Let A, B, C be subsets of U, then A⊆B, B⊆C ⇒ A⊆C.
Let A, B be subsets of U, then A=B ⇒ B=A.
Let A, B, C be subsets of U, then A=B and B=C ⇒ A=C.
Let A be any subset of U, then A⊆ϕ ⇒ A=ϕ.
Let A be any subset of U, then (i) A∪A=A and (ii) A∩A=A.
Let A, B be subsets of U, then (i) A∪B=B∪A and (ii) A∩B=B∩A.
Let A be any subset of U, then (i) A∪U=U (ii) A∩U=A (iii) A∪ϕ=A (iv) A∩ϕ=ϕ.
Let A, B be subsets of U, then (i) A⊆A∪B and (ii) A∩B⊆A.
Let A, B be subsets of U, then (i) (A∪B)∩A=A and (ii) (A∩B)∪A=A.
Let A, B, C be subsets of U, then (i) (A∪B)∪C=A∪(B∪C) and (ii)(A∩B)∩C=A∩(B∩C).
Let A, B, C be subsets of U, then (i) A∪(B∩C)=(A∪B)∩(A∪C) and (ii) A∩(B∪C)=(A∩B)∪(A∩C).
Let A be subset of U, then (i) A∪A'=U (ii) A∩A'=ϕ (iii) ϕ'=U (iv) U'=ϕ (v) A''=A.
State an prove De - Morgan's laws.
For any three sets A, B and C, (i) A-(B∪C)=(A-B)∩(A-C) (ii) A-(B∩C)=(A-B)∪(A-C).
For any two sets A and B, prove that A-B=A-(A∩B).
For any two sets A and B, prove that (A-B)∪(B-A)=(A∪B)-(A∩B).
For any two sets A and B, prove that A-B⊆A.
For any two sets A and B, prove that A-B=A∩B'
For any two sets A and B, prove that A-B⊆B'.
For any two sets A and B, prove that A∩B=ϕ ⇒A⊆B'.
For any two sets A and B, prove that A∩B=ϕ ⇒A∩B'=A.
For any two sets A and B, prove that A∩B=ϕ ⇒A∪B'=B'.
For any two sets A and B, prove that A⊆B ⇒B'⊆A'.
For any two sets A and B, prove that A⊆B⇒A∩B=A.
For any two sets A and B, prove that A⊆B⇒A∪B=B.
Discussion on Real numbers.
Properties of Real numbers.
Representation of numbers on real line.
Intervals of real line.
Definition and meaning of absolute value of a real number.
Solve the inequalities (i) 2x+1<3 (ii) -2≺3x-5<4.
Solve the inequality (x-2)(x+3)≥0.
Solve the inequality 5x²+12x≥4.
Solve the inequality 3x³-12x<0.
Solve the inequality x(x+2)⁄(x-1)≤0.
Find A∪B where A=(-2, 3) and B= (1, 5).
Find A∩B where A=(-2, 3) and B= (1, 5).
Find A-B where A=(-2, 3) and B= (1, 5).
Find A∪B where A=[-1, 5) and B= 3, 8].
15. (N) Find A-B where A=[-1, 5) and B= [3, 8].
Find A∪B where A=(-7, 1) and B= [-5, 3).
Find A∩B where A=(-7, 1) and B= [-5, 3).
Find A-B where A=(-7, 1) and B= [-5, 3).
Find A' and B' where A=(-7, 1) and B= [-5, 3).
Let x,y,z∈R. Prove that x+z=y+z⇒x=y.
Let x,y,z∈R. Prove that x≻y⇒x+z≻y+z.
Let x,y,z∈R. Prove that x≻y⇒x+z≻y+z.
Let x,y∈R. Prove that x.y=1⇒x=y=1⁄x.
Let x,y,z∈R. Prove that x≺y, y≺z⇒x≺z.
Let x,y,z∈R and z≻0. Prove that x≻y⇒xz≻yz.
Let x,y,z∈R and z≺0. Prove that x≻y⇒x⁄z≺y⁄z.
Let x∈R. Prove that ⅠxⅠ≥0.
Let x∈R. Prove that x≤ⅠxⅠ and -x≤ⅠxⅠ.
Let x∈R. Prove that -ⅠxⅠ≤x≤ⅠxⅠ.
Let x,y∈R. Prove that -Ⅰx+yⅠ≤ⅠxⅠ+ⅠyⅠ.
Let x,y∈R. Prove that ⅠxⅠ-ⅠyⅠ≤Ⅰx-yⅠ.
Let x,y∈R. Prove that |xy|=|x||y|.
32. (N) Let x,y∈R. Prove that |Ⅰx⁄y|=|x|⁄|y| , y≠0.
Prove that |x|≤a⇔-a≤x≤a for all x∈R and a>0.
34. (N) Rewrite |x-2|<3 without the absolute value sign.
35. (N) Rewrite |5x-2|>8 without the absolute value sign.
Rewrite |5x-2|≥3 without the absolute value sign.
Find x and y if (x-1, 3)=(2, x+2y).
Find A×B, if A={x∶ x=1,3,5} and B={y∶ y=2x}.
Find A² if A ={3, 5, 7}.
Find (a) ( A∩B)×(C∩D) and (b) (A×C)∩(A×D)∩(B×C)∩(B×D).
Prove that A×(B∪C)=(A×B)∪(A×C).
Definition of a Relation. Domain and range of a relation.
Let A={1,2,3} and B={1,3,7}. Find a relation R from A to B such that (i) x›y (ii) x=y (iii) x=y+3.
Let R = {(1, 5), (3, 2),(1, 1),(3, 1)} be a relation. Find its domain and range.
Let A={2, -2, 4} and B={2, 3, 6}. Find a relation R such that x∈A divides y∈B. Write the domain and range of R.
Let L be a set of all lines in a plane and R be a relation on L (x is parallel to y). Prove that R is an equivalence relation.
Definition
One to one, Onto and Bijective functions.
Constant, Identity, Equal, Linear and Quadratic functions.
Let R be a relation from A={1,2,3} to B={3,2,5} given by {(1,2),(2,2),(3,3),(2,5)}. Is R a function from A to B?
Let f∶ A→Z be a function from A={0,1,2} to the set of integers Z defined by f(x)=2x-3. Find the image 1 and 4.
Let f∶ R→R be a function defined by f(x)=x for x<0; x+1 for x=0; 1 for x>0. If h<0, find f(0+h) and f(0-h).
Let f∶ A→Z be a function from A={0,1,2,3} to the set of integers Z defined by f(x)=x-4. Find the range of the function.
Let f∶ Z→Z be a function defined by f(x)=x². Is f one to one⁇
Let f∶ Z→Z be a function defined by f(x)=x². Is f an onto function?
Let f∶ N→Z be a function defined by f(x)=x². Show that f is one to one but not onto.
Let f∶ N→A be a function defined by f(x)=x². If A is the set of squares of natural numbers, show that f is one to one and onto.
Let f∶ N→O be a function defined by f(x)=2x+1. If O is the set of odd integers, show that f is one to one and onto.
Let f∶ R→R be a function defined by f(x)=x³. Show that f is one to one and onto.
