CBSE Sample Papers for Class 12 Maths Paper 6

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

Board	CBSE
Class	XII
Subject	Maths
Sample Paper Set	Paper 6
Category	CBSE Sample Papers

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 and B are skew symmetric matrices of order n, then whether A + B is symmetric or skew symmetric matrix.

Question 2.
If α, β, γ be the direction angles of a line then find the value of sin²α + sin²β + sin²γ.

Question 3.
Evaluate \int { { 3 }^{ -2x }{ e }^{ -2x }dx }

Question 4.
Mean Value theorem for the function f(x) = \(\sqrt { { x }^{ 2 }-4 }\) in [2, 4] is verified, find the value of c.

SECTION B

Question 5.

Question 6.
If y = 3 cos (log x) + 4 sin (log x) show that x²y² + xy¹ + y = 0.

Question 7.
Find the equation of tangent to the curve y = x² + 2x + 3 at (0, 3).

Question 8.
Find k if f (x) is continuous at x = 5.

Question 9.

Question 10.
Find the general solution of the differential equation y log y dx – x dy = 0

Question 11.

Question 12.
Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\quad tanx\quad dx }\)

SECTION C

Question 13.

Question 14.

Question 15.

Question 16.

Question 17.

Question 18.

Question 19.

Question 20.

Question 21.
Solve the differential equation \(\frac { dy }{ dx }\) – 3y cot x = sin 2x.

Question 22.
In a hockey match both teams A and B scored same number of goals up to the end of the game. So to decide the winner the referee asked both the captains to throw a die alternatively and decided that the team whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

Question 23.
Let R be the set of all real numbers except 1 and * be an operation on R defined as a * b = a + b – ab ∀ a, b ∈ R. Prove that
(a) R is closed under *
(b) * is commutative and associative
(c) Find the identity element of (R, *)
(d) Find the invertible element of (R, *)

SECTION D

Question 24.
Verify A(adj A) = (adj A) A = |A| I for the matrix A = \(\left( \begin{matrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{matrix} \right)\)
OR
Amit purchases 3 pens, 2 bags and 1 instrument box and pays ₹ 41. From the same shop Anita purchases 2 pens, 1 bag and 2 instrument boxes and pays ₹ 29 while Anju purchases 3 pens, 2 bags and 2 instrument boxes and pays ₹ 46. Translate the problem into a system of equations. Solve the system of equations by matrix method and hence find the cost of 1 pen, 1 bag and 1 instrument box.

Question 25.
A farmer mixes two brands P and Q of cattle feed. Brand P costing ₹ 250 per bag contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C whereas brand Q costing ₹ 200 per bag contains 1.5 units of nutritional demerit A, 11.25 units of element B and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

Question 26.
Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one third that of the cone and greatest volume of cylinder is \(\frac { 4 }{ 27 } \pi { h }^{ 3 }{ tan }^{ 2 }\alpha\)
OR
Show that the volume of the largest cylinder that can be inscribed in a sphere of radius R is \(\frac { 1 }{ \surd 3 }\) times the volume of the sphere.

Question 27.
Find the area of the region {(x, y): y² ≤ 4x, 4x² + 4y² ≤ 9} using integration.

Question 28.
The compressors used in refrigerators are manufactured by three different factories at A, B, C. It is known that the factory A produces twice as many compressors as B, which produces the same number as C. Experience shows that 0.2% of compressors produced at A as well as at B are defective and 0.4% at C. A quality controller chooses a compressor and find, it is a defective one. What is the probability that it was produced at factory B?

Question 29.
Find the equation of the plane through the intersection of the planes 2x + y – 3z + 4 = 0 and 3x + 4y + 8z – 1 = 0 and making equal intercepts on the coordinate axes.

Solutions

Solution 1.
A’ = -A, B’ = -B
(A + B)’ = – (A + B)
So A + B is a skew symmetric matrix

Solution 2.
sin²α + sin²β + sin²γ = 3 – (cos²α + cos²β + cos²γ) = 3 – 1 = 2

Solution 3.

Solution 4.

Solution 5.

Solution 6.

Solution 7.
\(\frac { dy }{ dx }\) = 2x + 2, m = 2
Equation of tangent 2x – y + 3 = 0

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Solution 23.
a * b = a + b – ab, ∀ a, b ∈ R – {l}
(a) Let if possible a + b – ab = 1
(a – 1){b – 1) = 0
a = 1, b = 1
But a ≠ 1, b ≠ 1
So a + b – ab ∈ R – {1}
R is closed under *.
(b) Prove * is commutative and associative.
(c) Let identity element is e.
Here e = 0
(d) a * b = 0
a + b – ab = 0 ⇒ b = \(\frac { a }{ a-1 }\) ∈ R
Invertible element of R is \(\frac { a }{ a-1 }\) ∀ a ∈ R – {1}

Solution 24.

Solution 25.
Let x bags of brand P and y bags of brand Q are mixed.
Objective function at minimise cost Z = 250 x + 200 y
Subject to constraints are
3x + 1.5y ≥ 18
2.5x + 1125y ≥ 45
2x + 3y ≥ 24
x, y ≥ 0

Common feasible region is unbounded so ₹ 1950 may or may not be the minimum cost.
250x + 200y < 1950
Open half plane has no common point with the feasible region, so minimum value of cost is ₹ 1950 when 3 bags of brand P and 6 bags of brand Q to be mixed.

Solution 26.

Solution 27.

Solution 28.
Let E₁: Compressor is produced by factory A
E₂: Compressor is produced by factory B
E₃: Compressor is produced by factory C
E : getting a defective compressor
Let number of compressors produced by factory B or C = x
number of compressors produced by factory A = 2x

Solution 29.

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