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

Board |
CBSE |

Class |
XII |

Subject |
Maths |

Sample Paper Set |
Paper 4 |

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 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.

If A is a square matrix of order 3 such that |adj A| = 64, find the value of |A|.

Question 2.

Question 3.

If f(x) = x + 7, g(x) = x – 7, find (fog) (5)

Question 4.

If \(\vec { PO } +\vec { OQ } =\vec { QO } +\vec { OR }\), show that the point P, Q, R are collinear.

**SECTION B**

Question 5.

Using elementary row transformation, find the inverse of matrix \(\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}\)

Question 6.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground away from the ball at the rate of 2 cm/sec. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Question 7.

Question 8.

Using differentials, find the approximate value of √49.5.

Question 9.

Find the general solution of the differential equation xy \(\frac { dy }{ dx }\) = (x + 2)(y + 2)

Question 10.

Question 11.

A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

Question 12.

If \(\int _{ 0 }^{ 1 }{ \left( 3{ x }^{ 2 }+2x+k \right) } dx=0\), find the value of k.

**SECTION C**

Question 13.

Question 14.

Question 15.

A wire of length 20 metre is to be cut into two pieces. One of the piece will be bent into shape of a square and the other into an equilateral triangle. What should be the length of the two pieces so that the sum of the areas of square and triangle is minimum?

Question 16.

Find the intervals in which the function f(x) = (x – 1)(x – 2)^{2} is increasing or decreasing.

Question 17.

Find the equation of tangent to the curve x = a sin^{3}t, y = b cos^{3}t at a point where t = \(\frac { \pi }{ 4 }\)

Question 18.

Question 19.

Question 20.

Question 21.

Find the coordinates of the point where the line through the points (3, -4, -5) and (2, -3, 1) crosses the plane 3x + 2y + z + 14 = 0.

Question 22.

The probability that a teacher will give a surprise test during any class meeting is \(\frac { 1 }{ 5 }\). If a student is absent twice, what is the probability that he will miss at least one test? If a student remains absent quite often, what life skill is he lacking?

Question 23.

Two cards are drawn simultaneously (without replacement) from a well shuffled pack of 52 cards. Find mean, variance and standard deviation of the number of aces.

**SECTION D**

Question 24.

Question 25.

Show that the binary operation * on A = R – {-1} defined as a * b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.

Question 26.

Prove that the curves y^{2} = 4x and x^{2} = 4y divide the area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

**OR**

Using integration find the area of the region bounded by the lines 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0.

Question 27.

Question 28.

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Question 29.

Every gram of rice provides 0.1 gm of protein and 0.25 gm of carbohydrates and the corresponding values of wheat are 0.05 gm and 0.5 gm respectively. It is given that rice cost ₹ 40 per kg and wheat ₹ 60 per kg. The mini mum daily requirement of protein and carbohydrate for an average child are 50 gm and 200 gm respectively. In what quantities should rice and wheat be mixed in daily diet to provide minimum daily requirements of protein and carbohydrates at minimum cost. Formulate this as L.P.P and solve it.

**Solutions**

Solution 1.

|adj A| = |A|^{4-1}

|A| = 8

Solution 2.

Solution 3.

fog(x) = f[g(x)] = f{x – 7} = x – 7 + 7 = x.

(fog) (5) = 5

Solution 4.

\(\vec { PO } +\vec { OQ } =\vec { QO } +\vec { OR }\)

\(\vec { PQ } =\vec { QR }\) (by triangle law)

Vectors are parallel in which Q is common point, so P, Q, R are collinear.

Solution 5.

Solution 6.

Solution 7.

Solution 8.

Solution 9.

Solution 10.

Solution 11.

Solution 12.

Solution 13.

Solution 14.

Solution 15.

Let length of first piece of wire is x m from which equilateral triangle formed and length of second piece of wire is (20 – x) m from which square is formed.

Solution 16.

Solution 17.

Solution 18.

Solution 19.

Solution 20.

Solution 21.

Solution 22.

P(test miss) = \(\frac { 1 }{ 5 }\), student absent twice.

Req. probability = P(Test miss only one day) + P(Test miss in both days)

= P(happening test in one of day he was absent and not happening test on the day he was

If a student remains absent quite often then he will loose confidence and lack of knowledge.

Solution 23.

Let X denote the number of aces when 2 cards drawn without replacement. X : 0, 1, 2.

Solution 24.

Solution 25.

a * b = a + b + ab for all a, b ∈ A

* is commutative because a * b = b* a

* is associative because a * (b * c) = (a * b) * c

Let e be the identity element.

a * e = e * a = a

a + e + ae = e + a + ea = a

e(1 + a) = 0

e = 0 [a ≠ -1]

Let b is the inverse of a.

a * b = b * a = e

a + b + ab = b + a + ba = 0

a + b(1 + a) = 0

b = \(\frac { -a }{ 1+a }\) is the inverse of each element a of A.

Solution 26.

Solution 27.

Solution 28.

Required plane equation a(x + 1) + b(y – 3) + c(z – 2) = 0

a + 2b + 3c = 0

3a + 3b + c = 0

After solving \(\frac { a }{ -7 }\) = \(\frac { b }{ 8 }\) = \(\frac { c }{ -3 }\)

Required plane equation is 7x – 8y + 3z + 25 = 0

Solution 29.

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