Let f∶ A→B be a function from A={-2,0,2,3,5} and B={10,7,-2,0,3}. Test if f is one to one or onto or both or neither.
Definition
Let f∶ A→B be a function as shown in the diagram. Find (i) f¯¹(0) (ii) f¯¹(1) (iii) f¯¹(3) (iv) f¯¹(4) and f¯¹(1,3,4).
Let f∶ A→B be a function where A={0,1,2,3} and B={4,5,6,7}. If f(0)=7, f(1)=6, f(2)=5, f(3)=4, write f¯¹∶ B→A as a set of ordered pairs.
Let f∶ R→R be a function defined by f(x)=(x⁄2)-3. Find a formula for f¯¹.
Let f∶ R→R be a function defined by f(x)=x³+1. Find a formula for f¯¹.
Let f∶ R-{0}→R-{3} be a function defined by f(x)=(3x-2)⁄x. Show that f is a bijective function. Also show that f(f¯¹(x))=x=f¯¹(f(x)).
Let f∶ R-{3}→R-{2} be a function defined by f(x)=2x)⁄(x-3). Show that f is a bijective function. Also show that f(f¯¹(x))=x=f¯¹(f(x)).
Definition
Let f∶ R→R and g∶ R→R be defined by f(x)=3x+1 and g(x)=x-3. Find f ∘ g and g ∘ f.
Let f∶ R→R defined by f(x)=x². Find f²(x).
Let f∶ R→R and g∶ R→R be defined by f(x)=x⁄2 and g(x)=2⁄x. Find f ∘ g.
5. (N) Let f∶ R+→R and g∶ R→R be defined by f(x)=log(x) and g(x)=e^x². Find f ∘ g and f ∘ g.
Let f∶ R→R and g∶ R→R be defined by f(x)=x²-1 and g(x)=x³. Find f ∘ g and g ∘ f.
Let f∶ R→R and g∶ R→R be defined by f(x)=x³+1 and g(x)=2x-3. Find f ∘ g(2) and g ∘ f(-1).
Definition and examples
Definition and examples.
Find the domain and range of the function y=f(x)=1⁄√(x²-x-2).
Definitions
sinh x, cosh x, tanh x, coth x, cosech x, sech x
cosh²x+sinh²x=cosh(2x) cosh²x-sinh²x=1 tanh²x+sech²x=1 coth²x-cosech²x=1
cosh (x+y)=coshx coshy+sinhx sinhy.
cosh (x-y)=coshx coshy-sinhx sinhy.
sinh (x+y)=sinhx coshy + coshx sinhy.
sinh (x-y)=sinhx coshy-coshx sinhy.
tanh (x+y)=(tanhx+tanhy)⁄(1+tanhx tanhy).
tanh (x-y)=(tanhx-tanhy)⁄(1-tanhx tanhy).
coth (x-y)=(1-cothx cothy)⁄(cothy-cothx)
coth (x+y)=(cothx cothy+1)⁄(cothx+cothy).
sinh(2x)=2sinhx coshx
cosh(2x)=cosh²x+sinh²x
tanh(2x)=2tanhx⁄(1+tanh²x)
coth(2x)=(1+coth²x)⁄2cothx
Definition
(i) log1=0 (ii) loga_a=1 (iii) logx+logy=log(xy)
logx-logy=log(x⁄y)
logxⁿ=nlogx
logx_a=logx_b⁄loga_b
logx_a=logx⁄loga
a^(logx_a)=x
loga^x_a=x
Prove that log(x²y³⁄z⁴)=2logx+3logy-4logz.
Prove that log(x²⁄yz)+log(y²⁄zx)+log(z²⁄xy)=0.
Prove that log(1+2+3)⁴=log1⁴+log2⁴+log3⁴.
Prove that x^(logy-logz)× y^(logz-logx)× z^(logx-logy)=1
Prove that (xy)^(logy-logz)× (yz)^(logz-logx)× (zx)^(logx-logy)=1
Prove log[√a(√a(√a(√a²)))]=1.
If logx⁄(y-z)=logy⁄(z-x)=logz⁄(x-y), prove that x^xy^yz^z=1.
If a²+b²=2ab, prove that log(a+b)⁄2=(loga+logb)⁄2.
If a²+b²=6ab, prove that log(a-b)⁄2=(loga+logb)⁄2.
If a²+1⁄b²=6a⁄b, prove that log(ab-1)⁄2b=(loga-logb)⁄2.
If x=loga, y=log2a, z=log3a, prove that xyz+1=2yz
1. Reference: (i) origin (ii) points on axes 2. symmetry 3. increasing and decreasing 4. periodicity 5. asymptote
Derive equation of the ellipse whose major axis is the y axis.
Test whether the function f(x)=x²-5 is odd or even.
Test whether the function f(x)=x² sinx is odd or even.
Test whether the function f(x)=x cosx +sinx is odd or even.
Test whether the function f(x)=√(4+x) +√(4-x) is odd or even.
Test whether the function f(x)=√(4+x) - √(4-x) is odd or even.
Test whether the function f(x)=√(4+x²) +√(4-x²) is odd or even.
Test whether the function f(x)=2^(x)-2^(-x) is odd or even.
Test whether the function f(x)=[2^(x)-2^(-x)]⁄[2^(x)+2^(-x)]is odd or even.
Test whether the function f(x)=[2^(x)-2^(-x)]⁄[2^(x)-2^(-x)]is odd or even.
Test the symmetry of the function f(x)=x²sinx.
Test the symmetry of the function f(x)=√(4+x) +√(4-x).
Test periodicity of the function f(x)= cosbx.
Test periodicity of the function f(x)= sinx⁄2.
Test periodicity of the function f(x)= tan3x.
Test periodicity of the function f(x)= cos2x+tan2x.
Test periodicity of the function f(x)= cot3x+sinx.
Find the asymptotes of the curve y=2x.
Find the asymptotes of the curve y=x².
Find the asymptotes of the curve y=1⁄(x-1).
Find the asymptotes of the curve y=2^x.
Find the asymptotes of the curve y=3^(-x)+1.
Sketch the graph of the curve y=4x+2.
Sketch the graph of the curve y=(x²-4)⁄(x-2).
Sketch the graph of the curve y=x².
Sketch the graph of the curve y-4=(x-2)².
Sketch the graph of the curve y+2=(x-3)².
Sketch the graph of the curve y=-2x²+4x+6.
Sketch the graph of the curve y=x³.
Sketch the graph of the curve y+1=(x-1)³.
Sketch the graph of the curve y=x(x-1)(x+2).
Sketch the graph of the curve y=2^x.
Sketch the graph of the curve y=3^(-x).
Sketch the graph of the curve y=logx.
Sketch the graph of the curve y=log(-x).
Sketch the graph of the curve xy=1.
Formula and Properties
Formula and Properties
Properties
Means of arithmetic, geometric and harmonic sequences.
The A.M., G.M. and H.M. between the numbers a and b are given by (a+b)⁄2, √(ab) and 2ab⁄(a+b) respectively.
