Geometric Interpretation of Def.Int as Limit of Sum

Q. Define feasible region of a linear programming problem.

Q. Let $a$ be a positive real number such that ${\int }_0^a{e}^{x-\left[x\right]}dx=10e-9$ where $\left[x\right]$ is the greatest integer less than or equal to $x$. Then $a$ is equal to

1. $10+{\log }_{e}3$
2. $10-{\log }_e(1+e)$
3. $10+{\log }_e(1+e)$
4. $10+{\log }_{e}2$

Q. Let $I=\underset{a}{\overset{b}{\int }}(x^4-2x^2)dx$. If $I$ is minimum then the ordered pair $(a,b)$ is :

1. $(0,\sqrt{2})$
2. $(-\sqrt{2},0)$
3. $(-\sqrt{2},\sqrt{2})$
4. $(\sqrt{2},-\sqrt{2})$

Q. If $m=max\{||x-5|-|x-3||\}$ and $n=min\{|x-2|+|x-4|\},$
then the value of $m+n$ is

1. $4$
2. $3$
3. $5$
4. $6$

Q. If f(x) is a continuous function defined on [a, b] such that $f\left(x\right)\ge 0$ $x\in [a,b]$ then the area under the curve as the limit of a sum can be given as
(b - a) lim $n\to \infty$ $\frac{1}{n}\left[f\right(a)+f(a+h)...f(a+(n-1)h\left)\right]$
Where $h=\frac{b-a}{n}$ & $h\to 0$ as $n\to \infty$

1. True
2. False

Q. If $I$ is the greatest of the definite integrals
$I_1={\int }_0^1{e}^{-x}co{s}^{2}xdx,I_2{\int }_0^1{e}^{-x^2}co{s}^{2}xdx$
$I_3={\int }_0^1{e}^{-x^2}dx,I_4={\int }_0^1{e}^{-x^2/2}dx,$ then

1. $I=I_1$
2. $I=I_2$
3. $I=I_3$
4. $I=I_4$

Q. Does there exist a function which is continuous everywhere but not differentiable at exactly tow points? Justify your answer.

Q. Let $I=\underset{a}{\overset{b}{\int }}(x^4-2x^2)dx$. If $I$ is minimum then the ordered pair $(a,b)$ is :

1. $(-\sqrt{2},0)$
2. $(-\sqrt{2},\sqrt{2})$
3. $(0,\sqrt{2})$
4. $(\sqrt{2},-\sqrt{2})$

Q. If $I_1=\underset{0}{\overset{\pi /2}{\int }}\cos \left(\sin x\right)dx,I_2=\underset{0}{\overset{\pi /2}{\int }}\sin \left(\cos x\right)dx$ and $I_3=\underset{0}{\overset{\pi /2}{\int }}\cos xdx$, then

1. $I_1>I_3>I_2$
2. $I_3>I_1>I_2$
3. $I_{1_1}>I_2>I_3$
4. $I_3>I_2>I_1$

Q. If one of the diameters of the curve $x^2+y^2-4x-6y+9=0$ is a chord of a circle with centre $(1,1)$, the radius of this circle is

1. $3$
2. $2$
3. $\sqrt{2}$
4. $1$

Q. If $\left[x\right]$ denotes the greater integer less than or equal to x, then the value of ${\int }_1^5[|x-3|]$dx is

1. 1
2. 2
3. 4
4. 8

Q.

Let (3, 4, -1) and (-1, 2, 3) are the end points of a diameter of sphere. Then the radius of the sphere is equal to
[Orissa JEE 2003]

1. 1

2. 2

3. 3

4. 9

Q. Let $a,b,c$ be non – zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx.$
Then the quadratic equation $a{x}^2+bx+c=0$ has

1. No root in (0, 2)
2. At least one root in (0, 2)
3. A double root in (0, 2)
4. Two imaginary roots

Q. The integral of $\int \frac{dx}{x^{13}(x^6-1)}$ is:

1. $\frac{1}{6}\ln \left|\frac{x^6-1}{x^6}\right|+c$
2. $\frac{1}{6}[\ln \left|\frac{x^6-1}{x^6}\right|+x^{-6}+\frac{1}{2}{x}^{-12}]+c$
3. $\frac{1}{6}[\ln \left|\frac{x^6-1}{x^6}\right|+x^{-6}+x^{-12}]+c$
4. $\frac{1}{6}[\ln \left|\frac{x^6-1}{x^6}\right|+x^{-6}+\frac{1}{4}{x}^{-12}]+c$

Q. Let $a,b,c$ be non – zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx.$
Then the quadratic equation $a{x}^2+bx+c=0$ has

1. At least one root in (0, 2)
2. A double root in (0, 2)
3. No root in (0, 2)
4. Two imaginary roots

Q. The value of ${\int }_0^2(x-{\log }_{2}a)dx=2{\log }_2\left(\frac{2}{a}\right)$ for which of the following conditions?

