If \[C=2\,\,cos\,\,\theta ,\] then the value of the determinant \[\Delta =\left| \begin{matrix} C & 1 & 0 \\ 1 & C & 1 \\ 0 & 1 & C \\ \end{matrix} \right|\] is_______.
Let \[\overrightarrow{n}\] be a vector of magnitude \[2\sqrt{3}\] such that it makes equal acute angles with the coordinate axes. Find the vector forms of the equation of a plane passing through \[\left( 1,-1,2 \right)\]and normal to\[\overrightarrow{n}\].
The volume of the tetrahedron whose vertices are the points with position vectors \[\hat{i}-6\hat{j}+10\hat{k},\] \[-\hat{i}-3\hat{j}+7\hat{k},\] \[5\hat{i}-\hat{j}+\lambda \,\hat{k}\] and \[7\hat{i}-4\hat{j}+7\hat{k},\] is \[11\,cu.\]units, then the value of \[\lambda \] is _______.
Let \[\overrightarrow{a}\text{ =}\hat{i}+\hat{j}+\hat{k}\] and \[\overrightarrow{r}\] be a variable vector such that \[\overrightarrow{r}.\,\,\hat{i},\]\[\overrightarrow{r}.\,\,\hat{j},\] and \[\overrightarrow{r}.\,\,\hat{k}\] are positive integers. If \[\overrightarrow{r}.\,\,\overrightarrow{a}\le 12,\] then the total number of such vectors is _________.
Let P, Q and R try to hit the target simultaneously but independently. Their respective probabilities of hitting the targets are \[\frac{3}{4},\] \[\frac{1}{2}\]and\[\frac{5}{8}\] . The probability that the target is hit by A or B but not by C is ________.
If the plane \[2ax-3ay+4az+6=0\] passes through the mid-point of the line joining the centres of the spheres \[~{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-8y+6x-2z=13\] and \[~{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10x+4y-2z=8,\] then a equals to
If the function f(x) is differentiable at x = a, then \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{2}}f(a)-{{a}^{2}}f(x)}{x-a}\] is equal to ________.
If \[\sqrt{m}=a{{e}^{\theta \cot \alpha }}\] where a and \[\alpha \]are real numbers, then \[\frac{{{d}^{2}}m}{d{{\theta }^{2}}}-4m\,\,{{\cot }^{2}}\alpha \]is equal to _________.
The solution of the differential equation \[\frac{x+\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+.......}{1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+.......}=\frac{dx-dy}{dx+dy}\] is __________.
A is one of 6 horses entered for a race and is to be ridden by one of two jockeys B and C. It is 2:1 that B rides A, in which case all the horses are equally likely to win. If C rides A, his chance of winning is trebled. What are the odds against winning of A?
Two numbers b and c are chosen at random with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9. The probability that \[{{x}^{2}}+bx+c>0,\] \[\forall x\in R,\] is________.
In a binomial distribution \[B\left( n,p=\frac{1}{4} \right),\] if the probability of atleast one success is greater than or equal to \[\frac{9}{10},\] then n is greater than ______.
A person starts from a point A and travels 3 km eastwards to B and then turns left and travels thrice that distance to reach C. He again turns left and travels five times the distance he covered between A and B, and finally reaches his destination D. The shortest distance between his starting point and the destination is:
In a certain code '7 8 6' means 'bring me apple', '9 5 8' means 'peel green apple' and '6 4 5' means 'bring green fruit'. Which of the following is the code for 'me'?
If \[\to \] stands for 'addition', \[\leftarrow \] stands for 'subtraction', \[\uparrow \] stands for 'division', \[\downarrow \] stands for 'multiplication', \[\nearrow \] stands for 'equal to', then which of the following alternatives is correct?
Direction: Read the following passage carefully and answer the questions given below it.
A group of friends having seven members: A, B, C, D, E, F and G contains four men and three ladies. Each one of them has a different profession: stockbroker, lawyer, doctor, professor, engineer, businessman and banker and each one has passed out of a different college P, S, V, W, X, Y and Z, not necessarily in the same order. None of the ladies is a businessman or a stockbroker. C is a doctor and she has passed out from 'College X'. A is a 'College Y' passed out. B is not a professor. E is a banker and is 'College S' passed out. F is a stockbroker and has not studied in 'College P'. G is a businessman and has studied in 'College V?. The professor is 'College Z' passed out. The lawyer has studied in 'College P'. None of the ladies has studied in 'College Y' or 'College S'.
