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CBSE Sample Papers for Class 12 Maths Paper 6 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 6.

CBSE Sample Papers for Class 12 Maths Paper 6

Students who are going to appear for CBSE Class 12 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 6 of Solved CBSE Sample Paper for Class 12 Maths is given below with free PDF download solutions.

Time: 3 Hours
Maximum Marks: 100

General Instructions:

• All questions are compulsory.
• Questions 1-4 in section A are very short answer type questions carrying 1 mark each.
• Questions 5-12 in section B are short answer type questions carrying 2 marks each.
• Questions 13-23 in section C are long answer I type questions carrying 4 marks each.
• Questions 24-29 in section D are long answer II type questions carrying 6 marks each.

SECTION A

Question 1.
If A is a matrix of order 2 x 3 and B is of order 3 x 4, what is the order of (AB)’?

Question 2.
If \(\vec { P }\) is a unit vector and \(\left( \vec { x } -\vec { P } \right) \cdot \left( \vec { x } +\vec { P } \right) =80\), then find the value of \(\left| \vec { x } \right|\).

Question 3.
Evaluate \(\int { \sqrt { \frac { x }{ 1-{ x }^{ 3 } } } } dx\)

Question 4.
Find the point at which tangent to the curve y = x² makes an angle of 45° with x-axis.

SECTION B

Question 5.
Express the matrix \(A=\begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix}\) as the sum of symmetric and skew symmetric matrix.

Question 6.
If x^y = y^x, find \(\frac { dy }{ dx }\)

Question 7.
Verify Rolle’s theorem for f(x) = sin 2x in [0, \(\frac { \pi }{ 2 }\)] and find the value of ]0, \(\frac { \pi }{ 2 }\)[

Question 8.
Discuss continuity of the function at x = 0

Question 9.
Find the coordinate of point of intersection of lines

Question 10.
If P(A) = \(\frac { 1 }{ 4 }\), P(A | B) = \(\frac { 1 }{ 2 }\), P(B | A) = \(\frac { 2 }{ 3 }\) then find P(B).

Question 11.
If 3 tan^-1x + cot^-1x = π then find the value of x.

Question 12.

SECTION C

Question 13.

Question 14.

Question 15.

Question 16.

Question 17.

Question 18.
Solve the differential equation (1 + y²) dx = (tan^-1 y – x) dy ; y(0) = 0

Question 19.
The scalar product of the vector \(\hat { i } +\hat { j } +\hat { k }\) with a unit vector along the sum of vectors \(2\hat { i } +4\hat { j } -5\hat { k }\) and \(\lambda \hat { i } +2\hat { j } +3\hat { k }\) is equal to one. Find the value of λ.

Question 20.

Question 21.
A and B throw a die alternately till one of them gets a 5 and wins the game. F ind their respective probabilities of winning if A starts the game. Why gambling is not a good way of earning money?

Question 22.
In a bolt factory, machine A, B and C manufacture respectively 25%, 35% and 40% of the bolts. of their output, 5%, 4% and 2% are respectively defective bolts. A bolt is drawn at random from the total production and is found to be defective. Find the probability that it is manufactured by machine B.

Question 23.
A binary operation * on the set {0, 1, 2, 3, 4, 5} is defined as

Show that zero is the identity element for. this operation and each non-zero element ‘a’ of the set is invertible with 6 – a being the inverse of a.

SECTION D

Question 24.

Question 25.
Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
OR
An open box with square base is to be made out of a given quantity of sheet of area a² sq.units. Show that the maximum volume of the box is \(\frac { { a }^{ 3 } }{ 6\surd 3 }\) cubic units.

Question 26.
Using integration find the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.
OR
Find the area of the region in the first quadrant enclosed by the y-axis, the line y = x and the circle x² + y² = 32 using integration.

Question 27.
A dealer wishes to purchase a number of fans and sewing machines. He has only ₹ 5760 to invest and has space for at most 20 items. A fan cost him ₹ 360 and a sewing machine ₹ 240. On selling he get a profit of ₹ 22 on a fan and ₹ 18 on a sewing machine. Assuming that he can sell all the items that he store, how should he invest his money in order to maximize profit? Formulate this as L.P.P. and solve it graphically.

Question 28.
Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38.
OR
Find the equation of the plane passing through the points A(3, -1, 2), B (5, 2, 4) and C(-1, -1, 6). Also find the distance of the point P(6, 5, 9) from the plane.

