Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is\[\frac{3}{4},\] then which one of the following can be true?
If \[f\left( a \right)=2,\] \[f'\left( a \right)=1,\] \[g\left( a \right)=-\,1,\] \[g'\left( a \right)=2,\] then find the value of \[\underset{x\to a}{\mathop{\lim }}\,\,\,\frac{g\,(x)f\,(a)-g\,(a)f\,(x)}{x-a}\].
If \[{{C}_{r}}\]stands for \[{}^{n}{{C}_{r}},\] then the sum of the series\[\frac{2\,\left( \frac{n}{2} \right)!\left( \frac{n}{2} \right)!}{2!}[C_{0}^{2}-2\,C_{1}^{2}+3\,C_{2}^{2}-.......+{{(-1)}^{n}}\,(n+1)\,C_{n}^{2}]\]. Where n is an even positive integer, is equal to:
Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey; 80 played cricket and basketball and 40 played cricket and hockey; 24 played all the three games. The number of boys who did not play any game is
If \[\alpha \] and \[\beta \] are the roots of \[{{x}^{2}}+px+q=0\] and \[{{\alpha }^{4}},\] \[{{\beta }^{4}}\] are the roots of \[{{x}^{2}}-rx+s=0,\] then the equation \[{{x}^{2}}-4qx+2{{q}^{2}}-r=0\] has always:
If the straight line through the point P(3, 4) makes an angle \[\frac{\pi }{6}\] with the x-axis and meets the line\[~12x+5y+10=0\] at Q, then the length PQ is
If \[\frac{1-3p}{2},\frac{1+4p}{3}\] and \[\frac{1+p}{6}\] are the probabilities of three mutually exclusive and exhaustive events, then the set of all values of p is
Let \[\alpha ,\]\[\beta \] be the roots of the equation \[\left( x-a \right)\left( x-b \right)=c,\] \[c\ne 0\].Then the roots of the equation \[(x-\alpha )(x-\beta )+c=0\] are:
If a, b, c, d and p are distinct real numbers such that \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{p}^{2}}-2(ab+bc+cd)p+({{b}^{2}}+{{c}^{2}}+{{d}^{2}})\le 0\]then a, b, c, d:
If \[{{z}_{1}},\] \[{{z}_{2}}\] are two complex numbers such that \[\left| \frac{{{z}_{1}}-{{z}_{2}}}{{{z}_{1}}+{{z}_{2}}} \right|=1\] and \[i{{z}_{1}}=k{{z}_{2}},\] where \[K\in R,\] then the angle between \[{{z}_{1}}-{{z}_{2}}\] and \[{{z}_{1}}+{{z}_{2}}\] is
A five digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is:
If \[\cos \theta =\frac{8}{17}\] and \[\theta \] lies in the 1st quadrant, then the value of \[\cos (30{}^\circ +\theta )+\cos (45{}^\circ -\theta )+\]\[\cos (120{}^\circ -\theta )\] is
Axis of a parabola is y = x and vertex and focus are at a distance \[2\sqrt{2}\]and \[5\sqrt{2}\]respectively from the origin. Then equation of the parabola is:
In a plane there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides no three lines pass through one point, no line passes through both points A and B and no two are parallel. Then the number of intersection points the lines have is equal to
Let A and B be two events such that\[p\left( \overline{A\bigcup B} \right)=\frac{1}{6},\] \[p(A\bigcap B)=\frac{1}{4}\] and \[p(\overline{A)}=\frac{1}{4},\]where \[\overline{A}\] stands for the complement of the event A. Then the events A and B are
A ate grapes and pineapple; B ate grapes and oranges; C ate oranges, pineapple and apple; D ate grapes, apple and pineapple. After taking fruits, B and C fell sick. In view of above facts, it can be said that the cause of sickness was:
A person X was driving in a place where all roads ran either north-south or east-west, forming a grid. Roads are at a distance of 1 km from each other in a parallel. He started at the intersection of two roads, drove 3 km north, 3 km west and 4 km south. Which further route could bring him back to his starting point, if the same route is not repeated?
There are five hobby clubs in a college- photography, yachting, chess, electronics and gardening. The gardening group meets every second day, the electronics group meets every third day, the chess group meets every fourth day, the yachting group meets every fifth day and the photography group meets every sixth day. How many times do all the five groups meet on the same day within 180 days?
In a question paper there are five questions to be attempted and answer to each question has two choices - True (T) or False (F). It is given that no two candidates have given the answers to the five questions in an identical sequence. For this to happen the "maximum number of candidates is:
Directions: Study the information below and answer questions based on it. Each of the six friends. A, B, C, D, E and F scored different marks in an examination. C scored more than only A and E. D scored less than only B. E did not score the least. The one who scored the third highest marks scored 81 marks. E scored 62 marks.
Which of the following could possibly be C's score?
Directions: Study the information below and answer questions based on it. Each of the six friends. A, B, C, D, E and F scored different marks in an examination. C scored more than only A and E. D scored less than only B. E did not score the least. The one who scored the third highest marks scored 81 marks. E scored 62 marks.
The person who scored the maximum scored 13 marks more than F's marks. Which of the following can be D's score?
Let \[P(x)={{a}_{0}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{4}}+.......{{a}_{n}}{{x}^{2n}}\] be a polynomial in a real variable x with\[0<{{a}_{0}}<{{a}_{1}}<{{a}_{2}}<.......<{{a}_{n}}\]. The function P(x) has:
In a race, a competitor has to collect 6 apples which are kept in a straight line on a track and a bucket is placed at the beginning of the track which is a starting point. The condition is that the competitor can pick only one apple at a time, run back with it and drop it in the bucket. If he has to drop all the apples in the bucket, how much total distance he has to run if the bucket is 5 meters from the first apple and all other apples are placed 3 meters apart?
The equation of the circle passing through (1, 2) and the points of intersection of \[{{x}^{2}}+{{y}^{2}}+15x-5y+8=0\]and\[2{{x}^{2}}+2{{y}^{2}}+6x-9y-25=0\]is:
The angle between a pair of tangents drawn from a point P to the circle \[{{x}^{2}}+{{y}^{2}}+8x-10y+30\,{{\sin }^{2}}\alpha 0+34\,{{\cos }^{2}}\alpha =0\] is \[2\alpha \]. The equation of the locus of the point P is:
Let \[\frac{7}{{{2}^{1/2}}+{{2}^{1/4}}+1}=A+B{{.2}^{1/4}}+C{{.2}^{1/2}}+D{{.2}^{3/4}},\] then find the value of\[{{A}^{2}}+{{B}^{2}}+{{C}^{2}}-{{D}^{2}}\].
Given that n A.M.'s are inserted between two sets of numbers a, 2b and 2a, b, where a, \[b\in R\]. Suppose further that \[{{m}^{th}}\]mean between these two sets of numbers is same, then the ratio a : b equals