-
Find \[\lambda \] so that the vectors \[\vec{a}=2\hat{i}-\hat{j}+\hat{k},\] \[\vec{b}=\hat{i}+2\hat{j}-3\hat{k}\] and \[\vec{c}=3\hat{i}+\lambda \hat{j}+5\hat{k}\] are coplanar.\
View Answer play_arrow
-
Evaluate \[\int_{1}^{2}{{{\log }_{e}}[x]dx,}\] where \[\left[ . \right]\] denotes the greatest integer function.
View Answer play_arrow
-
Find \[\int{\frac{{{8}^{1\,+\,x}}+{{4}^{1\,-\,x}}}{{{2}^{x}}}\,}dx.\]
View Answer play_arrow
-
Find the values of a and b, if A = B, where \[A=\left[ \begin{matrix} a+4 & 3b \\ 8 & -\,6 \\ \end{matrix} \right]\] and \[B=\left[ \begin{matrix} 2a+2 & {{b}^{2}}+2 \\ 8 & {{b}^{2}}-5b \\ \end{matrix} \right].\]
View Answer play_arrow
-
If A is a skew-symmetric matrix of odd order n, then show that |A| = 0.
View Answer play_arrow
-
Using differentials, find the approximate value of \[{{(82)}^{1/4}}\] up to three places of decimal.
View Answer play_arrow
-
Find the angle between the vectors \[\hat{i}-2\hat{j}+3\hat{k}\] and \[3\hat{i}-2\hat{j}+\hat{k}.\]
View Answer play_arrow
-
An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airline must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs. 100000 and each flight of a model 535 plane costs Rs.150000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.
View Answer play_arrow
-
The random variable X can take only the values 0, 1, 2. Given that \[P(X=0)=P(X=1)=p\] and that \[E({{X}^{2}})=E(X),\] find the value of p.
View Answer play_arrow
-
Find the local maxima and local minima of the function \[f(x)=(\sin x-\cos x),\] where \[0<x<2\pi .\]
View Answer play_arrow
-
Evaluate \[\int{{{\{1+2\tan \,x(\tan \,x+\sec \,x)\}}^{1/2}}dx}\]
View Answer play_arrow
-
Find the value of k for which \[f(x)=\left\{ \begin{matrix} kx+5, & \text{when}\,\,x\le 2 \\ x-1, & \text{when}\,\,x>2 \\ \end{matrix} \right.\] Is continuous at x = 2.
View Answer play_arrow
-
Find the equation of the plane passing through the point (1, 1, 1) and containing the line \[\vec{r}=(-\,3\hat{i}+\hat{j}+5\hat{k})+\lambda (3\hat{i}-\hat{j}-5\hat{k}).\]
View Answer play_arrow
-
Find the minimum value of \[{{({{\sec }^{-1}}x)}^{2}}+{{(\text{cose}{{\text{c}}^{-1}}x)}^{2}}.\]
View Answer play_arrow
-
If \[\hat{a}\] and \[\hat{b}\] are unit vectors inclined at an angle \[\theta ,\] then prove that \[\sin \frac{\theta }{2}=\frac{1}{2}|\hat{a}-\hat{b}|.\]
View Answer play_arrow
-
A librarian has to accommodate two different types of books on a shelf. The books are 6 cm and 4 cm thick and weight 1 kg and \[1\frac{1}{2}\] kg each, respectively. The shelf is 96 cm long and at most can support a weight of 21 kg. How should the shelf be filled with the books of two types in order to include the greatest number of books? Make it as an LPP and solve it graphically.
View Answer play_arrow
-
In Arjun's school, annual sports meet with the title 'Sports for Healthy Life' was being organised. On the last day of the meet, there was the event of hurdle race. In this hurdle, it was decided that a player has to cross 10 hurdles. |
From the past games experience it can be said that the probability of a player to clear each hurdle will be 5/6. Find the probability that a player will knock down fewer than 2 hurdles. |
OR |
A couple has 2 children. Find the probability that both are boys, if it is known that |
(i) One of them is a boy, |
(ii) The older child is a boy. |
View Answer play_arrow
-
If \[x=\sqrt{{{a}^{{{\sin }^{-1}}t}}}\] and \[y=\sqrt{{{a}^{{{\cos }^{-1}}t}}},\,\,a>0\] and \[-\,1<t<1,\] then prove that \[\frac{dy}{dx}=\frac{-\,y}{x}.\] |
OR |
In a given function \[f(x)={{x}^{3}}+b{{x}^{2}}+ax,\] \[x\in [1,\,\,3],\] Rolle's theorem holds with \[c=2+\frac{1}{\sqrt{3}}.\] Find the values of a and b. |
View Answer play_arrow
-
Find the particular solution of the differential equation \[\frac{dy}{dx}=\frac{x(2\log \,x+1)}{\sin y+y\cos y'}\] given that \[y=\frac{\pi }{2},\] when x = 1.