Question 29.
Show that the differential equation x dy – y dx = \(\sqrt { { x }^{ 2 }+{ y }^{ 2 } }\) dx is homogeneous and hence solve it.

Solutions

Solution 1.
A = [a_ij]_2×3
B = [b_ij]_3×4
Order of AB = 2 x 4
Order of (AB)’ = 4 x 2

Solution 2.

Solution 3.

Solution 4.
Let the point is (x, y)
y = x²

Solution 5.

Solution 6.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.
y = sinpt
x = sint

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.
Let E₁ is the event the bolt is manufactured by machine A
E₂ is the event the bolt is manufactured by machine B
E₃ is the event the bolt is manufactured by machine C
A is the event bolt drawn is defective

Solution 23.

Solution 24.

Solution 25.

Solution 26.
x + 2y = 2 ……. (1)
y – x = 1 …… (2)
2x + y = 7 ……. (3)
From (1) and (2), (0, 1)
From (2) and (3), (2, 3)
From (1) and (3), (4, -1)

Solution 27.
Let dealer purchases x fans and y sewing machines.
Objective function is maximize profit Z = 22x + 18y

Solution 28.

Solution 29.

We hope the CBSE Sample Papers for Class 12 Maths Paper 6 help you. If you have any query regarding CBSE Sample Papers for Class 12 Maths Paper 6, drop a comment below and we will get back to you at the earliest.

CBSE Sample Papers for Class 12 Maths Paper 5 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 5.

CBSE Sample Papers for Class 12 Maths Paper 5

Students who are going to appear for CBSE Class 12 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 5 of Solved CBSE Sample Paper for Class 12 Maths is given below with free PDF download solutions.

Time: 3 Hours
Maximum Marks: 100

General Instructions:

• All questions are compulsory.
• Questions 1-4 in section A are very short answer type questions carrying 1 mark each.
• Questions 5-12 in section B are short answer type questions carrying 2 marks each.
• Questions 13-23 in section C are long answer I type questions carrying 4 marks each.
• Questions 24-29 in section D are long answer II type questions carrying 6 marks each.

SECTION A

Question 1.
What is the principal value of \({ tan }^{ -1 }\left( tan\frac { 2\pi }{ 3 } \right)\)

Question 2.
If A is a square matrix of order 3 x 3 and |3A| = k |A|, then find the value of k.

Question 3.
What is the distance of the point (p, q, r) from the x-axis?

Question 4.
Let f: R → R be defined by f(x) = 3x² – 5 and g : R → R be defined by g(x) = \(\frac { x }{ { x }^{ 2 }+1 }\). Find fog.

SECTION B

Question 5.
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b): |a – b| is even} is an equivalence relation.

Question 6.
Let l_i, m_i, n_i For i = 1, 2, 3 be the direction cosines of three mutually perpendicular vectors in space. Show that AA’ = I₃.
Where

Question 7.
Form the differential equation corresponding to the function y² = a(b – x) (b + x) by eliminating a and b.

Question 8.
If e^y (x + 1) = 1, prove that \(\frac { dy }{ dx }\) = -e^y

Question 9.
Find the Cartesian and vector equations of line which passes through the point (-2, 4, -5) and parallel to the line given by
\(\frac { x+3 }{ 3 } =\frac { 4-y }{ -5 } =\frac { 8-z }{ -6 }\)

Question 10.
Solve the following L.R.R graphically
Maximise Z = 3x + 4y
Subject to constraints x + y ≤ 4, x ≥ 0, y ≥ 0

Question 11.
If P (E) = 0.35, P(F) = 0.45, P(E ∪ F) = 0.65. Find (i) P (E|F) (ii) P(E ∩ \(\bar { F }\)).

Question 12.
Find the points on the curve y = x³ at which the slope of the tangent is equal to y-coordinate of the point.

SECTION C

Question 13.
Find the area of the region bounded by the y-axis, y = cos x and y = sin x, 0 ≤ x ≤ \(\frac { \pi }{ 4 }\).

Question 14.
Evaluate \(\int _{ 0 }^{ 4 }{ \left( 2{ x }^{ 2 }+5x \right) } dx\) dx as a limit of sum.

Question 15.

Question 16.

Question 17.
Use differentials to find the approximate value of \(\frac { 1 }{ { \left( 2.002 \right) }^{ 2 } }\)
OR
Find the intervals for which the function f(x) = sin x – cos x is increasing or decreasing in [0, 2π].

