#### NCERT Solutions Class 12 Mathematics Chapter 10 Vector Algebra

Exercise 10.1 1. Represent graphically a displacement of 40 km, 30° east of north. Solution Here, vector OP represents the displacement of 40 km, 30° East of North.  2. Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 metres north-west (iii)…

#### NCERT Solutions Class 12 Mathematics Chapter 11 Three Dimensional Geometry

Exercise 11.1  1. If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines. Solution Let direction cosines of the line be l, m, and n.  I = cos 90° = 0  m = cos 135° = -1/√2  n = cos 45…

#### NCERT Solutions Class 12 Mathematics Chapter 1 Relations and Functions

Exercise 1.1 Question1. Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation R in the set A = {1, 2, 3…13, 14} defined as R = {(x, y): 3x − y = 0} (ii) Relation R in the set N of natural numbers defined as R = {(x, y): y =…

#### NCERT Solutions Class 12 Mathematics Chapter 13 Probability

Exercise 13.1 1. 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  2. Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32 Solution It is…

#### NCERT Solutions Class 12 Mathematics Chapter 3 Matrices

Exercise 3.1 Find the value of the following: Question: 1 (i) The order of the matrix, (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23. Answer (i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the…

#### NCERT Solutions Class 12 Mathematics Chapter 12 Linear Programming

Exercise 12.1 Question1.  Maximize Z = 3x + 4y subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0. Answer : The feasible region determined by the constraints, x + y ≤ 4, x ≥ 0, y ≥ 0, is as follows. The corner…

#### NCERT Solutions Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions

Exercise 2.1 Find the principal values of the following: Question 1. sin-1(-1/2) Answer :  1. Let sin-1(−1/2) = y, then sin y = −1/2 = −sin(π/6) = sin(−π/6) Range of the principal value of sn-1 is [-π/2, π/2] and sin -π/6) = -1/2 Therefore, the principal value…

#### NCERT Solutions Class 12 Mathematics Chapter 7 Integrals

Exercise 7.1 1. Find an anti derivative (or integral) of the following functions by the method of inspection. sin 2x .  Answer : The anti derivative of sin 2x is a function of x whose derivative is sin 2x. It is known that, Therefore, the anti…

#### NCERT Solutions Class 12 Mathematics Chapter 5 Continuity and Differentiability

Exercise 5.1 Question. Prove that the function f (x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5. Answer : The given function is f(x) = 5x - 3  At x = 0, f(0) = 5× 0 -3 = -3  Question. Examine the continuity of the function f (x) = 2x2 – 1 at x…

#### NCERT Solutions Class 12 Mathematics Chapter 9 Differential Equations

Exercise 9.1 1. Determine order and degree (If defined) of differential equation d4y /dx4 + sin(y')  = 0  Solution d4y /dx4 + sin(y''')  = 0  ⇒ y''' + sin(y''') = 0  The highest order derivative present in the differential equation is y…

#### NCERT Solutions Class 12 Mathematics Chapter 4 Determinants

Exercise 4.1 1. Evaluate the determinants in Exercises 1 and 2. Solution 2. Evaluate the determinants in Exercises 1 and 2.  Solution 3. If A Solution 4. If A Solution 5. Evaluate the determinants  Solution 6. If A Solution 7.…

#### NCERT Solutions Class 12 Mathematics Chapter 6 Application of Derivatives

Exercise - 6.1 1. Find the rate of change of the area of a circle with respect to its radius r when  (i) r = 3 cm (ii) r = 4 cm  Solution The area of a circle (A) with radius (r) is given by,  A = πr2         Now,…

#### NCERT Solutions Class 12 Mathematics Chapter 8 Application of Integrals

Exercise 8.1 Question. Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x - axis.  Answer :  Question. Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in…