1. $a>0$
2. $a>2$
3. $a=4$
4. $a=8$

Q. Let $a,b,c$ be non – zero real numbers such that
${\int }_0^1(1+co{s}^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+co{s}^{8}x)(a{x}^2+bx+c)dx.$
Then the quadratic equation $a{x}^2+bx+c=0$ has

1. Two imaginary roots
2. No root in (0, 2)
3. At least one root in (0, 2)
4. A double root in (0, 2)

Q. If the value of $\int ({\sin }^{n}3x+{\cos }^{n}3x)dx=x+C,$ then the value of $n$ is
(where $C$ is constant of integration)

Q. ${\int }_0^{\frac{\pi }{2}}log(tanx+cotx)dx=$

1. $\pi log2$
2. $-\pi log2$
3. $-\frac{\pi }{2}log2$
4. $\frac{\pi }{2}log2$

Q. ${\int }_0^{\frac{\pi }{2}}log(tanx+cotx)dx=$

1. $\pi log2$
2. $-\pi log2$
3. $-\frac{\pi }{2}log2$
4. $\frac{\pi }{2}log2$

Q. If $I_1={\int }_0^{\pi /2}\cos \left(\sin x\right)dx$ ; $I_2={\int }_0^{\pi /2}\sin \left(\cos x\right)dx$ and $I_3={\int }_0^{\pi /2}\cos xdx$, then

1. $I_1>I_3>I_2$
2. $I_3>I_1>I_2$
3. $I_1>I_2>I_3$
4. $I_3>I_2>I_1$

Q. Let $f(a,b)={\int }_a^b(x^2-4x+3)dx,(b>a)$ then

1. $f(a,3)$ is least when $a=1$
2. $f(4,b)$ is an increasing function $\forall b\ge 4$
3. $f(0,b)$ is least for $b=2$
4. $min\left\{f(a,b)\right\}=-\frac{4}{3}\forall a,b\in R$

Q. Let a, b, c be non-zero real numbers such the : ${\int }_0^1(1+{\cos }^{8}x)(a{x}^2+bx+c)dx={\int }_0^2(1+{\cos }^{8}x)(a{x}^2+bx+c)dx,$ then the quadratic equation $a{x}^2+bx+c=0$ has

1. no root in $(0,2)$
2. atleast one root in $(0,2)$
3. a double root in $(0,2)$
4. none

Q. Consider the integral $I={\int }_0^{\pi }ln\left(\sin x\right)dx$.What is ${\int }_0^{\frac{\pi }{2}}$ ln $\left(\sin x\right)dx$ equal to?

1. $4I$
2. $2I$
3. $I$
4. $\frac{I}{2}$

Q. Let $a,b,c$ be non-zero real numbers such that ${\int }_0^1\left(1+{\cos }^{8}x\right)\left(a{x}^2+bx+c\right)dx={\int }_0^2\left(1+{\cos }^{8}x\right)\left(a{x}^2+bx+c\right)dx$ then the quadratic equation $a{x}^2+bx+c=0$ has-

1. a double root in $(0,2)$
2. atleast one root $(0,2)$
3. no roots in $(0,2)$
4. none

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2+x,& -1\le x<0\\ 1,& x=0\\ {\log }_{\frac{1}{2}}(x+\frac{1}{2}),& 0<x<\frac{3}{2}\end{array}$

1. No global minima occurs
2. No global maxima occurs
3. Global minima will occur at $x=-\frac{1}{2}$
4. Global maxima will occur at $x=0$ and equal to $1$

Q. ${\int }_0^{\frac{\pi }{2}}log(tanx+cotx)dx=$

1. $\pi log2$
2. $-\pi log2$
3. $-\frac{\pi }{2}log2$
4. $\frac{\pi }{2}log2$

Q.

Consider the integrals $I_1={\int }_0^1{e}^{-x}{\cos }^{2}xdx,I_2={\int }_0^1{e}^{-x^2}{\cos }^{2}xdx,I_3={\int }_0^1{e}^{-x}dx$ and $I_4={\int }_0^1{e}^{-\left(1/2\right)x^2}dx$. The greatest of these integrals is

1. $I_4$
2. $I_3$
3. $I_1$
4. $I_2$

Q. $\int \frac{dx}{\tan x+\cot x}=$

1. $-\frac{\cos 2x}{4}+c$
2. $\frac{\sin 2x}{4}+c$
3. $\frac{-\sin 2x}{4}+c$
4. $\frac{\cos 2x}{4}+c$

Q. If $f\left(x\right)$ is a continuous function defined on $[a,b]$ such that $f\left(x\right)\ge 0$ $x\in [a,b]$ then the area under the curve as the limit of a sum can be given as
$(b-a)\underset{n\to \infty }{lim}\frac{1}{n}\left[f\right(a)+f(a+h)...f(a+(n-1)h\left)\right]$
Where $h=\frac{b-a}{n}$ and $h\to 0$ as $n\to \infty$

1. True
2. False