CLASS 12 CBSE TERM2 MATHS SPECIMEN 2022

BOARD -

CLASS -

SUBJECT -

CBSE

12th

MATHEMATICS

Paper Pattern for Written Term-II

TIME -

MARKS -

2 hour

Visit CBSE OFFICIAL PAGE for Regulations and Syllabus of Class 12th CBSE

Solved Specimen Paper Semester-2 2022

Question 1 [MATHEMATICS Specimen Paper Semester-2 2022]

Find ∫
logx / (1+logx)²
dx

Solution

Find ∫
sin2x / √(9-cos⁴x)
dx

Solution

Question 2 [MATHEMATICS Specimen Paper Semester-2 2022]

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

Solution

Question 3 [MATHEMATICS Specimen Paper Semester-2 2022]

If 𝑎̂ and 𝑏̂ are unit vectors, then prove that |𝑎̂ + 𝑏̂| = 2𝑐𝑜𝑠
𝜃 / 2
, where 𝜃 is the angle between them.

Solution

Question 4 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the direction cosines of the following line:
3-x / -1
=
2y-1 / 2
=
z / 4

Solution

Question 5 [MATHEMATICS Specimen Paper Semester-2 2022]

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-byone without replacement.

Solution

Question 6 [MATHEMATICS Specimen Paper Semester-2 2022]

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?

Solution

Question 7 [MATHEMATICS Specimen Paper Semester-2 2022]

Find: ∫
x+1 / (x²+1)x
dx

Solution

Question 8 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the general solution of the following differential equation: x
dy / dx
= y-x sinx (
y / x
)

Solution

Find the particular solution of the following differential equation,given that y = 0 when x= 𝜋/4 :
dy / dx
+ ycotx =
2 / 1+sinx

Solution

Question 9 [MATHEMATICS Specimen Paper Semester-2 2022]

If 𝑎⃗ ≠ 0, ⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗

Solution

Question 10 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the shortest distance between the following lines:
𝑟⃗ = (𝑖̂+ 𝑗̂− 𝑘̂) + 𝑠(2𝑖̂+ 𝑗̂+ 𝑘̂)
𝑟⃗ = (𝑖̂+ 𝑗̂+ 2𝑘̂) + 𝑡(4𝑖̂+ 2𝑗̂+ 2𝑘̂)

Solution

Find the vector and the cartesian equations of the plane containing the point 𝑖̂+ 2𝑗̂− 𝑘̂ and parallel to the lines 𝑟⃗ = (𝑖̂+ 2𝑗̂+ 2𝑘̂) + 𝑠(2𝑖̂− 3𝑗̂+ 2𝑘̂) and 𝑟⃗ = (3𝑖̂+ 𝑗̂− 2𝑘̂) + 𝑡(𝑖̂− 3𝑗̂+ 𝑘̂)

Solution

Question 11 [MATHEMATICS Specimen Paper Semester-2 2022]

Evaluate: ∫ _-1 ² |𝑥³-3x² + 2| dx

Solution

Question 12 [MATHEMATICS Specimen Paper Semester-2 2022]

Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y² = x and the x-axis.

Solution

Using integration, find the area of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ √3𝑥, 𝑥² + 𝑦 ² ≤ 4}

Solution

Question 13 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.

Solution

Question 14 [MATHEMATICS Specimen Paper Semester-2 2022]

An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company’s 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?
(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?

Solution

CLASS 12 CBSE TERM2 MATHS SPECIMEN 2022

Paper Pattern for Written Term-II

Visit CBSE OFFICIAL PAGE for Regulations and Syllabus of Class 12th CBSE

Solved Specimen Paper Semester-2 2022

Question 1 [MATHEMATICS Specimen Paper Semester-2 2022]

Find ∫ logx / (1+logx)2 dx

Solution

Find ∫ sin2x / √(9-cos4x) dx

Solution

Question 2 [MATHEMATICS Specimen Paper Semester-2 2022]

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

Solution

Question 3 [MATHEMATICS Specimen Paper Semester-2 2022]

If 𝑎̂ and 𝑏̂ are unit vectors, then prove that |𝑎̂ + 𝑏̂| = 2𝑐𝑜𝑠 𝜃 / 2 , where 𝜃 is the angle between them.

