Board Paper of Class 12-Science 2018 Maths - Solutions

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. VIEW SOLUTION

Differentiate ${\tan }^{-1}\left(\frac{1+\cos x}{\sin x}\right)$ with respect to x. VIEW SOLUTION

Give $A=\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute A^–1 and show that 2A^–1 = 9I – A. VIEW SOLUTION

Prove that :
$3 {\sin }^{-1}x={\sin }^{-1} \left(3x-4x^3\right), x\in \left[-\frac{1}{2}, \frac{1}{2}\right]$ VIEW SOLUTION

A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. VIEW SOLUTION

If θ is the angle between two vectors $\hat{i}-2\hat{j}+3\hat{k} \mathrm{and} 3\hat{i}-2\hat{j}+\hat{k}$, find sin θ. VIEW SOLUTION

Find the differential equation representing the family of curves y = a e^bx+5, where a and b are arbitrary constants. VIEW SOLUTION

Evaluate: $\int \frac{\cos 2x+2{\sin }^2 x}{{\cos }^2 x}dx$ VIEW SOLUTION

If y = sin (sin x), prove that $\frac{d^{2}y}{d{x}^2}+\tan x \frac{dy}{dx}+y {\cos }^2 x= 0.$ VIEW SOLUTION

Find the particular solution of the differential equation e^x tan y dx + (2 – e^x) sec² y dy = 0, give that $y=\frac{\pi }{4}$ when x = 0.
 
Find the particular solution of the differential equation $\frac{dy}{dx}+2y \tan x=\sin x,$ given that y = 0 when $x=\frac{\pi }{3}$. VIEW SOLUTION

Find the shortest distance between the lines $\overrightarrow{r}=\left(4\hat{i}-\hat{j}\right)+\lambda \left(\hat{i}+2\hat{j}-3\hat{k}\right)$ and $\overrightarrow{r}=\left(\hat{i}-\hat{j}+2\hat{k}\right)+\mu \left(2\hat{i}+4\hat{j}-5\hat{k}\right).$ VIEW SOLUTION

Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. VIEW SOLUTION

Using properties of determinants, prove that
$\left|\begin{array}{ccc}1& 1& 1+3x\\ 1+3y& 1& 1\\ 1& 1+3z& 1\end{array}\right|=9\left(3xyz+xy+yz+zx\right)$ VIEW SOLUTION

Find the equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point (x₁, y₁), where x₁ = 2 and y₁ > 0.

OR

Find the intervals in which the function $f\left(x\right)=\frac{x^4}{4}-x^3-5x^2+24x+12$ is (a) strictly increasing, (b) strictly decreasing.

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Find :
$\int \frac{2 \cos x}{\left(1-\sin x\right) \left(1+{\sin }^2 x\right)}dx$ VIEW SOLUTION

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Let $\overrightarrow{a}=4\hat{i}+5\hat{j}-\hat{k}, \overrightarrow{b}=\hat{i}-4\hat{j}+5\hat{k} \mathrm{and} \overrightarrow{c}=3\hat{i}+\hat{j}-\hat{k}.$ Find a vector $\overrightarrow{d}$ which is perpendicular to both $\overrightarrow{c} \mathrm{and} \overrightarrow{b} \mathrm{and} \overrightarrow{d}·\overrightarrow{a}=21$. VIEW SOLUTION

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? VIEW SOLUTION

If ${\left(x^2+y^2\right)}^2=xy, \mathrm{find} \frac{dy}{dx}$. If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find $\frac{dy}{dx}$ when $\mathrm{\theta }=\frac{\pi }{3}$. VIEW SOLUTION

Evaluate :
$\underset{0}{\overset{\pi /4}{\int }}\frac{\sin x+\cos x}{16+9 \sin 2x}dx$
 
Evaluate :
$\underset{1}{\overset{3}{\int }}\left(x^2+3x+e^x\right)dx,$
as the limit of the sum. VIEW SOLUTION

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws 'A' while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture a packet of screws 'B'. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws 'A' at a profit of 70 paise and screws 'B' at a profit of Rs 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit. VIEW SOLUTION

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that

R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2].
 
Show that the function f : ℝ → ℝ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall x\in \mathrm{ℝ}$ is neither one-one nor onto. Also, if g : ℝ → ℝ is defined as g(x) = 2x – 1, find fog(x). VIEW SOLUTION

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32. VIEW SOLUTION

If $A=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$, find A^–1. Use it to solve the system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3.
 
Using elementary row transformations, find the inverse of the matrix $A=\left[\begin{array}{ccc} 1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$. VIEW SOLUTION

Find the distance of the point (–1, –5, –10) from the point of intersection of the line $\overrightarrow{r}=2\hat{i}-\hat{j}+2k+\lambda \left(3\hat{i}+4\hat{j}+2\hat{k}\right)$ and the plane $\overrightarrow{r}·\left(\hat{i}-\hat{j}+\hat{k}\right)=5$. VIEW SOLUTION