Question 18.
Solve the differential equation y dx + (x – y²) dy = 0
OR
Solve the differential equation \(\frac { dy }{ dx } =\frac { x+y+1 }{ x+y }\)

Question 19.
A bag contains (2n + 1) coins. It is known that n of these coins have a head on both its sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is \(\frac { 31 }{ 42 }\). Find the value of n.

Question 20.
A plane meets the coordinate axis in A, B and C such that the centroid of ΔABC is the point (α, β, γ). Show that the equation of the plane is \(\frac { x }{ \alpha } +\frac { y }{ \beta } +\frac { z }{ \gamma } =3\).

Question 21.
Two cards are drawn simultaneously without replacement from a well shuffled pack of 52 cards. Find the mean and variance of number of red cards.

Question 22.

Question 23.
If a 20 year old girl drives her car at 25 km/h, she has to spend ₹ 4/km on petrol. If she drives her car at 40 km/h, the petrol cost increases to ₹ 5/km. She has ₹ 200 to spend on petrol and wishes to find the maximum distance she can travel within one hour. Express the above problem as L.P.P. Write any one value reflected in the problem.

SECTION D

Question 24.

Question 25.

OR
Let N be the set of all natural numbers and let R be a relation in N x N defined by (a, b) R (c, d) ⇒ ad = bc
For all (a, b), (c, d) ∈ N x N, show that R is an equivalence relation on N x N.

Question 26.

Question 27.
Define skew lines. Using only vector approach, find the shortest distance between the following two skew lines

Question 28.

Question 29.
Show that the semivertical angle of a right circular cone of given surface area and maximum volume is \({ sin }^{ -1 }\left( \frac { 1 }{ 3 } \right)\)

Solutions

Solution 1.

Solution 2.
A is a square matrix of order 3 x 3
|3A| = k|A| ⇒ |3A| = 3³ |A| = 27 |A| = k |A|
So k = 27.

Solution 3.

Solution 4.

Solution 5.
A= {1,2, 3, 4, 5}
R = {(a, b): |a- b| is even}
Reflexive: (a, a): |a – a| = 0 is even
So (a, a) ∈ R, ∀ a ∈A
So R is reflexive.
Symmetric: (a, b): |a – b| is even, ∀ (a, b) ∈ R
= |b – a| is even
= (b, a) ∈ R
So R is symmetric.
Transitive: Let (a, b), (b, c) ∈ R
(a, b): |a – b| is even and (b, c): |b – c| is even.
a – b = ± 2m, b – c = ± 2n, ∀ m, n ∈ N
a – c = ± 2m ± 2n = ± 2 (m + n)
|a – c| is even.
So (a, c) ∈ R.
So R is transitive.
Since R is reflexive, symmetric and transitive, so R is an equivalence relation.

Solution 6.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.

Solution 16.

Solution 17.

Solution 18.

Solution 19.
Let E₁ is the event: Coin having head on both sides
E₂ is the event: Coin is fair.
A is the event: head comes upon tossing a coin

Solution 20.

Solution 21.
Let Random variable x denote the number of red cards drawn.
Possible values of x are 0, 1, 2
Two cards are drawn simultaneously without replacement.
Total cards = 52, Red cards = 26
Remaining cards = 26

Solution 22.

Solution 23.
Let distance covered with speed of 25 km/h is x km, distance covered with speed of 40 km/h is y km.
Objective function is maximize distance = Z = x + y,
Subject to constraints are
4x + 5y ≤ 200
x ≥ 0, y ≥ 0
Value: At slow speed consumption of petrol is low, so maintain normal speed and save petrol.

Solution 24.

Solution 25.

Solution 26.

Solution 27.
Skew lines: The lines which are neither intersecting nor parallel are called skew lines.

Solution 28.

Solution 29.

We hope the CBSE Sample Papers for Class 12 Maths Paper 5 help you. If you have any query regarding CBSE Sample Papers for Class 12 Maths Paper 5, drop a comment below and we will get back to you at the earliest.

CBSE Sample Papers for Class 12 Maths Paper 4 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 4.

CBSE Sample Papers for Class 12 Maths Paper 4

Students who are going to appear for CBSE Class 12 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 4 of Solved CBSE Sample Paper for Class 12 Maths is given below with free PDF download solutions.

