If A and B are two sets containing 4 and 9 elements respectively, then the minimum and the maximum value of number of elements in \[A\cup B\] are, respectively _______.
If \[f:R\to R\] satisfies \[f\left( x\text{+}y \right)=f\left( x \right)+f\left( y \right),\] \[\forall x,\] \[y\in R\] and f(1)= 11, then \[\sum\limits_{r=1}^{n}{f(r)}\] is _______.
At the foot of a mountain, the elevation of its peak is found to be \[\frac{\pi }{4},\] after ascending h metre towards the mountain up a slope of \[\frac{\pi }{6}\] inclination, the elevation is found to be \[\frac{\pi }{3}\]. Height of the mountain is ________.
The sides of a \[\Delta \,ABC\]are in AP, if the angles A and C are the greatest and the smallest angles respectively, then \[4\,\,\left( 1-cos\,A \right)\left( 1cos\,C \right)\] is equal to _______.
The sum of infinite terms of a decreasing G.P. is equal to the greatest value of the function \[f(x)={{x}^{3}}+3x-9\] in the interval \[\left[ -\,2,3 \right]\] and the difference between the first two terms is f'(0). Then, the common ratio of a G.P. is
If a, b and c are positive numbers in a G.P., then the roots of the quadratic equation \[({{\log }_{e}}a){{x}^{2}}-2({{\log }_{e}}b)x+({{\log }_{e}}c)=0\] are ______.
The number of points having both coordinates as integers that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0) is _______.
If p and q are respectively the perpendiculars from the origin upon the straight lines, whose equations are \[x\,\sec \theta +y\,\,\cos ec\theta =a\] and\[x\cos \theta -y\,\,\sin \theta =a\,\,\cos 2\theta \], then \[4{{p}^{2}}+{{q}^{2}}\] is equal to _________.
The abscissa of two points M and N are the roots of the equation \[{{x}^{2}}+2ax-{{b}^{2}}=0\] and their ordinates are the roots of the equation\[{{x}^{2}}+2px-{{q}^{2}}=0\]. The radius of the circle with MN as diameter is_________.
The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse \[{{x}^{2}}+2{{y}^{2}}=6\] which touches the ellipse \[{{x}^{2}}+4{{y}^{2}}=4\] is ________.
If A and B are any two events having \[p\,(A\,\,\cup \,\,B)=\,\,\frac{1}{2}\]and\[p(\overline{A})=\frac{2}{3},\]then the probability of \[\left( \overline{A}\,\,\cap \,\,B \right)\] is________.
The mean and variance of 7 observations are 8 and 16, respectively. If 5 of the observations are 2, 4, 10, 12 and 14, then find the remaining two observations.
Sanjeev walks 10 metres towards the South. Turning to the left, he walks 20 metres and then moves to his right. After moving a distance of 20 metres, he turns to the right and walks 20 metres. Finally, he turns to the right and moves a distance of 10 metres. How far and in which direction is he from the starting point?
How many such pairs of letters are there in the word OVERWHELM wherein each of which has as many letters between them in the word as in the English alphabet?
Direction: Study the following information carefully and answer the questions given below:
Eight family members Dhruv, Garima, Avinash, Varsha, Aakash, Deepti, Charu and Moksh are sitting around a square table in such a way that two persons sit on each of the four sides of the table facing the centre. Members sitting on opposite sides are exactly opposite to each other. Aakash and Garima are exactly opposite to each other. Deepti is immediately right to Garima. Dhruv and Moksh are sitting on the same side. Moksh is exactly opposite of Avinash who is to the immediate left of Varsha, Dhruv is towards right of Deepti.
Direction: Study the following information carefully and answer the questions given below:
Eight family members Dhruv, Garima, Avinash, Varsha, Aakash, Deepti, Charu and Moksh are sitting around a square table in such a way that two persons sit on each of the four sides of the table facing the centre. Members sitting on opposite sides are exactly opposite to each other. Aakash and Garima are exactly opposite to each other. Deepti is immediately right to Garima. Dhruv and Moksh are sitting on the same side. Moksh is exactly opposite of Avinash who is to the immediate left of Varsha, Dhruv is towards right of Deepti.
Who is next to Varsha in the anti-clockwise direction?
Direction: Study the following information carefully and answer the questions given below:
Eight family members Dhruv, Garima, Avinash, Varsha, Aakash, Deepti, Charu and Moksh are sitting around a square table in such a way that two persons sit on each of the four sides of the table facing the centre. Members sitting on opposite sides are exactly opposite to each other. Aakash and Garima are exactly opposite to each other. Deepti is immediately right to Garima. Dhruv and Moksh are sitting on the same side. Moksh is exactly opposite of Avinash who is to the immediate left of Varsha, Dhruv is towards right of Deepti.
Direction: Study the following information carefully and answer the questions given below:
Eight family members Dhruv, Garima, Avinash, Varsha, Aakash, Deepti, Charu and Moksh are sitting around a square table in such a way that two persons sit on each of the four sides of the table facing the centre. Members sitting on opposite sides are exactly opposite to each other. Aakash and Garima are exactly opposite to each other. Deepti is immediately right to Garima. Dhruv and Moksh are sitting on the same side. Moksh is exactly opposite of Avinash who is to the immediate left of Varsha, Dhruv is towards right of Deepti.
Which of the following statements is definitely true?
A circle of radius r is inscribed in a square. The mid-points of sides of the square have been connected by line segment and a new square resulted. The sides of the resulting square were also connected by segment so that a new square was obtained and so on, then the radius of the circle inscribed in the \[{{n}^{th}}\] square is ____.
If \[{{Z}_{1}}=a+i\,b\] and \[{{Z}_{2}}=c+i\,d\] are complex numbers such that \[\left| {{Z}_{1}} \right|=\left| {{Z}_{2}} \right|=1\] and \[\operatorname{Re}({{Z}_{1}}\overline{{{Z}_{2}}})=0,\] then the pair of complex numbers \[a+i\,c={{w}_{1}}\] and \[b+i\,d={{w}_{1}}\] satisfies ____.
Let \[a={{3}^{\frac{1}{223}}}+1,\] for all \[n\ge 3\] and let \[f(x)={}^{n}{{C}_{0}}{{a}^{n-1}}-{}^{n}{{C}_{1}}{{a}^{n-3}}+{}^{n}{{C}_{2}}{{a}^{n-3}}-.....+{{(-1)}^{n-1}}{}^{n}{{C}_{n-1}}{{a}^{0}}.\] If the value of \[f\left( 2007 \right)+f\left( 2008 \right)\text{ }=\text{ }{{3}^{k}},\] where \[k\in N,\] then the value of k is ____.
The two adjacent sides of a cyclic quadrilateral are 2 m and 5 m and the angle between them is \[60{}^\circ \]. If the area of the quadrilateral is \[4\sqrt{3}\,{{m}^{2}},\] then the remaining two sides (in m) are _______.
Let \[f:R\to [0,\infty )\] be such that \[\underset{x\to 5}{\mathop{\lim }}\,\,\,f(x)\] exists and \[\underset{x\to 5}{\mathop{\lim }}\,\,\,\,\frac{{{[f(x)]}^{2}}-9}{\sqrt{|x-5|}}=0\]. Then, \[\underset{x\to 5}{\mathop{\lim }}\,\,\,f(x)\] equals to ________.
A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these four members) for the team. If the team has to include almost one boy, the number of ways of selecting the team is __________.