Solution

Question 4 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the direction cosines of the following line: 3-x / -1 = 2y-1 / 2 = z / 4

Solution

Question 5 [MATHEMATICS Specimen Paper Semester-2 2022]

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-byone without replacement.

Solution

Question 6 [MATHEMATICS Specimen Paper Semester-2 2022]

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?

Solution

Question 7 [MATHEMATICS Specimen Paper Semester-2 2022]

Find: ∫ x+1 / (x2+1)x dx

Solution

Question 8 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the general solution of the following differential equation: x dy / dx = y-x sinx ( y / x )

Solution

Find the particular solution of the following differential equation,given that y = 0 when x= 𝜋/4 : dy / dx + ycotx = 2 / 1+sinx

Solution

Question 9 [MATHEMATICS Specimen Paper Semester-2 2022]

If 𝑎⃗ ≠ 0, ⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗

Solution

Question 10 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the shortest distance between the following lines: 𝑟⃗ = (𝑖̂+ 𝑗̂− 𝑘̂) + 𝑠(2𝑖̂+ 𝑗̂+ 𝑘̂) 𝑟⃗ = (𝑖̂+ 𝑗̂+ 2𝑘̂) + 𝑡(4𝑖̂+ 2𝑗̂+ 2𝑘̂)

Solution

Find the vector and the cartesian equations of the plane containing the point 𝑖̂+ 2𝑗̂− 𝑘̂ and parallel to the lines 𝑟⃗ = (𝑖̂+ 2𝑗̂+ 2𝑘̂) + 𝑠(2𝑖̂− 3𝑗̂+ 2𝑘̂) and 𝑟⃗ = (3𝑖̂+ 𝑗̂− 2𝑘̂) + 𝑡(𝑖̂− 3𝑗̂+ 𝑘̂)

Solution

Question 11 [MATHEMATICS Specimen Paper Semester-2 2022]

Evaluate: ∫ -1 2 |𝑥3-3x2 + 2| dx

Solution

Question 12 [MATHEMATICS Specimen Paper Semester-2 2022]

Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y2 = x and the x-axis.

Solution

Using integration, find the area of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ √3𝑥, 𝑥2 + 𝑦 2 ≤ 4}

Solution

Question 13 [MATHEMATICS Specimen Paper Semester-2 2022]

Find the foot of the perpendicular from the point (1, 2, 0) upon the plane x – 3y + 2z = 9. Hence, find the distance of the point (1, 2, 0) from the given plane.

Solution

Question 14 [MATHEMATICS Specimen Paper Semester-2 2022]

Solution

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CONTACT

Find ∫
logx / (1+logx)²
dx

Find ∫
sin2x / √(9-cos⁴x)
dx

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

If 𝑎̂ and 𝑏̂ are unit vectors, then prove that |𝑎̂ + 𝑏̂| = 2𝑐𝑜𝑠
𝜃 / 2
, where 𝜃 is the angle between them.

Find the direction cosines of the following line:
3-x / -1
=
2y-1 / 2
=
z / 4

Find: ∫
x+1 / (x²+1)x
dx

Find the general solution of the following differential equation: x
dy / dx
= y-x sinx (
y / x
)

Find the particular solution of the following differential equation,given that y = 0 when x= 𝜋/4 :
dy / dx
+ ycotx =
2 / 1+sinx

Find the shortest distance between the following lines:
𝑟⃗ = (𝑖̂+ 𝑗̂− 𝑘̂) + 𝑠(2𝑖̂+ 𝑗̂+ 𝑘̂)
𝑟⃗ = (𝑖̂+ 𝑗̂+ 2𝑘̂) + 𝑡(4𝑖̂+ 2𝑗̂+ 2𝑘̂)

Evaluate: ∫ _-1 ² |𝑥³-3x² + 2| dx

Using integration, find the area of the region in the first quadrant enclosed by the line x + y = 2, the parabola y² = x and the x-axis.

Using integration, find the area of the region {(𝑥, 𝑦): 0 ≤ 𝑦 ≤ √3𝑥, 𝑥² + 𝑦 ² ≤ 4}