Board Paper of Class 12-Science 2015 Maths Abroad(SET 2) - Solutions

Evaluate:
$\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\cos x}{1+e^x}dx$ VIEW SOLUTION

Three machines E₁, E₂ and E₃ in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E₁ and E₂ are defective and that 5% of those produced by machine E₃ are defective. If one bulb is picked up at random from a day's production, calculate the probability that it is defective.
 
Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X. VIEW SOLUTION

The two vectors $\hat{j}+\hat{k} \mathrm{and} 3\hat{i}-\hat{j}+4\hat{k}$ represent the two sides vectors $\overrightarrow{\mathrm{AB}} \mathrm{and} \overrightarrow{\mathrm{AC}}$ respectively of triangle ABC. Find the length of the median through A. VIEW SOLUTION

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$. VIEW SOLUTION

If 2 tan⁻¹ (cos θ) = tan⁻¹ (2 cosec θ), (θ ≠ 0), then find the value of θ.
If ${\tan }^{-1}\left(\frac{1}{1+1.2}\right)+{\tan }^{-1}\left(\frac{1}{1+2.3}\right)+...+{\tan }^{-1}\left(\frac{1}{1+n.\left(n+1\right)}\right)={\tan }^{-1} \mathrm{\theta }$, then find the value of θ. VIEW SOLUTION

If $\mathrm{A}=\left[\begin{array}{cc} 2& -1\\ -1& 2\end{array}\right]$ and I is the identity matrix of order 2, then show that A²= 4 A − 3 I. Hence find A⁻¹.
If $\mathrm{A}=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right] \mathrm{and} \mathrm{B}=\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& -1\end{array}\right] \mathrm{and} {\left(\mathrm{A}+\mathrm{B}\right)}^2={\mathrm{A}}^2+{\mathrm{B}}^2$, then find the values of a and b. VIEW SOLUTION

Using properties of determinants, prove the following :
$\left|\begin{array}{ccc}1& a& a^2\\ a^2& 1& a\\ a& a^2& 1\end{array}\right|={\left(1-a^3\right)}^2$ VIEW SOLUTION

Evaluate :

$\int \frac{\sin \left(x-a\right)}{\sin \left(x+a\right)}dx$

OR

Evaluate :

$\int \frac{x^2}{\left(x^2+4\right)\left(x^2+9\right)}dx$ VIEW SOLUTION

Find whether the following function is differentiable at x = 1 and x = 2 or not :
$f\left(x\right)=\left\{\begin{array}{ccc}x,& & x < 1\\ 2-x,& & 1\le x\le 2\\ -2+3x-x^2,& & x>2 \end{array}\right.$ VIEW SOLUTION

In a parliament election, a political party hired a public relations firm to promote its candidates in three ways − telephone, house calls and letters. The cost per contact (in paise) is given in matrix A as

$\mathrm{A} = \left[\begin{array}{c}140\\ 200\\ 150\end{array}\right]\begin{array}{c}\mathrm{Telephone}\\ \mathrm{House} \mathrm{Call}\\ \mathrm{Letters} \end{array}$

The number of contacts of each type made in two cities X and Y is given in the matrix B as

$$\begin{gathered} \begin{array}{ccc}\mathrm{Telephone}& \mathrm{House} \mathrm{Call}& \mathrm{Letters}\end{array} \\ \mathrm{B} =\left[\begin{array}{ccc} 1000 & 500& 5000\\ 3000 & 1000 & 10000\end{array}\right]\begin{array}{c}\mathrm{City} \mathrm{X}\\ \mathrm{City} \mathrm{Y}\end{array} \end{gathered}$$

Find the total amount spent by the party in the two cities.

What should one consider before casting his/her vote − party's promotional activity or their social activities ? VIEW SOLUTION

Evaluate :

$\int e^{2x}·\sin \left(3x+1\right) dx$ VIEW SOLUTION

Find the point on the curve 9y² = x³, where the normal to the curve makes equal intercepts on the axes. VIEW SOLUTION

$$\begin{gathered} \mathrm{If} y={\left(x+\sqrt{1+x^2}\right)}^n, \mathrm{then} \mathrm{show} \mathrm{that} \\ \left(1+x^2\right)\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}=n^{2}y. \end{gathered}$$ VIEW SOLUTION

Find the minimum value of (ax + by), where xy = c².
Find the coordinates of a point of the parabola y = x² + 7x + 2 which is closest to the straight line y = 3x − 3. VIEW SOLUTION

Maximise z = 8x + 9y subject to the constraints given below :
2x + 3y ≤ 6
3x − 2y ≤6
y ≤ 1
x, y ≥ 0 VIEW SOLUTION

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, −6. VIEW SOLUTION

Let f : N → ℝ be a function defined as f(x) = 4x² + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f. VIEW SOLUTION

Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x = $\sqrt{y}$ and y-axis. VIEW SOLUTION

Find the probability distribution of the number of doublets in four throws of a pair of dice. Also find the mean and variance of this distribution. VIEW SOLUTION

Solve the following differential equation :

$\left[y-x \cos \left(\frac{y}{x}\right)\right]dy+\left[y \cos \left(\frac{y}{x}\right)-2x \sin \left(\frac{y}{x}\right)\right]dx=0$ Solve the following differential equation :

$\left(\sqrt{1+x^2+y^2+x^2 y^2}\right) dx+xy dy = 0$ VIEW SOLUTION