Direction: Read the following passage carefully and answer the questions given below it.
A group of friends having seven members: A, B, C, D, E, F and G contains four men and three ladies. Each one of them has a different profession: stockbroker, lawyer, doctor, professor, engineer, businessman and banker and each one has passed out of a different college P, S, V, W, X, Y and Z, not necessarily in the same order. None of the ladies is a businessman or a stockbroker. C is a doctor and she has passed out from 'College X'. A is a 'College Y' passed out. B is not a professor. E is a banker and is 'College S' passed out. F is a stockbroker and has not studied in 'College P'. G is a businessman and has studied in 'College V?. The professor is 'College Z' passed out. The lawyer has studied in 'College P'. None of the ladies has studied in 'College Y' or 'College S'.
Which of the following groups represents female in the group of friends?
Direction: Read the following passage carefully and answer the questions given below it.
A group of friends having seven members: A, B, C, D, E, F and G contains four men and three ladies. Each one of them has a different profession: stockbroker, lawyer, doctor, professor, engineer, businessman and banker and each one has passed out of a different college P, S, V, W, X, Y and Z, not necessarily in the same order. None of the ladies is a businessman or a stockbroker. C is a doctor and she has passed out from 'College X'. A is a 'College Y' passed out. B is not a professor. E is a banker and is 'College S' passed out. F is a stockbroker and has not studied in 'College P'. G is a businessman and has studied in 'College V?. The professor is 'College Z' passed out. The lawyer has studied in 'College P'. None of the ladies has studied in 'College Y' or 'College S'.
Direction: Read the following passage carefully and answer the questions given below it.
A group of friends having seven members: A, B, C, D, E, F and G contains four men and three ladies. Each one of them has a different profession: stockbroker, lawyer, doctor, professor, engineer, businessman and banker and each one has passed out of a different college P, S, V, W, X, Y and Z, not necessarily in the same order. None of the ladies is a businessman or a stockbroker. C is a doctor and she has passed out from 'College X'. A is a 'College Y' passed out. B is not a professor. E is a banker and is 'College S' passed out. F is a stockbroker and has not studied in 'College P'. G is a businessman and has studied in 'College V?. The professor is 'College Z' passed out. The lawyer has studied in 'College P'. None of the ladies has studied in 'College Y' or 'College S'.
What should come next in the letter series given below? \[c~\,b\,~a~\,a\,~c\,~b\,~a\,~a\,~b\,~c\,~b~\,a~\,a~\,b~\,c\,~c\,~b\,~a\,~a\,~b\,~c\,~d\,~c\,~b\,~a\,~a\]
Let \[\omega \] be a complex cube root of unity with\[\omega \ne 1.\] A fair die is thrown three times. If \[{{n}_{1}},\]\[{{n}_{2}}\] and \[{{n}_{3}}\] are the numbers obtained on the die, then the probability that \[{{\omega }^{{{n}_{1}}}}+{{\omega }^{{{n}_{2}}}}+{{\omega }^{{{n}_{3}}}}=0\] is ________.
For any real number x, let |x| denotes the largest integer less than or equal to x. Let f be a real valued function defined on the interval [-10, 10] by \[f(x)=\left\{ \begin{matrix} x-[x], & if\,[x]\,\,is\,\,odd, \\ 1+[x]-x & if\,[x]\,\,is\,\,even. \\ \end{matrix} \right.\]. Then the value of \[\frac{{{\pi }^{2}}}{10}\int\limits_{-10}^{10}{f(x)\cos \pi x\,\,dx}\] is _______.
Let \[P=\left[ \begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \\ \end{matrix} \right]\] and \[l\] be the identity matrix of order 3. If \[Q=[{{q}_{ij}}]\] is a matrix, such that \[{{P}^{50}}-Q=I,\] then \[\frac{{{q}_{31}}-{{q}_{32}}}{{{q}_{21}}}\] equals to _______.
A cylindrical container is to be made from a certain solid material with the following constraints: it has a fixed inner volume of \[V\,\,m{{m}^{3}},\]has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum, when the inner radius of the container is 10 mm, then the value of V is _________.