1. Algebra
2. Trigonometry
3. Analytic Geometry
4. Vectors
5. Statistics
6. Calculus
7. Computational Methods
8. Mechanics
9. Mathematics for Economics
Definition and meaning of n!
Prove that n!=n⨯(n-1)! Also show that (r+3)!=(r+3)⨯(r+2)⨯(r+1)⨯r!
Prove that (n+1)! ∕ n!=n+1
Prove that n! ∕ n=(n-1)!
Prove that 0!= 1
Prove that n! ∕ r!=(r+1)⨯(r+2)⨯(r+3)⨯...⨯n
Prove that 2n! ∕ n!=1⨯3⨯5⨯...⨯(2n-1)⨯2ⁿ
Basic Principle of Counting
A stadium has 3 entrances and 4 exits. In how many ways can a viewer enter and leave the stadium?
Using the integers 3, 4, 6, 7 and 9 only once, how many 3 digit numbers can be formed?
Using the integers 3, 4, 6, 7 and 9 only once, how many 3 digit numbers less than 800 can be formed?
Using the integers 3, 4, 6, 7 and 9 only once, how many 3 digit even numbers can be formed?
Using the integers 3, 4, 6, 7 and 9 only once, how many 3 digit odd numbers can be formed?
Using the integers 3, 4, 6, 7 and 9 only once, how many numbers between 6K and 7K can be formed?
Meaning and notation of Permutation
Derivation of formula for P(n, r)
Find (i) P(5, 2) (ii) P(7, 0) and (iii) P(n, 2)
If P(20, r+5) ∕ P(18, r+2)=4180 ∕ 1, find the value of r.
Using the integers 3, 4, 6, 7 and 9 only once, how many 3 digit numbers can be formed?
In how many ways can the letters of the word 'TRIANGLE' be arranged?
Find the number of three digit numbers divisible by 5 using 1,2,3,4,5,6,7 only once.
In how many ways can 5 people be seated in a row of 5 seats?
In how many ways can 3 boys and 4 girls be seated in a row of 7 seats if boys and girls may sit anywhere?
In how many ways can 8 people be seated in a row of 8 seats if particular 2 people always sit together?
In how many ways can 3 boys and 4 girls be seated in a row of 7 seats if boys and girls alternate each other?
In how many ways can 4 boys and 4 girls be seated in a row of 8 seats if boys and girls alternate each other?
Derivation of formula for permutations with repetitions.
Find the number of permutation of the letters of the word 'CYCLE'.
In how many ways can the letters of the word 'AASIGNMENTS' be arranged?
In how many ways can the letters of the word 'INTERNET' be arranged?
In how many ways can the letters of the word 'INTERNET' be arranged so that two E's always come together?
In how many ways can the letters of the word 'INTERNET' be arranged so that two E's never come together?
Derivation of formula for Circular Permutations.
In how many ways 6 people can be seated around a round table?
In how many ways can 6 students and 6 teacher be seated in a round table if they may sit any where?
In how many ways can 6 students and 6 teacher be seated in a round table if they sit alternately?
In how many ways can 6 students and 6 teacher be seated in a round table if particular 3 teachers always sit together?
Derivation of formula for circular arrangements with flip.
In how many ways 8 beads of different colours be made into a bracelet?
In how many ways 1 flag of UNO, 2 flags of Nepal, 4 flags of US and 3 flags of Japan be flagged in a row?
In how many ways a packet of Kurkure, a packet of Oreo biscuit, a packet of Lays and a packet of Chewing gum be distributed among three students?
In how many ways 5 letters can be posted in 3 letter boxes?
In how many ways the letters of the word 'POPULAR' can be arranged so that all consonants are never together?
In how many ways can the letters of the word MOUNTAINS be arranged so that no two vowels come together?
Meaning of combination and derivation of formula for C(n, r).
Compute (i) C(5, 2) (ii) C(7, 0) (iii) C(n, 2)
Compute (i) C(n, 0) (ii) C(n, n).
Compute (i) C(n, 1) (ii) C(n, n-1).
C(n, r) in terms of r diminishing factors.
Prove C(n, r)=C(n, n-r). (Complementary combinations)
Compute: C(15, 13).
