1. A⃗=2i^+3j^−k^,B⃗=i^+2j^−k^,C⃗=i^+bj^+3k^ \vec{A}=2 \hat{i}+3 \hat{j}-\hat{k}, \vec{B}=\hat{i}+2 \hat{j}-\hat{k}, \vec{C}=\hat{i}+b \hat{j}+3 \hat{k} A=2i^+3j^−k^,B=i^+2j^−k^,C=i^+bj^+3k^
What do you mean by position vector?
If the component of vector B⃗ \vec{B} B along A⃗ \vec{A} A is perpendicular to C⃗ \vec{C} C, then find the value of b b b.
Find the angle between the vectors A⃗+B⃗ \vec{A}+\vec{B} A+B and A⃗×B⃗ \vec{A} \times \vec{B} A×B.
2. g(z)=mzsin−1z \mathrm{~g}(\mathrm{z})=\mathrm{mz} \sin ^{-1} \mathrm{z} g(z)=mzsin−1z is a function and x2 b2+y2a2=1 \frac{\mathrm{x}^{2}}{\mathrm{~b}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1 b2x2+a2y2=1 is a curve.
Find the value of ∫121zcos(lnz)dz \int_{1}^{2} \frac{1}{z} \cos (\ln \mathrm{z}) \mathrm{dz} ∫12z1cos(lnz)dz.
Integrate ∫g(x)dx \int g(x) d x ∫g(x)dx.
If b>a \mathrm{b}>\mathrm{a} b>a then find the half area of the region bounded by the curve given in the stem.
3. M=[1213−3−1210] M=\left[\begin{array}{ccc}1 & 2 & 1 \\ 3 & -3 & -1 \\ 2 & 1 & 0\end{array}\right] M=1322−311−10
If [2−xy−12]=[23+y42] \left[\begin{array}{cc}2 & -x \\ y-1 & 2\end{array}\right]=\left[\begin{array}{cc}2 & 3+y \\ 4 & 2\end{array}\right] [2y−1−x2]=[243+y2] then find (x,y) (x, y) (x,y).
Find M2−3M+MI M^{2}-3 M+M I M2−3M+MI. where I I I is an identity matrix.
If M−1 \mathbf{M}^{-1} M−1 exists, find it.
4. A, B⊂R,B=R−{13},g:A→B,g(x)= \ \mathrm{~A}, \mathrm{~B} \subset \mathbb{R}, \mathrm{B}=\mathbb{R}-\left\{\frac{1}{3}\right\}, \mathrm{g}: \mathrm{A} \rightarrow \mathrm{B}, \mathrm{g}(\mathrm{x})= A, B⊂R,B=R−{31},g:A→B,g(x)= x−53x+1 \frac{x-5}{3 x+1} 3x+1x−5 and h(x)=x2+1 h(x)=x^{2}+1 h(x)=x2+1.
Determine the differential co-efficient of sine1−x \sin \mathrm{e}^{\sqrt{1-\mathrm{x}}} sine1−x.
Show that (hog)(1)−(goh)(2)=2 (\mathrm{hog})(1)-(\mathrm{goh})(2)=2 (hog)(1)−(goh)(2)=2.
Verify the existence of g−1(x) \mathrm{g}^{-1}(\mathrm{x}) g−1(x), if g−1(x) \mathrm{g}^{-1}(\mathrm{x}) g−1(x) exist, then find it.
5. f(x)=lnx f(x)=\ln x f(x)=lnx and g(x)=ex g(x)=e^{x} g(x)=ex.
Find the value of limx→0e2x−1ax \lim _{x \rightarrow 0} \frac{e^{2 x}-1}{a x} limx→0axe2x−1.
Determine the maximum and minimum value of f(2x)x \frac{f(2 x)}{x} xf(2x) if it exists.
Evaluate ∫1e2f(x)xdx+∫12g(x)dx \int_{1}^{e^{2}} \frac{f(x)}{x} d x+\int_{1}^{2} g(x) d x ∫1e2xf(x)dx+∫12g(x)dx.
