Use matrix product |
to solve the system of equations |
\[x-y+2z=1,\] \[2y-3z=1\] |
and \[3x-2y+4z=2.\] |
OR |
The sum of three numbers is 6. Twice the third number when added to the first number gives 7. On adding the sum of the second and third numbers to thrice the first number, we get 12. Find the numbers, using matrix method. |
Verify Rolle's theorem for the function |
\[f(x)=\log \left\{ \frac{{{x}^{2}}+ab}{x(a+b)} \right\}\] on [a, b], where 0<a<b. |
OR |
Show that the function \[f(x)=\,\,|x+1|+|x-1|,,\] \[\forall \,x\in R\] is not differentiable at the points \[x=-\,1\] and x = 1. |
Solve the differential equation |
\[y{{e}^{y}}dx=({{y}^{3}}+2x{{e}^{y}})\,dy,\] y(0) = 1 |
OR |
Find the differential equation of the family of curves \[y={{e}^{2x}}(acos2x+b\sin 2x),\] where a and b are arbitrary constants. |
Show that the function f: \[R\to R\] defined by |
\[f(x)=\frac{3x-1}{2},\] \[x\in R\] is one-one and onto functions. Also, find the inverse of the function f. |
OR |
Examine which of the following is a binary operation and check whether the operation is commutative and associative? |
(i) On \[{{Z}^{+}},\] define \[a*b={{2}^{ab}}.\] |
(ii) On Q, define \[a*b=\frac{ab}{2}.\] |
Draw a rough sketch of \[{{y}^{2}}=x+1\] and \[{{y}^{2}}=-\,x+1\] and determine the area enclosed by the two curves. |
OR |
Draw a rough sketch of \[y=\sin 2x\] and determine the area enclosed by the two curves, X-axis and lines \[x=\frac{\pi }{4}\] and \[\frac{3\pi }{4}.\] |
From the point P (1, 2, 4), a perpendicular is drawn on the plane \[2x+y-2z+3=0.\] |
Find the equation, the length and the coordinates of the foot of the perpendicular. |
OR |
Show that the lines |
\[\vec{r}=(-\,3\hat{i}+\hat{j}+5\hat{k})+\lambda (-\,3\hat{i}+\hat{j}+5\hat{k})\] and |
\[\vec{r}=(-\,\hat{i}+2\hat{j}+5\hat{k})+\mu (-\,\hat{i}+2\hat{j}+5\hat{k})\] |
are coplanar. Also, find the equation of the plane containing these lines. |
To From | Cost (in Rs.) | ||
A | B | C | |
P | 160 | 100 | 150 |
Q | 100 | 120 | 100 |
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