Let a be a complex number such that |a| < 1 and \[{{z}_{1}},\,{{z}_{2}}.....\] be vertices of a polygon such that \[{{z}_{k}}=1+a+{{a}^{2}}+.........+{{a}^{k-1}}\].Then the vertices of the polygon lie within a circle
The interior angles of a polygon are in A.P. If the smallest angle be \[120{}^\circ \] and the common difference be \[5{}^\circ ,\] then the number of sides of the polygon
In a certain test there are n questions. In this test \[{{2}^{k}}\]students gave wrong answers to at least \[\left( n-k \right)\] questions, where k = 0, 1, 2, ......., n. If the total number of wrong answers is 4095, then value of n is
Value of x for which the sixth term of the expansion of \[E={{\left( {{3}^{{{\log }_{3}}\sqrt{{{9}^{|x-2|}}}}}+{{7}^{(1/5){{\log }_{7}}[(4){{.3}^{|x+2|}}-9]}} \right)}^{7}}\] is 567, are
Let \[S={{C}_{1}}-\left( 1+\frac{1}{2} \right)\,\,{{C}_{2}}+\left( 1+\frac{1}{2}+\frac{1}{3} \right)\,\,{{C}_{3}}-....+\] \[{{(-1)}^{n-1}}\left( 1+\frac{1}{2}....+\frac{1}{n} \right){{c}_{n}},\] then
If a variable line drawn through the point of intersection of straight lines \[\frac{x}{\alpha }+\frac{y}{\beta }=1\] and \[\frac{x}{\beta }+\frac{y}{\alpha }=1\] meets the coordinate axes in A and B, then the locus of the mid-point of A is
If \[{{P}_{1}}\,,\] \[{{P}_{2}}\] and \[{{P}_{3}}\]be the perpendiculars from the points \[({{m}^{2}},\,\,2m)\], \[\left( mm',m+m' \right)\] and \[(m{{'}^{2}},2m')\]respectively on the line \[x\cos \alpha +y\sin \alpha \]\[+\frac{{{\sin }^{2}}\alpha }{\cos \alpha }=0,\] then \[{{P}_{1}},\,\,{{P}_{2}}\,\,and\,\,{{P}_{3}}\]P. P, are in
The straight line \[3x+4y-5=0\] and \[4x=3y+15\] intersect at the point P. On these lines the points Q and R are chosen so that PQ = QR. The slopes of the lines QR passing through (1, 2) are
Let a circle touches to the directrix of a parabola \[{{y}^{2}}=2ax\] has its centre coinciding with the focus of the parabola. Then the point of intersection of the parabola and circle is
Direction: In a music school 28 students learn trumpet 30 students learn violin and 32 students learn guitar. 6 students learn trumpet and violin 8 students learn violin and guitar 10 students learn guitar and trumpet. The number of students who learn only one instrument is 54. Also 20 students learn only violin. Every student learn at least one instrument out of the three instruments.
Direction: In a music school 28 students learn trumpet 30 students learn violin and 32 students learn guitar. 6 students learn trumpet and violin 8 students learn violin and guitar 10 students learn guitar and trumpet. The number of students who learn only one instrument is 54. Also 20 students learn only violin. Every student learn at least one instrument out of the three instruments.
Find the number of students who learn trumpet and guitar but not violin?
In a certain coding system, 'rbm std bro pus' means 'the cat is beautiful', 'tnh pus dim std? means 'the dog is brown', 'pus dim bro pus cus' means 'the dog has the cat'. What is the code for 'has'?
Direction: Read the information carefully and answer the question based on it. Five persons are sitting in a row. One of the two persons at the extreme ends is intelligent and other one is fair. A fat person is sitting to the right of a weak person. A tall person is too left of the fair person and the weak person is sitting between the intelligent and the fat person.
Direction: Read the information carefully and answer the question based on it. Five persons are sitting in a row. One of the two persons at the extreme ends is intelligent and other one is fair. A fat person is sitting to the right of a weak person. A tall person is too left of the fair person and the weak person is sitting between the intelligent and the fat person.
Direction: Read the information carefully and answer the question based on it. Five persons are sitting in a row. One of the two persons at the extreme ends is intelligent and other one is fair. A fat person is sitting to the right of a weak person. A tall person is too left of the fair person and the weak person is sitting between the intelligent and the fat person.
Person to the left of weak person possesses which of the following characteristics?
Direction: In each question given below certain symbols are used with the following meaning.
P ?Q means 'P is greater than Q'.
P @ Q, means 'P is either greater than or equal to Q'.
P * Q means 'P is equal to Q'.
P # Q. means 'P is either smaller than or equal to Q'.
P $ Q means 'P is smaller than Q'.
Now, in each of the following questions, assuming the given statements to be true, find which of the two conclusions I & II given below them is/are definitely true.
Direction: In each question given below certain symbols are used with the following meaning.
P ?Q means 'P is greater than Q'.
P @ Q, means 'P is either greater than or equal to Q'.
P * Q means 'P is equal to Q'.
P # Q. means 'P is either smaller than or equal to Q'.
P $ Q means 'P is smaller than Q'.
Now, in each of the following questions, assuming the given statements to be true, find which of the two conclusions I & II given below them is/are definitely true.
Direction: In each question given below certain symbols are used with the following meaning.
P ?Q means 'P is greater than Q'.
P @ Q, means 'P is either greater than or equal to Q'.
P * Q means 'P is equal to Q'.
P # Q. means 'P is either smaller than or equal to Q'.
P $ Q means 'P is smaller than Q'.
Now, in each of the following questions, assuming the given statements to be true, find which of the two conclusions I & II given below them is/are definitely true.
Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statement/Expressions on the left with the Statements/Expressions on the right
a.
The number of permutations the word ENDEA is
(p)
5!
b.
The number of permutations in Which the letter E occurs in the first and the last positions is
(q)
\[2\times 5!\]
c.
The number of permutations in Which none of the letters D, L, N occurs in the last five positions is
(r)
\[7\times 5!\]
d.
The number of permutations in Which the letters A, E, 0 occur only in odd position is
If PQ is a double ordinate of hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies
Consider the following statements about be three arbitrary events \[{{E}_{1}},\,{{E}_{2}},\,{{E}_{3}}\]of a sample space S. Which of the following statements are correct?
A)
P (only one of them occurs)\[=P\,({{\bar{E}}_{1}}{{E}_{2}}{{E}_{3}}+{{E}_{1}}{{\bar{E}}_{2}}{{E}_{3}}+{{E}_{1}}{{E}_{2}}{{\bar{E}}_{3}})\]
doneclear
B)
P (none of them occurs)\[=P\,({{\bar{E}}_{1}}+{{\bar{E}}_{2}}+{{\bar{E}}_{3}})\]
doneclear
C)
P (at least one of them occurs)\[=P\,({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\]
doneclear
D)
P (all three occurs)\[=P\,({{E}_{1}}+{{E}_{2}}+{{E}_{3}})\]