Time: 3 Hours
Maximum Marks: 100

General Instructions:

• All questions are compulsory.
• Questions 1-4 in section A are very short answer type questions carrying 1 mark each.
• Questions 5-12 in section B are short answer type questions carrying 2 marks each.
• Questions 13-23 in section C are long answer I type questions carrying 4 marks each.
• Questions 24-29 in section D are long answer II type questions carrying 6 marks each.

SECTION A

Question 1.
Find the principal value of \({ cos }^{ -1 }\left( cos\frac { 7\pi }{ 6 } \right)\)

Question 2.
A and B are square matrices of order 3 each |A| = 2 and |B| = 3, find |3AB|.

Question 3.
If \(\vec { a } =2\hat { i } -\hat { j } +3\hat { k }\) and \(\vec { b } =6\hat { i } +\lambda \hat { j } +9\hat { k }\) and \(\vec { a } ||\vec { b }\), find the value of X.

Question 4.
If f(x) = x + 7 and g (x) = x – 7, find fog (7).

SECTION B

Question 5.
Differentiate cos^-1(e^x) with respect to e^x.

Question 6.
Find the sum of the order and the degree of the differential equation.

Question 7.

Question 8.
How many equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) are there in all?

Question 9.
Find the direction cosines of the normal to the plane 2x + 3y – z = 5 and also find the distance from the origin.

Question 10.
A shopkeeper sells only tables and chairs. He has only ₹ 6000 to invest and has space for atmost 20 items. A table costs him ₹ 400 and a chair ₹ 250. He can sell a table at a profit of ₹ 25 and a chair at a profit of ₹ 40. Supposing that he sell all his stock, formulate the problem as L.F.P.

Question 11.
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which its area increases when side is 10 cm long.

Question 12.
A couple has 2 children. Find the probability that both are boys if it is known that
(i) atleast one of them is a boy
(ii) the older child is a boy.

SECTION C

Question 13.

Question 14.
Evaluate \(\int _{ 0 }^{ 3 }{ \left( { x }^{ 2 }-5 \right) } dx\)

Question 15.
Find the area of the region bounded by the curve \(y=\sqrt { 1-{ x }^{ 2 } }\) and the line y = x bounded with positive x-axis.

Question 16.
Determine for what values of x the function f(x) = x³ + \(\frac { 1 }{ { x }^{ 3 } }\), x ≠ 0 is strictly increasing or strictly decreasing.
OR
Find the point on the curve y = x³ – 11x + 5 at which the tangent is y = x – 11.

Question 17.
It is given that for the function f(x) = x³ – 6x² + ax + b Rolle’s theorem verified in [1, 3] with c = 2 + \(\frac { 1 }{ \surd 3 }\). Find the value of a and b.

Question 18.
If A + B + C = π then find the value of

OR
Using properties of determinant prove that

Question 19.
Find the Cartesian as well as the vector equation of the plane passing through the intersection of the planes \(\vec { r } \cdot \left( 2\hat { i } +6\hat { j } \right) +12=0\) and \(\vec { r } \cdot \left( 3\hat { i } -\hat { j } +4\hat { k } \right) =0\) which are at unit distance from the origin.

Question 20.

Question 21.
For three persons A, B, C the chances of being selected as manager of a firm are in the ratio 4 : 1 : 2 respectively. The respective probabilities for them to introduce new changes in market strategy are 0.3, 0.8 and 0.5 respectively. If the changes does take place find the probability that it was due to appointment of B or C. Write the qualities that makes a person successful manager.

Question 22.
Using graphical method solve the following L.P.P.
Z = 4x + 8y
Subject to constraints
2x + y ≤ 30
x + 2y ≤ 24
x ≥ 3, y ≤ 9, y ≥ 0

Question 23.
Suppose that 80% of the people are right handed. What is the probability that at most 6 of at random samples of 10 people are right-handed?

SECTION D

Question 24.

Question 25.
Does the following trigonometric equation have any solution? If yes, obtain the solutions \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) =-{ tan }^{ -1 }7\)
OR
Determine whether the operation * define on Q is binary operation or not. a * b = ab + 1. If yes, check the commutative and associative properties. Also check the existence of identity element and the inverse of all elements in Q.

Question 26.

Question 27.
Find the shortest distance between the line x – y + 1 = 0 and the curve y² = x.

Question 28.
Find the distance of the point (-2, 3, -4) from the line \(\frac { x+2 }{ 3 } =\frac { 2y+3 }{ 4 } =\frac { 3z+4 }{ 5 }\) measured parallel to the plane 4x + 12y – 3z + 1 = 0.

