If \[{{a}_{r}}>0,\] \[r\in N\] and \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},.......{{a}_{2n}}\] are in AP, then \[\frac{{{a}_{1}}+{{a}_{2n}}}{\sqrt{{{a}_{1}}}+\sqrt{{{a}_{2}}}}+\frac{{{a}_{2}}+{{a}_{2n-1}}}{\sqrt{{{a}_{2}}}+\sqrt{{{a}_{3}}}}+\frac{{{a}_{3}}+{{a}_{2n-2}}}{\sqrt{{{a}_{3}}}+\sqrt{{{a}_{4}}}}+......+\frac{{{a}_{n}}+{{a}_{n+1}}}{\sqrt{{{a}_{n}}}+\sqrt{{{a}_{n+1}}}}\] is equal
Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is
Let the complex numbers \[{{z}_{1}},\] \[{{z}_{2}}\] and \[{{z}_{3}}\] be the vertices of an equilateral triangle. Let \[{{z}_{0}}\] be the circumventer of the triangle, then \[{{z}_{1}}^{2}+{{z}_{2}}^{2}+{{z}_{3}}^{2}=\]
If \[\sin \alpha +\sin \beta +\sin \gamma =0=\cos \alpha +\cos \beta +\cos \gamma ,\] then the value of \[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma \] is
If a denotes the number of permutations of \[x+2\]things taken all at a time, b the number of permutations of x things taken 11 at a time and c the number of permutations of \[x-11\] things taken all at a time such that a = 182bc, then the value of x is
In a city no two persons have identical sets of teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positions of the teeth, then the maximum population of the city is
Six points in a plane be joined in all possible ways by indefinite straight lines, and if no two of them be coincident or parallel, and no three pass through the same point (with the exception of the original 6 points). The number of distinct points of intersection is equal to
The value of \[{{\sin }^{2}}5{}^\circ +{{\sin }^{2}}10{}^\circ +{{\sin }^{2}}15{}^\circ \]\[+.......+{{\sin }^{2}}85{}^\circ +{{\sin }^{2}}90{}^\circ \] is equal to
The range of values of r for which the point\[\left( -5+\frac{r}{\sqrt{2}},-3+\frac{r}{\sqrt{2}} \right)\] is an interior point of the major segment of the circle \[{{x}^{2}}+{{y}^{2}}=16,\] cut off by the line \[x+y=2,\] is
PP' is a diameter of the ellipse \[{{b}^{2}}{{x}^{2}}+{{a}^{2}}{{y}^{2}}={{a}^{2}}{{b}^{2}}\]such that \[PP{{'}^{2}}\] is the arithmetic mean of the squares of the major and minor axes. Then the slope of PP' is
A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at random. Find the probability that among the balls drawn there is at least one ball of each colour.
If \[\cos \frac{\pi }{7},\] \[cos\frac{3\pi }{7},\] \[\cos \frac{5\pi }{7}\] be the roots of the equation \[8{{x}^{3}}-4{{x}^{2}}-4x+1=0,\] then find the equation whose roots are \[{{\sec }^{2}}\,\,\frac{\pi }{7},\] \[{{\sec }^{2}}\,\,\frac{3\pi }{7}\] and\[{{\sec }^{2}}\frac{5\pi }{7}\].
For \[a\in R\] (the set of all real numbers), \[a\ne -\,1,\] \[\underset{x\to \infty }{\mathop{\lim }}\,\,\,\frac{({{1}^{a}}+{{2}^{a}}+....+{{n}^{a}})}{{{(n+1)}^{a-1}}[(na+1)+(na+2)+....(na+n)]}=\frac{1}{60}\]
A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps he is one step away from the starting point.
Find the value of y, if \[y=\,\,6+{{\log }_{3/2}}\left[ \frac{1}{3\,\sqrt{2}}\sqrt{4-\frac{1}{3\,\sqrt{2}}\sqrt{4-\frac{1}{3\,\sqrt{2}}\sqrt{4-\frac{1}{3\,\sqrt{2}}.......,}}} \right]\]