Page 31.16 Ex. 31.1 Q1. Answer : Sample space, S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Consider the given events. A = An even number on the card B = A number more than 3 on the card Clearly, A = {2, 4, 6, 8, 10} B = {4, 5, 6, 7, […]

# CBSE Class 12th Mathematics, NCERT Solutions, Sample & Previous Year Papers

## Three Dimensional Geometry, Class 12 Mathematics R.D Sharma Solutions

DIRECTION COSINES & DIRECTION RATIOS Page 27.22 Ex. 27.1 Q1. Answer : Let the direction cosines of the line be l, m, n. Now, l = cos900 = 0m = cos600 = 12n = cos300 = 32Therefore, the direction cosines of the line are 0, 12, 32. Q2. Answer : Let the direction cosines of the line […]

## Vector Algebra, Class 12 Mathematics R.D Sharma Solutions

ALGEBRA OF VECTORS Page 23.4 Ex. 23.1 Q1. Answer : (i) The vector OP→ represents the required displacement vector. (ii) The vector OQ→ represents the required vector. (iii) The vector OR→ represents the required vector. Q2. Answer : The quantities which have only magnitude and which are not related to any fixed direction in space […]

## Differential Equations, Class 12 Mathematics R.D Sharma Solutions

Page 22.4 Ex. 22.1 Q1. Answer : In this differential equation, the order of the highest order derivative is 3 and its power is 1. So, it is a differential equation of order 3 and degree 1. It is a non-linear differential equation because the differential coefficient dxdt has exponent 2, which is greater than […]

## Applications Of Integrals, Class 12 Mathematics R.D Sharma Solutions

Page 21.13 Ex.21.1 Q1. y2=8x represents a parabola with vertex at origin and axis of symmetry along the +ve direction of x-axisx=2 is line parallel to y-axisLet (x, y) be a given point on the parabola , y2=8xSince parabola y2=8x is symmetric about x-axis ,∴Required area =2 area OCAO On slicing the area above […]

## Linear Programming, Class 12 Mathematics R.D Sharma Solutions

LINEAR PROGRAMMING Page 30.14 Ex.30.1 Q1. Answer : Let x and y number of gadgets A and B respectively being produced in order to maximize the profit. Since, each unit of gadget A takes 10 hours to be produced by machine A and 6 hours to be produced by machine B and each unit of […]

## Integrals, Class 12 Mathematics R.D Sharma Solutions

INDEFINITE INTEGRALS Page 19.4 Ex.19.1 Q1. Answer : (i) ∫x4dx=x4+14+1+C =x55+C (ii) ∫x54dx=x54+154+1+C=49×94+C (iii) ∫x-5dx=x-5+1-5+1+C=-14x-4+C=-14×4+C (iv) ∫dxx3/2=∫x-3/2dx=x-32+1-32+1+C=x-12-12+C=-2x+C (v) ∫3xdx=3xln 3+C (vi) ∫dxx23=∫dxx2/3=∫x-2/3 dx=x-23+1-23+1+C=3×13+C (vii) ∫32 log3xdx=∫3log3 x2dx=∫x2dx=x33+C (viii) ∫logxx dx=∫1·dx=x+C Q2. Answer : (i) ∫1+cos 2x2dx ∫2 cos2x2dx ∴1+cos2A=2cos2A=∫cosx dx=sin x+C (ii) ∫1-cos 2x2dx =∫2sin2x2dx ∴1-cos 2x=2sin2x=∫sinx dx=-cos x+C Q3. Answer : ∫e6 log x-e5 […]

## Applications of Derivatives, Class 12 Mathematics R.D Sharma Solutions

DERIVATIVE AS A RATE MEASURE Page 13.3 Ex.13.1 Q1. Answer : Let T be the total surface area of a cylinder. Then, T = 2πrr+h Since the radius varies, we differentiate the total surface area w.r.t. radius r. Now, dTdr=ddr2πrr+h⇒dTdr=ddr2πr2+ddr2πrh⇒dTdr=4πr+2πh⇒dTdr=2πr+h Q2. Answer : Let V and r be the volume and diameter of the sphere, […]

## Continuity and Differentiability, Class 12 Mathematics R.D Sharma Solutions

CONTINUITY Page 9.17 Ex.9.1 Q1. Answer : Given: fx=xx, x≠01, x=0 We observe (LHL at x = 0) =limx→0-fx = lim h→0f0-h = lim h→0f-h =limh→0-h-h=limh→0-hh =limh→0-1 =-1 (RHL at x = 0) =limx→0+fx = lim h→0f0+h= lim h→0fh =limh→0hh=limh→0hh=limh→01=1 ∴limx→0+fx ≠limx→0-fx Hence, fx is discontinuous at the origin. Q2. Answer : Given: fx=x2-x-6x-3, x≠35, […]

## Determinants, Class 12 Mathematics R.D Sharma Solutions

Page 6.11 Ex. 6.1 Q1. Answer : (i) M11=-1M21= 20Cij= -1i+jMijC11= -11+1-1 = -1C21 = -11+220 = -20D = -1×5-20×0=-5 (ii) M11= 3M21= 4Cij = -1i+jMijC11=-11+1M11= 3C21=-12+1M21=-4 = -4D=3 × -1 – 4 × 2 = -3 – 8 = -11 (iii) M11= -1252 =-2 – 10=-12M21 = -3252 = -6 – 10 = -16M31 […]

## Matrices, Class 12 Mathematics R.D Sharma Solutions

Page 5.6 Ex. 5.1 Q1. Answer : We know that if a matrix is of order m×n, then it has mn elements. The possible orders of a matrix with 8 elements are given below: 1×8, 2×4, 4×2, 8×1 Thus, there are 4 possible orders of the matrix. The possible orders of a matrix with 5 […]

## Inverse Trigonometric Functions, Class 12 Mathematics R.D Sharma Solutions

Page 4.7 Ex. 4.1 Q1. Answer : (i) Let tan-1-3=y Then, tany=-3 We know that the range of the principal value branch is -π2, π 2. Thus, tany=-3=-tanπ3=tan-π3⇒y=-π3∈-π2,π2 Hence, the principal value of tan-1-3 is -π3. (ii) Let cos-1-12=y Then, cosy=-12 We know that the range of the principal value branch is 0, π. Thus, […]

## Relations & Functions, Class 12 Mathematics R.D Sharma Solutions

RELATIONS Page 1.11 Ex. 1.1 Q1. Answer : (i) Reflexivity: Let x be an arbitrary element of R. Then,x∈R ⇒x and x work at the same place is true since they are the same.⇒x, x∈RSo, R is a reflexive relation. Symmetry: Let x, y∈R⇒x and y work at the same place ⇒y and x work […]

## Probability, Class 12 Mathematics NCERT Solutions

NCERT Solutions

## Differential Equation, Class 12 Mathematics NCERT Solutions

NCERT Solutions

## Integrals, Class 12 Mathematics NCERT Solutions

NCERT Solutions

## Application of Derivatives, Class 12 Mathematics NCERT Solutions

NCERT Solutions

## Continuity and Differentiability, Class 12 Mathematics NCERT Solutions

NCERT Solutions