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Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical PDF Download

Class-XII


Time: 120 Minutes


Max. Marks: 40

General Instructions :

  1. This question paper contains three section A, B and C. Each part is compulsory.
  2. Section - A has 6 short answer type (SA1) questions of 2 marks each.
  3. Section - B has 4 short answer type (SA2) questions of 3 marks each.
  4. Section - C has 4 long answer type questions (LA) of 4 marks each.
  5. There is an internal choice in some of the questions.
  6. Q14 is a case-based problem having 2 sub parts of 2 marks each.

Section - A

Q.1. Find Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
OR
Find : Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-MedicalClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
OR
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Put cos2x  = t
⇒ -2cos x sinx dx = dt
⇒ sin 2x dx = -dt
The given integral
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical


Q.2. Write the sum of the order and the degree of the following differential equation: 
(d/dx)(dy/dx) = 5

Order = 2

Degree = 1
Sum = 3


Q.3. If Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are unit vectors, then prove that Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical where θ is the angle between them.

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical 1 + 1 + 2cosθ
= 2(1 + cosθ) = 4cos2(θ/2)
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical = 2cos(θ/2)


Q.4. Find the direction cosines of the following line:
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

The given line is Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Its direction ratios are <1, 1, 4>
Its direction cosines are Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical


Q.5. A bag contains 1 red and 3 white balls. Find the probability distribution of the number of red balls if 2 balls are drawn at random from the bag one-by-one without replacement.

Let X be the random variable defined as the number of red balls.

Then X = 0, 1
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Probability distribution Table:
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical


Q.6. Two cards are drawn at random from a pack of 52 cards one-by-one without replacement. What is the probability of getting first card red and second card Jack?.

The required probability = P((The first is a red jack card and The second is a jack card) or (The first is a red non-jack card and The second is a jack card))

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Section - B

Q.7. Find Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

= 1/2(log)|x2 + 3x + 2|
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
= (1/2)log|x2  + 3x + 2| - (3/2) Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical


Q.8. Find the particular solution of the differential equation Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical given that y(0) = √3.
OR
Prove that x– y2 = C(x2 + y2)2 is the general solution of the differential equation (x3 – 3xy2) dx = (y3 – 3x2y) dy, where C is a parameter.

From the given, equation, Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Integrating, we get tan-1 y = tan-1x + C
As x  = 0, y =√3 so tan–1√3 = C or C= π/3
Now, the required solution
tan-1y = tan-1 x + (π/3)
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
y - x = √3 (1 + xy)
OR
x2 - y2 = C(x2 + y2)2 ...(i)
Differentiating above
2x - 2yy' = 2C (x2 + y2)(2x + 2yy')
or (x - yy') = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical  using eq . (i)
or(y2 + x2)(x - yy') = (x2- y2)(2x +2yy') - (y2 + x2)yy1 + x(y2 + x2) = (x2 - y2)2x + (x2 - y2)2yy1
[2y(x2 - y2) + y(y2 - x2)] (dy/dx) = 2x(x2 - y2) - x(y- x2)
or (y3  - 3x2y)(dy/dx) = (x3 - 3xy2)
or (y3 - 3x2y)dy = (x3 - 3xy2)dx
Hence x2 – y2 = C(x2 + y2)2 is the solution of given differential equation.


Q.9. If Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical respectively are the position vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are collinear or not.
OR
Find the area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2).

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Let required angle be θ.
Then Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
⇒ θ = 180° or π
Since θ = π so Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are collinear.
OR
We have, A(0, 4, 1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2).
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Since,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Thus quadrilateral formed is parallelogram.
∴ Area of quadrilateral ABCD
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
= 9 sq. units.


Q.10. If the lines Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are perpendicular, find the value of λ. Hence find whether the lines are intersecting or not.