Question 29.
If \(A=\left( \begin{matrix} 2 & 3 & 7 \\ 3 & -2 & -1 \\ 1 & 1 & 2 \end{matrix} \right)\) , find A^-1.
Using A^-1 solve the following system of equations
2x + 3y + 7z = 12
3x – 2y – z = 0
x + y + 2z = 4

Solutions

Solution 1.

Solution 2.
|3AB| = 3³ |A| |B| = 27 (2) (3) = 162

Solution 3.

Solution 4.
f(x) = x + 7, g(x) = x – 7
(fog) (x) = f(g(x)) = f(x – 7) = x – 7 + 7 = x
(fog) (7) = 7.

Solution 5.
Let y = cos^-1 (e^x) and t = e^x

Solution 6.

Solution 7.

Solution 8.
Possible equivalence relations are {(1, 2), (2, 1), (1, 1), (2, 2), (3, 3)} and {(1, 1), (2, 2), (3, 3), (1, 2),(2, 1), (1, 3), (3, 1), (2, 3), (3, 2)}
There are two possible equivalence relation.

Solution 9.

Solution 10.
Let shopkeeper sales x tables and y chairs.
Objective function is Maximize Profit Z = 25x + 40y
Subject to constraints are
400 x + 250 y ≤ 6000 (investment constraint)
x + y ≤ 20 (space constraint)
x, y ≥ 0 (Non-negative constraint)

Solution 11.

Solution 12.
As per order of birth two children may be (B₁B₂, G₁G₂, B₁G₂, G₁B₂}
Let E is the event both are boys ⇒ (B₁B₂)
E₁ is the event atleast one of them is a boy ⇒ (B₁B₂, B₁G₂, G₁B₂)
E₂ is the event the older child is a boy ⇒ (B₁B₂, B₁G₂)

Solution 13.

Solution 14.

Solution 15.

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Solution 23.

Solution 24.

Solution 25.

Solution 26.

Solution 27.

Solution 28.

Solution 29.

We hope the CBSE Sample Papers for Class 12 Maths Paper 4 help you. If you have any query regarding CBSE Sample Papers for Class 12 Maths Paper 4, drop a comment below and we will get back to you at the earliest.

CBSE Sample Papers for Class 12 Maths Paper 3 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 3.

CBSE Sample Papers for Class 12 Maths Paper 3

Students who are going to appear for CBSE Class 12 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 3 of Solved CBSE Sample Paper for Class 12 Maths is given below with free PDF download solutions.

Time: 3 Hours
Maximum Marks: 100

General Instructions:

• All questions are compulsory.
• Questions 1-4 in section A are very short answer type questions carrying 1 mark each.
• Questions 5-12 in section B are short answer type questions carrying 2 marks each.
• Questions 13-23 in section C are long answer I type questions carrying 4 marks each.
• Questions 24-29 in section D are long answer II type questions carrying 6 marks each.

SECTION A

Question 1.
State the reason why the Relation R = {(a, b) : a ≤ b²} on the set R of real numbers is not reflexive.

Question 2.
If A is a square matrix of order 3 and |2A| = k |A|. then find the value of k.

Question 3.
If \(\vec { a }\) and \(\vec { b }\) are two non-zero vectors such that \(\left| \vec { a } \times \vec { b } \right| =\vec { a } \cdot \vec { b }\), then find the angle between \(\vec { a }\) and \(\vec { b }\).

Question 4.
If * is a binary operation on the set R of real numbers defined by a * b = a + b – 2, then find the identity element for the binary operation *.

SECTION B

Question 5.
Simplify \({ cot }^{ -1 }\frac { 1 }{ \sqrt { { x }^{ 2 }-1 } }\) for x < -1.

Question 6.
If A and B are symmetric matrix then prove that AB – BA is skew symmetric.

Question 7.

Question 8.
If x changes from 4 to 4.01, then find the approximate change in log_e x.

Question 9.
Find \(\int { \left( \frac { 1-x }{ 1+{ x }^{ 2 } } \right) ^{ 2 } } { e }^{ x }dx\)

Question 10.
Obtain the differential equation of the family of circles passing through the points (a, 0) and (-a, 0).

Question 11.

Question 12.

SECTION C

Question 13.
If \(A=\begin{pmatrix} 1 & -2 \\ 2 & 1 \end{pmatrix}\), then using A^-1 solve the following system of equations: x – 2y = -1, 2x + y = 2.

