For the following questions answer them individually
For a $$3\times 3$$ matrix , let trace (M) denote the sum of all the diagonal elements of M. Let A be a $$3\times 3$$ matrix such that $$|A|=\frac{1}{2}$$ trace (A) =3.If B=adj(adj(2A)), then the value of $$|B|+$$ trace (B)equals:
In a group of 3 girls and 4 boys, there are two boys $$B_{1}\text{ and }B_{2}$$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $$B_{1}\text{ and }B_{2}$$ are not adjacent to each other, is :
Let $$\alpha,\beta,\gamma$$ and $$\delta$$ be the coefficients of $$x^{7},x^{5},x^{3}$$ and x respectively in the expansion of $$(x+\sqrt{x^{3}-1})^{5}+(x-\sqrt{x^{3}-1})^{5},x > 1$$.If u and v satisfy the equations $$\alpha u+\beta v=18\\ \gamma u+\delta v=20$$ then u+v equals:
Let a line pass through two distinct points P(−2,−1, 3) and , and be parallel to the vector $$3\widehat{i}+2\widehat{j}+2\widehat{k}$$. If the distance of the point Q from the point R(1, 3, 3) is 5 , then the square of the area of △PQR is equal to :
If and are two events such that P(A ∩ B) = 0.1. and P(A | B) and P(B ∣ A) are the roots of the equation $$12x^{2} − 7x + 1 = 0$$, then the value of $$\frac{P(\overline{A} \cup \overline{B})}{P(\overline{A} \cap \overline{B}}$$ is:
If $$\int_{}^{}e^{x}\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{sin^{-1}x}{(1-x^{2})^{3/2}}+\frac{x}{1-x^{2}}\right)dx=g(x)+C$$ where C is the constant of integration, then $$g(\frac{1}{2})$$ equals
The area of the region enclosed by the curves $$y=x^{2}-4x+4\text{ and }y^{2}=16-8x$$ is :
Let $$f(x)=\int_{0}^{x^{2}}\frac{t^{2}-8t+15}{e^{t}}dt,x\in R$$. Then the numbers of local maximum and local minimum points of f.respectively, are :
Let $$P(4, 4\sqrt{3})$$be a point on the parabola $$y^{2}=4ax$$ and and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to :
Let $$\overrightarrow{a}$$ and $$ \overrightarrow{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{3}$$. Tf $$\lambda \overrightarrow{a} +2\overrightarrow{b}\text{ and }3\overrightarrow{a}-\lambda \overrightarrow{b}$$ are perpendicular to each other, then the number of values of $$\lambda$$ in [-1,3] is :
