Board Paper of Class 12Humanities 2017 Maths Abroad(SET 3)  Solutions
General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1 4 in Section A are very shortanswer type questions carrying 1 mark each.
(iv) Questions 512 in Section B are shortanswer type questions carrying 2 marks each.
(v) Questions 1323 in Section C are longanswer I type questions carrying 4 marks each.
(vi) Questions 2429 in Section D are longanswer II type questions carrying 6 marks each.
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1 4 in Section A are very shortanswer type questions carrying 1 mark each.
(iv) Questions 512 in Section B are shortanswer type questions carrying 2 marks each.
(v) Questions 1323 in Section C are longanswer I type questions carrying 4 marks each.
(vi) Questions 2429 in Section D are longanswer II type questions carrying 6 marks each.
 Question 1
Write the distance of the point (3, –5, 12) from xaxis. VIEW SOLUTION
 Question 2
 Question 3
For what value of 'k' is the function $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{ll}\frac{\mathrm{sin}5\mathrm{x}}{3\mathrm{x}}+\mathrm{cos}\mathrm{x},& \mathrm{if}\mathrm{x}\ne 0\\ \mathrm{k},& \mathrm{if}\mathrm{x}=0\end{array}\right.$ is continuous at x = 0? VIEW SOLUTION
 Question 4
If A = 3 and ${\mathrm{A}}^{1}=\left[\begin{array}{rr}3& 1\\ \frac{5}{3}& \frac{2}{3}\end{array}\right]$, then write the adj A. VIEW SOLUTION
 Question 5
Find : $\int \frac{dx}{\sqrt{32x{x}^{2}}}$ VIEW SOLUTION
 Question 6
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
 Question 7
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
 Question 8
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 $\overrightarrow{\mathrm{r}}\xb7\left(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}5\hat{\mathrm{k}}\right)=7.$ Find the equation of the line in cartesian and vector forms. VIEW SOLUTION
 Question 9
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
 Question 10
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
 Question 11
If A and B are square matrices of order 3 such that A = –1, B = 3, then find the value of 2AB. VIEW SOLUTION
 Question 12
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
 Question 13
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
 Question 14
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 $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}and\overrightarrow{a}\xb7\overrightarrow{c}=6$. VIEW SOLUTION
 Question 15
Evaluate : $\underset{2}{\overset{1}{\int}}\left{x}^{3}x\rightdx$
OR
Find : $\int {e}^{2x}\mathrm{sin}\left(3x+1\right)dx$ VIEW SOLUTION
 Question 16
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
 Question 17
Find : $\int \frac{{e}^{x}}{\left(2+{e}^{x}\right)\left(4+{e}^{2x}\right)}dx$ VIEW SOLUTION
 Question 18
If xy = e^{(x – y)}, then show that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}\left(\mathrm{x}1\right)}{\mathrm{x}\left(\mathrm{y}+1\right)}.$
OR
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
 Question 19
Using properties of determinants show that
$\left\begin{array}{ccc}1& 1& 1+\mathrm{x}\\ 1& 1+\mathrm{y}& 1\\ 1+\mathrm{z}& 1& 1\end{array}\right=\mathrm{xyz}+\mathrm{yz}+\mathrm{zx}+\mathrm{xy}.$
OR
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
 Question 20
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
 Question 21
Find the general solution of the differential equation $\mathrm{x}\mathrm{cos}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}\mathrm{cos}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)+\mathrm{x}.$ VIEW SOLUTION
 Question 22
Prove that:
${\mathrm{tan}}^{1}\left(\frac{\sqrt{1+{\mathrm{x}}^{2}}+\sqrt{1{\mathrm{x}}^{2}}}{\sqrt{1+{\mathrm{x}}^{2}}\sqrt{1{\mathrm{x}}^{2}}}\right)=\frac{\mathrm{\pi}}{4}+\frac{1}{2}{\mathrm{cos}}^{1}{\mathrm{x}}^{2};\u20131\mathrm{x}1$ VIEW SOLUTION
 Question 23
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
 Question 24
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).
OR
Using integration, find the area of the region {(x, y) : x^{2} + y^{2} ≤ 1 ≤ x + y}. VIEW SOLUTION
 Question 25
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
 Question 26
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by $f\left(x\right)=\frac{x2}{x3},\forall \mathrm{x}\in A$. Show that f is bijective. Also, find
(i) x, if f^{−1}(x) = 4
(ii) f^{−1}(7)
OR
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
 Question 27
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 $\frac{x+2}{5}=\frac{y3}{4}=\frac{z}{5}$ or not.ORFind the image P' of the point P having position vector $\hat{i}+3\hat{j}+4\hat{k}$ in the plane $\overrightarrow{r}\xb7\left(2\hat{i}\hat{j}+\hat{k}\right)+3=0$. Hence find the length of PP'. VIEW SOLUTION
 Question 28
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
 Question 29
Find the particular solution of the differential equation $\left(1+{y}^{2}\right)+\left(x{e}^{{\mathrm{tan}}^{1}}y\right)\frac{dy}{dx}=0$, given that y = 0 when x = 1. VIEW SOLUTION
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