Board Paper of Class 12-Science 2015 Maths (SET 1) - Solutions

If $\overrightarrow{\mathrm{a}}=7\hat{\mathrm{i}}+\hat{\mathrm{j}}-4 \hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}} + 6 \hat{\mathrm{j}} + 3\hat{\mathrm{k}},$then find the projection of $\overrightarrow{\mathrm{a}} \mathrm{on}\overrightarrow{\mathrm{b}}$. VIEW SOLUTION

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

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ. VIEW SOLUTION

Write the element a₂₃ of a 3 ✕ 3 matrix A = (a_ij) whose elements a_ij are given by $a_{ij}=\frac{\left|i-j\right|}{2}.$ VIEW SOLUTION

Find the differential equation representing the family of curves $\mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}}$+ B, where A and B are arbitrary constants. VIEW SOLUTION

Find the integrating factor of the differential equation
$\left(\frac{{e^{-2}}^{\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right)\frac{dx}{dy}=1$. VIEW SOLUTION

If $\mathrm{A}=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)$ find ${\mathrm{A}}^2-5\mathrm{A}+4\mathrm{I}$ and hence find a matrix X such that ${\mathrm{A}}^2-5\mathrm{A}+4\mathrm{I}+\mathrm{X}=\mathrm{O}$

OR

If $\mathrm{A}=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right], \mathrm{find} {\left(\mathrm{A}\text{'}\right)}^{-1}.$ VIEW SOLUTION

If $f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|$, using properties of determinants find the value of f(2x) − f(x). VIEW SOLUTION

Find : $\int \frac{dx}{\sin x+\sin 2x}$
 
Integrate the following w.r.t. x
$\frac{x^2-3x+1}{\sqrt{1-x^2}}$ VIEW SOLUTION

Evaluate : $\underset{-x}{\overset{x}{\int }} {\left(\cos ax-\sin bx\right)}^2 dx$ VIEW SOLUTION

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black. An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained. VIEW SOLUTION

$\mathrm{If} \stackrel{\to }{\mathrm{r}}=\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}+\mathrm{z}\hat{\mathrm{k}}, \mathrm{find} \left(\stackrel{\to }{\mathrm{r}}\times \hat{\mathrm{i}}\right)·\left(\stackrel{\to }{\mathrm{r}}\times \mathrm{j}\right)+\mathrm{xy}$ VIEW SOLUTION

Find the distance between the point (−1, −5, −10) and the point of intersection of the line $\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$ and the plane x − y + z = 5. VIEW SOLUTION

If sin [cot⁻¹ (x+1)] = cos(tan⁻¹x), then find x.
If (tan⁻¹x)² + (cot⁻¹x)² = $\frac{5{\mathrm{\pi }}^2}{8}$, then find x. VIEW SOLUTION

If $y={\tan }^{-1} \left(\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}\right), x^2\le 1,$ then find $\frac{dy}{dx}$. VIEW SOLUTION

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that $y^2\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}+y=0.$ VIEW SOLUTION

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ? VIEW SOLUTION

Find : $\int \left(x+3\right)\sqrt{3-4x-x^2} \mathrm{dx}$ VIEW SOLUTION

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:
 

School
Article	A	B	C
Hand-fans	40	25	35
Mats	50	40	50
Plates	20	30	40

Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation. VIEW SOLUTION

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation. VIEW SOLUTION

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle $x^2+y^2=4 \mathrm{at} \left(1, \sqrt{3}\right)$

OR

Evaluate $\underset{1}{\overset{3}{\int }}\left(e^{2-3x}+x^2+1\right) dx$ as a limit of a sum.
 

VIEW SOLUTION

Solve the differential equation :

$\left({\tan }^{-1}y-x\right)dy=\left(1+y^2\right)dx.$

OR

Find the particular solution of the differential equation $\frac{dy}{dx}=\frac{xy}{x^2+y^2}$ given that y = 1, when x = 0. VIEW SOLUTION

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4} \mathrm{and} \frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then find the value of k and hence find the equation of the plane containing these lines. VIEW SOLUTION

If A and B are two independent events such that $\mathrm{P}\left(\overline{\mathrm{A}} \cap \mathrm{B}\right) =\frac{2}{15} \mathrm{and} \mathrm{P}\left(\mathrm{A} \cap \overline{\mathrm{B}}\right) = \frac{1}{6},$ then find P(A) and P(B). VIEW SOLUTION

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. VIEW SOLUTION

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

$$\begin{gathered} 2x + 4y \le 8 \\ 3x + y \le 6 \\ x + y \le 4 \\ x \ge 0, y\ge 0 \end{gathered}$$ VIEW SOLUTION