Board Paper of Class 12Commerce 2019 Math Delhi(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
If $3\mathrm{A}\mathrm{B}=\left[\begin{array}{cc}5& 0\\ 1& 1\end{array}\right]\mathrm{and}\mathrm{B}=\left[\begin{array}{cc}4& 3\\ 2& 5\end{array}\right]$, then find the matrix A. VIEW SOLUTION
 Question 2
Write the order and the degree of the following differential equation:
${x}^{3}{\left(\frac{{d}^{2}y}{d{x}^{2}}\right)}^{2}+x{\left(\frac{dy}{dx}\right)}^{4}=0$
VIEW SOLUTION
 Question 3
If f(x) = x + 1, find $\frac{d}{dx}\left(fof\right)\left(x\right)$. VIEW SOLUTION
 Question 4
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector $2\hat{i}+2\hat{j}3\hat{k}$. VIEW SOLUTION
 Question 5
Find : ∫ sin x ⋅ log cos x dx VIEW SOLUTION
 Question 6
Evaluate :$\underset{\mathrm{\pi}}{\overset{\mathrm{\pi}}{\int}}\left(1{x}^{2}\right)\mathrm{sin}x{\mathrm{cos}}^{2}xdx.$
OR
Evaluate : $\underset{1}{\overset{2}{\int}}\frac{\leftx\right}{x}dx.$ VIEW SOLUTION
 Question 7
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not ? VIEW SOLUTION
 Question 8
Find a matrix A such that 2A − 3B + 5C = O, where $B=\left[\begin{array}{ccc}2& 2& 0\\ 3& 1& 4\end{array}\right]\mathrm{and}C=\left[\begin{array}{ccc}2& 0& 2\\ 7& 1& 6\end{array}\right].$ VIEW SOLUTION
 Question 9
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number is even" and B be the event "number is marked red". Find whether the events A and B are independent or not. VIEW SOLUTION
 Question 10
Form the differential equation representing the family of curves y = e^{2x} (a + bx), where 'a' and 'b' are arbitrary constants. VIEW SOLUTION
 Question 11
A die is thrown 6 times. If "getting an odd number" is a "success", what is the probability of (i) 5 successes? (ii) atmost 5 successes?
OR
The random variable X has a probability distribution P(X) of the following form, where 'k' is some number.$\mathrm{P}\left(\mathrm{X}=x\right)=\left\{\begin{array}{lll}k& ,& \mathrm{if}x=0\\ 2k& ,& \mathrm{if}x=1\\ 3k& ,& \mathrm{if}x=2\\ 0& ,& \mathrm{otherwise}\end{array}\right.$
Determine the value of 'k'. VIEW SOLUTION
 Question 12
If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is $\sqrt{3}$.
OR
If $\overrightarrow{a}=2\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}2\hat{j}+\hat{k}\mathrm{and}\overrightarrow{c}=3\hat{i}+\hat{j}+2\hat{k},\mathrm{find}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$. VIEW SOLUTION
 Question 13
Using properties of determinants, prove the following:
$\left\begin{array}{ccc}a& b& c\\ ab& bc& ca\\ b+c& c+a& a+b\end{array}\right={a}^{3}+{b}^{3}+{c}^{3}3abc.$ VIEW SOLUTION
 Question 14
Solve: tan^{−1} 4x + tan^{−1} 6x = $\frac{\mathrm{\pi}}{4}$. VIEW SOLUTION
 Question 15
Show that the relation R on ℝ defined as R = {(a, b) : a ≤ b}, is reflexive, and transitive but not symmetric.ORProve that the function f : N → N, defined by f(x) = x^{2} + x + 1 is oneone but not onto. Find inverse of f : N → S, where S is range of f. VIEW SOLUTION
 Question 16
Find the equation of tangent to the curve $y=\sqrt{3x2}$ which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact. VIEW SOLUTION
 Question 17
If log (x^{2} + y^{2}) = $2{\mathrm{tan}}^{1}\left(\frac{y}{x}\right)$, show that $\frac{dy}{dx}=\frac{x+y}{xy}$.ORIf x^{y} − y^{x} = a^{b}, find $\frac{dy}{dx}$. VIEW SOLUTION
 Question 18
If y = (sin^{−1}x)^{2}, prove that $\left(1{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}x\frac{dy}{dx}2=0$. VIEW SOLUTION
 Question 19
Prove that ${\int}_{0}^{a}f\left(x\right)dx={\int}_{0}^{a}f\left(ax\right)dx$, hence evaluate ${\int}_{0}^{\mathrm{\pi}}\frac{x\mathrm{sin}x}{1+{\mathrm{cos}}^{2}x}dx$. VIEW SOLUTION
 Question 20
Find: $\int \frac{\mathrm{cos}x}{\left(1+\mathrm{sin}x\right)\left(2+\mathrm{sin}x\right)}dx$ VIEW SOLUTION
 Question 21
Solve the differential equation: $\frac{dy}{dx}\frac{2x}{1+{x}^{2}}y={x}^{2}+2$ORSolve the differential equation: $\left(x+1\right)\frac{dy}{dx}=2{e}^{y}1;y\left(0\right)=0$. VIEW SOLUTION
 Question 22
If $\hat{i}+\hat{j}+\hat{k},2\hat{i}+5\hat{j},3\hat{i}+2\hat{j}3\hat{k}$ and $\hat{i}6\hat{j}\hat{k}$ respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{CD}}$ are collinear or not. VIEW SOLUTION
 Question 23
Find the value of λ, so that the lines $\frac{1x}{3}=\frac{7y14}{\lambda}=\frac{z3}{2}$ and $\frac{77x}{3\lambda}=\frac{{\displaystyle y5}}{{\displaystyle 1}}=\frac{{\displaystyle 6z}}{{\displaystyle 5}}$ are at right angles. Also, find whether the lines are intersecting or not. VIEW SOLUTION
 Question 24
A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m^{3}. If building of tank costs ₹ 70 per square metre for the base and ₹ 45 per square metre for the sides, what is the cost of least expensive tank? VIEW SOLUTION
 Question 25
If $\mathrm{A}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 2\\ 3& 1& 1\end{array}\right],$ find A^{–1}. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.
OR
Find the inverse of the following matrix using elementary operations.
$\mathrm{A}=\left[\begin{array}{ccc}1& 2& 2\\ 1& 3& 0\\ 0& 2& 1\end{array}\right]$ VIEW SOLUTION
 Question 26
Prove that the curves y^{2} = 4x and x^{2} = 4y divide the area of the square bounded by sides x = 0, x = 4, y = 4 and y = 0 into three equal parts.
OR
Using integration, find the area of the triangle whose vertices are (2, 3), (3, 5) and (4, 4). VIEW SOLUTION
 Question 27
A manufacturer has employed 5 skilled men and 10 semiskilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semiskilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semiskilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many of items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit. VIEW SOLUTION
 Question 28
Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
OR
Find the vector equation of the plane that contains the lines $\overrightarrow{r}=\left(\hat{i}+\hat{j}\right)+\lambda \left(\hat{i}+2\hat{j}\hat{k}\right)$ and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained. VIEW SOLUTION
 Question 29
Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean and variance of the number of kings. VIEW SOLUTION

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