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CBSE Previous Year Question Papers Class 12 Maths 2018

CBSE Previous Year Question Papers Class 12 Maths 2018

Time allowed: 3 hours
Maximum marks : 100

General Instructions:

  • All questions are compulsory.
  • The question paper consists of 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.
  • All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
  • 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.
  • Use of calculators is not permitted. You may ask for logarithmic tables, if required.

**Answer is not given due to the change in present syllabus

Section – A

Question 1.
Find the value of tan-1 \sqrt{{3}} – cot-1 ( \sqrt{{-3}} ) [1]
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2018 1

Question 2.
If the matrix A = \left[\begin{array}{rrr}{0} & {a} & {-3} \\ {2} & {0} & {-1} \\ {b} & {1} & {0}\end{array}\right]is skew-symmetric, find the values of’a’ and ‘b’. [1]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 2

Question 3.
Find the magnitude of each of the two vectors \vec{a} \text { and } \vec{b}, having the same magnitude such that the angle between them is 60° and their scalar product is \frac{9}{2}. [1]
Solution:
Let \vec{a} \text { and } \vec{b}be two such vectors.
CBSE Previous Year Question Papers Class 12 Maths 2018 3
CBSE Previous Year Question Papers Class 12 Maths 2018 4

Question 4.
If a * b denotes the larger of ‘a’ and ‘b’ and if a o b = (a * b) + 3, then write the value of (5) o (10), where * and o are binary operations.** [1]

Section – B

Question 5.
Prove that: [2]
CBSE Previous Year Question Papers Class 12 Maths 2018 5
Solution:
R.H.S = sin-1(3x – 4x3)
Putting x = sin θ in R.H.S, we get
R.H.S = sin-1(3 sin θ – 4 sin3 θ)
= sin-1(sin 3θ)
CBSE Previous Year Question Papers Class 12 Maths 2018 6

Question 6.
Given A = \left[\begin{array}{rr}{2} & {-3} \\ {-4} & {7}\end{array}\right], compute A-1 and show that 2A-1 = 9I – A. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 7
CBSE Previous Year Question Papers Class 12 Maths 2018 8

Question 7.
Differentiate \tan ^{-1}\left(\frac{1+\cos x}{\sin x}\right)with respect to x. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 9

Question 8.
The total cost C(x) associated with the pro-duction of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 30s + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. [2]
Solution :
Cost function is given as
C(s) = 0.005x3 – 0.02x2 + 30s + 5000
Marginal cost (MC) = \frac{d}{d x}(\mathrm{C}(x))
= 0.005(3x2) – 0.02 (2x) + 30
= 0.015x2 – 0.04s+ 30
When x = 3,MC = 0.015(3)2 – 0.04(3) + 30
= 0.135 – 0.12 + 30
= 30.015

Question 9.
Evaluate:
CBSE Previous Year Question Papers Class 12 Maths 2018 10
Solution:
We have,
CBSE Previous Year Question Papers Class 12 Maths 2018 11

Question 10.
Find the differential equation representing the family of curves y = aebx + 5, where a and b are arbitrary constants. [2]
Solution:
Given curve is
y = aebx + 5 …(i)
Differentiating (i) w.r.t. s, we get
CBSE Previous Year Question Papers Class 12 Maths 2018 12

Question 11.
If θ is the angle between two vectors \hat{i}-2 \hat{j}+3 \hat{k}and 3 \hat{i}-2 \hat{j}+\hat{k}, find sin θ. [2]
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 13
CBSE Previous Year Question Papers Class 12 Maths 2018 14

Question 12.
A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. [2]
Solution :
The sample space has 36 Outcomes. Let A be event that the sum of observations is 8.
∴ A = {(2, 6), (3, 5), (5, 3), (4, 4), (6, 2)}
CBSE Previous Year Question Papers Class 12 Maths 2018 15

Section – C

Question 13.
Using properties of determinants, prove that: [4]
CBSE Previous Year Question Papers Class 12 Maths 2018 16
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 17
CBSE Previous Year Question Papers Class 12 Maths 2018 18

Question 14.
If (x2 + y2)2 = xy, find \frac{d y}{d x}. [4]
Solution:
We have,
(x2 + y2)2 = xy …(i)
CBSE Previous Year Question Papers Class 12 Maths 2018 19
OR
If x = a(2θ – sin 2θ) and y = a(1 – cos 2θ), find \frac{d y}{d x}when θ = \frac{\pi}{3}.
Solution:
Given, x = a (2θ – sin 2θ)
and y = a (1 – cos 2θ)
Differentiating x* and y w.r.t. θ, we get
CBSE Previous Year Question Papers Class 12 Maths 2018 20