View Answer play_arrow
-
Two schools A and B decided to award prizes to their students for three values honesty (x), punctuality (y) and obedience (z). School A decided to award a total of Rs. 11000 for the three values to 5, 4 and 3 students, respectively, while school B decided to award Rs. 10700 for the three values to 4, 3 and 5 students, respectively. If all the three prizes together amount to Rs. 2700, then (i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication. (ii) Is it possible to solve the system of equations, so obtained using matrices? (iii) Which value you prefer to be rewarded most and why?
View Answer play_arrow
-
Evaluate \[\int{\frac{\sin x+\cos x}{9+16\sin 2x}dx.}\] |
OR |
Evaluate \[\int{\frac{{{x}^{2}}+1}{{{(x-1)}^{2}}(x+3)}dx.}\] |
View Answer play_arrow
-
Prove that \[\int_{0}^{2\pi }{\frac{x{{\sin }^{2n}}x}{{{\sin }^{2n}}x+{{\cos }^{2n}}x}\,dx}={{\pi }^{2}}.\]
View Answer play_arrow
-
Bag A contains 2 white and 3 red balls and bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B
View Answer play_arrow
-
Find the equation of tangent to the curve give b by \[x=a{{\sin }^{3}}t\] and \[y=b{{\cos }^{3}}t\] at a point \[t=\pi /2.\] |
OR |
A telephone company in a town has 500 subscribers on its list and collects fixed charge subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Rs. 1, one subscriber will |
Discontinue the service. Find what increase will bring maximum profit? |
View Answer play_arrow
-
Find the area of the region bounded by the curves |
\[{{y}^{2}}=4ax\] and \[{{x}^{2}}=4ay.\] |
OR |
Using integration, find the area of the triangular region whose sides have the equation \[y=2x+1,\]\[y=3x+1\] and x = 4. |
View Answer play_arrow
-
Considering the Earth as a plane having equation \[\vec{r}(2\hat{i}+2\hat{k})=8\] and the Mars as a line having equation \[\vec{r}=\hat{i}-2\hat{j}+\hat{k}+\lambda (2\hat{i}+3\hat{j}+\hat{k})\] such that their common point is \[(\alpha ,\,\,\beta ,\,\,\gamma ).A\] monument is standing vertically on the Earth such that its peak is at the point \[(-\,2,\,\,-\,6,\,\,-\,11).\] (i) Find the distance between the peak and common point of Earth and Mars. (ii) How can we save our monuments?
View Answer play_arrow
-
Let \[f,\,g:R\to R\] be two functions defined as \[f(x)=|x|+x,\] \[g(x)=|x|-x,\] \[\forall x\in R.\] Then, find fog and gof.
View Answer play_arrow
-
Solve the given differential equation \[\sqrt{1-{{y}^{2}}}\,dx=({{\sin }^{-1}}y-x)\,dy,\] y(0) = 0.
View Answer play_arrow
-
Using properties of determinants, prove that |
\[\left| \begin{matrix} {{a}^{2}}+1 & ab & ac \\ ab & {{b}^{2}}+1 & bc \\ ca & ca & {{c}^{2}}+1 \\ \end{matrix} \right|=1+{{a}^{2}}+{{b}^{2}}+{{c}^{2}}.\]? |
OR |
Prove that |
\[\left| \begin{matrix} {{(b\,+c)}^{2}} & {{a}^{2}} & {{a}^{2}} \\ {{b}^{2}} & {{(c\,+a)}^{2}} & {{b}^{2}} \\ {{c}^{2}} & {{c}^{2}} & {{(a\,+b)}^{2}} \\ \end{matrix} \right|=2abc{{(a+b+c)}^{3}}.\] |
View Answer play_arrow