HSC Science (Computer Science) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

Let ${\displaystyle A\left(\overline{a}\right)}$ and ${\displaystyle B\left(\overline{b}\right)}$ are any two points in the space and ${\displaystyle \text{R}\left(\overline{\text{r}}\right)}$ be a point on the line segment AB dividing it internally in the ratio m : n, then prove that ${\displaystyle \overline{r}=\frac{m\overline{b}+n\overline{a}}{m+n}}$

Show that four points A, B, C and D whose position vectors are 

${\displaystyle 4\hat{i}+5\hat{j}+\hat{k},-\hat{j}-\hat{k}-\hat{k},3\hat{i}+9\hat{j}+4\hat{k}\text{and}4(-\hat{i}+\hat{j}+\hat{k})}$ respectively are coplanar.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is ${\displaystyle \frac{4r}{3}}$. Also find maximum volume in terms of volume of the sphere

If u and v are two functions of x then prove that

${\displaystyle \int uvdx=u\int vdx-\int [d\frac{u}{dx}\int vdx]dx}$

Hence evaluate, ${\displaystyle \int x{e}^{x}dx}$

Find : ${\displaystyle \int (x+3)\sqrt{3-4x-x^{2}dx}}$

Find: ${\displaystyle I=\int \frac{dx}{\sin x+\sin 2x}}$

Prove that: 

${\displaystyle \begin{array}{ll}{\int }_{-a}^{a}f\left(x\right)dx =2{\int }_0^{a}f\left(x\right)dx\text{,}& \text{If} f\left(x\right) \text{is an even function}\\ =0\text{,}& \text{if} f\left(x\right) \text{is an odd function}\end{array}}$

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after ${\displaystyle \frac{5}{2}}$ hours  ${\displaystyle (\text{Given} \sqrt{2}=1.414)}$

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Show that every homogeneous equation of degree two in x and y, i.e., ax² + 2hxy + by² = 0 represents a pair of lines passing through origin if h²−ab≥0.

Let ${\displaystyle A\left(\overline{a}\right)}$ and ${\displaystyle B\left(\overline{b}\right)}$ are any two points in the space and ${\displaystyle \text{R}\left(\overline{\text{r}}\right)}$ be a point on the line segment AB dividing it internally in the ratio m : n, then prove that ${\displaystyle \overline{r}=\frac{m\overline{b}+n\overline{a}}{m+n}}$

Show that four points A, B, C and D whose position vectors are 

${\displaystyle 4\hat{i}+5\hat{j}+\hat{k},-\hat{j}-\hat{k}-\hat{k},3\hat{i}+9\hat{j}+4\hat{k}\text{and}4(-\hat{i}+\hat{j}+\hat{k})}$ respectively are coplanar.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is ${\displaystyle \frac{4r}{3}}$. Also find maximum volume in terms of volume of the sphere

If u and v are two functions of x then prove that

${\displaystyle \int uvdx=u\int vdx-\int [d\frac{u}{dx}\int vdx]dx}$

Hence evaluate, ${\displaystyle \int x{e}^{x}dx}$

Find : ${\displaystyle \int (x+3)\sqrt{3-4x-x^{2}dx}}$

Find: ${\displaystyle I=\int \frac{dx}{\sin x+\sin 2x}}$

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

Given is X ~ B(n, p). If E(X) = 6, and Var(X) = 4.2, find the value of n.

Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.

Find the symbolic form of the given switching circuit. Construct its switching table and interpret your result.