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 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 = x^{2} 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^{-1}x + cot^{-1}x = 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 + y^{2}) 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 a^{2} 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 x^{2} + y^{2} = 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 = x^{2}

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 E_{1} is the event the bolt is manufactured by machine A

E_{2} is the event the bolt is manufactured by machine B

E_{3} 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.

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