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

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

Put, 1 – tan x = y
So that, –sec2x dx = dy
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
OR
Let Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Since,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
= – [0 – (–1)] + (2 – 0)
= –1 + 2
= 1


Q.2. Show that the function y = ax + 2a2 is a solution of the differential equation Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

y = ax + 2a2 ⇒ (dy/dx) = a

From L.H.S. of differential equation
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
= 2(a)2 + x(a) – (ax + 2a2) = 0
= R.H.S. 


Q.3. If Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical are mutually perpendicular unit vectors, then find value of Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Given Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical are mutually perpendicular unit vectors, i.e.,

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical ...(i)
and
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical ...(ii)
Now,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
[ ∵ Dot product is distributive over addition]
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical = 4(1) + 1 + 1 = 6
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical


Q.4. A line passes through the point with position vector Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical and makes angles 60°, 120°, and 45° with x, y and z-axis respectively. Find the equation of the line in the Cartesian form. 

D-Cosines of line are 1/2, -(1/2), 1/√2

Equation of line is : Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or 2x – 4 = – 2y – 6 = √2 ( z -4).


Q.5. Find the probability distribution of X, the number of heads in a simultaneous toss of two coins.

Let X be the number of heads.

Possible v alues 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)- 1 | Sample Papers for Class 12 Medical and Non-Medical


Q.6. The probability that it will rain on any particular day is 50%. Find the probability that it rains only on first 4 days of the week.

P(rain on any particular day) = 50%

= 50/100 = 1/2
P(rain on first four days of week)
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical.

Section - B

Q.7. Prove that: Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical hence evaluate Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Let Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Put a - x = t ⇒ dx = dt
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Now,
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

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

Put cos x = t ⇒ -sin x dx = dt
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
I is an even find in Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical


Q.8. Find the general solution of the following differential equation: x dy – (y + 2x2)dx = 0
OR
Solve the following differential equation. Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

The given differential equation can be written as Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Comparing above with (dy/dx) +Py = Q
Here P = -(1/x) and Q = 2x
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Hence, the required solutions is:
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
⇒ (y/x) = 2x + c
⇒ y = 2x2 + cx
OR
Given differential equation is Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
On integrating both sides, we get
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
On putting 1 + y2 = t and 1 + x2 = u2
or 2y dy = dt and 2x dx = 2u du
⇒ y dy  = dt/2 and x dx = u du
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical [put 1 + y2 = t]
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-MedicalClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-MedicalC
which is the required solution.


Q.9. Find the area of a rectangle having vertices A, B, C and D with position vectors Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical respectively.
The two vectors Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical represent the two sides AB and AC, respectively of DABC. Find the length of the median through A.

The position vectors of vertices A, B, C and D of rectangle ABCD are given as:

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

The adjacent sides Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical of the given rectangle are given as :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Now, it is known that the area of parallelogram whose adjacent sides are Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Therefore, the area of the given rectangle is Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
OR
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Now ABEC represent a parallelogram with AE as the diagonal.
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Now, Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical


Q.10. Write the sum of intercepts cut off by the plane Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical on the three axes.

Given, equation of plane is Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Put Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical we get Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
⇒ 2x + y - z = 5
or Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
On comparing it with standard equation of plane in intercept form Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
a = 5/2, b = 5 and c = -5
Now, required sum of intercepts cut off by the plane on the three axis = a + b + c
= (5/2) + 5 - 5 = 5/2 units.

Section - C

Q.11. Find : Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Put, [sinx = t or cosxdx = dt]
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical 
Put , 1,
1 = 4C , i.e., C = 1/4 Put 0 ,
0 = A + B + C, which gives A = 3/4
Therefore the required integral
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical + c
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-MedicalClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical


Q.12. Find the area bounded by lines x = 2y+  3, y – 1 = 0 and y + 1 = 0.
Or
Find the region bounded by the curve  y2 =4x, y-axis and the line y = 3.

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | 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)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
= [ 1 + 3 - 1 + 3]
= 6 sq. units
OR
The area bounded by the curve, y2 = 4x,  y-axis, and y = 3 is represented as :
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

Area of OAB = Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
= (1/12) x 27
= 9/4 sq. units.


Q.13. Find the equation of plane passing through the points A(3, 2, 1), B(4, 2, –2) and C(6, 5, –1) and hence find the value of λ for which A(3, 2, 1), B(4, 2, –2), C(6, 5, –1) and D(λ, 5, 5) are coplanar.

A plane which passes through A(3, 2, 1), B(4, 2, –2) and C(6, 5, –1) is

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
⇒ (x - 3)(0 + 9)-(y - 2)(-2 + 9) + (z - 1)(3 - 0) = 0
⇒ 9x - 7(y - 2) + 3(z - 1) = 0
⇒ 9(x - 3) 9x - 7y + 3z = 16
Thus, plane passing through point A, B and C is 9x – 7y + 3z = 16

Now, given A, B, C and D (λ, 5, 5) are coplanar. So, D  lies on the plane passing through A, B and C
∴ 9λ -  7(5) + 3(5)=16
⇒ 9λ = 36
⇒ λ = 4

Case-Based/Data Based 

Q.14. Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. Let E1 and E2 be the events that selecting a student with 100% attendance and selecting a student who is not regular, respectively.
Based on the above information, answer the following questions:
(i) Find the values of PClass 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
(ii) What is the probability that the student has 100% attendance.

Let E1 : Selecting a student with 100% attendance 

E2 : Selecting a student who is not regular
A  : selected student attains A grade.

Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
(i) Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
(ii) Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical
Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | Sample Papers for Class 12 Medical and Non-Medical

The document Class 12 Mathematics: CBSE Sample Question Paper- Term II (2021-22)- 1 | 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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