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
Determine the value of 'k' for which the following function is continuous at x = 3:
$f\left(x\right)=\left\{\begin{array}{ccc}\frac{{\left(x+3\right)}^{2}36}{x3}& ,& x\ne 3\\ k& ,& x=3\end{array}\right.$ VIEW SOLUTION
 Question 2
If for any 2 × 2 square matrix A, A(adj A) = $\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, then write the value of A. VIEW SOLUTION
 Question 3
Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20. VIEW SOLUTION
 Question 4
Find :
$\int \frac{{\mathrm{sin}}^{2}\mathrm{x}{\mathrm{cos}}^{2}\mathrm{x}}{\mathrm{sin}\mathrm{x}\mathrm{cos}\mathrm{x}}\mathrm{dx}$ VIEW SOLUTION
 Question 5
 Question 6
Two tailors, A and B earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP. VIEW SOLUTION
 Question 7
A die, whose faces are marked 1, 2, 3, in red and 4, 5, 6 in green, is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events. VIEW SOLUTION
 Question 8
The xcoordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, –2) is 4. Find its zcoordinate. VIEW SOLUTION
 Question 9
Show that the function f(x) = x^{3} – 3x^{2} + 6x – 100 is increasing on ℝ. VIEW SOLUTION
 Question 10
Find the value of c in Rolle's theorem for the function f(x) = x^{3} – 3x in $\left[\sqrt{3},0\right]$. VIEW SOLUTION
 Question 11
If A is a skewsymmetric matrix of order 3, then prove that det A = 0. VIEW SOLUTION
 Question 12
The volume of a sphere is increasing at the rate of 8 cm^{3}/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm. VIEW SOLUTION
 Question 13
There are 4 cards numbered 1, 3, 5 and 7, 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
Show that the points A, B, C with position vectors $2\hat{\mathrm{i}}\hat{\mathrm{j}}+\hat{\mathrm{k}}$, $\hat{\mathrm{i}}3\hat{\mathrm{j}}5\hat{\mathrm{k}}$ and $3\hat{\mathrm{i}}4\hat{\mathrm{j}}4\hat{\mathrm{k}}$ respectively, are the vertices of a rightangled triangle. Hence find the area of the triangle. VIEW SOLUTION
 Question 15
Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer. VIEW SOLUTION
 Question 16
If ${\mathrm{tan}}^{1}\frac{x3}{x4}+{\mathrm{tan}}^{1}\frac{x+3}{x+4}=\frac{\mathrm{\pi}}{4}$, then find the value of x. VIEW SOLUTION
 Question 17
Using properties of determinants, prove that
$\left\begin{array}{ccc}{a}^{2}+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right={\left(a1\right)}^{3}$
OR
Find matrix A such that
$\left(\begin{array}{cc}2& 1\\ 1& 0\\ 3& 4\end{array}\right)A=\left(\begin{array}{cc}1& 8\\ 1& 2\\ 9& 22\end{array}\right)$ VIEW SOLUTION
 Question 18
If x^{y} + y^{x} = a^{b}, then find $\frac{\mathrm{dy}}{\mathrm{dx}}.$
OR
If e^{y}(x + 1) = 1, then show that $\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}={\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}.$ VIEW SOLUTION
 Question 19
Evaluate :
$\underset{0}{\overset{\mathrm{\pi}}{\int}}\frac{x\mathrm{tan}x}{\mathrm{sec}x+\mathrm{tan}x}dx$
OR
Evaluate :
$\underset{1}{\overset{4}{\int}}\left\{\leftx1\right+\leftx2\right+\leftx4\right\right\}\mathrm{dx}$ VIEW SOLUTION
 Question 20
Solve the following linear programming problem graphically :
Maximise Z = 7x + 10y
subject to the constraints
4x + 6y ≤ 240
6x + 3y ≤ 240
x ≥ 10
x ≥ 0, y ≥ 0 VIEW SOLUTION
 Question 21
Find :
$\int \frac{{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}}{{\left({\mathrm{e}}^{\mathrm{x}}1\right)}^{2}\left({\mathrm{e}}^{\mathrm{x}}+2\right)}$ VIEW SOLUTION
 Question 22
If $\overrightarrow{a}=2\hat{i}\hat{j}2\hat{k}\mathrm{and}\overrightarrow{b}=7\hat{i}+2\hat{j}3\hat{k}$, then express $\overrightarrow{b}$ in the form of $\overrightarrow{b}=\overrightarrow{{b}_{1}}+\overrightarrow{{b}_{2}}$, where $\overrightarrow{{b}_{1}}$ is parallel to $\overrightarrow{a}\mathrm{and}\overrightarrow{{b}_{2}}$ is perpendicular to $\overrightarrow{a}$. VIEW SOLUTION
 Question 23
Find the general solution of the differential equation $\frac{dy}{dx}y=\mathrm{sin}x$. VIEW SOLUTION
 Question 24
Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A(4, 1), B(6, 6) and C(8, 4).Find the area enclosed between the parabola 4y = 3x^{2} and the straight line 3x – 2y + 12 = 0. VIEW SOLUTION
OR
 Question 25
Find the particular solution of the differential equation $\left(xy\right)\frac{dy}{dx}=\left(x+2y\right)$, given that y = 0 when x = 1. VIEW SOLUTION
 Question 26
Find the coordinates of the point where the line through the points (3, –4, –5) and (2, –3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, –3) and (0, 4, 3).ORA variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is $\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}}+\frac{1}{{z}^{2}}=\frac{1}{{p}^{2}}$. VIEW SOLUTION
 Question 27
Consider $\mathrm{f}:\mathrm{\mathbb{R}}\left\{\frac{4}{3}\right\}\to \mathrm{R}\left\{\frac{4}{3}\right\}\mathrm{given}\mathrm{by}\mathrm{f}\left(\mathrm{x}\right)=\frac{4\mathrm{x}+3}{3\mathrm{x}+4}$. Show that f is bijective. Find the inverse of f and hence find f^{–1} (0) and x such that f^{–1} (x) = 2.ORLet $\mathrm{A}=\mathrm{\mathbb{Q}}\times \mathrm{\mathbb{Q}}$ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∊ A. Determine, whether * is commutative and associative. Then, with respect to * on A
(i) Find the identity element in A.
(ii) Find the invertible elements of A. VIEW SOLUTION
 Question 28
If $\mathrm{A}=\left[\begin{array}{ccc}2& 3& 5\\ 3& 2& 4\\ 1& 1& 2\end{array}\right]$, then find A^{–1} and hence solve the system of linear equations $2x3y+5z=11,3x+2y4z=5\mathrm{and}x+y2z=3$. VIEW SOLUTION
 Question 29
A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening. VIEW SOLUTION
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