Question 14.

Question 15.
If x = a sin pt, y = b cos pt, then show that \(\left( { a }^{ 2 }-{ x }^{ 2 } \right) y\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ b }^{ 2 }=0\)

Question 16.
Find the equation of the normal to the curve 2y = x², which passes through the point (2, 1).
OR
Separate the interval [0, \(\frac { \pi }{ 2 }\)] into subintervals in which the function f(x) = sin⁴x + cos⁴x is strictly increasing or strictly decreasing.

Question 17.
A magazine seller has 500 subscribers and collects annual subscription charges of ₹ 300 per subscriber. She proposes to increase the annual subscription charges and it is believed that for every increase of ₹ 1, one subscriber will discontinue. What increase will bring maximum income to her? Make appropriate assumptions in order to apply derivatives to reach the solution. Write one important role of magazines in our lives.

Question 18.

Question 19.
Find the general solution of the differential equation (1 + tan y) (dx – dy) + 2xdy = 0.
OR

Question 20.

Question 21.
Find the values of‘a’so that the following lines are skew:

Question 22.
A bag contains 4 green and 6 white balls. Two balls are drawn one by one without replacement. If the second ball drawn is white, what is the probability that the first ball drawn is also white?

Question 23.
Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of diamond cards drawn. Also, find the mean and the variance of the distribution.

SECTION D

Question 24.
Let f: [0, ∞) → R be a function defined by fx) = 9x² + 6x – 5. Prove that f is not invertible. Modify, only the codomain of f to make f invertible and then find its inverse.
OR
Let * be a binary operation defined on Q x Q by (a, b) * (c, d) = (ac, b + ad), where Q is the set of rational numbers. Determine, whether * is commutative and associative. Find the identity element for * and the invertible elements of Q x Q.

Question 25.
Using properties of determinants, prove that

prove that atleast one of the following statements is true:
(a) p, q, r are in G.P.
(b) a is a root of the equation px² + 2qx + r = 0.

Question 26.
Using integration, find the area of the region bounded by the curves y = √(5 – x²) and y = |x – 1|.

Question 27.
Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { xsinx\quad cosx }{ { sin }^{ 4 }x+{ cos }^{ 4 }x } } dx\)
OR
Evaluate \(\int _{ 0 }^{ 4 }{ \left( x+{ e }^{ 2x } \right) } dx\)

Question 28.
Find the equation of the plane through the point (4, -3, 2) and perpendicular to the line of intersection of the planes x – y + 2z – 3 = 0 and 2x – y – 3z = 0. Find the point of intersection of the line \(\vec { r } =\vec { i } +2\vec { j } -\vec { k } +\lambda \left( \vec { j } +3\vec { j } -9\vec { k } \right)\) and the plane obtained above.

Question 29.
In a mid-day meal programme, an NGO wants to provide vitamin rich diet to the students of an MCD school. The dietician of the NGO wishes to mix two types of food in such a way that vitamin contents of the mixture contains atleast 8 units of vitamin A and 10 units of vitamin C. Food 1 contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C. Food 2 contains 1 unit per Kg of vitamin A and 2 units per kg of vitamin C. It costs ₹ 50 per kg to purchase Food 1 and ₹ 70 per kg to purchase Food 2. Formulate the problem as LPP and solve it graphically for the minimum cost of such a mixture?

Solutions

Solution 1.

Solution 2.
k = 2³ = 8

Solution 3.
sin θ = cos θ ⇒ θ = 45°

Solution 4.
e ∈ R is the identity element for * if a * e = e * a = a ∀ a ∈ R ⇒ a + e – 2 = a ⇒ e = 2

Solution 5.

Solution 6.
A, B is symmetric matrix so A’ = A, B’ = B
(AB – BA)’ = (AB)’ – (BA)’ = B’A’ – A’B’ = BA – AB = – (AB – BA)
So AB – BA is skew symmetric matrix.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.

Solution 16.
Let the normal be at (x₁, y₁) to the curve 2y = x². \(\frac { dy }{ dx }\) = x.
The slope of the normal at

Solution 17.
Increase in subscription charges = ₹ x, Decrease in the number of subscriber = x.
Obviously, x is a whole number.
Income is given by y = (500 – x) (300 + x).
Let us assume for the time being
0 ≤ x < 500, x ∈ R

y is maximum when x = 100, which is a whole number. Therefore, she must increase the subscription charges by ₹ 100 to have maximum income.
Magazines contribute, a great deal, to the development of our knowledge. Through valuable and subtle critical and commentary articles on culture, social civilization, new life style we learn a lot of interesting things. Through reading magazines, our mind and point of view are consolidated and enriched.