The A.M., G.M. and H.M. between any two positive numbers satisfy (a) (GM)²=AM⨉HM (b) AM≥GM≥HM
If n A.M’s be inserted between two numbers a and b, the common difference, d is given by d=(b-a)(n+1) and the sequence is given by a, a+d, a+2d, ...b.
If n A.M’s be inserted between two numbers a and b and k=(n+1)⁄2, prove that the kth mean is given by Ak=(a+b)⁄2.
Prove that the sum of n arithmetic means between a and b is given by n(a+b)⁄2.
In an AP, prove that t_p=t_n+(p-n)d where d=(t_m-t_n)⁄(m-n).
If x,y,z are the sums of the first p,q,r terms of an AP, prove that x(q-r)⁄p+y(r-p)⁄q+z(p-q)⁄r=0
If b+c, c+a, a+b are in AP, prove that a, b, c are also in AP.
In an AP, if S_n=S_m, prove that S_(n+m)=0.
If a², b², c² are in AP, prove that b+c, c+a, a+b are in HP.
If a, b, c are in HP, prove that (b+a)⁄(b-a)+(b+c)⁄(b-c)=2.
If H be the HM between a and b, prove that (H-2a)(H-2b)=H².
If a, A, b form an AP and a, G₁, G₂, b are in GP, prove that (G₁)²⁄G₂+(G₂)²⁄G₁=2A
If a, x, b form an AP; a, b, c form GP and b, y, c form an AP, prove that x, b, y form an HP.
If a, b, c are in GP, then prove that a+b, 2b, b+c are in HP.
If a+b, 2b, b+c are in HP, then prove that a, b, c are in GP.
If a, 2b, c are in HP, prove that a-b, b, c-b are in GP.
If a-b, b, c-b are in GP, prove that a, 2b, c are in HP.
If the mth term of an AP is n and the nth term is m, show that the (m+n)th term is zero.
If the mth term of an AP is n and the nth term is m, show that the pth term is equal to m+n-p.
If (x-y)⁄(y-z)=x⁄x or x⁄y or x⁄z prove that x, y, z are in AP or GP or HP.
If (x+y)⁄2, y, (y+z)⁄2 are in HP, prove that x, y, z are in GP.
If G is the GM between a and b prove that 1⁄(G²-a²) + 1⁄(G²-b²) = 1⁄G².
If H is the HM between a and b show that 1⁄(H-a) + 1⁄(H-b) = 1⁄a+1⁄b.
If A be the AM and H be the HM between a and b show that (a-A)⁄(a-H) + (b-A)(b-H) = A⁄H.
If x be the AM between y and z, y be the GM between z and x then prove that z will be the HM between x and y.
If b be the AM between a and c, c be the HM between b and a then prove that a will be the GM between c and b.
If a, b, c be in AP, b, c, d in GP and c, d, e in HP, prove that a, c, e are in GP.
Show that y² is greater than, equal to or less than zx according as x, y, z are in AP or GP or HP respectively.
Find the two numbers whose AM is p and GM is q.
Find the two numbers whose AM is 25 and GM is 20.
The AM between two numbers exceeds their GM by 2 and the GM exceeds the HM by 1.6. Find the numbers.
The sum of three positive numbers is 36. When the numbers are increased by 1, 4, and 43 respectively, the resulting numbers are in GP. Find the numbers.
If a^x = b^y = c^z and a, b, c are in GP, prove that x, y, z are in HP.
Sum formula
Which series has finite sum (i) 1.5+1.5²+1.5³+... (ii) 0.9-0.9²+0.9³-0.9⁴+…
Find the sum of the infinite series 3+3⁄2+3⁄4+3⁄8+....
Find the sum of the infinite series 2+√2+1+1⁄√2+...
Find the sum of the infinite series 2⁄3+5⁄3²+2⁄3³+5⁄3⁴+2⁄3⁵+5⁄3⁶+….
Find the sum of the infinite series a⁄x+b⁄x²+a⁄x³+b⁄x⁴+a⁄x⁵+b⁄x⁶+….
The sum to infinity of a GS is 25, and the first term is 5. Find its ratio.
The sum to infinity of a GS is 25, and 4 times each term is equal to the sum of all terms which follow it. Find the series.
The sum to infinity of a GS is 15, and the sum of their squares is equal to 45. Find the series.
The sum of first two terms of an infinite GS is 12 and each term is equal to twice the sum of all terms which follow it. Find a, r and the series.
Prove that 2^(1⁄3)×2^(1⁄9)×2^(1⁄27)....=√2
A ball, after striking the ground ascends to the rth fraction of its previous descent. If a be the initial descent of the ball, find the total distance covered before coming to rest.
A rubber ball is dropped from a height of 16ft. At each rebound it rises to a height which is 3⁄4th of the previous fall. What is the total distance covered before coming to rest.
A rubber ball is dropped from a height of 100ft At each rebound it rises to a height which is 2⁄3rd of the previous fall. What is the total distance covered before coming to rest.
Definition
Square matrix, Null matrix, Diagonal matrix, Scalar matrix, Unit matrix, Upper and lower triangular matrix, Transpose of matrix, Symmetric and skew - symmetric matrix,
Scalar multiplication
Addition and Subtraction
Matrix multiplication
For any given square matrix A, prove that A+A' is a symmetric matrix and A-A' is a skew symmetric matrix.
Find a 2⨉3 matrix A= (a_ij) where a_ij=|2i-3j|.
Find a 3⨉3 matrix A= (a_ij) where a_ij=2^i -3^j.
Introduction
Using Minors, Using Cofactors and Using Sarrus rule
Find the value of the given determinant by expanding about second row and third column and also by Sarrus rule.
Prove that the value of the given determinant is equal to zero.
Prove that the value of the given determinant is equal to zero.
Solve the matrix equation to find the value of x.
Without expanding, prove that the value of the given determinant is equal to zero.
Without expanding, prove that the value of the given determinant is equal to zero.
Without expanding, prove that the value of the given determinant is equal to zero.
Without expanding, prove that the value of the given determinant.
Without expanding, prove that the value of the given determinant.
Without expanding, find the value of the given determinant.
Without expanding, prove that the value of the given determinant is equal to (a-b)(b-c)(c-a).
Prove that the value of the determinant is equal to xyz(1+a⁄x+b⁄y+c⁄z)
Prove that the value of the determinant is equal to 2(x+y)(y+z)(z+x).
Prove Δ = (x²+y²+z²)(x+y+z)(x-y)(y-z)(z-x).
Find the adjoint of the given matrix.
Find the inverse of the given square matrix. of order 2
Verify that A(adj.A)=ΔI
Prove that given square matrices of order 2 are inverse of each other.
Prove that given square matrices of order 3 are inverse of each other.
Find the inverse of the given square matrix. of order 3.
Sum, Product, Quotient and reciprocal
Commutivity property
Associativity with respect to addition.
Associativity with respect to multiplication.
Distributive property
Prove: i²=-1.
Prove: (a, b)= a + ib.
Prove: a + i b = 0 iff a=0 and b = 0.
Prove: a + ib = c + id iff a=c and b=d.
Powers of i
Conjugate of a complex number.