If C(n,2)=235, find n.
Use of Pascal formula
Pascal formula
If C(n, 13)= C(n, 6), then find C(n, 16).
Relation between P(n, r) and C(n, r)
Selection of two at a time.
Binomial Theorem and proof of its formula.
Different forms in Binomial theorem.
Number of terms and General term in expansion of (a + x)ⁿ.
Middle term Formula.
Binomial coefficients and Relation I
Binomial coefficients and Relation II
Sum of Binomial coefficients with even and odd suffixes.
Find (i) the general term (ii) the term free of x in (x²-1∕x²)²ⁿ.
Find (i) the general term (ii) the term containing x⁻¹ in (x+1∕x)²ⁿ⁺¹.
I
C₀ , C₁ , C₂ , ..... are coefficients in (1+x)ⁿ.
C₀ , C₁ , C₂ , ... are coefficients in (1+x)ⁿ.
Binomial expansion
The Euler's Number.
Exponential series.
Series of e.
Series of 1∕e.
Prove that the value of e lies in between 2 and 3.
Prove that e is not rational.
Find the series for a^x.
Find the series for 1∕2 (e^x+e^-x).
Find the series for 1∕2 (e^x-e^-x).
Prove that (1+1∕1!+1∕2!+...)(1-1∕1!+1∕2!-...)=1.
Logarithmic series and special forms.
Prove that 1∕(1.2)+1∕(3.4)+1∕(5.6)+... to ∞ = log2.
Prove that 1∕(2.3)+1∕(4.5)+1∕(6.7)+... to ∞ = log(e∕2).
Prove that 1∕2-1∕(2.2²)+1∕(2.2³)+1∕(2.2⁴)+... to ∞ = log(3∕2).
Prove that (1∕3-1∕2)+1∕2(1∕3²+1∕2²)+..... to ∞ = 0.
Prove that 1+1∕(3.2²)+1∕(5.2⁴)+1∕(7.2⁶)+... to ∞ = log3.
Prove that 1∕n-1∕2n²+1∕3³-=1∕(n+1)²+1∕3(n+1)³+...
If y=x+x²∕2+x³∕3+.. to ∞ then prove that x = y-y²∕2!+y³∕3!-...to ∞
If x = y-y²∕2!+y³∕3!-...to ∞ then prove that y=x+x²∕2+x³∕3+.. to ∞
Complex number in Polar form.
Complex Number in Euler form.
Product of Complex Numbers in Polar form.
Product of Complex Numbers in Euler form.
Quotient of complex numbers in Polar form.
Quotient of Complex Numbers in Euler form.
De - Moivre's theorem. Positive integral power of Complex Numbers in Polar form.
Express √3+i in polar form.
Express √3+i in Euler form.
System of linear equations.
Classify the system of linear equations x + y=-1 and 3x+2y=0.
Classify the system of linear equations 3x+2y=5 and 6x+4y=15.
Classify the system of linear equations 3x+2y=5 and 6x+4y=10.
Steps involved in Row equivalent method for solving two variable linear equations.
Steps involved in Row equivalent method for solving three variable linear equations.
Solve 3x-2y=6 and 5x+y=23 by row equivalent matrix.
Solve x+2y-4z=3, 2x-y+2z=6 and 3x+y+z=12 by row equivalent matrix.
Matrix Inversion method for the solution of system of 2 variable linear equations.
Matrix Inversion method for the solution of system of 3 variable linear equations.
Sum of the first n natural numbers.
Sum of the first n even natural numbers.
Sum of the first n odd natural numbers.
Sum of squares of the first n natural numbers.
Sum of squares of the first n even natural numbers.
Sum of squares of the first n odd natural numbers.
Sum of cubes of the first n natural numbers.
Sum of cubes of the first n even natural numbers.
Sum of cubes of the first n odd natural numbers.
Find the sum to n terms of the series whose nth term is 2n+3.
State cosine laws and prove one of them.
State and prove sine law.
State and prove the projection laws.
In any ΔABC, prove that cos(A⁄2)=√[s(s-a)⁄bc].
In any ΔABC, prove that sin(A⁄2)=√[(s-b)(s-c)⁄bc].