Solution 18.

Solution 19.

Solution 20.

Solution 21.
As 2 : 3 : 4 ≠ 5 : 2 : 1, the lines are not parallel
Any point on the first line is (2λ + 1, 3λ + 2, 4λ + a)
Any point on the second line is (5μ + 4, 2μ + 1, μ)
Lines will be skew, if, apart from being non-parallel, they do not intersect. There must not exist a pair of values of λ, μ, which satisfy the three equations simultaneously:
2λ + 1 = 5μ + 4, 3λ + 2 = 2μ + 1, 4λ + a = μ
Solving the first two equations, we get λ = -1, μ = -1
These values will not satisfy the third equation if a ≠ 3

Solution 22.
Let E₁ = First ball drawn is white,
E₂ = First ball drawn is green,
A = Second ball drawn is white
The required probability, by Bayes’ Theorem

Solution 23.

Solution 24.

Solution 25.

Solution 26.

Solution 27.

Solution 28.

Solution 29.
Let x kg of Food 1 be mixed with y kg of Food 2. Then to minimize the cost, C = 50x + 70y subject to the following constraints:
2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0

In the half plane 50x + 70y < 380, there is no point common with the feasible region. Hence, the minimum cost is ₹ 380.

We hope the CBSE Sample Papers for Class 12 Maths Paper 3 help you. If you have any query regarding CBSE Sample Papers for Class 12 Maths Paper 3, drop a comment below and we will get back to you at the earliest.

CBSE Sample Papers for Class 12 Maths Paper 2 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 2.

CBSE Sample Papers for Class 12 Maths Paper 2

Students who are going to appear for CBSE Class 12 Examinations are advised to practice the CBSE sample papers given here which is designed as per the latest Syllabus and marking scheme as prescribed by the CBSE is given here. Paper 2 of Solved CBSE Sample Paper for Class 12 Maths is given below with free PDF download solutions.

Time: 3 Hours
Maximum Marks: 100

General Instructions:

• All questions are compulsory.
• Questions 1-4 in section A are very short answer type questions carrying 1 mark each.
• Questions 5-12 in section B are short answer type questions carrying 2 marks each.
• Questions 13-23 in section C are long answer I type questions carrying 4 marks each.
• Questions 24-29 in section D are long answer II type questions carrying 6 marks each.

SECTION A

Question 1.
Find λ when projection of \(\bar { a } =\lambda \hat { i } +\hat { j } +4\hat { k }\) on \(\bar { b } =2\hat { i } +6\hat { j } +3\hat { k }\) is 4 units.

Question 2.
Evaluate \(\int _{ 2 }^{ 3 }{ \frac { 1 }{ 2x } } dx\)

Question 3.
Write the cofactor of the element a₃₁ in \(A=\left( \begin{matrix} 3 & 2 & 6 \\ 5 & 0 & 7 \\ 3 & 8 & 5 \end{matrix} \right)\)

Question 4.
For which values of x, f(x) = \(\left| x-1 \right| +\left| x-2 \right|\) is not differentiable?

SECTION B

Question 5.
For what value of c, Mean Value theorem is applicable for the function f(x) = x + \(\frac { 1 }{ x }\) on [1, 3] ?

Question 6.
Find \(\frac { dy }{ dx }\), if y = \({ tan }^{ -1 }\left( \frac { 1-cosx }{ 1+cosx } \right)\)

Question 7.
For the matrix \(A=\begin{pmatrix} 3 & 1 \\ 7 & 5 \end{pmatrix}\) find the value of x and y so that A² + xI = yA

Question 8.
Using derivative, find the approximate percentage increase in the area of a circle if its radius is increased by 2%.

Question 9.
The random variable x has a probability distribution P(x) of the following form, where k is some number

Find the value of (i) k (ii) P(x ≥ 2)

Question 10.
Find the equation of line passing through the point with position vector \(\hat { i } +\hat { j } +\hat { k }\) and perpendicular to the plane \(\left( 2\hat { i } +\hat { j } +\hat { 3k } \right) \cdot \hat { r } =5\) in both vector form and Cartesian form.

