CBSE Sample Papers for Class 12 Maths Paper 7 are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 7.

## CBSE Sample Papers for Class 12 Maths Paper 7

Board |
CBSE |

Class |
XII |

Subject |
Maths |

Sample Paper Set |
Paper 7 |

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 7 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 for any 2 x 2 square matrix A, A(adj A) = \(\begin{pmatrix} 1 & 0 \\ 0 & 8 \end{pmatrix}\) then write the value of |A|.

Question 2.

Determine the value of ‘k’ for which the following function is continuous at x = 3:

Question 3.

Question 4.

Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.

**SECTION B**

Question 5.

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

Question 6.

Find the value of c in Rolle’s theorem for the function f(x) = x^{3} – 3x in [-√3, 0].

Question 7.

The volume of a cube is increasing at the rate of 9 cm^{3}/s. How fast is its surface area increasing when the length of an edge is 10 cm?

Question 8.

Show that the function f(x) = x^{3} – 3x^{2} + 6x – 100 is increasing on R.

Question 9.

The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, -2) is 4. Find its z-coordinate.

Question 10.

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “number obtained is red”. Find if A and B are independent events.

Question 11.

Two tailors, A and B, earn ₹ 300 and ₹ 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.

Question 12.

**SECTION C**

Question 13.

Question 14.

Question 15.

Question 16.

Question 17.

Question 18.

Solve the differential equation (tan^{-1}x – y) dx = (1 + x^{2}) dy.

Question 19.

Show that the points A, B, C with position vectors \(2\hat { i } -\hat { j } +\hat { k }\), \(\hat { i } -3\hat { j } -5\hat { k }\) and \(3\hat { i } -4\hat { j } -4\hat { k }\) respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.

Question 20.

Find the value of λ, if four points with position vectors \(3\hat { i } +6\hat { j } +9\hat { k }\), \(3\hat { i } +2\hat { j } +3\hat { k }\), \(2\hat { i } +3\hat { j } +\hat { k }\) and \(4\hat { i } +6\hat { j } +\lambda \hat { k }\) are coplanar.

Question 21.

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.

Question 22.

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.

Question 23.

Maximise Z = x + 2y

subject to the constraints

x + 2y ≥ 100

2x – y ≤ 0

2x + y ≤ 200

x, y ≥ 0

Solve the above L.P.P. graphically.

**SECTION D**

Question 24.

Question 25.

Question 26.

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

Question 27.

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).

**OR**

Find the area enclosed between the parabola 4y = 3x^{2} and the straight line 3x – 2y + 12 = 0.

Question 28.

Find the particular solution of the differential equation (x – y) \(\frac { dy }{ dx }\) = (x + 2y), given that y = 0 when x = 1.

Question 29.

Find the coordinates of the point where the line through the points (3, -4, -5) and (2, -3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, -3) and (0, 4, 3).

**OR**

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is \(\frac { 1 }{ { x }^{ 2 } } +\frac { 1 }{ { y }^{ 2 } } +\frac { 1 }{ { z }^{ 2 } } =\frac { 1 }{ { p }^{ 2 } }\)

**Solutions**

Solution 1.

|A| = 8

Solution 2.

k = 12

Solution 3.

– log |sin 2x| + c or log |sec x| – log |sin x| + c.

Solution 4.

Writing the equations as

2x – y + 2z = 5

2x – y + 2z = 8

⇒ Distance = 1 unit

Solution 5.

Solution 6.

f'(c) = 3c^{2} – 3 = 0

c^{2} = 1 ⇒ c = ±1.

Rejecting c = 1 as it does not belong to (-√3, 0)

We get c = -1.

Solution 7.

Solution 8.

f(x) = x^{3} – 3x^{2} + 6x – 100

f'(x) = 3x^{2} – 6x + 6 = 3 [x^{2} – 2x + 2] = 3 [(x – 1)^{2} + 1]

Since f'(x) > 0 ∀ x ∈ R

f(x) is increasing on R.

Solution 9.

Solution 10.

Solution 11.

Let A works for x days and B for y days.

L.P.P is Minimise C = 300 x + 400 y

Subject to :

6x + 10y ≥ 60

4x + 4y ≥ 32

x ≥ 0, y ≥ 0

Solution 12.

Solution 13.

Solution 14.

Solution 15.

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Let E_{1} : Selecting a student with 100% attendance

E_{2} : Selecting a student who is not regular

A : Selected student attains A grade.

No, regularity is required everywhere.

Solution 23.

Z = x + 2y

x + 2y ≥ 100, 2x – y ≤ 0, 2x + y ≤ 200, x, y ≥ 0

For correct graph of three lines For correct shading

Z(A) = 0 + 400 = 400

Z(B) = 50 + 200 = 250

Z(C) = 20 + 80 = 100

Z(D) =0 + 100 = 100

Max (= 400) at x = 0, y = 200

Solution 24.

Solution 25.

Solution 26.

Solution 27.

Solution 28.

Solution 29.

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