Modulus of a complex number
Real and imaginary parts of a complex number.
Conjugate of sum and product of complex numbers.
Conjugate of squares equals to Square of conjugate.
Double conjugate of a complex number z equals to z.
The roots of the equation ax²+bx+c=0.
Nature of roots of quadratic equations
Show that a quadratic equation cannot have more than two roots.
Explore the nature of the roots of 3x²+2x-1=0.
Explore the nature of the roots of x²+√x=0.
If the roots of equation x²-kx+1=0 are equal, find k.
If the roots of (a²+b²)x²-2(ac+bd)x+(c²+d²)=0 are equal, prove that a∕b=c∕d.
If a,b,c are rational and a+b+c=0 prove that the roots of (b+c-a)x²+(c+a-b)x+(a+b-c)=0 are rational.
If the roots of qx²+2px+2q=0 are real and unequal, prove that the roots of (p+q)x²+2qx+(p-q)=0 are imaginary.
Find the relation between roots and coefficients of a quadratic equation.
Express a quadratic equation expressed in terms of its roots.
Symmetric relation of roots of a quadratic equation.
Find the quadratic equation whose roots are 2 and 5.
Find the quadratic equation whose one root is 2-√3.
Find the quadratic equation whose one root is 2-3i.
If α and β are the roots of 2x²-9x+10=0 then find the quadratic equation whose roots are α² and β².
If α and β are the roots of ax²+bx+c=0 then find the quadratic equation whose roots are α+h and β+h.
Symbolic representation of Inverse trigonometric functions
Inverse sine function
Inverse cosine function
Inverse tangent function
Inverse cosecant function
Inverse secant function
Inverse cotangent function
For any angle x, prove that sin⁻¹sinx=x.
For real number y, prove that sinsin⁻¹y=y.
For real number x, prove that sin⁻¹x=cosec⁻¹1∕x.
For real number x, prove that cosec⁻¹x=sin⁻¹1∕x.
The general solution of sinx = sinα is given by x=nπ+(-1)ⁿα, n∈Z.
The general solution of sinx = 0 is given by x=nπ, n∈Z.
The general solution of sinx = 1 is given by x=(4n+1)π∕2, n∈Z.
The general solution of cosx = cosα is given by x=2nπ±α.
The general solution of cosx =1 is given by x=2nπ.
Solve the equation sin2x=0 for its general solution.
Solve the equation sin2x=1 for its general solution.
Solve the equation sin3x=-1 for its general solution.
Solve the equation sin2x=1∕2 for its general solution.
Solve the equation sin6x-cos3x=0 for its general solution.
Solve the equation sin6x-sin3x=0 for its general solution.
The general solution of cosx =1 is given by x=2nπ.
Length of perpendicular from a point (x₁, y₁) on a line x cosα + y sinα=p (normal form).
Length of perpendicular from a point (x₁, y₁) on a line ax + by + c=0 (general form).
5. (N) Equation of bisectors of angles between two lines a₁x + b₁y + c₁=0 and a₂x + b₂y + c₂ = 0.
Check whether the points (0, 0) and (-4, 1) lie on same or opposite sides of 2x+3y-6=0.
Find the distance between the point (3, 4) and the line 4x-3y+5=0.
Find the distance between the point (3, 4) and the line 4x-6=0.
Find the value of k if the distance of the point (-3, 2) from the line kx-4y+7=0 is equal to 2.
Find the distance between the parallel lines 2x-3y+9=0 and 2x-3y=4.
Find the distance between the parallel lines 3x-4y+9=0 and 6x-8y=2.
Find the points on x - axis which are at a distance a from the line x⁄a + y⁄b = a.
Find the equation of the line parallel to x+3y=5 at a distance √10 from the origin.
Find the equation of the line perpendicular to x+3y=5 at a distance √10 from the origin.
If p and p' be the length of the perpendiculars from O on the lines x secθ+y cosecθ=a and x cosθ - y sinθ = a cos2θ respectively, prove that 4p²+p'²=a².
Find the height of a triangle whose vertex is at (-3, 1) and equation of base is 3x-4y=2.
Find the points on the line x+2y=5 which are 2 units apart from the line 5x-12y=7.
If p be the length of the perpendicular dropped from origin on the line x⁄a + y⁄b=1, prove that 1⁄a²+1⁄b²=1⁄p².
The lengths of the perpendiculars drawn from the points (cosθ, sinθ) and (-secθ, cosecθ) on the line x secθ + y cosecθ are p and p' respectively, prove that 4⁄p²-p'²=4.
Find the equation of the line through (a, 0) at a distance a from the point (2a, 2a).
one corner of a square is at the origin and two of its sides are given by y=-2x and y=-2x+3. Find the equation of the other sides.
Prove that ax²+2hxy+by²=0 represents a line pair through the origin.
Find the angle between the line pair ax²+2hxy+by²=0. (1st method)
Find the angle between the line pair ax²+2hxy+by²=0. (2nd method)
Find the perpendicularity and coincidence conditions for the line pair ax²+2hxy+by²=0.
Find the equation of the bisector of angles between the line pair ax²+2hxy+by²=0.
7. (N) Find the equation of the bisector of angles between the line pair ax²+2hxy+by²=0. (Geometrical approach)
Find the condition that the general equation of second degree ax²+2hxy+by²+2gx+2fy+c=0 may represent a line pair.
Find the equation of the line pair through the origin and parallel to the line pair given by the equation ax²+2hxy+by²+2gx+2fy+c=0.
If the equation ax²+2hxy+by²+2gx+2fy+c=0 represents a pair of parallel lines then prove that a⁄h=h⁄b=g⁄f.
If the equation ax²+2hxy+by²+2gx+2fy+c=0 represents a pair of parallel lines, prove that the product of the perpendiculars from the origin on these lines is equal to c⁄√[(a-b)²+4h²].
Find the product of the length of the perpendicular from (x₁,y₁) on the line ax²+2hxy+by²=0.
Determine the lines represented by x²-8xy+15y²=0.
Determine the lines represented by x²+y²+2xy secθ=0.
Determine the lines represented by x²-y²+2xy cotθ=0.
Determine the lines represented by x²+2xy+y²-2x-2y-15=0.
Determine the lines represented by x²-5xy+4y²+x+2y-2=0.
Determine the lines represented by 2x²+7xy+3y²-4x-7y+2=0.
Determine the lines represented by 6x²-xy-12y²-8x+29y-14=0.
If the equation 2x²+kxy+3y²-4x-7y+2=0 represents a line pair, find the value of k.
If the equation 2x²+7xy+3y²-4x-7y+k=0 represents a line pair, find the value of k.
Show that the pair of lines x²+2xy+y²-2x-2y-15=0 are parallel to each other. Also find the distance between them.
Show that the pair of lines 6x²+5xy-6y²-4x+7y-2=0 are perpendicular to each other.
Find the angle between the pair of lines x²+9xy+14y²=0.
If the equation (c²-a²m²)x²+2a²mxy+(c²-a²)y²=0 represents perpendicular line pair, then prove that 2c²=a²(1+m²).
