Board Paper of Class 12-Science 2017 Maths Abroad(SET 3) - Solutions

Find : $\int \frac{dx}{\sqrt{3-2x-x^2}}$ VIEW SOLUTION

A company produces two types of goods A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of golds while that of type B requires 1 g of silver and 2 g of gold. The company can procure a maximum of 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, formulate LPP to maximize profit. VIEW SOLUTION

If P(A) = 0·4, P(B) = p, P(A ⋃ B) = 0·6 and A and B are given to be independent events, find the value of 'p'. VIEW SOLUTION

A line passes through the point with position vector $2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}$ and is perpendicular to the plane $\overrightarrow{\mathrm{r}}·\left(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)=7.$ Find the equation of the line in cartesian and vector forms. VIEW SOLUTION

Show that the function f given by f(x) = tan^–1 (sin x + cos x) is decreasing for all $\mathrm{x}\in \left(\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right).$ VIEW SOLUTION

Find $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{t}=\frac{2\mathrm{\pi }}{3}$ when x = 10 (t – sin t) and y = 12 (1 – cos t). VIEW SOLUTION

If A and B are square matrices of order 3 such that |A| = –1, |B| = 3, then find the value of |2AB|. VIEW SOLUTION

The radius r of a right circular cylinder is increasing uniformly at the rate of 0·3 cm/s and its height h is decreasing at the rate of 0·4 cm/s. When r = 3·5 cm and h = 7 cm, find the rate of change of the curved surface area of the cylinder. $\left[\mathrm{Use} \mathrm{\pi }=\frac{22}{7}\right]$ VIEW SOLUTION

There are 4 cards numbered 1 to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X. VIEW SOLUTION

If $\overrightarrow{a}=2\hat{i}+\hat{j}-\hat{k}, \overrightarrow{b}=4\hat{i}-7\hat{j}+\hat{k}$, find a vector $\overrightarrow{c}$ such that $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b} and \overrightarrow{a}·\overrightarrow{c}=6$. VIEW SOLUTION

Evaluate : $\underset{-2}{\overset{1}{\int }}\left|x^3-x\right|dx$

OR

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

In a shop X, 30 tins of pure ghee and 40 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 50 tins of pure ghee and 60 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration? VIEW SOLUTION

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

If xy = e^{(x – y)}, then show that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}\left(\mathrm{x}-1\right)}{\mathrm{x}\left(\mathrm{y}+1\right)}.$
 
If logy = tan^–1 x, then show that $\left(1+{\mathrm{x}}^2\right)\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}+\left(2\mathrm{x}-1\right)\frac{\mathrm{dy}}{\mathrm{dx}}=0.$ VIEW SOLUTION

Using properties of determinants show that

$\left|\begin{array}{ccc}1& 1& 1+\mathrm{x}\\ 1& 1+\mathrm{y}& 1\\ 1+\mathrm{z}& 1& 1\end{array}\right|=\mathrm{xyz}+\mathrm{yz}+\mathrm{zx}+\mathrm{xy}.$

 
Find matrix X so that $\mathrm{X}\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)=\left(\begin{array}{rrr}-7& -8& -9\\ 2& 4& 6\end{array}\right)$. VIEW SOLUTION

Solve the following LPP graphically :
Maximise Z = 105x + 90y
subject to the constraints
x + y ≤ 50
2x + y ≤ 80
x ≥ 0, y ≥ 0. VIEW SOLUTION

Find the general solution of the differential equation $\mathrm{x} \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y} \cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)+\mathrm{x}.$ VIEW SOLUTION

Prove that:

${\tan }^{-1}\left(\frac{\sqrt{1+{\mathrm{x}}^2}+\sqrt{1-{\mathrm{x}}^2}}{\sqrt{1+{\mathrm{x}}^2}-\sqrt{1-{\mathrm{x}}^2}}\right)=\frac{\mathrm{\pi }}{4}+\frac{1}{2} {\cos }^{-1}{\mathrm{x}}^2;–1<\mathrm{x}<1$ VIEW SOLUTION

Using vectors, find the area of triangle ABC, with vertices A (1, 2, 3), B (2, –1, 4) and C (4, 5, –1). VIEW SOLUTION

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices area A(1, 2), B (2, 0) and C (4, 3).
Using integration, find the area of the region {(x, y) : x² + y² ≤ 1 ≤ x + y}. VIEW SOLUTION

A wire of length 34 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a rectangle whose length is twice its breadth. What should be the lengths of the two pieces, so that the combined area of the square and the rectangle is minimum? VIEW SOLUTION

Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by $f\left(x\right)=\frac{x-2}{x-3}, \forall \mathrm{x} \in A$. Show that f is bijective. Also, find
(i) x, if f⁻¹(x) = 4
(ii) f⁻¹(7)
Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
(i) Show that * is commutative on A.
(ii) Show that * is associative on A.
(iii) Find the identity element of * in A. VIEW SOLUTION

Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0. Hence find whether the plane thus obtained contains the line $\frac{x+2}{5}=\frac{y-3}{4}=\frac{z}{5}$ or not. Find the image P' of the point P having position vector $\hat{i}+3\hat{j}+4\hat{k}$ in the plane $\overrightarrow{r} · \left(2\hat{i}-\hat{j}+\hat{k}\right)+3=0$. Hence find the length of PP'. VIEW SOLUTION

If $\mathrm{A}=\left[\begin{array}{ccc}1& -2& 0\\ 2& 1& 3\\ 0& -2& 1\end{array}\right]$, find A^–1 and hence solve the system of equations x – 2y = 10, 2x + y + 3z = 8 and –2y + z = 7. VIEW SOLUTION

Find the particular solution of the differential equation $\left(1+y^2\right)+\left(x-e^{{\tan }^{-1}} y\right) \frac{dy}{dx}=0$, given that y = 0 when x = 1. VIEW SOLUTION