CBSE Class 11th Maths, NCERT & R.D Sharma Solutions & Sample Papers

Sample Papers

Limits and Derivatives, Class 11 Mathematics R.D Sharma Solutions

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 […]

Introduction to Three Dimensional Geometry, Class 11 Mathematics R.D Sharma Solutions

Page 28.6 Ex 28.1 Q1. Answer : (i) The x-coordinate, the y-coordinate and the z-coordinate of the point (5, 2, 3) are all positive. Therefore, this point lies in XOYZ octant. (ii) The x-coordinate, the y-coordinate and the z-coordinate of the point (−5, 4, 3) are negative, positive and positive, respectively. Therefore, this point lies […]

Conic Sections, Class 11 Mathematics R.D Sharma Solutions

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 Solutions

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 Solutions

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 Solutions

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 Solutions

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 […]

Sequence and Series, Class 11 Mathematics R.D Sharma Solutions

ARITHMETIC PROGRESSION Page 19.4 Ex 19.1 Q1. Answer : Given : an = n2-n+1For n=1, a1 = 12-1+1 = 1For n=2, a2 = 22-2+1 = 3For n=3, a3 = 32-3+1 = 7For n=4, a4 = 42-4+1 =13For n=5, a5 = 52-5+1 = 21 Thus, the first five terms of the sequence are 1, 3, 7, […]

Permutations & Combinations, Class 11 Mathematics R.D Sharma Solutions

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 Solutions

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 Solutions

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 […]

Mathematical Induction, Class 11 Mathematics R.D Sharma Solutions

Page 12.2 Ex – 12.1 Q1. Answer : We have: P(n): n(n + 1) is even. Now, P(3) = 3(3 + 1) = 12 (Even) Therefore, P(3) is even. Page 12.3 Ex – 12.1 Q2. Answer : We have: P(n): n3+n is divisible by 3.Thus, we have:P(3) =33+3 =27+3=30 It is divisible by 3.Hence, P(3) is […]

Trigonometric Functions, Class 11 Mathematics R.D Sharma Solutions

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 […]

Relations & Functions, Class 11 Mathematics R.D Sharma Solutions

Page 2.8 Ex – 2.1 Q1. Answer : (i) a3+1, b-23=53, 13 By the definition of equality of ordered pairs, we have: a3+1, b-23=53, 13 ⇒a3+1 = 53 and b-23=13⇒a3= 53-1 and b = 13+23⇒a3=23 and b = 1⇒a = 2 and b = 1 (ii) (x + 1, 1) = (3, y − 2) By the […]

Sets, Class 11 Mathematics R.D Sharma Solutions

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. […]

Axiomatic Probability, Class 11 Mathematics Sample Papers

1. There will be total 10 MCQ in this test. 2. Please keep a pen and paper ready for rough work but keep your books away. 3. The test will consist of only objective type multiple choice questions requiring students to mouse-click their correct choice of the options against the related question number. 4. No negative […]