Question 11.
Evaluate \(\int _{ 0 }^{ 1 }{ x{ e }^{ { x }^{ 2 } } } dx\)

Question 12.
Anil wants to invest at most ₹ 12000 in bonds A and B. He wants to invest atleast ₹ 2000 in bond A and atleast ₹ 4000 in bond B. If rate of interest on bond A is 8% per annum and rate of interest on bond B is 9% per annum. Formulate it as L.P.P. to find maximum yearly income.

SECTION C

Question 13.
Two balls are drawn at random from a bag containing 2 white, 3 red, 5 green and 4 black balls one by one without replacement. Find the probability that both the balls are of different colours.

Question 14.
The probabilities of two students A and B coming to the school in time are \(\frac { 3 }{ 7 }\) and \(\frac { 5 }{ 7 }\) respectively. Assuming that the events A coming in time and B coming in time are independent. Find the probability of only one of them coming to the school in time. Write atleast one advantage of coming to the school in time.

Question 15.

Question 16.

Question 17.

Question 18.
Evaluate \(\int { \frac { sin\phi d\phi }{ \sqrt { { sin }^{ 2 }\phi +2cos\phi +3 } } }\)

Question 19.
Evaluate \(\int _{ 0 }^{ \pi }{ \frac { x\quad tanx }{ secx+tanx } } dx\)
OR
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\quad sinx\quad dx }\)

Question 20.

Question 21.
Find the particular solution for the differential equation

Question 22.
If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is √3.

Question 23.
Solve the following L.P.P. graphically:
Minimize and maximize Z = 3x + 9y
Subject to constraint
x + 3y ≤ 60
x + y ≥ 10
x ≤ y
x, y ≥ 0

SECTION D

Question 24.
Find the distance of the point P(3, 4, 4) from the point where the line joining the points A (3, -4, -5) and B (2, -3, 1) intersect the plane 2x + y + z = 7.
OR
It lines \(\frac { x-1 }{ 1 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 }\) and \(\frac { x-3 }{ 1 } =\frac { y-k }{ 2 } =\frac { z }{ 1 }\) intersect, then find the value of k and hence find the equation of the plane containing these lines.

Question 25.
Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d)
Show that R is an equivalence relation.
OR
A binary operation * is defined on the set X = R – {-1} by x * y = x + y + xy ∀ x, y ∈ X. Check whether * is commutative and associative. Find the identity element and also find the inverse of each element of X.

Question 26.
Given the sum of the perimeter of a square and a circle. Show that sum of their areas is least when the side of the square is equal to the diameter of the circle.
OR
Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is \(\frac { 2R }{ \surd 3 }\). Also find the maximum volume.

Question 27.
Determine the product \(\left( \begin{matrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{matrix} \right) \left( \begin{matrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{matrix} \right)\) and use it to solve the system of equations
x – y + z = 4
x – 2y – 2z = 9
2x + y + 3z = 1

Question 28.
Find the area of the region {(x, y): x² + y² ≤ 4, x + y ≥ 2} using integration.

Question 29.
Solve the following differential equation

Solutions

Solution 1.

Solution 2.

Solution 3.

Solution 4.
At x = 1 and x = 2, f(x) is not differentiable.

Solution 5.

Solution 6.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.
Let amount invested in bond A is ₹ x, and amount invested in bond B is ₹ y.
Objective function is maximise income Z = \(\frac { 8x }{ 100 } +\frac { 9y }{ 100 }\)
Subject to constraints are
x + y ≤ 12000 (Investment constraint)
x ≥ 2000
y ≥ 4000

Solution 13.
Bag contains- 2 White balls, 3 Red balls, 5 Green balls, 4 Black balls
Total balls = 14
2 balls are drawn without replacement.
P (both balls are of different colours) = 1 – P(both balls are of same colour)
= 1 – [P(both white) + P (both red) + P (both green) + P (both black)]

Solution 14.
Let E₁ is the event: A coming in time
and E₂ is the event: B coming in time

Solution 15.

Solution 16.
Using elementary row transformation
A = IA

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Solution 23.

Solution 24.
Equation of line joining the points A (3, -4, -5) and B (2, -3, 1) is

Solution 25.
R be a relation on N x N defined by
(a, b) R (c, d) if ad (b + c) = bc (a + d)

Solution 26.

Solution 27.

Solution 28.

Solution 29.

We hope the CBSE Sample Papers for Class 12 Maths Paper 2 help you. If you have any query regarding CBSE Sample Papers for Class 12 Maths Paper 2, drop a comment below and we will get back to you at the earliest.