Question 15.
If y = sin(sin x), prove that: [4]
CBSE Previous Year Question Papers Class 12 Maths 2018 21
Solution:
Given, y = sin (sin x)
Differentiating y w.r.t. x, we get
\frac{d y}{d x}= cos (sin x) • cos x
Differentiating again w.r.t. x, we get
\frac{d^{2} y}{d x^{2}}= cos (sin x) (- sin x) + cos x [-sin (sin x)-cos x]
CBSE Previous Year Question Papers Class 12 Maths 2018 22

Question 16.
Find the equations of the tangent and the normal to the curve 16x2 + 9y2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0. [4]
Solution:
Given curve is
16x2 + 9y2 = 145 ….(i)
Since (x1, y1) lies on equation (i),
CBSE Previous Year Question Papers Class 12 Maths 2018 23
CBSE Previous Year Question Papers Class 12 Maths 2018 24
CBSE Previous Year Question Papers Class 12 Maths 2018 25
OR
Find the intervals in which the function
CBSE Previous Year Question Papers Class 12 Maths 2018 26
(a) strictly increasing,
(b) strictly decreasing.
Solution:
Given function is
CBSE Previous Year Question Papers Class 12 Maths 2018 27

Question 17.
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ? [4]
Solution:
Let the length, breadth and height of the open tank be x, x and y units respectively.
Then, Volume (V) = x2y …(i)
CBSE Previous Year Question Papers Class 12 Maths 2018 28
CBSE Previous Year Question Papers Class 12 Maths 2018 29
Area is minimum, thus cost is minimum when x = 2y.
i.e., depth of tank is half of the width.
Value : Any relevant value.

Question 18.
Find: [4]
CBSE Previous Year Question Papers Class 12 Maths 2018 30
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 31
CBSE Previous Year Question Papers Class 12 Maths 2018 32

Question 19.
Find the particular solution of the differential equation ex tan ydx + (2 – ex) sec2 ydy = 0, given that y = \frac{\pi}{4}when x = 0. [4]
Solution:
Given differential equation is,
CBSE Previous Year Question Papers Class 12 Maths 2018 33
OR
Find the particular solution of the differential equation \frac{d y}{d x}+ 2y tan x = sin x, given than y = 0 when x = \frac{\pi}{3}.
Solution:
Given differential equation is,
CBSE Previous Year Question Papers Class 12 Maths 2018 34
Hence, particular solution is
y sec2 x = sec x – 2
or y = cos x – 2 cos2x

Question 20.
Let \vec{a}=4 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=\hat{i}-4 \hat{j}+5 \hat{k} \text { and } \vec{c}=3 \hat{i}+\hat{j}-\hat{k}Find a vector d which is perpendicular to both \vec{c} \& \vec{b} \text { and } \vec{d} \cdot \vec{a}=21. [4]
Solution:
Given,
CBSE Previous Year Question Papers Class 12 Maths 2018 35
CBSE Previous Year Question Papers Class 12 Maths 2018 36

Question 21.
Find the shortest distance between the lines \vec{r}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k})and \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k})[4]
Solution:
Given lines are
CBSE Previous Year Question Papers Class 12 Maths 2018 37
CBSE Previous Year Question Papers Class 12 Maths 2018 38

Question 22.
Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’, is obtained. If she obtained exactly one “tail7, what is the probability that she threw 3, 4, 5 or 6 with the dice? [4]
Solution :
Let E1 be the event that girl gets 1 or 2 on the roll and E2 be the event that girl gets 3, 4, 5, or 6 on the roll of a die.
CBSE Previous Year Question Papers Class 12 Maths 2018 39
Let A be the event that she gets exactly one tail. If she tossed coin 3 times and gets exactly one tail then possible outcomes are HTH, HHT, THH
CBSE Previous Year Question Papers Class 12 Maths 2018 40
= \frac{8}{11}

Question 23.
Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. [4]
Solution:
First five positive integers are 1, 2, 3, 4, 5. We select two positive numbers in 5 × 4 = 20 ways. Out of these, two numbers are selected at random. Let X denote larger of the two selected numbers. Then, X can have values 2, 3, 4 or 5.
P(X = 2) – P (larger no. is 2) = {(1, 2) and (2, 1)}
CBSE Previous Year Question Papers Class 12 Maths 2018 41
CBSE Previous Year Question Papers Class 12 Maths 2018 42
Therefore, mean and variance are 4 and 1 respectively.

Section – D

Question 24.
Let A = {x ϵ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b): a, b ϵ A, | a – b | is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]. [6]
Solution:
Given, R = {(a, b): a, b ϵ A, | a – b | is divisible by 4}
Reflexivity: For any a ϵ A
| a – a | = 0, which is divisible by 4
(a, a) ϵ R
So, R is reflexive.

