# Class 12

## CLASS 12 MATHS CHAPTER 13-PROBABILITY

Exercise 13.1 Solve The Following Questions. Question1. Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E). Solution :It is given that P(E) = 0.6, P(F) = 0.3, and P(E ∩ F) = 0.2 Question2. Compute P(A|B), if P(B) = 0.5 …

## CLASS 12 MATHS CHAPTER 12-LINEAR PROGRAMMING

Exercise 12.1 Solve The Following Questions. Solve the following Linear Programming Problems graphically: Question1.  Maximize Z = 3x + 4y subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0. Solution : The feasible region determined by the constraints, x + y ≤ 4, x ≥ 0, y ≥ 0, is as follows. The corner points of the feasible region …

## CLASS 12 MATHS CHAPTER 11-THREE DIMENSIONAL GEOMETRY

Exercise 11.1 Solve The Following Questions. Question1.If a line makes angles 90°, 135°, 45° with x ,y and z axes respectively, find its direction cosines. Solution : A-line makes 90°, 135°, 45°with x, y and z  axes respectively. Therefore, Direction cosines of the line are cos 90°, cos135°, and cos45° ⇒ Direction cosines of the …

## CLASS 12 MATHS CHAPTER 10-VECTOR ALGEBRA

Exercise 10.1 Solve The Following Questions. Question1. Represent graphically a displacement of 40 km, 30° east of north. Solution :Displacement 40 km, 30° East of North Here, vector represents the displacement of 40 km, 30° East of North. Question2. Check the following measures as scalars and vectors: (i) 10 kg    (ii) 2 meters north-west   (iii) 40° (iv) 40 …

## CLASS 12 MATHS CHAPTER 9-DIFFERENTIAL EQUATIONS

Exercise 9.1 Solve The Following Questions. Determine order and degree (if defined) of differential equations given in Questions 1 to 10: Question1.  Solution :Given: The highest order derivative present in the differential equation is y”” and its order is 4. The given differential equation is not a polynomial equation in its derivatives. Hence, its degree …

## CLASS 12 MATHS CHAPTER 8-APPLICATIONS OF INTEGRALS

Exercise 8.1 Solve The Following Questions. Question 1. Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.Solution : The area of the region bounded by the curve, y2 = x, the lines, x = 1 and x = 4, and the x-axis is the area ABCD.Question 2. Find the area of the region bounded by y2 = 9x, x = …

## CLASS 12 MATHS CHAPTER 7-INTEGRALS

Exercise 7.1 Solve The Following Questions. Find an antiderivative (or integral) of the following functions by the method of inspection in Exercises 1 to 5. Question 1. sin 2xSolution : The anti derivative of sin 2x is a function of x whose derivative is sin 2x. It is known that, herefore, the anti derivative of sin 2x is -1/2 …

## CLASS 12 MATHS CHAPTER 6-APPLICATION OF DERIVATIVES

Exercise 6.1 Solve The Following Questions. Question1. Find the rate of change of the area of a circle with respect to its radius r when (a)r  = 3 cm (b)r = 4 cm Solution :The area of a circle (A) with radius (r) is given by, A = πr2 Now, the rate of change of …

## CLASS 12 MATHS CHAPTER 5-CONTINUITY & DIFFERENTIABILITY

Exercise 5.1 Solve The Following Questions. Question1. Prove that the function f(x) = 5x – 3 is continuous at x = 0, at  x = – 3 and x = 5 Solution : Question2. Examine the continuity of the function f(x) = 2×2 – 1 at x = 3 Solution : Thus, f is continuous at …

## CLASS 12 MATHS CHAPTER 4-DETERMINANTS

Exercise 4.1 Solve The Following Questions. Evaluate the following determinants in Exercise 1 and 2. Question1.  Solution :  = 2(-1) – 4(-5) = -2 + 20 = 18 Question2. (i)  (ii)  Solution :(i)  = (cosθ)(cosθ) – (-sinθ) (sinθ)= cos2 θ + sin2 θ= 1 (ii)  = (x2 − x + 1)(x + 1) − (x − 1)(x + 1) = x3 − x2 + x + x2 − x + 1 − (x2 − 1) = x3 + 1 …

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