In the following [x] denotes the greatest integer less than or equal to x.
Match the functions in Column I with the properties Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$x|x|$$ | (P) | continuous in ($$-1,1$$). |
| (B) | $$\sqrt{|x|}$$ | (Q) | differentiable in ($$-1,1$$) |
| (C) | $$x+[x]$$ | (R) | strictly increasing in ($$-1,1$$) |
| (D) | $$|x-1|+|x+1|$$ | (S) | not differentiable at least at one point in ($$-1,1$$) |
Match the integrals in Column I with the values in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $$ | (P) | $${1 \over 2}\log \left( {{2 \over 3}} \right)$$ |
| (B) | $$\int\limits_0^1 {{{dx} \over {\sqrt {1 + {x^2}} }}} $$ | (Q) | $$2\log \left( {{2 \over 3}} \right)$$ |
| (C) | $$\int\limits_2^3 {{{dx} \over {1 + {x^2}}}} $$ | (R) | $${\pi \over 3}$$ |
| (D) | $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $$ | (S) | $${\pi \over 2}$$ |
A resistance of 2 $$\Omega$$ is connected across one gap of a metre-bridge (the length of the wire is 100 cm) and an unknown resistance, greater than 2 $$\Omega$$, is connected across the other gap. When these resistance are interchanged, the balance point shifts by 20 cm. Neglecting any corrections, the unknown resistance is
In an experiment to determine the focal length (f) of a concave mirror by the u-v method, a student places the object pin A on the principal axis at a distance x form the pole P. The student looks at the pin and its inverted image form a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right, oh the object pin. Then,
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