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

Find λ, if the vectors ${\displaystyle \overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k} \text{and} \overrightarrow{c}=\lambda \hat{j}+3\hat{k}}$  are coplanar.

If the lines ${\displaystyle \frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}}$ and ${\displaystyle \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}}$ intersect each other then find value of k

Find the co-ordinates of the foot of the perpendicular drawn from the point (0, 2, 3) to the line ${\displaystyle \frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}}$.

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

Find the derivative of the inverse of function y = 2x³ – 6x and calculate its value at x = −2

If y = cos(m cos^–1x), then show that ${\displaystyle (1-x^2)\frac{{\text{d}}^{2}y}{\text{d}{x}^2}-x\frac{\text{d}y}{\text{d}x}+{\text{m}}^{2}y}$ = 0

The displacement of a particle at time t is given by s = 2t³ – 5t² + 4t – 3. Find the velocity when 𝑡 = 2 sec

Find ${\displaystyle \int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx}$

Choose the correct alternative:

${\displaystyle \int \frac{x+2}{2x^2+6x+5}\text{d}x=\text{p}\int \frac{4x+6}{2x^2+6x+5}\text{d}x+\frac{1}{2}\int \frac{1}{2x^2+6x+5}\text{d}x}$, then p = ?

If the population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousand to 60 thousand in 40 years, what will be the population in another 20 years? ${\displaystyle (\text{Given}\sqrt{\frac{3}{2}}=1.2247)}$

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

If θ is the measure of acute angle between the pair of lines given by ${\displaystyle a{x}^2+2hxy+b{y}^2=0,}$ then prove that ${\displaystyle \tan \theta =\left|\frac{2\sqrt{h^2-ab}}{a+b}\right|,a+b\ne 0}$

Find λ, if the vectors ${\displaystyle \overrightarrow{a}=\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}-\hat{j}-\hat{k} \text{and} \overrightarrow{c}=\lambda \hat{j}+3\hat{k}}$  are coplanar.

If the lines ${\displaystyle \frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}}$ and ${\displaystyle \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}}$ intersect each other then find value of k

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and  B at a profit of Rs 4. Find the production level per day for maximum profit graphically.

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  ${\displaystyle \frac{dy}{dx}}$at t = ${\displaystyle \frac{\pi }{4}}$

If y = f(x) is a differentiable function of x such that inverse function x = f^–1 (y) exists, then prove that x is a differentiable function of y and ${\displaystyle \frac{dx}{dy}=\frac{1}{\left(\frac{dy}{dx}\right)}\text{where}\frac{dy}{dx}\ne 0}$

If ${\displaystyle \sqrt{1-x^2} +\sqrt{1-y^2}}$ =  a(x − y), show that dy/dx = ${\displaystyle \sqrt{\frac{1-y^2}{1-x^2}}}$

Find ${\displaystyle \int \frac{\sqrt{x}}{\sqrt{a^3-x^3}}dx}$

Evaluate :

${\displaystyle {\int }_{-\pi }^{\pi } {(\cos ax-\sin bx)}^2 dx}$