Find the equation of the pair of lines joining the origin and the point of intersection of the line y=mx+c and the curve x²+y²=a².
Find the equation of the pair of lines joining the origin and the point of intersection of the line y=3x+2 and the curve x²+3y²+2xy+4x+8y-11=0.
Prove that lines joining the origin and the point of intersection of the line x⁄a + y⁄b=1 and the curve x²+y²=c² are at right angles if 1⁄a²+1⁄b²=2⁄c².
Find the equation of the bisectors of the angle between x²-pxy-y²=0.
If the pair of lines x²-2pxy-y²=0 and x²-2qxy-y²=0 be such that each pair bisects the angle between the other pair. Prove that pq+1=0.
If ax²+2hxy+by²=0 and a'x²+2h'xy+b'y²=0 are lines having same bisectors, prove that h(a'-b')=h'(a-b).
Find the equation of the lines through the origin and right angles to the pair of lines ax²+2hxy+by²=0.
Find the distance of (2, -3, 4) from the origin.
Find the condition that a point P(x,y,z) is always 5 units far from the point (1,2,3).
Find the section point that divides the join of (-2,-1,2) and (2,3,6) internally in the ratio of 1∕3.
Find the section point that divides the join of (-2,-1,2) and (2,3,6) externally in the ratio of 1∕3.
In what ratio the point (0,1,5) divides the join of (-4,-5,3) and (6,10,8)?
Find a point on x axis which divides the join of (-3,-4,-8) and (4,3,6).
Find a point on xy plane that divides the line through (3,2,1) and (-2,-3,-4).
Find the mid point of the sides of the triangle formed by points (3,2,5), (-1,6,-1) and (7,-4,7).
Find the mid point of the sides of the quadrilateral formed by points (3,2,5), (-1,6,-1), (7,-4,7) and (3,4,9).
Find the centroid of the triangle formed by points (3,2,6), (-1,6,-1) and (7,-5,7).
Show that the points (1,4,2), (3,1,3), (5,0,8) and (3,3,7) form a parallelogram.
Four points A,B,C and D form a parallelogram. If A(1,4,2), B(3,1,3) and C(5,0,8), find the coordinates of D.
The centroid of ∆ABC is at (3,1,4). If A(3,2,6) and B(-1,6,-1), find the coordinates of point C.
Show that the points (0,7,10), (-1,6,6) and (-4,9,6) form an isosceles right triangle.
Find the point where the line joining (-2,7,2) and (2,3,6) cuts the plane 2x+y+z=7.
Relation of direction cosines of a line.
Angle between two lines in terms of direction cosines
If a line makes angles α, β and γ with coordinate axes, prove that cos2α+cos2β+cos2γ=-1.
If a line makes angles α, β and γ with coordinate axes, prove that sin²α+sin²β+sin²γ=2.
If the direction cosines of a line are 2∕3, 1∕3 and 2∕c, then find the value of c.
If a line makes angle of 60° to both of x and y axes then find its angle with z axis.
Find the direction cosines of a line equally inclined to the axes.
If two angles that a line makes with axes are 30° and 60° then find the third angle.
If two angles that a line makes with axes are 10° and 80° then find the third angle.
Find the direction cosines of the line through the points (1,3,2) and (2,5,4).
If P(1,5,3) and Q(4,8,k) are two points such that line PQ is equally inclined to the axes, then find the value of k.
Find the angle between the lines whose direction cosines are proportional to 3, -1, 5 and 2, 1, 3 respectively.
Verify that the diagonal AC and BD of the rhombus ABCD are perpendicular to each other where A(5, -1, 1), B(7, -4, 7), C(1, -6, 10) and D(-1, -3, 4).
If the join of A(5,k,1) and B(1, -6, 10) is perpendicular to the join of C(7,-4,7) and D(-1,-3,4), find the value of k.
If the line joining A(5,k,2) & B(1,6,10) is parallel to the line joining C(4,-4,3) & D(2,-3,7), find the value of k.
Find the angle between the diagonals of a cube.
Find the angle between lines whose direction cosines satisfy l + m + n = 0 and l²+m² = n².
Find the angle between lines whose direction cosines satisfy l + m + n = 0 and lm + mn = 2nl.
Find the projection on the coordinate axes of the line segment joining (1, 1, 5) and (2, -3, 7).
Prove that lim (xⁿ-aⁿ)⁄(x-a)=naⁿ-¹ as x→a.
Evaluate lim x²⁄(x-1) when x→0.
Evaluate lim(x²-x-6)⁄(x-3) when x→3.
Evaluate lim(x²+x-6)⁄(2x²-x-6) when x→2.
Evaluate lim [2⁄(x-1) - 4⁄(x²-1)] when x→1.
Evaluate lim[(2x²-4x-24)⁄(x²-16) - 4⁄(4-x)] when x→4.
Evaluate lim (x-5)⁄[√(2x-1) - 3] when x→5.
Evaluate lim (2x²-32)⁄[√(2x+1) - 3] when x→4.
Evaluate lim (x-√(8-x²)) ⁄[√(x²+12) -4] when x→2.
Evaluate lim [√x-√(x-1)] when x→∞.
Evaluate lim [√(x-a)-√(x-b)] when x→∞.
Evaluate lim √x[√(x-a)-√(x-b)] when x→∞.
Evaluate lim [√(ax)-√(bx)] when x→∞.
Evaluate lim [√x-√2b)] when x→∞.
Find the limit of [⁶√x-3]/[∛x-9] as x→729.
Find the derivative dy⁄dx if y=(3t² -5)^(2’3)and t=√(8x-1).
Prove that Limit of sinθ⁄θ as θ→0 is equal to 1.
Find the limit of sinax⁄x as x→0.
Find the limit of tanx⁄x as x→0.
Find the limit of sinax⁄sinbx as x→0.
Find the limit of sinmx⁄tannx as x→0.
Find the limit of tan(x-p)⁄(x²-p²) as x→0.
Find the limit of [cos(ax)-cos(bx)]⁄x² as x→0.
Find the limit of [sin(ax)-sin(bx)]⁄x as x→0.
Find the limit of [sin(ax) cos(bx)]⁄sincx as x→0.
Find the limit of (1-cosax)⁄(1-cosbx) as x→0.
Find the limit of x cosx as x→0.
Find the limit of x cotx as x→0.
Find the limit of [1-cosx)⁄x² as x→0.
Find the limit of [1-cosx)⁄(x-π)² as x→π.
Find the limit of [sinx+tanx]⁄x as x→0.
Find the limit of [tanx-sinx]⁄x³ as x→0.
Find the limit of [tan2x-sin2x]⁄x³ as x→0.
Find the limit of [tan2x-sin2x]⁄x³ as x→0.
Find the limit of [cosecx-cotx]⁄x as x→0.
Find the limit of [1-cos7x)⁄x² as x→0.
Find the limit of tan(x-a)⁄(√x-√a) as x→a.
Find the limit of [cosx-cosy]⁄(x-y) as x→y.
Find the limit of [xcosθ-θcosx]⁄(x-θ) as x→θ.
Find the limit of [(x+z)sec(x+z)-zsecz)]⁄x as x→0.