In any ΔABC, prove that tan(A⁄2)=√[(s-b)(s-c)⁄s(s-a)].
In any ΔABC, prove that the area is given by Δ=1⁄2 bc sinA.
In any ΔABC, prove that Δ=√[s(s-a)(s-b)(s-c)].
In any ΔABC, prove that Δ=abc⁄4R.
Prove that area of a tringle, Δ =√[2a²b²+2b²c²+2c²a²-a⁴-b⁴-c⁴]
If two angles of a triangle are 60° and 30°, find the ratios of the sides.
If two angles of a triangle are 75° and 60°, find the ratios of the sides.
In a triangle, if a=√3+1, A=75° and B=60°, solve the triangle.
Definition of circle.
Derive the equation of circle in standard form.
Derive the equation of circle in central form.
Equation of circle in parametric form.
Derive equation of circle in diameter form.
Find the equation of a circle with centre at the origin and radius 1.
Find the equation of a circle with centre at (3, 4) and radius 5.
Find the equation of a circle with centre at (a, a) and radius a.
Find the centre and radius of the circle x²+y²+2ax+2ay+a²=0.
Using properties that tagent is perpendicular to radius.
Using the property that radius passes through centre.
You are given a line x+2y-1=0 and a circle x²+y²-2x-10+1=0. (a) Find the point of intersection of the line and circle. (b) Find the length of the intercept. (c) Find the equation of the circle of which the intercept is a diameter.
Definition of a parabola.
Derive the equation of parabola in standard form.
Derive the equation of parabola in standard form. (axis y axis)
Equations and elements of parabola.
Derive the equation of parabola whose directrix is parallel to y axis, vertex at (h, k) and latus rectum of length 4a.
Derive the equation of parabola whose directrix is parallel to x axis, vertex at (h, k) and latus rectum of length 4a.
Equations and elements of parabola not in standard form.
Find the equation of the parabola in standard position whose focus is at (5, 0).
Find the equation of the parabola in standard position whose focus is at (0, 3).
Find the equation of the parabola in standard position whose directrix is x-3=0.
26. (N) Find the coordinates of (i) vertex (ii) focus (iii) point of intersection of axis and directrix on the parabola y²=-12x. Also find the equation of (iv) axis and (v) directrix.
Definition of ellipse and Elements of an ellipse.
Derive the equation of ellipse whose major axis is the x axis.
Existence of two foci and two directrices of an ellipse.
Length of latus rectum when a ≻ b.
Length of latus rectum when b ≻ a.
Equation of ellipse in central form whose major axis is parallel to x axis.
Find the eccentricity of the ellipse x²∕16+y²∕25=1.
Find the length of latus rectum of the ellipses (i) x²∕25+y²∕16=1 and (ii) x²∕16+y²∕25=1 .
In an ellipse the distance between the vertices is 15 and the distance between foci is 10, find the distance between its directrices.
Introduction to Hyperbola.
Derive equation of hyperbola with transverse axis along x - axis.
Derive equation of hyperbola with transverse axis along y - axis.
Length of latus rectum of hyperbola whose transverse axis is along x axis.
Length of latus rectum of hyperbola whose transverse axis is along y axis.
Equation of hyperbola in central form with transverse axis along y - axis.
Find the eccentricities of the following hyperbola.
Find the centers of the following hyperbola.
Find the foci of the following hyperbola.
Find the equations of the directrices of the following hyperbola.
Definition of vector product.
Magnitude of a⨯b.
Relation of vectors a and b and with their cross product.
Cross product in different forms.
Geometrical interpretation of cross product.
Area of triangle and parallelogram in terms of cross product.
Non commutivity of cross product.
Distributive property of cross product.
Scalar multiple of cross product.
Cross Product of unit coordinate vector with it self.
Interpretation of value of correlation coefficient r.
If variances are 9 and 49 and covariance is -14, find the correlation coefficient.
Find r if ∑(x-x⁻)²=25 , ∑(x-x⁻)²= 81 and ∑(x-x⁻)(y-y⁻)=-35.
Find r if n=25, σ(x)=6.3, σ(y)=3.6 and ∑(x-x⁻)(y-y⁻)=405.
Find r if n=12, x⁻=7, y⁻=4, ∑x²=600, ∑y²=203 and ∑xy=340.
