Chapter 14

The equation of the tangent to the curve \(y=x+\frac{4}{x^{2}}\), which is parallel to the \(x\)-axis, is (A) \(y=1\) (B) \(y=2\) (C) \(y=3\) (D) \(y=0\)

Let \(f: R \rightarrow R\) be defined by \(f(x)= \begin{cases}k-2 x, & \text { if } x \leq-1 \\ 2 x+3 & \text { if } x>-1\end{cases}\) If \(f\) has a local minimum at \(x=-1\), then apossible value of \(k\) is (A) 0 (B) \(-\frac{1}{2}\) (C) \(-1\) (D) 1

A spherical balloon is filled with \(4500 \pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72 \pi\) cubic meters per minute, then the rate (in meters per minute) at which theradius of the balloon decreases 49 minutes after the leakage began is (A) \(\frac{9}{7}\) (B) \(\frac{7}{9}\) (C) \(\frac{2}{9}\) (D) \(\frac{9}{2}\)

Let the real values \(a, b\) be such that the function \(\mathrm{f}\) given by \(f(x)=\ln |x|+b x^{2}+a x, x \neq 0\) has extreme values at \(x=-1\) and \(x=2\). Statement \(\mathbf{1}: f\) has local maximum at \(x=-1\) and at \(x=2\). Statement 2: \(a=\frac{1}{2}\) and \(b=\frac{-1}{4}\) (A) Statement 1 is false, statement 2 is true (B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1 (C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1 (D) Statement 1 is true, statement 2 is false

If the functions \(f\) and \(g\) are differentiable functions on \([0,1]\) satisfying \(f(0)=2=g(1), g(0)=0\) and \(f(1)=6\) then for some \(c \in] 0,1[\) (A) \(2 f^{\prime}(C)=g^{\prime}(c)\) (B) \(2 f^{\prime}(C)=3 g^{\prime}(c)\) (C) \(f^{\prime}(C)=g^{\prime}(c)\) (D) \(f^{\prime}(C)=2 g^{\prime}(c)\)

If \(x=-1\) and \(x=2\) are extreme points of \(f(x)=\alpha \log |x|+\beta x^{2}+x\), then (A) \(\alpha=-6, \beta=\frac{1}{2}\) (B) \(\alpha=-6, \beta=-\frac{1}{2}\) (C) \(\alpha=2, \beta=-\frac{1}{2}\) (D) \(\alpha=2, \beta=\frac{1}{2}\)

The normal to the curve, \(x^{2}+2 x y-3 y^{2}=0\), at $(1,1): (A) meets the curve again in the second quadrant. (B) meets the curve again in the third quadrant. (C) meets the curve again in the fourth quadrant. (D) does not meet the curve again.

A wire of length 2 units is cur into two parts which are bent respectively to form a square of side \(=x\) units and a circle of radius \(=\mathrm{r}\) units. If the sum of the areas of the square and the circle so formed is minimum, then: (A) \(2 x=r\) (B) \(2 x=(\pi+4) r\) (C) \((4-\pi) x=\pi r\) (D) \(x=2 r\)

Consider \(f(x)=\tan ^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right), \quad x \in\left(0, \frac{\pi}{2}\right)\). A normal to \(y=f(x)\) at \(x=\frac{\pi}{6}\) also passes through the point: (A) \(\left(\frac{\pi}{4}, 0\right)\) (B) \((0,0)\) (C) \(\left(0, \frac{2 \pi}{3}\right)\) (D) \(\left(\frac{\pi}{6}, 0\right)\)