Board Paper of Class 12Commerce 2019 Math Abroad(Set 3)  Solutions
General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.
(v) Use of calculators is not permitted. You may ask logarithmic tables, if required.
 Question 1
Find the acute angle between the planes $\overrightarrow{r}.\left(\hat{i}2\hat{j}2\hat{k}\right)=1\mathrm{and}\overrightarrow{r}.\left(3\hat{i}6\hat{j}+2\hat{k}\right)=0$ .
ORVIEW SOLUTION
Find the length of the intercept, cut off by the plane 2x + y − z = 5 on the xaxis
 Question 2
If y = log (cos e^{x}) then find $\frac{dy}{dx}$. VIEW SOLUTION
 Question 3
A is a square matrix with ∣A∣ = 4. then find the value of ∣A. (adj A)∣. VIEW SOLUTION
 Question 4
Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A. VIEW SOLUTION
 Question 5
 Question 6
Solve the following differential equation :
$\frac{dy}{dx}+y=\mathrm{cos}x\mathrm{sin}x$ VIEW SOLUTION
 Question 7
Find :
$\underset{\frac{\mathrm{\pi}}{4}}{\overset{0}{\int}}\frac{1+\mathrm{tan}x}{1\mathrm{tan}x}dx$ VIEW SOLUTION
 Question 8
Let * be an operation defined as * : R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too. VIEW SOLUTION
 Question 9
X and Y are two points with position vectors $3\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}3\overrightarrow{b}$ respectively. Write the position vector of a point Z which divides the line segment XY in the ratio 2 : 1 externally.
ORVIEW SOLUTION
Let $\overrightarrow{a}=\hat{i}+2\hat{j}3\hat{k}$ and $\overrightarrow{b}=3\hat{i}j+2\hat{k}$ be two vectors. Show that the vectors $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and$\left(\overrightarrow{a}\overrightarrow{b}\right)$ are perpendicular to each other.
 Question 10
Out of 8 outstanding students of a school, in which there are 3 boys and 5 girls, a team of 4 students is to be selected for a quiz competition. Find the probability that 2 boys and 2 girls are selected.
ORVIEW SOLUTION
In a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
 Question 11
The probabilities of solving a specific problem independently by A and B are $\frac{1}{3}\mathrm{and}\frac{1}{5}$ respectively. If both try to solve the problem independently, find the probability that the problem is solved. VIEW SOLUTION
 Question 12
For the matrix A = $\left[\begin{array}{cc}2& 3\\ 5& 7\end{array}\right]$, find (A + A') and verify that it is a symmetric matrix. VIEW SOLUTION
 Question 13
A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall. VIEW SOLUTION
 Question 14
Prove that :
${\mathrm{cos}}^{1}\left(\frac{12}{13}\right)+{\mathrm{sin}}^{1}\left(\frac{3}{5}\right)={\mathrm{sin}}^{1}\left(\frac{56}{65}\right)$ VIEW SOLUTION
 Question 15
Prove that $\underset{0}{\overset{a}{\int}}$f(x) dx = $\underset{0}{\overset{a}{\int}}$ f(a − x) dx, and hence evaluate $\underset{0}{\overset{1}{\int}}$ x^{2} (1 − x)^{n} dx. VIEW SOLUTION
 Question 16
If x = sin t, y = sin pt, prove that $\left(1{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}x\frac{dy}{dx}+{p}^{2}y=0$.VIEW SOLUTION
OR
Differentiate tan^{−1$\left[\frac{\sqrt{1+{x}^{2}}\sqrt{1{x}^{2}}}{\sqrt{1+{x}^{2}}+\sqrt{1{x}^{2}}}\right]$ }with respect to cos^{−1}x^{2}.
 Question 17
Integrate the function $\frac{\mathrm{cos}\left(x+a\right)}{\mathrm{sin}\left(x+b\right)}$ w.r.t. x. VIEW SOLUTION
 Question 18
Let A = R − (2) and B = R − (1). If f : A ⟶ B is a function defined by $f\left(x\right)=\frac{x1}{x2},$ show that f is oneone and onto. Hence, find f^{−1}.
OR
Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation. VIEW SOLUTION
 Question 19
Solve the differential equation $\frac{dy}{dx}$ = 1 + x^{2} + y^{2} + x^{2}y^{2}, given that y = 1 when x = 0.
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
 Question 20
Using properties of determinants, find the value of x for which
$\left\begin{array}{ccc}4x& 4+x& 4+x\\ 4+x& 4x& 4+x\\ 4+x& 4+x& 4x\end{array}\right=0$. VIEW SOLUTION
 Question 21
Find the vector equation of the plane which contains the line of intersection of the planes $\overrightarrow{r}.\left(\hat{i}+2\hat{j}+3\hat{k}\right),4=0,\overrightarrow{r}.\left(2\hat{i}+\hat{j}\hat{k}\right)+5=0\phantom{\rule{0ex}{0ex}}$ and which is perpendicular to the plane $\overrightarrow{r}.\left(5\hat{i}+3\hat{j}6\hat{k}\right)+8=0$ VIEW SOLUTION
 Question 22
Find the value of x such that the four point with position vectors, $A(3\hat{i}+2\hat{j}+\hat{k}),B\left(4\hat{i}+x\hat{j}+5\hat{k}\right),C\left(4\hat{i}+2\hat{j}2\hat{k}\right)\mathrm{and}D\left(6\hat{i}+5\hat{j}\hat{k}\right)$ are coplanar. VIEW SOLUTION
 Question 23
If y = (log x)^{x} + x^{log x}, find $\frac{dy}{dx}.$ VIEW SOLUTION
 Question 24
Find the vector equation of a line passing through the point (2, 3, 2) and parallel to the line $\overrightarrow{r}=\left(2\hat{i}+3\hat{\mathit{j}}\right)+\lambda \left(2\hat{i}3\hat{j}+6\hat{k}\right).$ Also, find the distance between these two lines.
ORVIEW SOLUTION
Find the coordinates of the foot of the perpendicular Q drawn from P(3, 2, 1) to the plane 2x − y + z + 1 = 0. Also, find the distance PQ and the image of the point P treating this plane as a mirror
 Question 25
Using elementary row transformation, find the inverse of the matrix
$\left[\begin{array}{ccc}2& 3& 5\\ 3& 2& 4\\ 1& 1& 2\end{array}\right]$
OR
Using matrices, solve the following system of linear equations :
x + 2y − 3z = −4
2x + 3y + 2z = 2
3x − 3y − 4z = 11 VIEW SOLUTION
 Question 26
Using integration, find the area of the region bounded by the parabola y^{2 }= 4x and the circle 4x^{2} + 4y^{2} = 9.
OR
Using the method of integration, find the area of the region bounded by the lines 3x − 2y + 1 = 0, 2x + 3y − 21 = 0 and x − 5y + 9 = 0 VIEW SOLUTION
 Question 27
An insurance company insured 3000 cyclists, 6000 scooter drivers and 9000 car drivers. The probability of an accident involving a cyclist, a scooter driver and a car driver are 0⋅3, 0⋅05 and 0⋅02 respectively. One of the insured persons meets with an accident. What is the probability that he is a cyclist? VIEW SOLUTION
 Question 28
Using integration, find the area of the smaller region bounded by the ellipse $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1$ and the line $\frac{x}{3}+\frac{y}{2}=1.$ VIEW SOLUTION
 Question 29
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of ₹ 35 per package of nuts and ₹ 14 per package of bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates each machine for atmost 12 hours a day? Convert it into an LPP and solve graphically. VIEW SOLUTION

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