Find r if n=15, ∑x=35, ∑y=37, ∑x²=122, ∑y²=94 and ∑xy=82.
Marks in Math: 4 2 6 5 8 Marks in Physics: 4 3 5 ? 7
Find the rank correlation coefficient for repeated rank data.
Regression equations
Formula and Relation on Regression coefficients.
Check the validity of b(x, y)=0.98 and b(y, x)=1.5.
Check the validity of b(x, y)=0.52 and b(y, x)=-0.25.
Find r if b_xy=0.52 and b_yx=0.25.
Find r if b_xy=-0.52 and b_yx=-0.25.
Find regression coefficients if r= 0.6, σ(x)= 9 and σ(y)= 2.7.
Find the mean values from the regression equations: x-2y+6=0 and 3x-2y-6=0.
Identify regression equation of y on x from given equations.
Find regression and correlation coefficients from the equations: 3x-2y-6=0 and x-2y+6=0.
Probability formula
If a dice is rolled once, find the probability of getting a 2.
There are 10 boys and 12 girls in a math class. If 3 students are selected at random, find the probability of selecting 2 boys and 1 girl.
Find P(A∕B) and P(B∕A) if P(A)=0.42, P(B)=0.29 and P(A∩B)=0.14.
If a respondent is selected at random, find the probability that it is a smoker.
If a respondent is selected at random, find the probability that it is a male smoker.
If a female respondent is selected at random, find the probability that she is smoker.
Define derivative of a function.
Find, from definition, the derivative of e^x².
Find, from definition, the derivative of e^√x.
Find, from definition, the derivative of e^cosx.
Find, from definition, the derivative of e^tanx.
Find, from definition, the derivative of sin⁻¹x.
Find, from definition, the derivative of sec⁻¹x.
Find, from definition, the derivative of cos⁻¹x.
Find, from definition, the derivative of tan⁻¹x.
Find, from definition, the derivative of cot⁻¹x.
Actual and approximate change in values of function.
Calculate error of approximation on the value of f(x)=x² when x changes from 1 to 1.05.
Calculate % error of approximation of the change in surface area of a cube if its sides change from 25cm to 25.01cm.
Geometrical meaning of dy⁄dx.
Equation of tangent to a curve of y=f(x).
Find the slope of the tangent to the curve y=x³+3x+1 at (1,5).
Find the inclination of the tangent to the curve y=x³+3x+1 at (1,5).
At what angle does the the curve y=x³+3x-4 cut the x - axis?
Find the equation of the tangent to the curve x²+y²=5 at (2,1).
Find the point on the curve y=x²-2x+3 where tangent is parallel to x - axis.
Find the points on the curve y=3x⁴-8x³+6x²-1 where tangent are parallel to x - axis.
Find the points on the curve x²+y²=4 where tangent are parallel to y - axis.
Find the angle of intersection of the curves y=x² and x=y².
Equation of normal to a curve.
15
L - Hospital rule.
Prove ∫1∕√(a²-x²)dx=sin⁻¹(x∕a)+c.
Prove ∫1∕(a²-x²)dx=1∕2a log(a+x)∕(a-x)+c.
Prove ∫1∕(a²-x²)dx=1∕2a log(a+x)∕(a-x)+c.
Prove ∫1∕(a²-x²)dx=1∕2a log(a+x)∕(a-x)+c.
Prove ∫1∕(a²-x²)dx=1∕2a log(a+x)∕(a-x)+c.
Prove ∫1∕√(x²-a²)dx=log(x+√(x²-a²)+c=cosh⁻¹(x∕a)+c.
Prove ∫1∕√(x²-a²)dx=log(x+√(x²+a²)+c=sinh⁻¹(x∕a)+c.
Solve dy∕dx=y∕x
Solve dy∕dx=x∕y
Solve dy∕dx=y.
Solve dy∕dx=(1+y²)∕(1+x²).
Solve dy=(1+y²)dx.
Solve dy∕dx=(e^x+x)∕y.
Solve (1∕2)xdy∕dx=(e^2x-x).
Solve y(1+x)dy∕dx=x(1+y).