Find the limit of [xcotθ-θcotx]⁄(x-θ) as x→θ.
Limit of log(1+x)/x as x→0.
Limit of (e^x-1)/x as x→0.
Limit of (a^x-1)/x as x→0.
Evaluate the limit of (e^ax-1)/x as x→0.
Find the value of the limit x. 3^(x+2)/(e^3x - 1) as x tends to 0.
Find the value of the limit (e^2x - 1)/x.2^(x+1) as x tends to 0.
Find the value of the limit (a^x+b^x - 2)/x as x tends to 0.
Find the value of the limit (e^ax-e^bx)/x as x tends to 0.
Find the value of the limit log(x-a+1)/(x-a) as x tends to a.
Find the value of the limit log(x-3)/(x-4) as x tends to 4.
Find the value of the limit cosx / log(x+1-pi/2) as x tends to pi/2.
Find the right and left hand limits of the given function.
Find the right and left hand limits of the given function as x tends to 1.
Find the right and left hand limits of the given function as x tends to 2.
Find the right and left hand limits of the absolute value function.
Find the right and left hand limits of the absolute value function as x tends to 0.
Find the right and left hand limits of the absolute value function f(x)=|x-3|/(x-3) as x tends to 3.
Evaluate the right and left hand limits of √(x-1) as x tends to 1.
Continuity and types of discontinuity.
Test the continuity of f(x)=x+2 at x=1.
Test the continuity of f(x)=x²+1 at x=0.
Test the continuity of f(x)=x²/(x-1) x=1.
Test the continuity of f(x)=x²/(x-1) at x≠1.
Test the continuity of f(x)=(x²-4)/(x-2) at x=2.
Test the continuity at x=2 of f(x)=x+2 for x<2 ; f(x)=3x+2 for x≥2.
Test the continuity at x=2 of f(x)=4x for x<2 ; f(x)=8 for x=2; f(x)=x²+4 for x>2.
Test the continuity of the given function at x=1. If it is not continuous, can you make it continuous at x=1?
Derivative of the quotient f(x)/g(x)
Using definition, find the derivative of 2x² -3x+1.
Using definition, find the derivative of 1/(x-2).
Using definition, find the derivative of 1 ⁄ (3-x).
Using definition, find the derivative of 1 ⁄ (3x+5).
Using definition, find the derivative of √x+x.
Using definition, find the derivative of 1 ⁄ √x.
Using definition, find the derivative of 1 ⁄ √(4x+3).
Find the derivative of x³-3x²+3x-1.
Find the derivative of (x³-3x²+3x-1) ⁄ 2x².
Find the derivative of the given product.
Find the derivative of the (x^(3⁄2)-x^(-1⁄2))².
Find the derivative of the (x²-5x)(3x³-2x²).
Find the derivative of the (b-5√x)(b+3√x ).
Find the derivative of the [b+x^(3/4)][b-x^(1/4)]
Find the derivative of the x/(x+2 ).
Find the derivative of the (x²-b²)⁄(x²+b²). 06.39.17.
Find the derivative of the √x⁄(2+√x). 06.44.18.
Find the derivative of the (x-1)⁄(4-4x-x²).
Find the derivative of the (3x-5)².
Find the derivative of the (2x²-x+2) ⁵.
Find the derivative of the (1-2x-3x²)¯⁵.
Find the derivative of the x³ with respect to x².
Find the derivative of the x² with respect to x³.
Find the derivative of the1⁄√(ax²+bx+c).
Find the derivative of the(b-c)⁄[√(ax+b)- √(ax+c)].
Find the derivative of the b²⁄[√(x²+b²) -√(x²-b²)].
Find the derivative of the b²⁄[x -√(x²-b²).
Find the derivative of the √[(x²-b²)⁄(x²-b²)].
Find the derivative dy⁄dx if y=t² and t=x².
Find the derivative dy⁄dx if y=2t² -3t+2and t=2x².
Find the derivative dy⁄dx if y=2u⁄(u² -9) and u=x²+3.
Find the derivative dy⁄dx if x²+y²=a².
Find the derivative dy⁄dx if x²⁄a²+y²⁄b²=1.
Find the derivative dy⁄dx if xy²+x²y=a³.
Find the derivative dy⁄dx if x²+y²=x²y².
Find the derivative dy⁄dx if x³+y³=3xy².
Find the derivative dy⁄dx if x³+y³=3x²y.
Find the derivative of x^m with respect to x^n.
Find the derivative of(2x-5)³ with respect to (2x-5).
Find the derivative of 5x⁷-3x⁶+x⁴-x²+1 with respect to x³.
Find the derivative of(5x+3)³ with respect to (3x+5).
Differentiate e^(cos2t) with respect to e^(sin2t).
Using definition, find the derivative of sin6x.
Using definition, find the derivative of sin(ax+b).
Using definition, find the derivative of sin(5x∕2).
Using definition, find the derivative of sin² 3x.
Using definition, find the derivative of √(sin2x).
Using definition, find the derivative of √(cosecx).
Using definition, find the derivative of tan(3x∕5).
8. Worked out example 7 Using definition, find the derivative of cos² 3x.
Find the derivative of sin(ax+b).
Find the derivative of cos(ax+b).
Find the derivative of sec³ √(tanx).
Find the derivative of (1+tanx) ∕ (1-tanx).
Find the derivative of sin(2nx) sin(2mx).
Find the derivative of sin(6x) cos(4x).
Find the derivative of cos(5x) cos(3x).
Find the derivative of sin(5x)sin(3x).
Worked out example 17 Find dy/dx if x+y=siny.
Find dy∕dx if x+y=sin(x-y).
Find dy∕dx if x+y=tan(xy).
Find dy∕dx if x²y=tan(xy²).
Find dy∕dx if x²+y²=sin(xy).
Find dy∕dx if x²+y²=sin(xy).
Find the derivative of sin³ (3x).
Find the derivative of cos⁴ (4x).
Find the derivative of sec³ (4x-2)∕2.
Find the derivative of cosec⁷ (ax-b)∕c.
Find the derivative of √tan(7x-3).
Find the derivative of sinⁿ[(mx+c)∕d].
Find the derivative of sin{tan(2x)}.
Find the derivative of tan⁴{sin(2x-3)}.
Find the derivative of cos²{sin(3x)}.
Find the derivative of (x²+5x) sin(3x).
Find the derivative of sin√x∕√x.
Find the derivative of (1-2sin²x)∕cos²x.
Find the derivative of sin2nx∕cos²nx.
Find the derivative of (sinax – cosbx)∕(cosax + cosbx).
Find the derivative of √[(1-cosx)∕(1+cosx).
Find the derivative of √[(1-sinx)∕(1+sinx).
Find the derivative of cos2x∕(1-sin2x).
Find the derivative of (cos2x+1)∕sin2x.
Find the derivative of 1∕(secx-tanx).
Find the derivative of (secx+tanx)∕(secx-tanx).
Find the derivative of x³e^(5x).
Find the derivative of x³ log(x+1).
Find the derivative of [1∕(a²+b²)]e^(ax)[asinbx-bcosbx].
