Class 12 Mathematics Sample Paper With Solutions Set B

Please refer to Class 12 Mathematics Sample Paper With Solutions Set B below. These Class 12 Mathematics Sample Papers will help you to get more understanding of the type of questions expected in the upcoming exams. All sample guess papers for Mathematics Class 12 have been designed as per the latest examination pattern issued by CBSE. Please practice all CBSE Sample Papers for Mathematics in Standard 12.

Sample Paper Class 12 Mathematics With Solutions Set B

PART – A
Section – I

Class 12 Mathematics Sample Paper With Solutions Set B

Answer

2. Solution of the differential equation dy/dx = 1− x + y − xy is

Answer

3. Find the projection of a̅ = 2î – ĵ + k̂ on b̅ = î – 2ĵ + k̂

Answer

4. What is the degree of the differential equation

(a) 1
(b) 2
(c) 3
(d) 4

Answer

Section – II

Case study-based question is compulsory. Attempt any 4 sub parts. Each sub-part carries 1 mark.
5. A rumour on whatsapp spreads in a population of 5000 people at a rate proportional to the product
of the number of people who have heard it and the number of people who have not. Also, it is
given that 100 people initiate the rumour and a total of 500 people know the rumour after 2 days.
Based on the above information, answer the following questions.

(i) If y(t) denote the number of people who know the rumour at an instant t, then maximum value of y(t) is
(a) 500
(b) 100
(c) 5000
(d) none of these

Answer

(ii) dy/dt is proportional to
(a) (y – 5000)
(b) y(y – 500)
(c) y(500 – y)
(d) y(5000 – y)

Answer

(iii) The value of y(0) is
(a) 100
(b) 500
(c) 600
(d) 200

Answer

(iv) The value of y(2) is
(a) 100
(b) 500
(c) 600
(d) 200

Answer

(v) The value of y at any time t is given by

Answer

PART – B
Section – III

6. Evaluate :

Answer.

8. Find the equation of a plane with intercepts 2, 3 and 4 on the X, Y and Z-axes respectively.
Answer. As the plane has intercepts 2, 3 and 4 on X, Y and Z axes respectively.
∴ The required equation of the plane is

9. If P(A) = 3/8, P(B) =5/8 and P(A ∪ B) = 3/4, then find P(A | B)
Answer. P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

10. Find the area enclosed by the ellipse

Answer. We know that, the area enclosed by the ellipse

Section – IV

11. if a̅ = 2î + 3ĵ – k̂ and b̅ = î + 2ĵ + 2k̂, find a̅ x b̅ and |a̅ x b̅|
Answer. Given, a̅ = 2î + 3ĵ – k̂ and b̅ = î + 2ĵ + 3k̂,

12. Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.
OR
Find the cartesian equation of the plane passing through a point having position vector 2î + 3ĵ + 4k̂ and perpendicular to the vector 2î + ĵ – 2k̂
Answer. Vector equation of the line passing through (1, 2, –1) and parallel to the line
5x – 25 = 14 – 7y = 35z

Here, (x₁, y₁, z₁) = (2, 3, 4), a = 2, b = 1, c = –2
Cartesian equation is
a(x – x₁) + b(y – y₁) + c(z – z₁) = 0
⇒ 2(x – 2) + 1(y – 3) –2(z – 4) = 0
⇒ 2x – 4 + y – 3 – 2z + 8 = 0
⇒ 2x + y – 2z = –1

13. If A and B are events such that P(A) = 1/3, P(B) = 1/4 and P(A ∩ B) = 1/12, then find P(not A and not B).
Answer.

14. A box contains N coins, of which m are fair and the rest are biased. The probability of getting head when a fair coin is tossed is 1/2, while it is 2/3 when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. Find the probability that the coin drawn is fair.
Answer. Let E be the event that the coin tossed twice shows first head and then tail and F be the event that the coin drawn is fair.

Section – V

Answer.

16. Vectors  a̅, b̅ and c̅ are such that a̅ + b̅ + c̅ = 0 and |a̅| = 3, |b̅| = 5 and |c̅| = 7 Find the angle between a̅ + b̅
Answer. a̅ + b̅ + c̅ = 0 and |a̅| = 3, |b̅| = 5 and |c̅| = 7
We have a̅ + b̅ + c̅ = 0
⇒ a̅ + b̅ = – c̅ ⇒ |a̅, b̅|²= |-c̅|²
⇒ |a̅|² + |b̅|² + 2(a̅ + b̅) = |-c̅|²
⇒ 9 + 25+ 2|a̅||b̅| cosθ = 49
⇒ ⇒ 2 × 3 × 5 × cosθ = 49 – 34 = 15

17. Three machines E₁, E₂ and E₃ in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E₁ and E₂ are defective and that 5% of those produced by machine E₃ are defective. If one bulb is picked up at random from a day’s production, calculate the probability that it is defective.
Answer. Let A be the event that the bulb is defective

Section – VI

18. If the planes x – cy – bz = 0, cx – y + az = 0 and bx + ay – z = 0 pass through a straight line, then find the value of a² + b²+ c² + 2abc.
OR
Find the equation of the plane passing through the points (2, 2, –1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.
Answer. Given planes are
x – cy – bz = 0 …(i)
cx – y + az = 0 …(ii)
bx + ay – z = 0 …(iii)
The d.r.,s of normal to plane (i), (ii) and (iii) are (1, –c, –b), (c, –1, a) and (b, a, –1) respectively.
All planes pass through same line, then the line is perpendicular to each of the three normals. The d.r’s. of line from planes (i) and (ii) are

The equation of a plane passing through (2, 2, –1) is
a(x – 2) + b(y – 2) + c(z + 1) = 0 …(i)
This plane also passes through (3, 4, 2).
∴ a(3 – 2) + b(4 – 2) + c(2 + 1) = 0
⇒ a + 2b + 3c = 0 …(ii)
Now, plane (i) is parallel to the line whose direction ratios are 7, 0, 6
Therefore, 7a + 0(b) + 6c = 0 …(iii)
Solving (ii) and (iii) by cross-multiplication method, we get

Substituting the values of a, b, c in (i), we get
12λ(x – 2) + 15λ(y – 2) – 14λ(z + 1) = 0
⇒ 12x – 24 + 15y – 30 – 14z – 14 = 0 [∵ λ ≠ 0]
⇒ 12x + 15y – 14z = 68, which is the required equation of plane.

19. 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.
OR
Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x = √y and y-axis.
Answer. The given equation of the circle is x² + y² = 32 and the line is y = x
These intersect at A(4, 4) in the first quadrant. The required area is shown shaded in the figure. Points B(4, 0) and C (4 √2, 0)