Solve √(1- x²)dy=√(1- y²)dx
Solve x(1+ y²)dx=y(1+ x²)dy=0
Test the consistency and solve using Gauss elimination: x+2y+2z=1; 2x+y+3z=4; 5x-3y-3z=5.
Test the consistency and solve using Gauss elimination: 3x+y+2z=0; 2x+3y+4z=6; 2x-3y-4z=-10.
Test the consistency and solve using Gauss elimination: 3x+y+2z=0; 2x+3y+4z=6; 5x+6y-4z=6.
Test the consistency and solve using Gauss elimination: x+2y+2z=1; 2x+y+3z=4; 5x-2y+6z=2.
Use Gauss elimination with partial pivoting to solve x+2y+2z=9; 2x+y+3z=11; 5x-2y-2z=9.
Diagonally dominant and not diagonally dominant.
Mathematical model of LPP relating to Maximization and Minimization.
Identify all solutions of the system of linear equations x+2y+r=6 and 2x+y+r=9.
Reformulate the maximization of LPP into standard form.
Reformulate the minimization of LPP into standard form.
Use simplex method to find the optimal solution of LPP ⁚ Maximize Z =2x-3y subject to x+y≦90, x-y≦10, x,y≥0.
Use simplex method to find the optimal solution of LPP ⁚ Maximize F =15x₁+10x₂ subject to 2x₁+x₂≦10, x₁+3x₂≦10, x₁, x₂≥0.
Equations of motion
A particle starts from rest and moves with a uniform acceleration of 5 cm∕s². What will be its velocity at the end of 10 seconds?
Find the pull of the earth on a mass of 1kg. (g=9.8N∕kg)
Find the pull of the moon on a mass of 1kg. (g=9.8N∕kg)
Find the change in the momentum of a body of mass 9.81 kg if its velocity changes from 40km∕h to 60 km∕h. Also find the impulse of the force on the mass.
A train of 150 metric ton starts from rest and move along an incline. Find the constant force necessary to move the train through a distance of 1.8km in 60 seconds.
A bullet of mass 8.125gm is fired from a gun of mass 1.25kg. If the velocity of the recoil of the gun is 3.25m∕s, find the velocity of the bullet.
A bullet of mass 16gm is fired from a gun of mass 2kg with a velocity of 325m∕s. Find the velocity of the recoil of the gun.
Two heavenly bodies of masses 2⨯10⁷ kg and 1.6⨯10⁶ kg are moving in a straight line to collide each other. If they are moving with equal speed of 100m/s, find the speed of the body when they stick together.
A gun of mass 150 kg horizontally fires a shot of 0.8kg with a velocity of 900m∕s. Find the force to stop the recoil of the gun in 2s.
A gun of mass 150 kg horizontally fires a shot of 0.8kg with a velocity of 900m∕s. Find the force to stop the recoil of the gun in 2m.
A gun of mass 24 metric ton resting on an incline of 3 in 5, fires a shot of 60 kg horizontally with a velocity of 500m∕s. Find the distance it moves up the incline.
Resultant of two like parallel forces.
Resultant of two unlike parallel forces.
Find the ratio of forces from the given figure.
Find the forces from the given figure.
Find the ratio of forces from the given figure.
Find the forces from the given figure.
If R=100N, find the like parallel forces P and Q.
If R=100N, find the unlike parallel forces P and Q.
If for like forces P= 320N and Q= 220N, find the resultant.
For unlike forces P= 550N and Q= 220N, find the resultant.
A gun mounted on a cliff of height h abbe the see level. If the muzzle velocity u of the shot is given, prove that the maximum range R at see level measured from the foot is given by R = (u/g)√(u²+2gh) and for it the angle of projection α is given by tanα = u²/(gR).
Qd and Qs are the demand and supply functions of a good. Find the time path of the price P given that initial price is 5. Find the value of P when t=3. Is P(3) close to equilibriuum?
If y(t)=c(t)+I(t), c(t)=200+0.3y(t-1), I(t)=500. Express the given simple national income model as a difference equation. Find the general as well as particular solution when y(0)=1500. Interpret the time path.
All 11 MCQs and Answers.
