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# Board Paper of Class 12 2020 Mathematics - Solutions

This 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) (i) Determine whether the binary operation ∗ on R defined by ab = |ab| is commutative. Also, find the value of ( ̶ 3)∗2.

(b) (ii) Prove that: tan2(sec–1 2) + cot2 (cosec–1 3) = 11.

(c) (iii) Without expanding at any stage, find the value of the determinant:

$∆=\left|\begin{array}{ccc}20& a& b+c\\ 20& b& a+c\\ 20& c& a+b\end{array}\right|$

(d) (iv) If

(e) (v) Find

(f) (vi) The edge of a variable cube is increasing at the rate of 10 cm/sec. How fast is the volume of the cube increasing when the edge is 5 cm long?

(g) (vii) Evaluate: $\underset{4}{\overset{5}{\int }}\left|x-5\right|dx$

(h) (viii) Form a differential equation of the family of the curves y2 = 4ax.

(i) (ix) A bag contains 5 white, 7 red and 4 black balls. If four balls are drawn one by one with replacement, what is the probability that none is white?

(j) (x) Let A and B be two events such that find ‘p’ if A and B are independent events. VIEW SOLUTION

• Question 2
If the function f : R → R be defined as show that (gof )(x) = ( f og)(x). VIEW SOLUTION

• Question 3
(a) If ${\mathrm{cos}}^{-1}\frac{x}{2}+{\mathrm{cos}}^{-1}\frac{y}{3}=\theta$, then prove that 9x2 – 12xy cos θ + 4y2 = 36 sin2 θ

OR

(b) Evaluate: VIEW SOLUTION

• Question 4
Using properties of determinants, show that
$\left|\begin{array}{ccc}x& p& q\\ p& x& q\\ q& q& x\end{array}\right|=\left(x-p\right)\left({x}^{2}+px-2{q}^{2}\right)$ VIEW SOLUTION

• Question 5
Verify Rolle’s theorem for the function, f (x) = −1 + cos x in the interval [0, 2π] VIEW SOLUTION

• Question 6
If $y={e}^{m{\mathrm{sin}}^{-1}x}$, prove that $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}={m}^{2}y$ VIEW SOLUTION

• Question 7
(a) The equation of tangent at (2, 3) on the curve y2 = px3 + q is y = 4x − 7. Find the values of  ‘p’ and ‘q’.

OR

(b) Using L’Hospital’s rule, evaluate : $\underset{x\to 0}{\mathrm{lim}}\frac{x{e}^{x}-\mathrm{log}\left(1+x\right)}{{x}^{2}}$ VIEW SOLUTION

• Question 8
(a) Evaluate : $\int \frac{dx}{\sqrt{5x-4{x}^{2}}}$

OR

(b) Evaluate : VIEW SOLUTION

• Question 9
Solve the differential equation $\left(1+{x}^{2}\right)\frac{dy}{dx}=4{x}^{2}-2xy$ VIEW SOLUTION

• Question 10
Three persons A, B and C shoot to hit a target. Their probabilities of hitting the target are respectively. Find the probability that:
(i) Exactly two persons hit the target.
(ii) At least one person hits the target. VIEW SOLUTION

• Question 11
Solve the following system of linear equations using matrices:
x − 2y =10, 2x − y − z = 8, − 2y + z = 7 VIEW SOLUTION

• Question 12
(a) Show that the radius of a closed right circular cylinder of given surface area and maximum volume is equal to half of its height.

OR

(b) Prove that the area of right-angled triangle of given hypotenuse is maximum when the triangle is isosceles. VIEW SOLUTION

• Question 13
(a) Evaluate: $\int {\mathrm{tan}}^{-1}\sqrt{\frac{1-x}{1+x}}dx$

OR

(b) Evaluate: $\int \frac{2x+7}{{x}^{2}-x-2}dx$ VIEW SOLUTION

• Question 14
The probability that a bulb produced in a factory will fuse after 150 days of use is 0·05.
Find the probability that out of 5 such bulbs:
(i) None will fuse after 150 days of use.
(ii) Not more than one will fuse after 150 days of use.
(iii) More than one will fuse after 150 days of use.
(iv) At least one will fuse after 150 days of use. VIEW SOLUTION

• Question 15
(a) Write a vector of magnitude of 18 units in the direction of the vector $\stackrel{^}{i}-2\stackrel{^}{j}-2\stackrel{^}{k}$.

(b) Find the angle between the two lines:

(c) Find the equation of the plane passing through the point (2, ̶ 3, 1) and perpendicular to the line joining the points ( 4, 5, 0) and ( 1,  ̶ 2, 4). VIEW SOLUTION

• Question 16
(a) Prove that

OR

(b) Using vectors, find the area of the triangle whose vertices are:
A (3,  ̶ 1, 2), B (1,  ̶ 1,  ̶ 3) and C( 4,  ̶ 3, 1)
VIEW SOLUTION

• Question 17
(a) Find the image of the point (3, ̶ 2, 1) in the plane 3x  ̶  y + 4z = 2

OR

(b) Determine the equation of the line passing through the point ( ̶ 1, 3,  ̶ 2) and perpendicular to the lines:

VIEW SOLUTION

• Question 18
Draw a rough sketch of the curves y2 = x and y2 = 4 – 3x and find the area enclosed between them. VIEW SOLUTION

• Question 19
(a) The selling price of a commodity is fixed at ₹ 60 and its cost function is C(x) = 35x + 250
(i) Determine the profit function.
(ii) Find the break even points.

(b) The revenue function is given by R(x) = 100xx2 x3 . Find
(i) The demand function.
(ii) Marginal revenue function.

(c) For the lines of regression 4x – 2y = 4 and 2x – 3y + 6 = 0, find the mean of ‘x’ and the mean of ‘y’. VIEW SOLUTION

• Question 20
(a) The correlation coefficient between x and y is 0.6. If the variance of x is 225 , the variance of y is 400, mean of x is 10 and mean of y is 20 , find (i) the equations of two regression lines (ii) the expected value of y when x = 2

OR

(b) Find the regression coefficients byx , bxy and correlation coefficient ‘r’ for the following data : (2, 8), (6, 8 ), (4, 5), (7, 6), (5, 2) VIEW SOLUTION

• Question 21
(a) The marginal cost of the production of the commodity is 30 + 2x , it is known that fixed costs are ₹ 200, find
(i) The total cost.
(ii) The cost of increasing output from 100 to 200 units.

OR

(b) The total cost function of a firm is given by $\mathrm{C}\left(x\right)=\frac{1}{3}{x}^{3}-5{x}^{2}+30x-15$ where the selling price per unit is given as ₹ 6.
Find for what value of x will the profit be maximum.
VIEW SOLUTION

• Question 22
A company uses three machines to manufacture two types of shirts, half sleeves and full sleeves. The number of hours required per week on machine M1 , M2 and M3 for one shirt of each type is given in the following table :  M1 M2 M3 Half sleeves 1 2 8/5 Full sleeves 2 1 8/2

None of the machines can be in operation for more than 40 hours per week . The profit on each half sleeve shirt is ₹ 1 and the profit on each full sleeve shirt is ₹1·50. How many of each type of shirts should be made per week to maximize the company’s profit? VIEW SOLUTION
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