Point of Inflection

Q.

The least possible value of $a$ for which the equation, $2x^2+\left(a-10\right)x+\frac{33}{2}=2a$ has real root is

Q. The period of the function $f\left(x\right)=3\sin 2\sqrt{3}x+2\cos 5\sqrt{3}x$ is

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

Q.

The area (in sq units) bounded by the curves $y^2=4x$ and $x^2=4y$ in the plane is

1. $\frac{8}{3}$

2. $\frac{16}{3}$

3. $\frac{32}{3}$

4. $\frac{64}{3}$

Q.

The function $f\left(x\right)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at

1. $x=-2$

2. $x=0$

3. $x=1$

4. $x=2$

Q. if x =2t-3t^2 and y=6t^3 then dy/dx at point (-1, 6) is

Q.

The area (in sq units) enclosed between the curves $y=x$ and $y=2x-x^2$ is

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

2. $\frac{1}{6}$

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

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

Q. Period of the function $f\left(x\right)=\tan \left(\frac{x}{3}\right)+\sin 2x$ is

1. $4\pi$
2. $3\pi$
3. $5\pi$
4. $2\pi$

Q.

The ends of the base of an isosceles triangle are at $\left(2a,0\right)$ and $\left(0,a\right)$. The equation of one side is $x=2a$.The equation of the other side is

1. $x+2y-a=0$

2. $x+2y=2a$

3. $3x+4y-4a=0$

4. $3x-4y+4a=0$

Q. The number of point(s) of discontinuity of $f\left(x\right)=\left[5\sin x\right],x\in [0,\pi ]$ is

Q. Let $f\left(x\right)={\cos }^{-1}\left(\cos x\right)$ and $g\left(x\right)=x^2-2\pi x+{\pi }^2+a.$ Then the equation $f\left(x\right)=g\left(x\right)$ has

1. exactly $2$ solutions if $a<\pi$
2. No solution if $a=4\pi$
3. exactly one solution if $a=\pi$
4. more than two solutions for $a<0$

Q. The function $y=\frac{\mathrm{ax}+b}{\left(x-1\right)\left(x-4\right)}$ has a turning point or a point of inflexion at P(2, -1). Then, which of the following is(are) true about a and b?

1. a = 0
2. a = 1
3. b = 1
4. b = 0

Q. Let $f:R\to R$ be defined as $f\left(x\right)=(|x-1|+|4x-11|)[x^2-2x-2],$ where $[.]$ denotes the greatest integer function. Then the number of points of discontinuity of $f\left(x\right)$ in $(\frac{1}{2},\frac{5}{2})$ is

Q.

The area (in sq units) bounded by the curves $y=\sqrt{x}$, $2y-x+3=0$, $x$-axis and lying in the first quadrant is

1. $9$

2. $6$

3. $18$

4. $\frac{27}{4}$

Q. For which of the following functions Rolle's theorem is not applicable :

1. $f\left(x\right)=\sqrt{4-x^2}$ in $[-2,2]$
2. $f\left(x\right)=\left[x\right]$ in $[-1,1]$
   $[.]$ denotes the greatest integer function
3. $f\left(x\right)=x^2+3x-4$ in $[-4,1]$
4. $f\left(x\right)=\cos 2x$ in $[0,\pi ]$

Q. The function $f\left(x\right)=\sin \frac{\pi x}{2}+2\cos \frac{\pi x}{3}-\tan \frac{\pi x}{4}$ is periodic with period

1. $6$
2. $3$
3. $4$
4. $12$

Q. The period of the function $f\left(x\right)=\sin (2\pi x+\frac{\pi }{8})+2\sin (3\pi x+\frac{\pi }{3})$ is

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

Q. If $\sin 2A=\frac{1}{2}$ and $\sin 2B=-\frac{1}{2}$, then which one of the following is false

1. sin (A + B) may be 0
2. cos (A - B) may be zero
3. sin(A + B) or cos (A - B) is zero
4. sin (A + B) = 0

Q. The function(s) which are not having $2\pi$ as it's period is/are

1. $f\left(t\right)=\sin (2\pi t+\frac{\pi }{3})+2\sin (3\pi t+\frac{\pi }{4})+3\sin 5\pi t$
2. $g\left(t\right)=\sin \frac{\pi }{3}t+\sin \frac{\pi }{4}t$
3. $h\left(t\right)=\sin t+\cos 2t$
   
4. $k\left(t\right)=\sin \frac{2\pi }{3}t+\sin \frac{\pi }{4}t$

Q. Let $f\left(x\right)$ be a twice differentiable function and has no critical points. Let $g\left(x\right)=(x+6{)}^{2009}(x+1{)}^{2010}(x+2{)}^{2011}(x-3{)}^{2012}(x-4{)}^{2013}(x-5{)}^{2014}$ be such that $f\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)+f^{\prime \prime }\left(x\right)=0.$ Then the function $h\left(x\right)=f^2\left(x\right)+(f^{\prime }\left(x\right){)}^2$

1. is monotonically increasing in $(-2,4)$.
2. has exactly three points of inflection.
3. has exactly two points of local maxima.
4. has a negative point of local minimum.

Q.

If the centre of the circle $2x^2+pxy+q{y}^2+2gx+2fy+3$ = 0 is (1, -3), then the radius of the circle is

1. 

2. 

3. 

4. 17

Q.

Point of inflection of the function $f\left(x\right)=x^3$ is

1. 0

2. 1

3. 2

4. -1

Q. Thr period of the function : $f\left(x\right)=3\sin 2\left\{3x\right\}+5\cos 3\left\{2x\right\}$ is (where $\left\{\right\}$ denotes fraction part function)

Q. The graph of $f\left(x\right)=\frac{x^5}{20}-\frac{x^4}{12}+5$ has

1. one local maximum and one local minimum
2. one local minimum and one point of inflection.
3. one local maximum, one local minimum and one point of inflection.
4. one local maximum, one local minimum and two points of inflection.

Q. Which of the following is (are) true about the function $f\left(x\right)=4x^3–18x^2+27x–7$ ?

1. Has a maxima at x = 3/2
2. x = 3/2 is a point of inflexion
3. Has a minima at x = 3/2
4. Neither has a maxima nor minima

Q. If $(a,\frac{1}{a}),(b,\frac{1}{b}),(c,\frac{1}{c})\&(d,\frac{1}{d})$ are four distinct points on a circle of radius 4 units then, abcd is equal to

1. 16
2. 4
3. 
4. 1

Q. The number of integral value(s) of $a$ for which $f\left(x\right)=\{\begin{array}{cc}{a}^2-3+\sin x,& 0<x<\frac{\pi }{2}\\ 1+\cos x,& \frac{\pi }{2}\le x<\pi \end{array}$ has a point of local maxima at $x=\frac{\pi }{2},$ is

Q.

Point of inflection is the point in the domain where function changes from convex to concave.

1. True

2. False

Q. The are bounded by the curve $y=f\left(x\right)$ and the lines $x=0,y=0$ and $x=t$, lies in the interval

1. $(\frac{21}{64},\frac{11}{16})$
2. $(\frac{3}{4},3)$
3. $(9,10)$
4. $(0,\frac{21}{64})$

Q. Which of the following function(s) has/have point of inflection?

1. $f\left(x\right)=x^{6/7}$
2. $f\left(x\right)=x^6$
3. $f\left(x\right)=\cos x+2x$
4. $f\left(x\right)=x|x|$

Q. If P(a)=2/3 and P(b)=3/4 and P(a intersection b)=1/2 then find P(a' intersection b') and P(a' u b')