Two finite sets have p and q elements. The total number of subsets of the first set is 112 more than total number of subsets of second set. Find the value of p + q.
If \[\tan \,\,\alpha =\frac{a}{b},\] where \[\alpha =12\beta \] and \[\alpha \] being an acute angle, then \[\frac{1}{2}\]\[[a\,\cos ec\,4\beta -b\,\sec \,4\beta ]\] is equal to ????.
The upper \[{{\left( \frac{3}{4} \right)}^{th}}\]portion of a vertical pole subtends an angle tan \[{{\tan }^{-1}}\left( \frac{3}{5} \right)\] at a point in the horizontal plane through its foot and at a distance of 40 m from the foot. The possible height of the vertical pole is _______.
A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?
The sum of the coefficients of the binomial expansion of \[{{\left( \frac{1}{x}+2x \right)}^{n}}\] is equal to 6561. The constant term in the expansion is _________.
Let S be the focus of the parabola \[{{y}^{2}}=8x\] and PQ be the common chord of the circle \[{{x}^{2}}+{{y}^{2}}-2x-4y=0\] and the given parabola. The area of \[\Delta \,PQS\]is ________.
Let the equation of an ellipse be \[\frac{{{x}^{2}}}{144}+\frac{{{y}^{2}}}{25}=1\]. Then, the radius of the circle with centre \[\left( 0,\,\sqrt{2} \right)\] and passing through the foci of the ellipse is ________.
The mean of the data set comprising of 16 observations is 16. If one of the observations valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is _______.
In a certain code language, 'col tip mot' means 'singing is appreciable', 'mot bajmin' means 'dancing is good' and 'tip nopbaj' means 'singing and dancing', which of the following means 'good' in that code language?
Facing the East, Rajesh turns left and walks 10 metres, then he turns to his left again and walks 10 m. He then turns \[45{}^\circ \] towards his right and goes straight to cover 25 metres. Now, he is in which direction from his starting point?
In a row of boys, Aman is 16th from the left and Vivek is 18th from the right. Gagan is 11th from Aman towards the right and 3rd from Vivek towards the right end. How many number of boys are there in the row?
Direction: Study the information given below and answer the questions that follow:
There is a family of six persons A, B, C, D, E and F. They are lawyer, doctor, teacher, salesman, engineer and accountant. There are two married couples in the family. D, the salesman, is married to the lady teacher. The doctor is married to the lawyer. F, the accountant, is the son of B and brother of E. C, the lawyer, is the daughter-in-law of A. E is an unmarried engineer. A is the grandmother of F.
Direction: Study the information given below and answer the questions that follow:
There is a family of six persons A, B, C, D, E and F. They are lawyer, doctor, teacher, salesman, engineer and accountant. There are two married couples in the family. D, the salesman, is married to the lady teacher. The doctor is married to the lawyer. F, the accountant, is the son of B and brother of E. C, the lawyer, is the daughter-in-law of A. E is an unmarried engineer. A is the grandmother of F.
Direction: Study the information given below and answer the questions that follow:
There is a family of six persons A, B, C, D, E and F. They are lawyer, doctor, teacher, salesman, engineer and accountant. There are two married couples in the family. D, the salesman, is married to the lady teacher. The doctor is married to the lawyer. F, the accountant, is the son of B and brother of E. C, the lawyer, is the daughter-in-law of A. E is an unmarried engineer. A is the grandmother of F.
Direction: Study the information given below and answer the questions that follow:
There is a family of six persons A, B, C, D, E and F. They are lawyer, doctor, teacher, salesman, engineer and accountant. There are two married couples in the family. D, the salesman, is married to the lady teacher. The doctor is married to the lawyer. F, the accountant, is the son of B and brother of E. C, the lawyer, is the daughter-in-law of A. E is an unmarried engineer. A is the grandmother of F.
N, the set of natural numbers is partitioned into subsets \[{{S}_{1}}=\{1\},\] \[{{S}_{2}}=\{2,\,3\},\] \[{{S}_{3}}=\{4,\,5,\,6\},\] \[{{S}_{4}}=\{7,\,8,\,9,\,10\}\] and so on. Find the sum of elements of the subset\[{{S}_{30}}\].
In a \[\Delta \,ABC\], the maximum value of \[\frac{a\,{{\cos }^{2}}\frac{A}{2}+b\,{{\cos }^{2}}\frac{B}{2}+c\,{{\cos }^{2}}\frac{C}{2}}{a+b+c}\] is ________.
Let two fair six-faced dice A and B be thrown simultaneously. If \[{{E}_{1}}\]is the event that die A shows up four, \[{{E}_{2}}\] is the event that die B shows up two and \[{{E}_{3}}\]is the event that the sum of numbers on both dice is odd, then which one of the following statements is not true?
Find the value(s) of all primes p for which the system of equation \[p+1=2{{a}^{2}}\] and \[{{p}^{2}}+1=2{{b}^{2}}\] has a solution in positive integers a, b.