Find the derivative of [1∕(a²+b²)]e^(ax)[acosbx+bsinbx].
Find the derivative of logx∕cosx.
Find the derivative of logx∕sinx.
Find the derivative of sinax∕e^(ax).
Find the derivative of xⁿ∕e^(ax+b).
Find the derivative of log(cotx).
Find the derivative of log(logx).
Find the derivative of log(x+tanx).
Find the derivative of log(e^(x)+e^(-x)).
Find the derivative of sec(log(tanx))).
Find the derivative of sin(log(sine^x)).
Find the derivative of log(x+√(a²+x²)).
Find dy⁄dx if x²+y²=log(x+y).
Find dy⁄dx if x²+y²=log(xy).
Find dy⁄dx if e^(xy)=xy.
Find dy⁄dx if x= logt+sint and y= e^t +cost.
Show that the function f(x)= x³-3x²+1 is increasing at x=3 but decreasing at x=1.
Find the interval where the function f(x)= x²-x-2 is increasing or decreasing.
Find the interval where the function f(x)= 12x-x³ is increasing or decreasing.
Find the absolute maximum and minimum of function f(x)= 3x²-6x+2 in the interval [-2, 3].
Find the absolute maximum and minimum of function f(x)= 2x³ -15x²+36x+10 in the interval [-2, 3].
Find the local maximum and minimum of function f(x)= 3x²-6x+1.
Find the local maximum and minimum of function f(x)= 4x³ -6x²-9x+1.
Find the local maximum and minimum of function f(x)= x+ 100∕x.
Show that f(x)= x³ has no maxima and minima.
Show that f(x)= x³-6x²+24x has neither maxima nor minima.
Show that f(x)= x³-6x²+12x has neither maxima nor minima.
Find the interval where the graph of the function f(x)= x⁴-2x³-12x² is concave up or down.
Find the interval where the graph of the function f(x)= x(x-1)(x+2) is concave up or down.
Government is planning to cover a rectangular portion of forest by using a fencing of 100km. What is the maximum area than can be enclosed?
Show that the rectangle of maximum possible area for a given perimeter is a square.
A window is in the form of a rectangle surmounted by a semi-circle. If the total perimeter is 9m, find the radius of the semi – circle for the greatest window area.
Introduction
Find the antiderivatives of (i) 3x² (ii) 5x¯⁶ (iii) (1∕3)x^(¯1∕3)
Find the antiderivative of 3x² + 5x¯⁶ - (1∕3)x^(¯1∕3).
Find the antiderivative of (ax+b)(cx+d).
Find the antiderivative of (1∕√ x) + √ x.
Find the antiderivative of √x(1-x²).
Find the antiderivative of (x²-1)².
Find the antiderivative of (1-2x+3x²-4x³)∕x.
Find the antiderivatives of (i) (x+1)² (ii) (3-4x)⁵.
Find the antiderivative (x+1)∕(x-1).
Find the antiderivative (2x+1)∕(x-3).
Find the antiderivative (x²+5x+2)∕(x+2).
Find the antiderivative (x²+2)∕(x+3).
Find the antiderivative x√(x+3).
Find the antiderivative (x+1)√(5x+3).
Find the value of the integral ∫ (3x+2)√(5x+4)dx.
Find the value of the integral ∫ (3x+2) ∕√(x+4)dx.
Find the value of the integral ∫ (3x+2) ∕√(5x+4)dx.
Find the value of the integral ∫ 2∕[√(5x+4)- √(5x-7)]dx.
Find the value of the integral ∫ cos3x dx.
Evaluate the integral (i) ∫ cos²x dx and (ii) ∫ sin²x dx.
Find the value of the integral ∫sin⁴x dx.
Find the value of the integral ∫sin⁴nx dx.
Find the value of the integral ∫1∕[cos²x sin²x] dx.
Find the value of the integral ∫1∕[sec²x tan²x] dx.
Find the value of the integral ∫√[1+cosnx] dx.
Find the value of the integral ∫√[1-cosnx] dx.
Find the value of the integral ∫√[1+sin2nx] dx.
Find the value of the integral ∫1∕[1+cosnx] dx.
Find the value of the integral ∫1∕[1-cosnx] dx.
Find the value of the integral ∫1∕[1+sinnx] dx.
Find the value of the integral ∫1∕[1-sinnx] dx.
Find the value of the integral ∫sin4x cos2x dx.
Find the value of the integral ∫sinx cos4x dx.
Find the value of the integral ∫cos4x cos2x dx.
Find the value of the integral ∫sin4x sin2x dx.
Evaluate (i) ∫[e^(ax) +e^(bx)] and (ii) ∫[e^(ax) +e^(bx)]².
Find the value of the integral ∫(1+√x)³ ∕ 2√x dx.
Find the value of the integral ∫xdx∕(x²-a²)³.
Find the value of the integral ∫ (x+1) ∕√(x²+2x+3)dx.
Find the value of the integral ∫ (3x+1) ∕√(3x²+2x+1)dx.
Find the value of the integral ∫ (2x+2) sec² (x²+2x+3)dx.
Find the value of the integral ∫ cos(logx) dx ∕x.
Find the value of the integral ∫ sin⁴x cosx dx.
Find the value of the integral ∫ sin² x cos³x.
Find the value of the integral ∫ cos⁵ x sin³x.
Find the value of the integral ∫ cotx log(sinx)dx.
Find the value of the integral ∫ tan² x sec²x.
Find the value of the integral ∫ cot³’²xcosec⁴x.
Integrate ∫sinx∕[1-cosx]ⁿ dx.
Integrate ∫(sinnx-cosnx)∕(sinnx-cosnx) dx.
Integrate ∫tanx dx.
Integrate ∫cotx dx.
Integrate ∫secx dx.
Integrate ∫cosecx dx.
Integrate ∫tan³x dx.
Integrate ∫tan⁴x dx.
Find the antiderivative of (1+1∕x²) e^(x-1∕x) dx.
Integrate ∫e^sin2x cos2x dx.
Integrate ∫e^(sinx cosx) cos2x dx.
Integrate ∫[sin√x∕√x]dx.
Integrate ∫e^sin²x sin2x dx.
Integrate ∫[x e^x∕sin² (xe^x-e^x)]dx.
Integrate ∫[x e^x∕cos² (xe^x-e^x)]dx.
Integrate ∫[e^2x∕(1+e^x)]dx.
Integrate ∫[e^(-x)∕(1+e^x)]dx.
Integrate ∫[(e^x-1)∕(e^x+1)]dx.
Integrate ∫1∕(1+x²) dx.
Integrate ∫1∕(1+x²)² dx.
Integrate ∫1∕√(1-x²) dx.
Integrate ∫1∕√(x²-1) dx.
Integrate ∫1∕(a²-x²)³’² dx.
Integrate ∫1∕x√(a²+x²) dx.
Derivation of formula
Integrate ∫x logx dx.
Integrate ∫ logx dx.
Integrate ∫ x sinx dx.
4 Integrate ∫(2x-1) logx dx.
Integrate ∫xⁿ logx dx.
Integrate ∫x e^x dx.
Integrate ∫x e^3x dx.
