CBSE Solutions, PDF Book, MCQ, Revision Notes, Important Questions, Sample Paper, Previous Year Paper

Class 6

Class 7

Class 8

Class 9

Class 10

Class 11

Class 12

Maths Class 12 Applications of Derivatives, R.D Sharma Question Answer

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, […]

Maths Class 12 Continuity and Differentiability, R.D Sharma Question Answer

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, […]

Maths Class 12 Determinants R.D Sharma Question Answer

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 Question Answer

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 Question Answer

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 Question Answer

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 […]

Limits and Derivatives, Class 11 Mathematics R.D Sharma Question Answer

LIMITS Page 29.11 Ex 29.1 Q1. Answer : limx→0 xx Left hand limit: limx→0- xx Let x=0-h, where h→0.⇒limh→0 0-h0-h=limh→0 -hh=-1 Right hand limit: limx→0+ xxLet x=0+h, where h→0.limh→0 0+h0+h=limh→0 hh=1 Left hand limit ≠ Right hand limit Thus,lim x→0 xx does not exist. Q2. Answer : fx=2x+3,x≤2x+k,x>2Left hand limit:limx→2- fx=limx→2- 2x+3Let x=2-h, where h→0.limh→0 […]

Conic Sections, Class 11 Mathematics R.D Sharma Question Answer

CIRCLES Page 24.16 Ex 24.1 Q1. Answer : Let (h, k) be the centre of a circle with radius a. Thus, its equation will be x-h2+y-k2=a2. (i) Here, h = −2, k = 3 and a = 4 ∴ Required equation of the circle: x+22+y-32=42 ⇒x+22+y-32=16 (ii) Here, h = a, k = b and radius […]

Binomial Theorem, Class 11 Mathematics R.D Sharma Question Answer

Page 18.11 Ex 18.1 Q1. Answer : (i) (2x + 3y)5 =C05(2x)5(3y)0+C15(2x)4(3y)1+C25(2x)3(3y)2+C35(2x)2(3y)3+C45(2x)1(3y)4+C55(2x)0(3y)5 =32×5+5×16×4×3y+10×8×3×9y2+10×4×2×27y3+5×2x×81y4+243y5=32×5+240x4y+720x3y2+1080x2y3+810xy4+243y5 (ii) (2x − 3y)4 =C04(2x)4(3y)0-C14(2x)3(3y)1+C24(2x)2(3y)2-C34(2x)1(3y)3+C44(2x)0(3y)4=16×4-4×8×3×3y+6×4×2×9y2-4×2x×27y3+81y4=16×4-96x3y+216x2y2-216xy3+81y4 (iii) x-1×6=C06 x61x0-C16 x51x1+C26 x41x2-C36 x31x3+C46 x21x4-6C5 x11x5+C66 x01x6=x6-6 x5×1x+15 x4×1×2-20×3×1×3+15×2×1×4-6 x×1×5+1×6=x6-6×4+15×2-20+15×2-6×4+1×6 (iv) (1 − 3x)7 =C07(3x)0-C17(3x)1+C27(3x)2-C37(3x)3+C47(3x)4-C57(3x)5+C67(3x)6-C77(3x)7=1-7×3x+21×9×2-35×27×3+35×81×4-21×243×5+7×729×6-2187×7=1-21x+189×2-945×3+2835×4-5103×5+5103×6-2187×7 (v) (ax-bx)6=C06(ax)6(bx)0-C16(ax)5(bx)1+C26(ax)4(bx)2-C36(ax)3(bx)3+C46(ax)2(bx)4-C56(ax)1(bx)5+C66(ax)0(bx)6 =a6x6-6a5x5×bx+15a4x4×b2x2-20a3b3×b3x3+15a2x2×b4x4-6ax×b5x5+b6x6=a6x6-6a5x4b+15a4x2b2-20a3b3+15a2b4x2-6ab5x4+b6x6 (vi) xa-ax6=C06xa6ax0-C16xa5ax1+C26xa4ax2-C36xa3ax3+C46xa2ax4-C56xa1ax5+C66xa0ax6=x3a3-6x2a2+15xa-20+15ax-6a2x2+a3x3 (vii) x3-a36=C06(x3)6(a3)0-C16(x3)5(a3)1+C26(x3)4(a3)2-C36(x3)3(a3)3+C46(x3)2(a3)4-C56(x3)1(a3)5+C66(x3)0(a3)6=x2-6×5/3a1/3+15×4/3a2/3-20xa+15×2/3a4/3-6×1/3a5/3+a2 (viii) (1+2x-3×2)5Consider 1-2x and 3×2 as two separate entities and apply the binomial […]

