Using MO theory, predict which of the following species has the shortest bond length?

The two functional groups present in a typical carbohydrate are

Let $$A$$ be $$a2\times 2$$ matrix with real entries. Let $$I$$ be the $$2\times 2$$ identity matrix. Denote by tr$$(A)$$, the sum of diagonal entries of $$a$$. Assume that $$a^2=I.$$ Statement-1 : If $$A\ne I$$ and $$A\ne -I$$, then det$$(A)=-1$$ Statement- 2 : If $$A\ne I$$ and $$A\ne -I$$, then tr $$(A)$$ $$\ne 0$$.

Let $$a,b,c$$ be any real numbers. Suppose that there are real numbers $$x,y,z$$ not all zero such that $$x=cy+bz,$$ $$y=az+cx,$$ and $$z=bx+ay.$$ Then $$a^2+b^2+c^2+2abc$$ is equal to :

Let $$A$$ be a square matrix all of whose entries are integers. Then which one of the following is true?

It is given that the events $$A$$ and $$B$$ are such that P\left( A \right) = {1 \over 4},P\left( {A|B} \right) = {1 \over 2} and P\left( {B|A} \right) = {2 \over 3}. Then $$P(B)$$ is :

A die is thrown. Let $$A$$ be the event that the number obtained is greater than $$3.$$ Let $$B$$ be the event that the number obtained is less than $$5.$$ Then $$P\left(A\cup B\right)$$ is :

The solution of the differential equation {{dy} \over {dx}} = {{x + y} \over x} satisfying the condition $$y(1)=1$$ is :

The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept -4. Then a possible value of k is :

$$AB$$ is a vertical pole with $$B$$ at the ground level and $$A$$ at the top. $$A$$ man finds that the angle of elevation of the point $$A$$ from a certain point $$C$$ on the ground is $$60^{\circ }$$. He moves away from the pole along the line $$BC$$ to a point $$D$$ such that $$CD=7$$ m. From $$D$$ the angle of elevation of the point $$A$$ is $$45^{\circ }$$. Then the height of the pole is :

Statement - 1 : $$\sum _{r=0}^n\left(r+1\right){}^n{C}_r=\left(n+2\right)2^{n-1}.$$ Statement - 2 : $$\sum _{r=0}^n\left(r+1\right){}^n{C}_r{x}^r={\left(1+x\right)}^n+nx{\left(1+x\right)}^{n-1}.$$

The value of \sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} is

Let $$f:N\to Y$$ be a function defined as f(x) = 4x + 3 where Y = { y $$\in$$ N, y = 4x + 3 for some x $$\in$$ N }. Show that f is invertible and its inverse is

A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at :

The conjugate of a complex number is {1 \over {i - 1}} then that complex number is :

The point diametrically opposite to the point $$P(1,0)$$ on the circle $$x^2+y^2+2x+4y-3=0$$ is :

The differential equation of the family of circles with fixed radius $$5$$ units and centre on the line $$y=2$$ is :

A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is {{1 \over 2}}. Then the length of the semi-major axis is :

Let R be the real line. Consider the following subsets of the plane $$R\times R$$ : $$S=\left\{(x,y):y=x+1and0<x<2\right\}$$ $$T=\left\{(x,y):x-yisan\mathrm{int}eger\right\}$$, Which one of the following is true ?

Let f\left( x \right) = \left\{ {\matrix{ {\left( {x - 1} \right)\sin {1 \over {x - 1}}} & {if\,x \ne 1} \cr 0 & {if\,x = 1} \cr } } \right. Then which one of the following is true?