MCQ
Choose the correct answer from the given four options: If $y = x^4 - 10$ and if $x$ changes from $2$ to $1.99,$ what is the change in $y:$
- ✓$0.32$
- B$0.03.2$
- C$5.68$
- D$5.968$
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Let $f, g:[-1,2] \rightarrow R$ be continuous functions which are twice differentiable on the interval $(-1,2)$. Let the values of $f$ and $g$ at the points $-1.0$ and $2$ be as given in the following table:
| $x=-1$ | $x=0$ | $x=2$ | |
| $f(x)$ | $3$ | $6$ | $0$ |
| $g(x)$ | $0$ | $1$ | $-1$ |
In each of the intervals $(-1,0)$ and $(0,2)$ the function $(f-3 g)^{\prime \prime}$ never vanishes. Then the correct statement(s) is(are)
$(A)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly three solutions in $(-1,0) \cup(0,2)$
$(B)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly one solution in $(-1,0)$
$(C)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly one solution in $(0,2)$
$(D)$ $f^{\prime}(x)-3 g^{\prime}(x)=0$ has exactly two solutions in $(-1,0)$ and exactly two solutions in $(0,2)$
| Column $I$ | Column $II$ |
| $(A)$ Interval contained in the domain of definition of non-zero solutions of the differential equation $(x-3)^2 y^{\prime}+y=0$ | $(p)$ $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ |
|
$(B)$ Interval containing the value of the integral $\int_1^5(x-1)(x-2)(x-3)(x-4)(x-5) d x$ |
$(q)$ $\left(0, \frac{\pi}{2}\right)$ |
| $(C)$ Interval in which at least one of the points of local maximum of $\cos ^2 x+\sin x$ lies | $(r)$ $\left(\frac{\pi}{8}, \frac{5 \pi}{4}\right)$ |
| $(D)$ Interval in which $\tan ^{-1}(\sin x+\cos x)$ is increasing | $(s)$ $\left(0, \frac{\pi}{8}\right)$ |
| $(t)$ $(-\pi, \pi)$ |