Select Board & Class

# Board Paper of Class 12 2018 Mathematics - Solutions

The Question Paper consists of three sections A, B and C.
Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C
Section A: Internal choice has been provided in three questions of four marks each and two questions of six marks each.
Section B: Internal choice has been provided in two questions of four marks each.
Section C: Internal choice has been provided in two questions of four marks each.
All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables and graph papers are provided.

• Question 1
(a) The binary operation ∗ : R × R → R is defined as ab = 2a + b. Find (2 ∗ 3) ∗ 4.

(b) If and A is symmetric matrix, show that 𝑎 = b

(c) Solve: 3 tan–1x + cot–1x = π

(d) Without expanding at any stage, find the value of:

$\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$

(e) Find the value of constant ‘k’ so that the function f(x) defined as:

$f\left(x\right)=\left\{\begin{array}{ll}\frac{{x}^{2}-2x-3}{x+1},& x\ne -1\\ k,& x=-1\end{array}\right\$

is continuous at x = −1.

(f) Find the approximate change in the volume ‘𝑉’ of a cube of side x metres caused by decreasing the side by 1%.

(g) Evaluate : $\int \frac{{x}^{3}+5{x}^{2}+4x+1}{{x}^{2}}dx.$

(h) Find the differential equation of the family of concentric circles x2 + ya2

(i) If A and B are events such that  and $P\left(A\cap B\right)=\frac{1}{4},$ then find:
(a) P(AB)
(b) P(BA)

(j) In a race, the probabilities of A and B winning the race are $\frac{1}{3}$ and $\frac{1}{6}$ respectively. Find the probability of neither of them winning the race. VIEW SOLUTION

• Question 2
If the function $f\left(x\right)=\sqrt{2x-3}$ is invertible then find its inverse. Hence prove that (𝑓𝑜𝑓–1) (x) = x. VIEW SOLUTION

• Question 3
If tan–1 𝑎 + tan–1 b + tan–1 𝑐 = π, prove that a + b + c = abc. VIEW SOLUTION

• Question 4
Use properties of determinants to solve for x:

VIEW SOLUTION

• Question 5
Show that the function $f\left(x\right)=\left\{\begin{array}{ll}{x}^{2},& x\le 1\\ \frac{1}{x},& x>1\end{array}\right\$is continuous at x = 1 but not differentiable.

OR

Verify Rolle’s theorem for the following function: f(x) = esin x on [0, π] VIEW SOLUTION

• Question 6
If  prove that $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+\left(2x-a\right)\frac{dy}{dx}=0$ VIEW SOLUTION

• Question 8
Find the points on the curve y = 4x3 – 3x + 5 at which the equation of the tangent is parallel to the x-axis.

OR

Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi }{4}$ at the uniform rate of 2 cm2/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water. VIEW SOLUTION

• Question 9
Solve: $\mathrm{sin}x\frac{dy}{dx}-y=\mathrm{sin}x·\mathrm{tan}\frac{x}{2}$

OR

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.
VIEW SOLUTION

• Question 10
Using matrices, solve the following system of equations:
2x – 3y + 5z = 11
3x + 2y – 4z = −5
x + y – 2z = – 3

OR

Using elementary transformation, find the inverse of the matrix:

$\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$ VIEW SOLUTION

• Question 11
A speaks truth in 60% of the cases, while B in 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? VIEW SOLUTION

• Question 12
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height. VIEW SOLUTION

• Question 13
Evaluate: $\int \frac{x-1}{\sqrt{{x}^{2}-x}}dx$

OR

Evaluate: VIEW SOLUTION

• Question 14
From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find:
(a) The probability distribution of X
(b) Mean of X
(c) Variance of X VIEW SOLUTION

• Question 15
(a) Find λ if the scalar projection of $\stackrel{\to }{a}=\lambda \stackrel{^}{i}+\stackrel{^}{j}+4\stackrel{^}{k}$ on $\stackrel{\to }{b}=2\stackrel{^}{i}+6\stackrel{^}{j}+3\stackrel{^}{k}$ is 4 units.

(b) The Cartesian equation of a line is: 2x – 3 = 3y + 1 = 5 – 6z. Find the vector equation of a line passing through (7, −5, 0) and parallel to the given line.

(c) Find the equation of the plane through the intersection of the planes $\stackrel{\to }{r}·\left(\stackrel{^}{i}+3\stackrel{^}{j}-\stackrel{^}{k}\right)=9$ and $\stackrel{\to }{r}·\left(2\stackrel{^}{i}-\stackrel{^}{j}+\stackrel{^}{k}\right)=3$ and passing through the origin. VIEW SOLUTION

• Question 16
If ABC are three non-collinear points with position vectors  respectively, then show that the length of the perpendicular from C on AB is $\frac{\left|\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right)+\left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)+\left(\stackrel{\to }{b}×\stackrel{\to }{a}\right)\right|}{\left|\stackrel{\to }{b}-\stackrel{\to }{a}\right|}.$

OR

Show that the four points ABC and D with position vectors  and $4\left(-\stackrel{^}{i}+\stackrel{^}{j}+\stackrel{^}{k}\right)$ respectively, are coplanar. VIEW SOLUTION

• Question 17
Draw a rough sketch of the curve and find the area of the region bounded by curve y2 = 8x and the line x = 2.

OR

Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = −6 and x = 0. VIEW SOLUTION

• Question 18
Find the image of a point having position vector : $3\stackrel{^}{i}-2\stackrel{^}{j}+\stackrel{^}{k}$ in the Plane $\stackrel{\to }{r}·\left(3\stackrel{^}{i}-\stackrel{^}{j}+4\stackrel{^}{k}\right)=2.$ VIEW SOLUTION

• Question 19
(a) Given the total cost function for x units of a commodity as:
$C\left(x\right)=\frac{1}{3}{x}^{3}+3{x}^{2}-16x+2.$
Find:
(i) Marginal cost function
(ii) Average cost function

(b) Find the coefficient of correlation from the regression lines:
x – 2y + 3 = 0 and 4x – 5y + 1 = 0.

(c) The average cost function associated with producing and marketing x units of an item is given by $AC=2x–11+\frac{50}{x}.$ Find the range of values of the output x, for which AC is increasing. VIEW SOLUTION

• Question 20
Find the line of regression of y on x from the following table.
 x 1 2 3 4 5 y 7 6 5 4 3

Hence, estimate the value of y when x = 6.

OR

From the given data:  Variable x y Mean 6 8 Standard Deviation 4 6
​ ​
and correlation coefficient: $\frac{2}{3}$. Find:
(i) Regression coefficients byx and bxy
(ii) Regression line x on y
(iii) Most likely value of x when y = 14 VIEW SOLUTION

• Question 21
A product can be manufactured at a total cost $C\left(x\right)=\frac{{x}^{2}}{100}+100x+40,$ where is the number of units produced. The price at which each unit can be sold is given by $\mathrm{P}=\left(200-\frac{x}{400}\right).$ Determine the production level x at which the profit is maximum. What is the price per unit and total profit at the level of production?

OR

A manufacturer’s marginal cost function is $\frac{500}{\sqrt{2x+25}}.$ Find the cost involved to increase production from 100 units to 300 units. VIEW SOLUTION

• Question 22
A manufacturing company makes two types of teaching aids and of Mathematics for Class X. Each type of requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of ₹ 80 on each piece of type and ₹ 120 on each piece of type B. How many pieces of type and type should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch. VIEW SOLUTION
What are you looking for?

Syllabus