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Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | 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 sections 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. Q.14 is a case-based problem having 2 sub parts of 2 marks each.

Section - A

Q.1. Find the value of Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
OR
Evaluate : Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

Let I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical+ Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
When f(x) is an even function, then,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
and if f(x) is an odd function, then
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical= 0
∴ I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = π
OR
Let,
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical 
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Also, put ex = t, ⇒ exdx = dt
⇒ I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= tan-1t + C
= tan-1(ex) + C


Q.2. Show that Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 0 is the solution of y= e–x (A cos x + B sin x).

Given that, y = e–x (A cos x + B sin x)
On differentiating both sides w.r.t., x we get
dy/dx = -e-x(A cos x + B sin x) + e–x (-A sin x + B cos x)
dy/dx = -y + e-x(-A sin x + B cos x)
Again, differentiating both sides w.r.t. x, we get
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical (-A sin x + B cos x)
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 0

(Hence Proved 1)


Q.3. Find the projection of vector Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical on the vector Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical.

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 4 + 6 + 2 = 12
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or p = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical  Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 3
= 12/3 = 4


Q.4. If the lines Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical are perpendicular to each other, then find the value of p.

Using formula for perpendicular condition,
l1l2 + m1m2 + n1n2 = 0
or – 8p + 6p – 28 = 0
or – 2p = 28
∴ p = 14


Q.5. If P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6, then P(A ∪ B).

Here,
P(A) = 0.4, P(B) = 0.8 and P(B/A) = 0.6
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
⇒ P(B ∩ A) = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 0.6 x 0.4 = 0.24
∵ P(A ∪ B) = P(A) + P(B) - P(B ∩ A)
= 0.4 + 0.8 - 0.24
= 1.2 - 0.24
= 0.96


Q.6. Find the probability distribution of X, the number of heads is a simultaneous toss of two coins.

Let X be the number of heads
Possible values of X are 0, 1, 2.
P(x = 0) = 1/4, P(x = 1) = 1/2, P(x = 2) = 1/4
The probability distribution of X is :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

Section - B

Q.7.Evaluate: Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

I =Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical        ..(i)
Apply the property Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical        ..(ii)
Adding eqn. (i) and (ii), we get
or 2I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
So, I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical


Q.8. Solve Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 0 subject to the initial condition y(0) = 0.
OR
Find the particular solution of the following differential equation :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical ; y = 0 when x = 0

Given differential equation can be written as: 

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Comparing with
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
⇒ P = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical, Q = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I.F. (Integrating factor)
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 1 + x2
∴ General solution is :
y(1 + x2) = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or, y(1 + x2) = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Putting x = 0 and y = 0, we get C = 0
∴ Solution is:
y = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
OR
Given equation can be written as
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
⇒ -log|2 - ey| + logc = log|x + 1|
⇒ (2 - ey)(x + 1) = c
When x= 0 , y = 0 ⇒ c =  1
∴ The required solution is (2 - ey) ( x +1) = 1


Q.9. Find the area of the parallelogram whose diagonals are represented by the vectors Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical andClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical.
OR
Find λ and μ if Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

The vector equation for diagonals are Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical andClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Now, Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = √4 + 16 + 16 = √36 = 6
Area of the parallelogram
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 3 sq. units
OR
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = 0
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical= 0
or 3μ + 9λ = 0 ...(i)
or μ – 27 = 0 ...(ii)
or – λ – 9 = 0 ...(iii)
From eqn. (ii) and (iii),
μ = 27
and λ = 9


Q.10. Find the value of λ, so that the lines  Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medicaland Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

are at right angles. Also, find whether the lines are intersecting or not.

Given lines are :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medicaland Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
As lines are perpendicular,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical= 0
⇒ λ =7
So, lines are
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Consider
Δ = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical 
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= -63
Since, as ∆  0 ⇒ lines are not intersecting.

Section - C

Q.11. Show that: Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-MedicalClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical                      ...(i)
By applying property
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical                      ...(ii)
Adding eqn. (i) & (ii)
∴ 2I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
=Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
⇒ 2I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
I = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical


Q.12. Find the area of the region bounded by the parabola y2 = x and the line 2y = x.
OR
Find the area of the region bounded by the parabola y2 = 16x and the line x = 4.

When y2 =x  and 2y = x
Solving we get y2 =2y
⇒ y = 0, 2  and when y = 2, x = 4 and y = 0
⇒ x = 0
So, points of intersection are (0, 0) and (4, 2).
Graphs of parabola y2 = x and 2y = x are as shown in the adjoining figure :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
From the figure, area of the shaded region,
A = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 4/3 sq. units
Parabola y2 = 16x and line x = 4
at x = 4, y2 = 64 ⇒ y = ±8
Hence, the point of intersection (4, 8) and (4, –8)
OR
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Area = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 16/3 x 8
= 128/3 sq. units


Q.13. If Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical are unit vectors such that Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical  and the angle between Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical and 

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical is π/6, then prove that: Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

As given, Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medicalboth Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical and Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical are unit vectors
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Let, Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
then Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical = |λ|
or  |λ| = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 2
∴ λ = ±2
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical

Case-Based/Data Based

Q.14. Bag I contains 1 white, 2 black and 3 red balls; Bag II contains 2 white, 1 black and 1 red balls; Bag III contains 4 white, 3 black and 2 red balls. A bag is chosen at random and two balls are drawn from it with replacement. They happen to be one white and one red.
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
Based on the above information, answer the following questions:
(i) What is the probability that they came from Bag III?
(ii) What is the probability that they will come from Bag I?

Let E1 = Bag I is chosen, E2 = Bag II is chosen, E3 = Bag III is chosen, A = The two balls drawn from the chosen bag are one white and one red.
P(E1) = 1/3
= P(E2) = P(E3),
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
(i) By Bayes’ Theorem, Required probability
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical x Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 64/199
(ii) By Bayes' theorem Required probability
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | Sample Papers for Class 12 Medical and Non-Medical
= 54/199

The document Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 5 | 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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