Integrate ∫x² sinx dx.
Integrate ∫x cos²x dx.
Integrate ∫sec³x dx.
Integrate ∫x³ e^(x²) dx.
Evaluate ∫₀¹x² dx
Evaluate ∫cos²x dx from x=0 to x=π.
Evaluate ∫(2x+1)² dx from x=1 to x=3.
Evaluate ∫√ (1+cosx) dx from x=0 to x=π.
Evaluate ∫x logx dx from x=1 to x=e.
Evaluate ∫1∕ (1+sinx) dx from x=0 to x=π∕2.
Solve x³-4x-9=0 bisecting [2,3] correct to 3 places of decimals.
Apply bisection method to find square root of 3 within 2 places of decimals.
Use Newton method to find the value of √(124).
Use Newton Raphson to solve x+2-e^x=0.
Use rectangular rule to compute ∫(x³+x)dx from x=0 to x=4.
Use trapezoidal rule to compute ∫(x³+x)dx from x=0 to x=4.
Composite trapezoidal rule to compute ∫f(x)dx from x=a to x=b.
Use composite trapezoidal rule to compute ∫(x³+x)dx from x=0 to x=4.
Simpson's one third rule to compute ∫f(x)dx from x=a to x=b.
Use Simpson's one third rule to compute ∫(x³+x)dx from x=0 to x=4.
Skewness: Meaning and formula
For a group of 10 items, ∑x=110, ∑x²=1650 and mode=10. Find Karl Pearson's coefficient of skewness and interpret the result.
For a group of 50 items, ∑x=150, ∑x²=600 and median=3.15. Find Karl Pearson's coefficient of skewness and interpret the result.
Karl Pearson's coefficient of skewness of a distribution is 0.32 and its mean and standard deviation are 29.6 and 6.5 respectively. Find the mode and median of the distribution.
The mean, mode and coefficient of skewness of a certain distribution are 25.5, 24.5 and 0.35 respectively. Calculate the mean, standard deviation and coefficient of variation of the distribution.
The lower quartile, median and upper quartile of a distribution are 8.6, 25.2 and 33 respectively. Find the coefficient skewness of the distribution.
The sum and difference of two quartiles of a distribution are 80 and 10 respectively. Calculate the coefficient skewness if its median is 40.
Compare the two distributions with respect to degree of variation and skewness.
Find the coefficient of skewness from the given data:
Find the coefficient of skewness from the given frequency distribution:
Find the coefficient of skewness from the given frequency distribution:
Find the coefficient of skewness based on quartiles from the given frequency distribution:
Standard deviation: Definition and formula.
Combined mean, combined standard deviation, combined variance and combined coefficient of variation.
The standard deviation and mean of a group are 5 and 150 respectively. Find the coefficient of variation of the group.
For a group of 50 items, ∑x=150 and ∑x²=600, find the coefficient of variation of the group.
For a group of 10 items, ∑x=110 and ∑x²=1650, find the coefficient of variation of the group.
The standard deviation and coefficient of variation of a group are 2 and 5% respectively. Find the mean of the distribution.
The means and standard deviations of two groups A and B are given in the following table. Compare the distributions of the groups with respect to variations.
The sizes, means and standard deviations of two groups A and B are given in the table. Compute the combined mean and standard deviation.
Find the standard deviation of the temperatures from the following data
Find the standard deviation of the daily income from the following data:
Find the standard deviation of the ages of plants from the following data:
Find the mean, standard deviation and coefficient of variation of life of electric bulbs from the following data:
Following are the marks obtained by X and Y. If consistency is the criterion for getting the prize, who should get the prize?
The arithmetic mean and standard deviation of a series of 20 items were calculated by a student as 20cm and 5cm. But while calculating them an item 13 was misread as 30. Find the correct mean and standard deviation.
For a number of 51 observations, the mean and standard deviation are 58.5 and 11 respectively. It was found after the calculation were made that one of the observations recorded as 15 was in correct. Find the correct mean and standard deviation.
The mean and standard deviation of 10 items were found to be 9.5 and 2.5 respectively. Later on, an additional observation became available and was included. Find the mean and standard deviation of the 11 items.
Probability : Basic terms
Find the sample space when two coins are tossed together.
Find the sample space when two dice are rolled together.
Find the sample space when a die and a coin are rolled together.
Three coins are tossed simultaneously. Find the following probabilities. (1) all heads (2) one tail only (3) at least 2 heads (4) no heads (5) a head on first coin (6) exactly 2 heads
A dice is thrown once. Find the following probabilities: (1) a number greater than or equal to 2 (2) a prime number (3) a composite number.
Two dice are rolled simultaneously. Find the following probabilities. (a) an odd number as a sum (b) an even number as a sum (c) a prime number as a sum (d) a composite number as a sum (e) a total of at least 10 (f) a doublet of primes.
A card is drawn from a pack of cards. Find the following probabilities. (a) a card of spade (b) a number card (c) non faced diamond card (d) face or ace card (e) heart or face card (f) red or face card
If P(A)=0.7 and P(B)=0.55 and P(A∪B)=0.87. Are events A and B independent?
For two independent events, If P(A)=0.42. Find the following probabilities. (i) P(A∩B) (ii) P(A∪B) (iii) P(A∪B)’ (iv) P(A’∩B’).
Tickets are number from 1 to 10. A ticket is drawn randomly, find the probability of getting a multiple of 3 or 5.
A problem is given to two students A and B. If P(A)=0.4 and P(B)=0.7 then find probability that (i) both of them solve (ii) at least one solve (iii) none of them solve (iv) exactly one of them solve it.
A problem is given to three students A, B and C. If P(A)=27, P(B)=14 and P(C)=15 then find probability that (i) the problem is solved (ii) the problem is solved by one of them only.
Find the probability of getting 5 Sundays in the month of June.
Find the probability of getting 53 Sundays in a normal year.
Definition of collinear vectors.
Show that given three points (1,-2,3), (2, 3, -4) and (0, -7, 10) are collinear.
Show that given three points (2, 5, -8), (4, -1, 6) and (3, 2, -1) are collinear.
Find the value of if the three points (1, 3, 2), (5, λ, 8) and (-1, -1, -1) are collinear.
Show that given three vectors (2, 5, -8), (4, -1, 6) and (3, 2, -1) are coplanar.
Any space vector r can be expressed as a linear combination of three non - coplanar vectors a, b and c.
Find the value of λ, if the following three vectors are coplanar.
Express the vector (3, 5) as a linear combination of vectors (5, 7) and (-2, 1).
Express the vector (3, 5, 2) as a linear combination of vectors (1, 2, 0), (0, -2, 2) and (1, 2, 3).
Check whether the given three vectors are linearly dependent or independent.
Check whether the given three vectors are linearly dependent or independent.
If three points (a,0), (x,y) and (0,b) are collinear, prove that x⁄a+y⁄b=1.
If four points (a,0,0), (0,b,0) (0,0,c)and (x,y,z) are coplanar, prove that x⁄a+y⁄b+z⁄c=1.
Administrator
MATH VISION PVT LTD
Mathematics Class 11 | Explained in Nepali
