question_answer4) Evaluate \[\int_{-\pi /6}^{\pi /6}{{{x}^{3}}\,{{\cos }^{2}}x\,dx.}\]
View Answer play_arrowquestion_answer6) Verify Lagrange's mean value theorem for \[f(x)=\log \,\,x\] in [1, 2].
View Answer play_arrowquestion_answer12) Evaluate \[\int_{\pi /4}^{\pi /2}{\sqrt{1-\sin 2x}\,dx}.\]
View Answer play_arrowquestion_answer13) Evaluate \[\int{\frac{\sin x}{\sqrt{1+\sin x}}}\,dx.\]
View Answer play_arrowIf \[x,\,\,y,\,\,z\in [-\,1,\,\,1],\] such that |
\[{{\sin }^{-1}}x+{{\sin }^{-1}}y+{{\sin }^{-1}}z=\frac{-\,3\pi }{2},\] find the value of \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}.\] |
OR |
Prove that |
\[2{{\tan }^{-1}}\left( \frac{1}{5} \right)+{{\sec }^{-1}}\left( \frac{5\sqrt{2}}{7} \right)+2{{\tan }^{-1}}\left( \frac{1}{8} \right)=\frac{\pi }{4}.\] |
Using properties of determinants, prove the following |
\[={{(1+{{a}^{2}}+{{b}^{2}})}^{3}}.\] |
OR |
If x, y, z are in GP, using properties of determinants, show that |
where \[x\ne y\ne z\]and p is any real number. |
Evaluate \[\int{(\sqrt{\tan \,\,x}\,+\,\sqrt{\cot \,x})}\,dx.\] |
OR |
Evaluate \[\int{{{e}^{x}}\left( \frac{1+\sin x}{1+\cos x} \right)}\,dx.\] |
question_answer18) Solve \[({{x}^{3}}-3x{{y}^{2}})\,dx=({{y}^{3}}-3{{x}^{2}}y)\,dy.\]
View Answer play_arrowLet * be the binary operation on N given by a * b = LCM of a and b. Find |
(1) 5 * 7, 20 * 16. |
(ii) Is * commutative? |
(iii) Is * associative? |
(iv) Find the identity of * in N. |
(v) Which elements of N are invertible for the operation *? |
OR |
Let N denote the set of all natural numbers and R be the relation on \[N\times N\] defined by d(a, b) R (c, d), if \[ad(b+c)\text{ }=\text{ }bc(a+b).\] Show that R is an equivalence relation. |
Find the image of line |
\[\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-\,5}\] in the plane \[2x-y+z+3=0.\] |
OR |
Find the distance of the point (3, 4, 5) from the plane \[x+y+z=2\] measured parallel to the line\[2x=y=z\]. |
Find the equations of tangent and normal to the curve \[y=\frac{(x-7)}{(x-2)(x-3)}\] at the point, where it cut the X-axis. |
OR |
Show that the equation of normal at any point on the curve x = 3 |
\[\cos \theta -{{\cos }^{3}}\theta ,\] \[y=3\sin \theta -{{\sin }^{3}}\theta \] is |
\[4(y\,{{\cos }^{3}}\theta -x\,{{\sin }^{3}}\theta )=3\,\sin 4\theta .\] |
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