The equations of the given lines are:
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
On comparing these lines with Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical we get a1 = –3, b1 = 2λ, c1 = 2 and a2 = 3λ, b2 = 2, c2 = –5
Since, lines are perpendicular, a1a2 + b1b2 + c1c2 = 0
So, (–3) (3λ) + (2λ) (2) + (2) (–5) = 0
⇒ –9λ + 4λ – 10 = 0
⇒ –5λ  – 10 = 0
⇒ –5λ = 10
⇒ λ = –2
Hence, for λ = –2 the given lines are perpendicular.
Now, given lines can be written as after substituting value of λ = –2
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
The coordinate of any point on first line are given by Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical (say)
or
x = -3r + 1, y = -4r + 2, z = 2r + 3
So, the coordinates of a general point on first line are (–3r + 1, –4r + 2, 2r + 3).
The coordinates of any point on second line are given by Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical (say)
or x = -6s + 1, y = 2s + 1, z = -5s + 6
So, the coordinates of a general point on second line are (–6s + 1, 2s + 1, –5s + 6)
If the lines intersect, then they have a common point. So, for some value of λ and μ, we must have –3r + 1 = –6s + 1, –4r + 2 = 2s + 1, 2r + 3 = –5s + 6
or r = 2s, –4r – 2s = –1, 2r + 5s = 3
Solving first two of these two equations, we get r = 1/5 and s = 1/10. These values of r and s do not satisfy the third equation.
Hence, the given lines do not intersect.

Section - C

Q.11. A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

Let E1 =Selecting bag A and E2 = Selecting bag B
∴ Probability for selecting bag  when die is thrown P(E1) = 1/3 and probability for selecting bag  when die  is thrown P(E2) = 2/3

Let A = Getting one red and one black ball
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical = 22/45


Q.12. Find the area of the region bounded by the circle x2 + y2 = 1.
OR
Find the area of the region bounded the curve y = x + 1 and the lines x = 2 and x = 3.

We have, x2 + y2 = 1, which is a circle having centre at (0, 0) and radius ‘1’ unit.

⇒ y= 1 - x2
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

From the figure, area of the shaded region, 

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
= π sq. units
OR
Given : Curve y = x + 1 and lines x = 2, x = 3
If x  = 2 ⇒ y = 3
If x  = 3 ⇒ y = 4
So, intersection points (2, 3) and (3, 4)
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

From the figure, area of the shaded region,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
= (7/2) sq. units


Q.13. If the vector Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are coplanar, then for a, b, c ≠ 1, then show that. Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical

Since the vector Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical are coplanar,

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
i.e., Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
R2 →  R2 → R1 and R→ R3 → R1,
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
or a(b - 1)(c - 1)-1(1 - a)(c - 1)-1(1 - a)(b - 1) = 0
i.e., a (1 - b)(1 - c) + (1 - a)(1 - c) + (1 - a)(1 - b) = 0
Dividing both the sides by (1 – a)(1 – b)(1 – c), we get
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
i.e., Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
i.e., Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical


Case-Based/Data Based
Q.14. The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number triples in 5 hours.
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
Based on the above information, answer the following questions:
(i) If ‘N’ is the number of bacteria, then find the differential equation. Also, find the general solution of obtained differential equation.
(ii) If N0 is the initial count of bacteria, the find the bacteria count after 10 hours.

(i) Given that N is the number of bacteria.

dN/dt ∝ N
⇒ dN/dt ∝ KN is the  required differential  equation.
Now, dN/N ∝ K dt
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical
⇒ log|N| = Kt + c,
which is the general solution of differential equation. ...(1)
(ii) Given, when t = 0, N = N0, from (1)
log|N0| = c
∴ (1) ⇒ log|N| = Kt + log|N0|
⇒ log|N/N0| = Kt ...(2)
Given, when t = 5, N = 3N0
From (2), log|3| = 5K
⇒ K = (1/5) log3
∴ Particular solution is
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical ...(3)
when t = 10
log|N/N0| =2 log 3 = log 9
⇒ N/N0 = 9
⇒ N = 9N0.

The document Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 2 | Sample Papers for Class 12 Medical and Non-Medical is a part of the Class 12 Course Sample Papers for Class 12 Medical and Non-Medical.
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