Symmetry: Let (a, b) ϵ R
⇒ | a – b | is divisible by 4
⇒ | b – a | is divisible by 4 [ ∵ | a – b | = | b – a | ]
⇒ (b, a) ϵ R
So, R is symmetric.

Transitive : Let (a, b) ϵ R and (b, c) ϵ R
⇒ | a – b | is divisible by 4
⇒ | a – b | = 4k
∴ a – b = ± 4k, k ϵ Z …(i)
Also, |b – c| is divisible by 4
⇒ |b – c| = 4m
∴ b – c = ±4m, m ϵ Z …(ii)
Adding equations (i) and (ii)
a – b + b – c = ±4 (k + m)
⇒ a – c = ±4 (k + m)
|a – c | is divisible by 4,
⇒ (a, c) ϵ R
So, R is symmetric.
⇒ R is reflexive, symmetric and transitive.
∴ R is an equivalence relation.
Let x be an element of R such that (x, 1) ϵ R
Then | x – 1| is divisible by 4
x – 1 = 0, 4, 8, 12,
⇒ -x = 1, 5, 9 (∵ x ≤ 12)
∴ Set of all elements of A which are related to 1 are {1, 5, 9}.
Equivalence class of 2 i.e.
[2] = {(a, 2): a ϵ A, |a – 2| is divisible by 4}
⇒ | a – 2| = 4k(k is whole number, k ≤ 3)
⇒ a = 2, 6, 10
Therefore, equivalence class [2] is {2, 6, 10]
OR
Show that the function f : R → R defined by f(x) = \frac{x}{x^{2}+1}, ∀x ϵ R is neither one-one nor onto. Also,if g: R → R is defined as g(x) = 2x – 1, find fog(x).
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 43
CBSE Previous Year Question Papers Class 12 Maths 2018 44

Question 25.
If A = \left[\begin{array}{rrr}{2} & {-3} & {5} \\ {3} & {2} & {-4} \\ {1} & {1} & {-2}\end{array}\right], find A-1. Use it to solve the system of equation: [6]
2x – 3y + 5z = 11
3x + 2y – 4z = -5
x + y – 2z = -3
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 45
CBSE Previous Year Question Papers Class 12 Maths 2018 46
CBSE Previous Year Question Papers Class 12 Maths 2018 47
OR
Using elementary row transformations, find the inverse of the matrix:
CBSE Previous Year Question Papers Class 12 Maths 2018 48
Solution:
We know that, A = IA
CBSE Previous Year Question Papers Class 12 Maths 2018 49

Question 26
Using integration, find the area of the region in the first quadrant enclosed by the X-axis, the line y = x and the circle x2 + y2= 32. [6]
Solution:
Given curve is
CBSE Previous Year Question Papers Class 12 Maths 2018 50
CBSE Previous Year Question Papers Class 12 Maths 2018 51

Question 27.
Evaluate: [6]
CBSE Previous Year Question Papers Class 12 Maths 2018 52
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 53
CBSE Previous Year Question Papers Class 12 Maths 2018 54
OR
Evaluate \int_{1}^{3}\left(x^{2}+3 x+e^{x}\right) d xas the limit of the sum.
Solution:
CBSE Previous Year Question Papers Class 12 Maths 2018 55
CBSE Previous Year Question Papers Class 12 Maths 2018 56
CBSE Previous Year Question Papers Class 12 Maths 2018 57

Question 28.
Find the distance of the point (-1, -5, -10) from the point of intersection of the line \vec{r}=2 \hat{i}-\hat{j}+2 \hat{k}+\lambda(3 \hat{i}+4 \hat{j}+2 \hat{k})and the plane \vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=5. [6]
Solution:
Equation of line is
CBSE Previous Year Question Papers Class 12 Maths 2018 58
CBSE Previous Year Question Papers Class 12 Maths 2018 59
CBSE Previous Year Question Papers Class 12 Maths 2018 60

Question 29.
A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand- operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of ₹1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit [6]
Solution:
Let the number of packets of screw ‘A’ manufactured in a day be x and that of screw B be y.
CBSE Previous Year Question Papers Class 12 Maths 2018 61
Plotting the points on the graph, we get the feasible region OABC as shown (Shaded).
CBSE Previous Year Question Papers Class 12 Maths 2018 62
CBSE Previous Year Question Papers Class 12 Maths 2018 63
Hence, profit will be maximum if company produces 30 packets of screw A and 20 packets of screw B and maximum profit = ₹ 41.

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