CBSE Sample Papers for Class 12 Maths Paper 1

These Sample papers are part of CBSE Sample Papers for Class 12 Maths. Here we have given CBSE Sample Papers for Class 12 Maths Paper 1.

CBSE Sample Papers for Class 12 Maths Paper 1

Board	CBSE
Class	XII
Subject	Maths
Sample Paper Set	Paper 1
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 1 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.
For what value of k the matrix \(\begin{pmatrix} 4 & k \\ 2 & 1 \end{pmatrix}\) has no inverse?

Question 2.
Show that the points (2, 3, 4), (-1, -2, 1) and (5, 8, 7) are collinear.

Question 3.
Evaluate \(\int { { sin }^{ -1 }\left( cosx \right) dx }\)

Question 4.
Find the point on the curve y² = x where the tangent makes an angle of \(\frac { \pi }{ 4 }\)with x-axis.

SECTION B

Question 5.

Question 6.

Question 7.
Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Question 8.

Question 9.
Find the equation of the plane which is parallel to the x-axis and has intercepts 5 and 7 on the y-axis and z-axis respectively.

Question 10.
Form the differential equation representing the given family of curves (y – b)² = 4 (x – a) by eliminating arbitrary constants a and b.

Question 11.

Question 12.
Evaluate \(\int { sin4x\quad sin8x\quad dx }\)

SECTION C

Question 13.
If the sum of two unit vectors \(\hat { a }\) and \(\hat { a }\) is a unit vector, show that the magnitude of their difference is √3.

Question 14.
Using vectors find the value of k such that the points (k, -10, 3) (1, -1, 3) and (3, 5, 3) are collinear.

Question 15.

Question 16.

Question 17.

Question 18.

Question 19.
Find the real solution of the equation

Question 20.
Find the general solution of the differential equation \(\frac { dy }{ dx }\) + 1 = e^x+y

Question 21.

Question 22.
Three persons A, B and C fire a target turnwise starting with A. Their probability of hitting the target are 0.5, 0.3 and 0.2 respectively. Find the probability of at most one hit. In life, we must set a target. To achieve the target we need to follow some values. Mention any such two values.

Question 23.
Let R be a relation on N x N defined by (a, b) R (c, d) ⇒ ad (b + c) = bc (a + d). Check whether R is an equivalence relation on N x N.

SECTION D

Question 24.

Question 25.
Rohit owns a field of 1000 sq.metre. He wants to plant trees in it. He has a sum of ₹ 1400 to purchase young trees. He has the choice of two types of trees. Type A require 10 sq m. of ground per tree and cost ₹ 20 per tree. Type B require 20 sq. m of ground per tree and cost ₹ 25 per tree. When fully grown; type A produces an average of 20 kg of fruits which can be sold at a profit of ₹ 2 per kg and type B produces an average of 40 kg of fruits which can be sold at a profit of ₹ 1.5 per kg. How many trees of each type should be planted to achieve maximum profit when they are fully grown? What is the maximum profit? Formulate the above LPP and solve it.
OR
Two tailors A and B earn ₹ 150 and ₹ 200 per day respectively. A can stitch 6 caps and 4 pants while B can stitch 10 caps and 4 pants per day. How many days shall each work if it is desired to produce atleast 60 caps and 32 pants at a minimum labour cost? Formulate the above LPP and solve it.

Question 26.
Prove that the right circular cone of maximum volume which can be inscribed in a sphere of radius r has altitude equal to \(\frac { 4r }{ 3 }\) and show that the maximum volume is \(\frac { 8 }{ 27 }\) of the volume of a sphere.

Question 27.
Find the area enclosed by the curve y = -x² and the straight line x + y + 2 = 0 using integration.
OR
Find the area of region bounded by the triangle whose vertices are (-1, 1) (0, 5) and (3, 2) using integration.

Question 28.
A discrete random variable X has the following probability distribution

Find the value of c. Also find mean, variance and standard deviation of this distribution.

Question 29.

Solutions

Solution 1.

Solution 2.
Direction ratios of AB = -3, -5, -3
Direction ratios of AC = 3, 5, 3
because direction ratios of AB and AC are in ratio, so they are parallel but there is point A common in them.
So A, B, C are collinear.

Solution 3.

Solution 4.

Solution 5.
R₁ → R₂ + sinθ R1
R₃ → R₃ + R₁ and expand
∆ = 2 (1+ sin²θ) ∀ θ, -1 ≤ sinθ ≤ 1
0 ≤ 2 sin²θ ≤ 2 ∀ θ, 0 ≤ sin²θ ≤ 1
2 ≤ 2 + 2 sin²θ ≤ 4 or 2 ≤ ∆ ≤ 4

Solution 6.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.
Multiply R₁ by a, R₂ by b, R₃ by c then taking abc common from C₃
then C₂ ⇔ C₃ and C₁ ⇔ C₂
OR

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

Solution 23.

Solution 24.

Solution 25.
Let x trees of type A and y trees of type B are planted.
Objective function is maximise profit Z = ₹ 40x + 60y
Subject to constraints are
10x + 20y ≤ 1000
20x + 25y ≤ 1400
x, y ≥ 0

Hence maximum profit is ₹ 3200 when he planted 20 trees of type A and 40 trees of type B.
Let tailor A work for x days and tailor B work for y days.
Objective function is minimum.
Labour cost Z = ₹ 150x + 200y.
Subject to constraints are
6x + 10y ≥ 60
4x + 4y ≥ 32
x, y ≥ 0

because feasible region is unbounded, so minimum value may or may not be ₹ 1350.
Take 150x + 200y < 1350
Open half plane determined by 150x + 200y < 1350 do not have a point common, so ₹ 1350 is the minimum labour cost when tailor A works for 5 days and tailor B for 3 days.

Solution 26.

Solution 27.

Solution 28.

Solution 29.

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