Board Paper of Class 12Commerce 2017 Maths (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
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of zaxis. VIEW SOLUTION
 Question 2
Evaluate : $\underset{2}{\overset{3}{\int}}{3}^{x}dx.$ VIEW SOLUTION
 Question 3
Determine the value of the constant 'k' so that function $\mathrm{f}\left(x\right)=\left\{\begin{array}{ll}\frac{kx}{\leftx\right},& \mathrm{if}x0\\ 3,& \mathrm{if}x\ge 0\end{array}\right.$ is continuous at x = 0. VIEW SOLUTION
 Question 4
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^{–1}) = (det A)^{k}. VIEW SOLUTION
 Question 5
Prove that if E and F are independent events, then the events E and F' are also independent. VIEW SOLUTION
 Question 6
A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?
It is being given that at least one of each must be produced. VIEW SOLUTION
 Question 7
Find $\int \frac{\mathrm{d}x}{{x}^{2}+4x+8}$ VIEW SOLUTION
 Question 8
Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z. VIEW SOLUTION
 Question 9
Show that the function $f\left(x\right)=4{x}^{3}18{x}^{2}+27x7$ is always increasing on $\mathrm{\mathbb{R}}$. VIEW SOLUTION
 Question 10
The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm. VIEW SOLUTION
 Question 11
Show that all the diagonal elements of a skew symmetric matrix are zero. VIEW SOLUTION
 Question 12
If $y={\mathrm{sin}}^{1}\left(6x\sqrt{19{x}^{2}}\right),\frac{1}{3\sqrt{2}}x\frac{1}{3\sqrt{2}}$, then find $\frac{dy}{dx}$. VIEW SOLUTION
 Question 13
Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\text{\hspace{0.17em}}\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}$ and $\overrightarrow{\mathrm{c}}={\mathrm{c}}_{1}\hat{\mathrm{i}}+{\mathrm{c}}_{2}\hat{\mathrm{j}}+{\mathrm{c}}_{3}\hat{\mathrm{k}},$ then
(a) Let c_{1} = 1 and c_{2} = 2, find c_{3} which makes $\overrightarrow{\mathrm{a}},\text{\hspace{0.17em}}\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ coplanar.
(b) If c_{2} = –1 and c_{3} = 1, show that no value of c_{1} can make $\overrightarrow{\mathrm{a}},\text{\hspace{0.17em}}\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ coplanar. VIEW SOLUTION
 Question 14
If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are mutually perpendicular vectors of equal magnitudes, show that the vector $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ is equally inclined to $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$. Also, find the angle which $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$ makes with $\overrightarrow{a}\mathrm{or}\overrightarrow{b}\mathrm{or}\overrightarrow{c}$. VIEW SOLUTION
 Question 15
The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that $\mathrm{\Sigma}{p}_{\mathit{i}}{x}_{i}^{2}=2\mathrm{\Sigma}{p}_{\mathit{i}}{x}_{\mathit{i}}$, find the value of p. VIEW SOLUTION
 Question 16
Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
Do you also agree that the value of truthfulness leads to more respect in the society? VIEW SOLUTION
 Question 17
Using properties of determinants, prove that $\left\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right=9{y}^{2}\left(x+y\right).$
OR
Let $A=\left(\begin{array}{cc}2& 1\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}5& 2\\ 7& 4\end{array}\right),C=\left(\begin{array}{cc}2& 5\\ 3& 8\end{array}\right)$, find a matrix D such that CD − AB = O. VIEW SOLUTION
 Question 18
Differentiate the function ${\left(\mathrm{sin}x\right)}^{x}+{\mathrm{sin}}^{1}\sqrt{x}$ with respect to x.
OR
If ${x}^{m}{y}^{n}={\left(x+y\right)}^{m+n}$, prove that $\frac{{d}^{2}y}{d{x}^{2}}=0$. VIEW SOLUTION
 Question 19
Evaluate : $\underset{0}{\overset{\mathrm{\pi}}{\int}}\frac{x\mathrm{sin}x}{1+{\mathrm{cos}}^{2}x}\mathrm{d}x$
OR
Evaluate : $\underset{0}{\overset{3/2}{\int}}\leftx\mathrm{sin}\pi x\right\mathrm{d}x$ VIEW SOLUTION
 Question 20
Solve the following L.P.P graphically:
Maximise Z = 20x + 10y Subject to the following constraints x + 2y ≤ 28,3x + y ≤ 24,x ≥ 2,x, y ≥ 0
 Question 21
Show that the family of curves for which $\frac{dy}{dx}=\frac{{x}^{2}+{y}^{2}}{2xy}$, is given by x^{2} – y^{2} = cx. VIEW SOLUTION
 Question 22
Find : $\int \frac{\left(3\mathrm{sin}x2\right)\mathrm{cos}x}{13{\mathrm{cos}}^{2}x7\mathrm{sin}x}dx$ VIEW SOLUTION
 Question 23
Solve the following equation for x:
$\mathrm{cos}\left({\mathrm{tan}}^{1}x\right)=\mathrm{sin}\left({\mathrm{cot}}^{1}\frac{3}{4}\right)$ VIEW SOLUTION
 Question 24
Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).
OR
Find the area bounded by the circle x^{2} + y^{2} = 16 and the line $\sqrt{3}\mathrm{y}=x$ in the first quadrant, using integration. VIEW SOLUTION
 Question 25
Find the equation of the plane through the line of intersection of $\underset{r}{\to}\xb7\left(2\hat{i}3\hat{j}+4\hat{k}\right)=1$ and $\underset{r}{\to}\xb7\left(\hat{i}\hat{j}\right)+4=0$ and perpendicular to the plane $\underset{r}{\to}\xb7\left(2\hat{i}\hat{j}+\hat{k}\right)+8=0$. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.
ORFind the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines $\frac{x8}{3}=\frac{\mathrm{y}+19}{16}=\frac{\mathrm{z}10}{7}$ and $\frac{x15}{3}=\frac{\mathrm{y}29}{8}=\frac{\mathrm{z}5}{5}$. VIEW SOLUTION
 Question 26
Consider f : R_{+} → [−5, ∞), given by f(x) = 9x^{2} + 6x − 5. Show that f is invertible with ${\mathrm{f}}^{1}\left(y\right)\left(\frac{\sqrt{y+6}1}{3}\right)$.
Hence Find
(i) f^{−1}(10)
(ii) y if ${\mathrm{f}}^{1}\left(y\right)=\frac{4}{3},$
where R_{+} is the set of all nonnegative real numbers.
OR
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A. VIEW SOLUTION
 Question 27
If the sum of lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum, when the angle between them is $\frac{\mathrm{\pi}}{3}.$ VIEW SOLUTION
 Question 28
If $\mathrm{A}=\left(\begin{array}{rrr}2& 3& 1\\ 1& 2& 2\\ \u20133& 1& 1\end{array}\right)$, find A^{–1} and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8. VIEW SOLUTION
 Question 29
Find the particular solution of the differential equation
$\mathrm{tan}x\xb7\frac{dy}{dx}=2x\mathrm{tan}x+{x}^{2}y;\left(\mathrm{tan}x\ne 0\right)$ given that y = 0 when $x=\frac{\pi}{2}$. VIEW SOLUTION
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