Rolle's Theorem

Q. solve\sqrt{2x+5}+\sqrt{x-1}>8

Q. If the circle x² + y² + 2ax + 8y + 16 = 0 touches x-axis, then the value of a is
(a) ± 16
(b) ± 4
(c) ± 8
(d) ± 1

Q. The equation of a circle with radius 5 and touching both the coordinate axes is
(a) x² + y² ± 10x ± 10y + 5 = 0
(b) x² + y² ± 10x ± 10y = 0
(c) x² + y² ± 10x ± 10y + 25 = 0
(d) x² + y² ± 10x ± 10y + 51 = 0

Q.

If $f\left(x\right)=\left|\begin{array}{ccc}sinx& sina& sinb\\ cosx& cosa& cosb\\ tanx& tana& tanb\end{array}\right|,$

where $0<a<b<\frac{\pi }{2}$
then the equation
$f^{\prime }\left(x\right)=0$ has in the interval $(a,b)$

1. Atleast one root

2. Atmost one root

3. No root

4. exactly one root

Q. Let the sets be A = {x : x is a real root of equation (x2 – 5x + 6)·(x2 – 12x + 35) = 0} and B = {1, 2, 3, 4, 5}, then the number of ordered pairs in (A × B) ∩ (B × A) is

Q.

If a function f(x )is continuous in [2, 5] , differentiable in (2, 5) and f(2) = f(5) then how many value x can have where f'(x) nullifies for sure?

1. zero

2. More than one

3. At most one

4. At least one

Q. If family of straight lines $ax+by+c=0$ always passes through a fixed point $(\frac{3}{2},1),$ then equation $36a{x}^2+8bx+2c=0$ has

1. at least one root in $[0,1]$
2. atleast one root in $[\frac{-1}{2},\frac{1}{2}]$
3. atleast one root in $[-1,2]$
4. atleast one root in $[0,\frac{1}{2}]$

Q. If Rolle's theorem holds true for the function $f\left(x\right)=2x^3+b{x}^2+cx,x\in [-1,1]$ at the point $x=\frac{1}{2},$ then $(2b+c)$ is equal to

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

Q. Write the number of integral solutions of $\frac{x+2}{x^2+1}>\frac{1}{2}$.

Q. If $c$ is a point at which Rolle's theorem holds for the function, $f\left(x\right)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $a\in R$, then $f"\left(c\right)$ is equal to:

1. $-\frac{1}{24}$
2. $\frac{-1}{12}$
3. $\frac{\sqrt{3}}{7}$
4. $\frac{1}{12}$

Q. If √x+1 + √x-1 = 2 , then the value of x is

Q. Let A = (3, –4), B = (1, 2) and P = (2λ – 1, 2λ + 1). If the sum PA + PB is minimum, then the value of λ is