For the following questions answer them individually
Let $$\alpha$$ and $$\beta$$ be the roots of the equation $$x^{2}+2ax+\left(3a+10\right)=0$$ such that $$\alpha < 1 < \beta$$. Then the set of all possible values of a is :
Let A= {2, 3, 5, 7, 9}. Let R be the relation on A defined by x R y if and only if $$2x\leq3y$$. Let l be the number of elements in R, and m be the minimum number of elements required to be added in R to make it a symmetric relation. Then l + m is equal to:
If the system of equations
3x + y + 4z = 3
$$2x+\alpha y-z = -3$$
x+ 2y + z = 4
has no solution, then the value of $$\alpha$$ is equal to :
If the area of the region $$\left\{\left(x,y\right): 1-2x \leq y \leq4-x^{2}, x\geq 0, y\geq0 \right\} \text{is} \frac{\alpha}{\beta} , \alpha,\beta \epsilon N, \text{gcd} \left(\alpha,\beta\right)=1, \text{then the value of } \left(\alpha+\beta\right) \text{is}$$
Let $$f: R\rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$ for all $$x\in R$$ and f'(a-1)=0, where a is a real number. Let g(x)= $$f(\tan^{2}x- 2\tan x+a)$$, $$0 < x < \frac{\pi}{2}$$.
Consider the following two statements :
(I) $$\text{g is increasing in } \left(0, \frac{\pi}{4} \right)$$
(II) $$\text{g is deceasing in } \left( \frac{\pi}{4} , \frac{\pi}{2} \right)$$
Then,
For the matrices $$A=\begin{bmatrix}3 -4 \\1 -1 \end {bmatrix}$$ and $$B=\begin{bmatrix}-29 49 \\-13 18 \end{bmatrix}$$, if $$\left(A^{15} + B \right) \begin{bmatrix}x \\y\end{bmatrix} = \begin{bmatrix}0 \\0 \end{bmatrix}, \text{then among the following which one is true ? }$$
Let one end of a focal chord of the parabola $$y^{2}=16x$$ be (16,16). If $$P\left(\alpha,\beta\right)$$ divides this focal chord internally in the ratio 5 : 2, then the minimum value of $$\alpha+\beta$$ is equal to :
Let $$A =\left\{x: |x^{2}-10|\leq6 \right\}$$ and $$B= \left\{x:|x-2|>1 \right\}$$. Then
If the line $$\alpha x+4y=\sqrt{7}$$, where $$\alpha \epsilon R$$, touch the ellipse $$3x^{2}+4y^{2}=1$$ at the point P in the first quadrant, then one of the focal distances of P is:
Let y = y(x) be the solution of the differential equation $$\sec x \frac{dy}{dx}-2y=2+3\sin x, x\epsilon \left(-\frac{\pi}{2}, \frac{\pi}{2} \right), y(0)=-\frac{7}{4}$$. Then $$y\left(\frac{\pi}{6}\right)$$ is equal to:
