Please refer to Class 12 Mathematics Sample Paper Term 2 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 Term 2 CBSE Sample Papers for Mathematics in Standard 12.

**Sample Paper Term 2 Class 12 Mathematics With Solutions Set B**

**SECTION β A**

**1. Find β«ππππ₯ /(1+ππππ₯) ^{2}ππ₯**

**Answer :**

**OR**

**Find β«π ππ2π₯ / β9βπππ ^{4}π₯ππ₯**

**Answer :**

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

Degree = 1

Sum = 3

**3. If πΜ and πΜ are unit vectors, then prove that****|πΜ+πΜ|=2πππ ΞΈ/2 where π is the angle between them.****Answer :** (πΜ+πΜ).(πΜ+πΜ)=|πΜ|^{2}+|πΜ|^{2}+2(πΜ.πΜ)

|πΜ+πΜ|^{2}=1+1+2πππ π

=2(1+πππ π)=4πππ ^{2}π/2

β΄|πΜ+πΜ|=2πππ π/2

**4. Find the direction cosines of the following line:****3βx/β1 = 2yβ1/2 = z/4****Answer :** The given line is π₯β3/1=π¦β1/2/1=π§/4

Its direction ratios are <1, 1, 4>

Its direction cosines are γ1/3β2, 1/3β2, 4/3β2γ

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

Then X = 0, 1

P(X=0) = 3/4 Γ 2/3 = 6/12 = 1/2

P(X=1) =1/4 Γ 3/3 + 3/4 Γ 1/3 = 6/12 = 1/2

Probability Distribution Table:

**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?****Answer :** 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))

= 2/52 Γ 3/51 + 24/52 Γ 4/51 = 1/26

**SECTION β B**

**7. Find: **

**Answer :**

**8. Find the general solution of the following differential equation: ****π₯ππ¦/ππ₯=π¦βπ₯π ππ(π¦/π₯)****Answer :** We have the differential equation: ππ¦/ππ₯=π¦π₯βπ ππ(π¦/π₯)

The equation is a homogeneous differential equation.

Putting π¦ = π£π₯ β ππ¦/ππ₯ = π£ + π₯ππ£/ππ₯

The differential equation becomes π£+π₯ππ£/ππ₯=π£βπ πππ£

β ππ£/π πππ£ = βππ₯/π₯ β πππ πππ£ππ£ =βππ₯/π₯

Integrating both sides, we get

πππ|πππ πππ£βπππ‘π£|=βπππ|π₯|+ππππΎ,πΎ>0 (Here, ππππΎ is an arbitrary constant.)

βπππ|(πππ πππ£βπππ‘π£)π₯|=ππππΎ

β|(πππ πππ£βπππ‘π£)π₯|=πΎ

β(πππ πππ£βπππ‘π£)π₯=Β±πΎ

β(πππ πππ¦/π₯βπππ‘π¦/π₯)π₯=πΆ, which is the required general solution.

**OR **

**Find the particular solution of the following differential equation, given that y = 0 when x = π/4:****dy/dx + ycot x = 2/1+sin x****Answer :** The differential equation is a linear differential equation

I F = π^{β«πππ‘π₯ππ₯} = π^{ππππ πππ₯} = π πππ₯

The general solution is given by

**9. If πββ 0,βββ πβ.πββ=πβ.πβ,πβΓπββ=πβΓπβ, then show that πββ=πβ.****Answer :** We have aΜ
.(bΜ
βcΜ
)=0

β(bΜ
βcΜ
)=0ββ or aΜ
β₯(bΜ
βcΜ
)

βbΜ
=cΜ
or aΜ
β₯(bΜ
βcΜ
)

Also, aΜ
Γ(bΜ
βcΜ
)=0ββ

β(bΜ
βcΜ
)=0ββ or aΜ
β₯(bΜ
βcΜ
)

βbΜ
=cΜ
or aΜ
β₯(bΜ
βcΜ
)

aΜ
πππ πππ‘ ππ πππ‘β ππππππππππ’πππ π‘π (bΜ
βcΜ
) πππ ππππππππ π‘π (bΜ
βcΜ
)

Hence, bΜ
=cΜ
.

**10. Find the shortest distance between the following lines:****πβ=(πΜ+πΜβπΜ)+π (2πΜ+πΜ+πΜ)****πβ=(πΜ+πΜ+2πΜ)+π‘(4πΜ+2πΜ+2πΜ)****Answer : **

**OR**

**Find the vector and the cartesian equations of the plane containing the point πΜ+2πΜβπΜ and parallel to the lines****πβ=(πΜ+2πΜ+2πΜ)+π (2πΜβ3πΜ+2πΜ)=0 and πβ=(3πΜ+πΜβ2πΜ)+π‘(πΜβ3πΜ+πΜ)=0****Answer :** Since, the plane is parallel to the given lines, the cross product of the vectors 2πΜβ3πΜ+2πΜ and πΜβ3πΜ+πΜ will be a normal to the plane

The vector equation of the plane is πβ.(3πΜβ3πΜ)=(πΜ+2πΜβπΜ).(3πΜβ3π)Μ

or, πβ.(πΜβπΜ)=2

and the cartesian equation of the plane is x β z β 2 = 0

**SECTION β C**

**11. Evaluate: β«|π₯ ^{3}β3π₯^{2}+2π₯|ππ₯.**

**Answer :**The given definite integral = β«

_{β1}

^{2}|π₯(π₯β1)(π₯β2)|ππ₯

**12. Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y 2 = x and the x-axis.****Answer :**

**OR**

**Using integration, find the area of the region {(π₯,π¦):0β€π¦β€β3π₯,π₯ ^{2}+π¦^{2}β€4}**

**Answer :**Solving π¦=β3π₯ πππ π₯

^{2}+π¦

^{2}=4 , we get the points of intersection as (1, β3) and (-1, ββ3)

**13. Find the foot of the perpendicular from the point (1, 2, 0) upon the plane****x β 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.****Answer :** The equation of the line perpendicular to the plane and passing through the point (1, 2, 0) is

π₯β1/1=π¦β2/β3=π§/2

The coordinates of the foot of the perpendicular are (π+1,β3π+2,2π) for some π

These coordinates will satisfy the equation of the plane. Hence, we have π+1β3(β3π+2)+2(2π)=9 βπ=1

The foot of the perpendicular is (2, -1, 2).

Hence, the required distance = β(1β2)^{2}+(2+1)^{2}+(0β2)^{2}=β14 π’πππ‘π

**14. CASE-BASED/DATA-BASED**

Fig 3 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The companyβs statistics show that an accident-prone person will have an accident at sometime within a fixed one-year period with probability 0.6, whereas this probability is 0.2 for a person who is not accident prone. The company knows that 20 percent of the population is accident prone.

**Based on the given information, answer the following questions.**

**(i) what is the probability that a new policyholder will have an accident within a year of purchasing a policy?****Answer :** Let E_{1} = The policy holder is accident prone.

E_{2} = The policy holder is not accident prone.

E = The new policy holder has an accident within a year of purchasing a policy.

(i) P(E)= P(E_{1})Γ P(EβE_{1}) + P(E_{2})Γ P(EβE_{2})

= 20/100 Γ 6/10 + 80/100 Γ 2/10 = 7/25

**(ii) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?****Answer : **