Geometrical Interpretation

Q.

Why do we use a constant in Integration?

Q. If the length of the subnormal at any point of the curve is constant, then the eccentricity of this curve is

1. $e=\sqrt{2}$
2. $e>1$
3. $0<e<1$
4. $e=1$

Q.

Let $f$ be any function continuous on $\left[a,b\right]$ and twice differentiable on $\left(a,b\right)$. If for all $x\in \left(a,b\right)$, $f\left(x\right)>0$ and $f"\left(x\right)<0$, then for any $c\in \left(a,b\right)$ , $\left(f\left(c\right)-f\left(a\right)]\right)÷\left(f\left(b\right)-f\left(c\right)]\right)$ is greater than

1. $\frac{\left(b-c\right)}{\left(c-a\right)}$

2. $1$

3. $\frac{\left(c-a\right)}{\left(b-c\right)}$

4. $\frac{\left(b+a\right)}{\left(b-a\right)}$

Q. XOZ -plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio

1. 3 : 7

2. 2 : 7

3. -3 : 7

4. -2 : 7

Q. A continuous and differentiable function will have a unique derivative and a unique anti derivative

1. True
2. False

Q.

What are the applications of various maths functions?

Q.

What is the geometrical interpretation of indefinite integral?

1. The equation $\int f\left(x\right)dx=F\left(x\right)+C$, represents a family of curves.
2. These members cannot be obtained by shifting any one of the curves parallel to itself.
3. The different values of $C$ will correspond to different members of this family.
4. None of these

Q. Integrate the function: $x\log 2x$

Q. The ratio in which the line joining (2, 4, 5), (3, 5, – 4) is divided by the yz-plane is :

1. 3 : 2
2. 4 : – 3
3. 2 : 3
4. – 2 : 3

Q. Let g(x) be an antiderivative for f(x). Then In $(1+(g\left(x\right){)}^2)$ is an antiderivate for

1. $\frac{2f\left(x\right)g\left(x\right)}{1+\left(f\right(x){)}^2}$
2. $\frac{2f\left(x\right)g\left(x\right)}{1+\left(g\right(x){)}^2}$
3. $\frac{2f\left(x\right)}{1+\left(f\right(x){)}^2}$
4. None

Q.

If $\int f\left(x\right)dx=F\left(x\right)$, then $\int 5\times f\left(x\right)dx=\frac{F\left(x\right)}{5}$, because integration is the inverse process of differentiation

1. False

2. True

Q. If $\int f\left(x\right)dx=F\left(x\right)$, then $\int 5\times f\left(x\right)dx$ =

1. $5F\left(x\right)$
2. $\frac{F\left(x\right)}{5}$
3. $F\left(x\right)$

Q. If n is any positive integer, show that the integral part of $(3+\sqrt{7}{)}^n$ is an odd number.

Q. Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

Q.

The area shaded in the given figure can be calculated by which of the following definite integral?

1. 

2. 

3. 

4. 

Q. $\stackrel{e^2}{\underset{1}{\int }}\left[lo{g}_{e}x\right]dx,x>0$ and $[\cdot ]$ is greatest integer function, is equal to

1. $e-1$
2. $e^2-e$
3. $e^2-1$
4. $e^2+1$

Q. The anti – derivative of the function (3x + 4) |sin x|, when $0<x<\pi$, is given by

1. $3sinx+(3x+4)cosx+c$
2. $3sinx-(3x+4)cosx+c$
3. None of these
4. $-3sinx-(3x+4)cosx+c$

Q. If there is an error of a% in measuring the edge of a cube, then percentage error in its surface is
(a) 2a%

(b) $\frac{a}{2}\%$

(c) 3a%

(d) none of these

Q. The solution of $\frac{dy}{dx}=\left(\frac{ax+b}{cy+d}\right)$ represents a parabola if :-

1. $a=0,c=0$
2. $a=1,c=2$
3. $a=0,c\ne 0$
4. $a=1,c=1$

Q.

Integration of an integrable function f(x) gives a family of curves which differ by a constant value.

1. True

2. False

Q. If the relative error in measuring the radius of a circular plane is α, find the relative error in measuring its area.

Q. Find the antiderivative of f(x) given by $f\left(x\right)=4x^3-\frac{3}{x^4}$ such that f(2)=0

Q. If $f\left(x\right)=\underset{x}{\overset{x^2}{\int }}\frac{1}{\log t}dt$ (where $x>0$), then the value of $f^{\prime }\left(x\right)$ is

1. $\frac{x^2}{\log x}$
2. $\frac{x}{\log x}$
3. $\frac{x-1}{\log x}$
4. $\frac{x+1}{\log x}$

Q. Integration of an integrable function f(x) gives a family of curves which differ by a constant value.

1. True
2. False

Q. Among the given options, for what value of $dx$ will the area under the curve be most precise to the true value

1. 
   $dx=1$
2. 
   $dx=0.1$
3. 
   $dx=0.01$
4. 
   $dx=0.001$

Q. Among the given options, for what value of $dx$ will the area under the curve be most precise to the true value

1. 
   $dx=1$
2. 
   $dx=0.1$
3. 
   $dx=0.01$
4. 
   $dx=0.001$