Probability, Class 11 Mathematics R.D Sharma Question Answer

Probability Page No.33.6 Ex.33.1 Q1. Answer : If a coin is tossed once, the possible outcomes are a head (H) and tail (T). Therefore, the sample space of this experiment is S = { H, T}. Q2. Answer : If a coin is tossed twice, the possible outcomes are HH, HT, TH, TT. Therefore, the […]

Mathematical Reasoning, Class 11 Mathematics R.D Sharma Question Answer

Mathematical Reasoning Page No.31.4 Ex.31.1 Q1. Answer : (i) Listen to me, Ravi! It is an exclamatory sentence. Therefore, it is not a statement. (ii) Every set is a finite set. It is a false assertive sentence because there are some sets that are infinite like the set of all real numbers. Therefore, it is […]

Straight Lines, Class 11 Mathematics R.D Sharma Question Answer

Page 23.13 Ex 23.1 Q1. Answer : (i) θ=-π4 ∴ Slope of the line=m=tanθ⇒Slope of the line=tan-π4=-tanπ4=-1 Hence, the slope of the line is -1. (ii) θ=2π3 ∴ Slope of the line=m=tanθ⇒Slope of the line=tan2π3=-tanπ3=-3 Hence, the slope of the line is -3. (iii) θ=3π4 ∴ Slope of the line=m=tanθ⇒Slope of the line=tan3π4=-tanπ4=-1 Hence, the […]

Permutations & Combinations, Class 11 Mathematics R.D Sharma Question Answer

PERMUTATIONS Page. 16.4 Ex 16.1 Q1. Answer : (i) 30!28! = 30×29×28!28! ∵ n!= n(n-1)! =30×29 = 870 (ii) 11!-10!9!=11×10×9! – 10×9!9! ∵ n! =n(n-1)! =9!(110-10)9! =100 (iii) LCM of (6!,7! and 8!): n! = n(n-1)! Therefore, (6!,7! and 8!) can be rewritten as: 8! = 8×7×6! 7! = 7×6! 6! = 6! ∴ LCM […]

Linear Inqualities, Class 11 Mathematics R.D Sharma Question Answer

Page 15.10 Ex 15.1 Q1. Answer : We have, 12x<50⇒x<5012 Dividing both the sides by 12⇒x<256(i) x∈RThen, the solution of the given inequation is -∞,256.(ii) x∈ZThen, the solution of the given inequation is ……….-3, -2, -1, 0, 1, 2, 3, 4.(iii) x∈NThen, the solution of the given inequation is 1, 2, 3, 4. Q2. Answer […]

Complex Numbers & Quadratic Equations, Class 11 Mathematics R.D Sharma Question Answer

COMPLEX NUMBERS Page 13.1 Ex – 13.1 Q1. Answer : i i457 =i4×114+1 =i4114×i =i ∵i4=1ii i528 =i4×132 =i4132 =1 ∵i4=1iii 1i58=1i4×14+2 =1i414×i2 =1i2 ∵i4=1 =-1 ∵i2=-1iv i37+1i67=i4×9+1+1i4×16+3 =i49×i+1i416×i3 =i-1i ∵i3=-i =i-1i×ii =i-ii2 =i–i ∵i2=-1 =2i v i41+1i2579 =i4×10+1+1i4×64+19 =i410×i+1i464×i9 =i+1i9 ∵ i4=1 =i+ii29 =i-i9 ∵i2=-1=0 vi i77+i70+i87+i4143=i4×19+1+i4×17+2+i4×21+3+i4×103+23 =i419×i+i417×i2+i421×i3+i4103×i2 =i-1-i-13 ∵i4=1,i3=-i and i2=-1 =-23 =-8vii i30+i40+i60=i4×7+2+i4×10+i4×15 […]

Maths Trigonometric Functions, Class 11 R.D Sharma Question Answer

 1. MEASUREMENT OF ANGLES Page 4.12 Ex – 4.1 Q1. Answer : We have:π rad =180°∴1 rad=180π° i 18π5=180π×9π5° =36×9° =324° ii-5π6=180π×-5π6° =-30×5° =-150° iii 18π5c=180π×18π5° =36×18° =648° iv -3c=180π×-3° =18022×7×-3° =-378022° =-1711822° =-171°1822×60′ =-171°49111′ =-171°49’111×60” =-171°49’5.45” ≈-171°49’5” (v) 11c=180π×11° =18022×7×11° =630° vi 1c=180π×1° =18022×7×1° =63011° =57311° =57°311×60′ =57°16411′ =57°16’411×60” =57°16’21.81”≈57°16’22” Q2. Answer : We have:180°=π rad∴1°=π180 […]

