Differentiation & Applications · JEE Main previous year questions

If $f(1)=1$, $f^{\prime }(1)=3$, then the derivative of $f(f(f(x)))+(f(x){)}^2$ at x = 1 is :

1. A.33
2. B.12
3. C.9
4. D.15

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

1. A.$\sqrt{3}$
2. B.$2\sqrt{3}$
3. C.$\sqrt{6}$
4. D.$\frac{2}{3}\sqrt{3}$

The maximum value of $3\cos \theta +5\sin \left(\theta -\frac{\pi }{6}\right)$ for any real value of $\theta$ is :

1. A.$\sqrt{34}$
2. B.$\sqrt{31}$
3. C.$\sqrt{19}$
4. D.$\frac{\sqrt{79}}{2}$

If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point (x, y) on it as $\frac{x^2-2y}{x}$, then the curve also passes through the point :

1. A.$(-1,2)$
2. B.$(-\sqrt{2},1)$
3. C.$(\sqrt{3},0)$
4. D.$(3,0)$

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $\frac{x^m{y}^n}{(1+x^{2m})(1+y^{2n})}$ is :

1. A.$\frac{1}{2}$
2. B.$\frac{1}{4}$
3. C.$\frac{m+n}{6mn}$
4. D.1

The maximum volume (in cu.m) of the right circular cone having slant height $3\text{ m}$ is :

1. A.$2\sqrt{3}\pi$
2. B.$3\sqrt{3}\pi$
3. C.$6\pi$
4. D.$\frac{4}{3}\pi$

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in ${\text{cm}}^2$) of this cone is :

1. A.$6\sqrt{2}\pi$
2. B.$6\sqrt{3}\pi$
3. C.$8\sqrt{2}\pi$
4. D.$8\sqrt{3}\pi$

Let $f(x)=x^2+\frac{1}{x^2}$ and $g(x)=x-\frac{1}{x}$, $x\in R-\{-1,0,1\}$. If $h(x)=\frac{f(x)}{g(x)}$, then the local minimum value of h(x) is

1. A.$2\sqrt{2}$
2. B.$3$
3. C.$-3$
4. D.$-2\sqrt{2}$

The tangent at the point $(2,-2)$ to the curve, $x^2{y}^2-2x=4(1-y)$ does not pass through the point :

1. A.$(4,\frac{1}{3})$
2. B.$(8,5)$
3. C.$(-4,-9)$
4. D.$(-2,-7)$

Consider a rectangle whose length is increasing at the uniform rate of $2m/sec$, breadth is decreasing at the uniform rate of $3m/sec$ and the area is decreasing at the uniform rate of $5{\mathrm{m}}^2/sec$. If after some time the breadth of the rectangle is $2\mathrm{m}$ then the length of the rectangle is

1. A.$2\mathrm{m}$
2. B.$4\mathrm{m}$
3. C.$1\mathrm{m}$
4. D.$3\mathrm{m}$

Let $f:R\to R$ be defined by $$f(x)=\{\begin{array}{ll}k-2x,\text{ if }& x\le -1\\ 2x+3,\text{ if }& x>-1\end{array}$$ If $f$has a local minimum at $x=-1$, then a possible value of $k$ is

1. A.$0$
2. B.$-\frac{1}{2}$
3. C.$-1$
4. D.$1$

The equation of the tangent to the curve $y=x+\frac{4}{x^2}$, that is parallel to the $x$-axis, is

1. A.$y=1$
2. B.$y=2$
3. C.$y=3$
4. D.$y=0$

If $x$ is real, the maximum value of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is

1. A.$\frac{1}{4}$
2. B.41
3. C.1
4. D.$\frac{17}{7}$

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

1. A.$\frac{3}{2}{x}^2$
2. B.$\sqrt{\frac{x^3}{8}}$
3. C.$\frac{1}{2}{x}^2$
4. D.$\pi x^2$

If $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$ then the difference between the maximum and minimum values of $u^2$ is given by :

1. A.$(a-b{)}^2$
2. B.$2\sqrt{a^2+b^2}$
3. C.$(a+b{)}^2$
4. D.$2(a^2+b^2)$

The maximum distance from origin of a point on the curve $x=a\sin t-b\sin \left(\frac{at}{b}\right)$ $y=a\cos t-b\cos \left(\frac{at}{b}\right)$, both $a,b>0$ is

1. A.$a-b$
2. B.$a+b$
3. C.$\sqrt{a^2+b^2}$
4. D.$\sqrt{a^2-b^2}$

If $f(x+y)=f(x).f(y)\forall$ x, y and f(5) = 2, f'(0) = 3, then f'(5) is

1. A.0
2. B.1
3. C.6
4. D.2