Geometrical Interpretation of Derivative

Q.

Image of the point $\left(-1,3\right)$ w.r.t. the line $y=2x$ is

1. $\left(-1,-5\right)$

2. $\left(1,3\right)$

3. $\left(3,1\right)$

4. $\left(5,1\right)$

Q. Find the slope of the tangent to the curve $f\left(x\right)=2x^6+x^4-1$ at x=1.

Q. $\text{Statement 1:}$ The maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ (where $1\le x\le 3)$ is $36.$
$\text{Statement 2:}$ The maximum distance between the points $(5,-4)$ and the point on the circle $(x-2{)}^2+y^2=1$ is $6$

1. Both the statements are true but $\text{Statement 2}$ is not the correct explanation of $\text{Statement 1}$
2. Both the statements are true and $\text{Statement 2}$ is the correct explanation of $\text{Statement 1}$
3. $\text{Statement 1}$ is true and $\text{Statement 2}$ is false
4. $\text{Statement 1}$ is false and $\text{Statement 2}$ is true

Q. If the function $f\left(x\right)=x^4+b{x}^2+8x+1$ has a horizontal tangent and a point of inflection for the same value of $x,$ then the value of $b$ is equal to

1. $-1$
2. $1$
3. $6$
4. $-6$

Q.

In the above figure first derivative (f’(x)) of the function y = f(x) at point P will be equal to

1. sin(ψ)

2. cos(ψ)

3. tan (ψ)

4. None of the above

Q. $\text{Statement 1:}$ The maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ (where $1\le x\le 3)$ is $36.$
$\text{Statement 2:}$ The maximum distance between the points $(5,-4)$ and the point on the circle $(x-2{)}^2+y^2=1$ is $6$

1. Both the statements are true and $\text{Statement 2}$ is the correct explanation of $\text{Statement 1}$
2. Both the statements are true but $\text{Statement 2}$ is not the correct explanation of $\text{Statement 1}$
3. $\text{Statement 1}$ is true and $\text{Statement 2}$ is false
4. $\text{Statement 1}$ is false and $\text{Statement 2}$ is true

Q.

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$).
If ${\alpha }_1<{\beta }_1<\alpha <\beta$, then

1. f(x) has a maxima in $[{\alpha }_1,{\beta }_1]$ and a minima is $[\alpha ,\beta ]$

2. f(x) has a minima in $({\alpha }_1,{\beta }_1)$ and a maxima in $(\alpha ,\beta )$

3. f'(x) > 0 wherever defined

4. f'(x) < 0 wherever defined

Q. The greatest distance of the point $P(10,7)$ from the circle $x^2+y^2-4x-2y-20=0$ is-

1. None of these
2. $15$
3. $5$
4. $10$

Q.

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$). Then if ${\alpha }_1<\alpha <{\beta }_1<\beta$, then

1. f(x) is increasing in ${\alpha }_1,{\beta }_1$

2. f(x) is decreasing in $\alpha ,\beta$

3. f(x) is decreasing in ${\beta }_1,\beta$

4. f(x) is decreasing in $(-\infty ,\alpha )$

Q. $\begin{array}{cccccc}& \text{Column I}& & \text{Column II}& & \text{Column III}\\ \left(1\right)& x^2+y^2=a^2& \left(A\right)& y=mx+\frac{a}{m}& \left(P\right)& x^2+y^2=2a^2\\ \left(2\right)& y^2=4ax& \left(B\right)& y=mx\pm \sqrt{a^2{m}^2+b^2}& \left(Q\right)& x^2+y^2=a^2+b^2\\ \left(3\right)& \frac{x^2}{a^2}+\frac{y^2}{b^2}=1& \left(C\right)& y=mx\pm \sqrt{a^2{m}^2-b^2}& \left(R\right)& x^2+y^2=a^2-b^2\\ \left(4\right)& \frac{x^2}{a^2}-\frac{y^2}{b^2}=1& \left(D\right)& y=mx\pm a\sqrt{1+m^2}& \left(S\right)& x=-a\end{array}$

$a)$ Column $\text{I}$ represents the equation of the conic.
$b)$ Column $\text{II}$ represents the slope form of the tangent.
$a)$ Column $\text{III}$ represents the director circle of the conic from column $\text{I}$

Which of the following is only correct combination?

1. $\left(1\right)\to \left(A\right)\to \left(P\right)$
2. $\left(2\right)\to \left(A\right)\to \left(P\right)$
3. $\left(3\right)\to \left(B\right)\to \left(Q\right)$
4. $\left(4\right)\to \left(C\right)\to \left(Q\right)$

Q.

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$). Then if ${\alpha }_1<\alpha <{\beta }_1<\beta$, then

1. f(x) is increasing in ${\alpha }_1,{\beta }_1$

2. f(x) is decreasing in $\alpha ,\beta$

3. f(x) is decreasing in ${\beta }_1,\beta$

4. f(x) is decreasing in $(-\infty ,\alpha )$

Q. $\text{Statement 1:}$ The maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ (where $1\le x\le 3)$ is $36.$
$\text{Statement 2:}$ The maximum distance between the points $(5,-4)$ and the point on the circle $(x-2{)}^2+y^2=1$ is $6$

1. Both the statements are true and $\text{Statement 2}$ is the correct explanation of $\text{Statement 1}$
2. Both the statements are true but $\text{Statement 2}$ is not the correct explanation of $\text{Statement 1}$
3. $\text{Statement 1}$ is true and $\text{Statement 2}$ is false
4. $\text{Statement 1}$ is false and $\text{Statement 2}$ is true

Q.

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$).
If the equations $x^2$ + bx + c = 0 and $x^2+b_{1}x+c_1=0$ do not have real roots, then

1. f'(x) = 0 has real and distinct roots

2. f'(x) = 0 has real and equal roots

3. f'(x) = 0 has imaginary roots

4. nothing can be said

Q. The maximum distance of the point $(4,4)$ from the circle $x^2+y^2-2x-15=0$ is

1. $10$
2. $9$
3. $5$
4. none of these

Q. The sum of the miinimum and maximum distances of the point $(4,-3)$ to the circle
$x^2+y^2+4x-10y-7=0$ is

1. $10$
2. $16$
3. $12$
4. $20$