Sets, Class 11 Mathematics R.D Sharma Question Answer

Page 1.2 Ex – 1.1 Q1. Answer : Well-defined collections are sets. Example: The collection of good teachers in a school is not a set, It is a collection. Thus, we can say that every set is a collection, but every collection is not necessarily a set. The collection of vowels in English alphabets is a set. […]

Statistics, Class 9 Mathematics R.D Sharma Solutions

Example 1 – Statistics (R.D. Sharma Maths Class 9th) Example 10, Page No. 18.22 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example 12, Page No. 18.23 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example 13, Page No. 18.23 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example […]

Surface Areas and Volumes, Class 9 Maths R.D Sharma Solutions

Example 13, Page No. 18.9 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example 14, Page No. 18.9 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example 15, Page No. 18.9 – Surface Areas & Volumes (R.D. Sharma Maths Class 9th) Example 17, Page No. 18.10 – Surface Areas & Volumes […]

Circles, Class 9 Mathematics R.D Sharma Solutions

Example 3, Circles “Page No. 16.13” (R.D. Sharma Maths Class 9th) Example 4, Circles “Page No. 16.13” (R.D. Sharma Maths Class 9th) Example 6, Circles “Page No. 16.14” (R.D. Sharma Maths Class 9th) Example 8, Circles “Page No. 16.15” (R.D. Sharma Maths Class 9th) Example 14, Circles “Page No. 16.19” (R.D. Sharma Maths Class 9th) […]

Areas of Parallelograms and Triangles, Class 9 Maths R.D Sharma Solutions

Example 8, Page No. 15.18, Areas of Parallelograms and Triangles” (R.D. Sharma Class 9th) https://youtube.com/watch?v=3He-BAXirLc Example 23, Page No. 15.27, Areas of Parallelograms and Triangles (R.D. Sharma Maths Class 9th) https://youtube.com/watch?v=YC6VqS4mRs8 Example 38, Page No. 15.34, Areas of Parallelograms and Triangles (R.D. Sharma Maths Class 9th) https://youtube.com/watch?v=LuJrOBQn_Eo Example 40, Page No. 15.35, Areas of Parallelograms […]

Triangles, Class 9 Mathematics R.D Sharma Solutions

Example 5 Page No 10.6 – Triangles – R.D Sharma Class 9th Maths Example 14, Page No 10.10 – Triangles – R.D Sharma Class 9th Maths Q 1, Ex 10.1, Page No 10.12 – Triangles – R.D Sharma Class 9th Maths Q 3, Ex 10.1, Page No 10.12 – Triangles – R.D Sharma Class 9th […]

Lines and Angles, Class 9 Mathematics R.D Sharma Solutions

Q 3, Ex. 6.3, Page No 107, Lines & Angles – Class 9th R D Sharma Solutions https://www.youtube.com/watch?v=FtvS-MzJEAc Q 6 & Q 9, Ex. 8.1, Page No 8.7, Lines & Angles – R.D Sharma Class 9th Maths Q 12, Ex. 8.1, Page No 8.7 – Lines & Angles, R.D Sharma CBSE Maths Class 9th Example […]

Linear Equations in Two Variables, Class 9 Maths R.D Sharma Solutions

Example 6 – Linear Equations – Page No 13.5, R.D Sharma Maths Class 9th Example 7 – Linear Equations – Page No 13.6, R.D Sharma Maths Class 9th Q 6, Ex. 13.2, Page No 13.7, Linear Equations – R.D Sharma Maths Class 9th Example 4, Page No 13.10, Linear Equations – R.D Sharma Class 9th […]

Polynomials, Class 9 Mathematics R.D Sharma Solutions

Example 1, Page No 4.1, Polynomials, R.D Sharma Maths Class 9th   Example 2, Page No 4.2, Polynomials, R.D Sharma Class 9th Maths   Example 3, Page No 4.3, Polynomials, R.D Sharma Maths Class 9th https://youtube.com/watch?v=P6jbo2W617Y   Example 4, Page No 4.4, Polynomials, R.D Sharma Maths Class 9th Solutions https://youtube.com/watch?v=2nSOSJPoAg4   Example 6 & 8, […]