Second Derivative Test for Local Maximum

Q. Let $a_1,a_2,a_3,\dots$ be an A.P. If $\frac{a_1+a_2+\cdots +a_{10}}{a_1+a_2+\cdots +a_p}=\frac{100}{p^2},$ $p\ne 10,$ then $\frac{a_{11}}{a_{10}}$ is equal to

1. $\frac{19}{21}$
2. $\frac{100}{121}$
3. $\frac{21}{19}$
4. $\frac{121}{100}$

Q. Find the maximum area of an isosceles triangle inscribed in the ellipse with its vertex at one end of the major axis.

Q.

The value of the limit $\underset{\theta \to 0}{\lim }\frac{\left[\tan \left({\mathrm{\pi cos}}^2\left(\mathrm{\theta }\right)\right)\right]}{\left[\sin (2\mathrm{\pi }{\sin }^2\left(\mathrm{\theta }\right))\right]}$ is equal to

1. $-\frac{1}{2}$

2. $-\frac{1}{4}$

3. $0$

4. $\frac{1}{4}$

Q.

Evaluate $\tan \theta \sin \left(\frac{\mathrm{\pi }}{2}+\theta \right)\cos \left(\frac{\mathrm{\pi }}{2}-\theta \right)$

1. $1$

2. $0$

3. $\frac{1}{\sqrt{2}}$

4. None of these

Q. Let $a_1,a_2,a_3,...$ be an A.P. with $a_6=2.$Then the common difference of this A.P., which maximises the product $a_1{a}_4{a}_5,$ is :

1. $\frac{2}{3}$
2. $\frac{8}{5}$
3. $\frac{6}{5}$
4. $\frac{3}{2}$

Q. The area of the region bounded by the curve $y=x^2$ and the line $y=16$ is

1. $\frac{32}{3}\text{sq. units}$
2. $\frac{256}{3}\text{sq. units}$
3. $\frac{64}{3}\text{sq. units}$
4. $\frac{128}{3}\text{sq. units}$

Q.

A rectangular area is to be enclosed by a wall on one side and fencing on the other three sides. If $18m$ of fencing are used, what is the maximum area that can be enclosed?

Q. The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point :

1. $(2,9)$
2. $(2,2)$
3. $(0,0)$
4. $(3,\frac{21}{2})$

Q. Let $f\left(x\right)$ be a polynomial function such that $f\left(x\right)+f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)=x^5+64$. Then, the value of $\underset{x\to 1}{lim}\frac{f\left(x\right)}{x-1}$ is equal to:

Q. If $\alpha ,\beta$ are the distinct roots of $x^2+bx+c=0$, then $\underset{x\to \beta }{lim}\frac{e^{2(x^2+bx+c)}-1-2(x^2+bx+c)}{{(x-\beta )}^2}$ is equal to

1. $b^2-4c$
2. $b^2+4c$
3. $2(b^2+4c)$
4. $2(b^2-4c)$

Q.

$\left(\frac{\sin \theta }{1-\cot \theta }\right)+\left(\frac{\cos \theta }{1-\tan \theta }\right)=$

1. $0$

2. $1$

3. $\cos \theta –\sin \theta$

4. $\cos \theta +\sin \theta$

Q.

A square lawn is bounded on three sides by a path of $4m$ wide. If the area of the path is $\frac{7}{8}$ that of the lawn, find the dimensions of the lawn.

Q. Maximum slope of the curve $y=–x^3+3x^2+9x–27$ is

1. \N
2. 12
3. 16
4. 32

Q.

Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere $?$

Q. If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f\left(x\right)=9x^4+12x^3-36x^2+25,x\in R,$ then:

1. $S_1=\{-2,0\};S_2=\left\{1\right\}$
2. $S_1=\{-2,1\};S_2=\left\{0\right\}$
3. $S_1=\{-1\};S_2=\{0,2\}$
4. $S_1=\{-2\};S_2=\{0,1\}$

Q.

An open topped box is to be constructed by removing equal squares from each corner of a $3$ metre by $8$ metre rectangular sheet of aluminium and folding up the sides. Find the volume of the largest such box.

Q.

The area of the rectangle is given by the product of its length and breadth. The length of a rectangle is two-thirds of its breadth. Find its area, if its breadth is $x$.

Q. If one corner of a long rectangular sheet of paper of width $1$ unit is folded over, so as to reach the opposite edge of the sheet, then

1. minimum length of the crease is $\frac{3\sqrt{3}}{4}$
2. minimum length of the crease is $\frac{\sqrt{3}}{4}$
3. reduced width of the paper is $\frac{1}{2}$
4. reduced width of the paper is $\frac{1}{4}$

Q. The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

1. $32$
2. $20\sqrt{2}$
3. $36$
4. $18\sqrt{3}$

Q. 3.Find the local maxima and local minima, if any, of the following functions. Findalso the local maximum and the local minimum values, as the case may be(i) f(x) = x2(iii) h (x) = sin x + cos x, 0 < x <(iv) f(x)-sin-cos x, 0 < x < 2π(v) f(x)=x3-6x2+9+15 (vi)(vii) g(x)=x2+2(ii) g(x)=x3-3xg(x)=-+-, x>0(viii)f(x)=W1-х, О < x <1Vill

Q.

The perimeters of two squares are $40$ and $96$ metres respectively. Find the perimeter of another square equal in area to the sum of the first two squares.

Q.

If $\tan (\theta +i\varphi )=\sin (x+iy)$, then the value of $\cot h\left(y\right)\sin h\left(2\varphi \right)$ is

1. $\tan \left(x\right)\cot \left(2\theta \right)$

2. $\tan \left(x\right)\sin \left(2\theta \right)$

3. $\cot \left(x\right)\sin \left(2\theta \right)$

4. none of these

Q.

The area of a rectangle trampoline is $112f{t}^2.$

The length of the trampoline is $6ft$ greater than the width of the trampoline, this situation can be represented by the equation $w^2+6w-112=0$

What is the length of the trampoline in feet?

1. $7$

2. $16$

3. $8$

4. $14$

Q.

The value of $\sin \left(\theta \right)+\cos \left(\theta \right)$ will be greatest when

1. $\theta =30°$

2. $\theta =45°$

3. $\theta =60°$

4. $\theta =90°$

Q. Find two positive numbers x and y such that their sum is 35 and the product x 2 y 5 is a maximum

Q. Let $L{L}^{\prime }$ be the latus rectum of $y^2=4ax$ and $P{P}^{\prime }$ be a double ordinate drawn between the vertex and the latus rectum. If the area of trapezium $P{P}^{\prime }L{L}^{\prime }$ is maximum when the distance $P{P}^{\prime }$ from the vertex is $\frac{a}{k}$, then the value of $k$ is

Q. Let $f\left(x\right)$ be a polynomial of degree $6$ in $x$, in which the coefficient of $x^6$ is unity and it has extrema at $x=–1$ and $x=1$. If $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x^3}=1$ then $5.f\left(2\right)$ is equal to

Q. Find the maximum profit that a company can make, if the profit function is given by p ( x ) = 41 − 72 x − 18 x 2

Q. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Q. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show that the minimum length of the hypotenuse is