
D = b² - 4ac
Substituting the values:
Now, the discriminant (D) should be greater
than or equal to zero:
9k² -32k≥0
To find the possible values of k, we need to
solve this inequality:
9k² -32k ≥ 0
k(9k - 32) ≥ 0
The values of k for which the inequality
holds true are:
k≤ 0 (non-negative integral values)
9k - 32 ≥ 0
k≥ 32/9 = 3.56
the valid range for k is -13 ≤ k ≤ 13.
Now, let's find the sum of all possible
non-negative integral values of k within this
valid range:
Possible non-negative integral values of k: 0,
1, 2, 3, 4, ..., 13
The sum of these values is the sum of the
first 14 natural numbers:
Sum = (14*15) / 2 = 105
Thus, the sum of all possible non-negative
integral values of k is 105.