Let $$f(x) = {{{x^2} - 6x + 5} \over {{x^2} - 5x + 6}}$$.
Match the conditions/expressions in Column I with statements in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | If $$ - 1 < x < 1$$, then $$f(x)$$ satisfies | (P) | $$0 < f(x) < 1$$ |
| (B) | If $$1 < x < 2$$, then $$f(x)$$ satisfies | (Q) | $$f(x) < 0$$ |
| (C) | If $$3 < x < 5$$, then $$f(x)$$ satisfies | (R) | $$f(x) > 0$$ |
| (D) | If $$x > 5$$, then $$f(x)$$ satisfies | (S) | $$f(x) < 1$$ |
Let $$(x,y)$$ be such that $${\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}$$.
Match the statements in Column I with the statements in Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | If $$a=1$$ and $$b=0$$, then $$(x,y)$$ | (P) | lies on the circle $$x^2+y^2=1$$ |
| (B) | If $$a=1$$ and $$b=1$$, then $$(x,y)$$ | (Q) | lies on $$(x^2-1)(y^2-1)=0$$ |
| (C) | If $$a=1$$ and $$b=2$$, then $$(x,y)$$ | (R) | lies on $$y=x$$ |
| (D) | If $$a=2$$ and $$b=2$$, then $$(x,y)$$ | (S) | lies on $$(4x^2-1)(y^2-1)=0$$ |
Match the statements in Column I with the properties Column II.
| Column I | Column II | ||
|---|---|---|---|
| (A) | Two intersecting circles | (P) | have a common tangent |
| (B) | Two mutually external circles | (Q) | have a common normal |
| (C) | Two circles, one strictly inside the other | (R) | do not have a common tangent |
| (D) | Two branches of a hyperbola | (S) | do not have a common normal |
A student performs an experiment to determine the Young's modulus of a wire, exactly 2 m long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be 0.8 mm with an uncertainty of $$\pm0.05\;\mathrm{mm}$$ at a load of exactly 1.0 kg. The student also measures the diameter of the wire to be 0.4 mm with an uncertainty of $$\pm0.01\;\mathrm{mm}$$. Take g = 9.8 m/s2 (exact). The Young's modulus obtained from the reading is
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