Question Number 12 a) Write the number of the total terms in the expansion of ((x-1/x)² )²⁵ . [1] b) Write the middle term in the expansion of (x+a)ⁿ when n is even. [1] c) What is the sum of the binomial coefficients in the expansion of (1+x)ⁿ? [1] d) Write log(1+x) in the series form. [base e] [1] e) Write e¯x in series form. [1]
13. a) Find the value of (1-ω+ω² )+(1+ω-ω² )⁴, where ω and ω² are imaginary cube roots of unity. [2] b) Solve the following system of equations using inverse matrix method. [3] x+2y+3z=20, 5x=2y+4, 3z=4x+4.
14. a) If 1/(p+r)=3/(p+q+r)-1/(q+r) in a triangle PQR, prove that ∠R=60°. [3] b) Find the eccentricity and foci of the ellipse 9x²+4y²-18x-16y-11=0 [2]
15. a) Find the equations of the tangent and normal to the circle x²+y²=13 at the point (2, 3). [3] b) In a rhombus, two of the diagonals are perpendicular to each other. Verify it by taking vector dot product of two vectors. [2]
16. a) Write the order of the differential equation ((d³ y)/(dx³ ))³+(dy/dx)²+5=0 [1] b) Write the derivative of sinhx with respect to x. [1] c) Write an example of exact differential equation in x and y. [1] d) Write the integral of ∫ 1/(x²-a² ) dx [1] e) State L – Hospital rule. [1]
17. The supply and price of a commodity for the last six years is given below. Price in Rs. Per kg 100 110 112 115 120 140 Supply in kg 30 40 45 20 55 55 a) Find the coefficient of correlation between price and supply. [2] b) Estimate supply in kg on which rate of price is Rs. 150. [3]
18. a) Integrate: ∫dx/(3 sinx-4 cosx ) [2] b) Solve dt/dx=(e^tan¯¹ x -t)/(1+x² ) [3]
20. a) If (1+x)^n=C₀+C₁ x+C₂ x²+⋯+⋯+Cn xⁿ, prove that C₁+2C₂+3C₃+⋯+⋯+nCn-1/2 (n.2ⁿ )=0. [3] b) Find the square roots of 1-√3 i using De – Moivre’s theorem. [2] c) Use principle of mathematical induction to prove 1+3+5+7+⋯+(2n-1)=n². [3]
21. a) Find the equation of the parabola whose focus is at the point (-3,4) and the directrix is 2x+5=y. [3] b) Find the area of the parallelogram whose diagonals are represented by the vectors 2i ⃗+3j ⃗-4k ⃗ and 3i ⃗-5j ⃗+2k ⃗. [3] c) In a triangle ABC, a=2, b=√6 and ∠A=45°. Solve the triangle. [2]
22. a) Water flows into an inverted conical tank at the rate of 36 cm3/min. When the depth of water is 12 cm, how fast is level rising, if the radius of the base and height of the tank is 21 cm and 35 cm respectively. [3] b) The concept of antiderivative is necessary for solving a differential equation. Justify the statement with an example. [2] c) A differential equation of the first degree is homogenous if it satisfies the condition dy/dx=f(y/x). Justify the statement with an example. [3]
About the properties of cube roots of unity.
Sum of squares of first 20 natural numbers.
Equation of a plane thrrough a point and parallel to a given plane.
To find the eccentricity of a hyperbola.
Testing the consistency of linear equations.
Define binomial expansion. Find the middle term. Find the sum of binomial coefficients.
Use of mathematical induction to prove (n+1)⁴<10ⁿ+¹.
(a) To find equation of plane. (b) To prove sine law by vector method.
a) Find derivative of (tanx)^logx. b) Solve the differential equation: (1 + x) dy = (1 + xy - x) dx.
(a) To find regression coefficients. (b) To find correlation coefficient. (c) To find mean values of variables.
Integrate by using partial fraction.
Use Simplex method to solve the LPP Max Z = x + y subject to 2x + 3y ≤ 22 2x + y ≤ 14 x , y ≥ 0.
Matrix based solution of linear word problem.
Diagonals of a cube in vector form
a) Expression of the problem in differential equation form. b) Solution of the problem. c) To find the percentages of the infected people after 10 days.
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Mathematics Class 12 | Explined in Nepali
