Date: March 2015

Write the value of `vec a .(vecb xxveca)`

Chapter: [0.1] Vectors

If `veca=hati+2hatj-hatk, vecb=2hati+hatj+hatk and vecc=5hati-4hatj+3hatk` then find the value of `(veca+vecb).vec c`

Chapter: [0.1] Vectors

Write the direction ratios of the following line :

`x = −3, (y−4)/3 =( 2 −z)/1`

Chapter: [0.11] Three - Dimensional Geometry

If `A=[[2,3],[5,-2]]` then write A^{-1}

Chapter: [0.03] Matrices

Find the differential equation representing the curve y = cx + c^{2}.

Chapter: [0.09] Differential Equations

Write the integrating factor of the following differential equation:

(1+y^{2}) dx−(tan^{−1} y−x) dy=0

Chapter: [0.09] Differential Equations

Using the properties of determinants, prove the following:

`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|=6x^2(1-x^2)`

Chapter: [0.04] Determinants

If x=α sin 2t (1 + cos 2t) and y=β cos 2t (1−cos 2t), show that `dy/dx=β/αtan t`

Chapter: [0.05] Continuity and Differentiability

Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`

Chapter: [0.05] Continuity and Differentiability

Find the derivative of the following function f(x) w.r.t. x, at x = 1 :

`f(x)=cos^-1[sin sqrt((1+x)/2)]+x^x`

Chapter: [0.05] Continuity and Differentiability

Evaluate :`int_0^(pi/2)(2^(sinx))/(2^(sinx)+2^(cosx))dx`

Chapter: [0.07] Integrals

Evaluate `∫_0^(3/2)|x cosπx|dx`

Chapter: [0.07] Integrals

To raise money for an orphanage, students of three schools A, B and C organised an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs 20, Rs 15 and Rs 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags, 15 scrap-books and 28 pastel sheets while School C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school.

By such exhibition, which values are generated in the students?

Chapter: [0.03] Matrices

Prove that :

`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`

Chapter: [0.02] Inverse Trigonometric Functions

Solve the following for *x* :

`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`

Chapter: [0.02] Inverse Trigonometric Functions

If `A=[[2,0,1],[2,1,3],[1,-1,0]]` , find A^{2} − 5 A + 16 I.

Chapter: [0.03] Matrices

Show that four points A, B, C and D whose position vectors are

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Chapter: [0.11] Three - Dimensional Geometry

Show that the following two lines are coplanar:

`(x−a+d)/(α−δ)= (y−a)/α=(z−a−d)/(α+δ) and (x−b+c)/(β−γ)=(y−b)/β=(z−b−c)/(β+γ)`

Chapter: [0.11] Three - Dimensional Geometry

Find the acute angle between the plane 5*x* − 4*y* + 7*z* − 13 = 0 and the *y*-axis.

Chapter: [0.11] Three - Dimensional Geometry

A and B throw a die alternatively till one of them gets a number greater than four and wins the game. If A starts the game, what is the probability of B winning?

Chapter: [0.13] Probability

A die is thrown three times. Events A and B are defined as below:

A : 5 on the first and 6 on the second throw.

B: 3 or 4 on the third throw.

Find the probability of B, given that A has already occurred.

Chapter: [0.13] Probability

Evaluate :

`int(sqrt(cotx)+sqrt(tanx))dx`

Chapter: [0.07] Integrals

Find:

`int(x^3-1)/(x^3+x)dx`

Chapter: [0.07] Integrals

Using integration, find the area of the region bounded by the lines *y *= 2 + *x*, *y *= 2 – *x *and *x *= 2.

Chapter: [0.08] Applications of the Integrals

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.

Chapter: [0.09] Differential Equations

Show that the differential equation `2xydy/dx=x^2+3y^2` is homogeneous and solve it.

Chapter: [0.09] Differential Equations

Find the direction ratios of the normal to the plane, which passes through the points (1, 0, 0) and (0, 1, 0) and makes angle π/4 with the plane *x* + *y* = 3. Also find the equation of the plane

Chapter: [0.11] Three - Dimensional Geometry

If the function *f* : R → R be defined by *f*(*x*) = 2*x* − 3 and *g* : R → R by *g*(*x*) = *x*^{3} + 5, then find the value of (fog)^{−1} (*x*).

Chapter: [0.01] Relations and Functions

Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (*a*, *b*) * (*c*, *d*) = (*ac*, *b* + *ad*), for all (*a*, *b*) (*c*, *d*) ∈ A.

Find

(i) the identity element in A

(ii) the invertible element of A.

(iii)and hence write the inverse of elements (5, 3) and (1/2,4)

Chapter: [0.01] Relations and Functions

If the function f(x)=2x^{3}−9mx^{2}+12m^{2}x+1, where m>0 attains its maximum and minimum at *p* and *q* respectively such that p^{2}=q, then find the value of *m*.

Chapter: [0.06] Applications of Derivatives

The postmaster of a local post office wishes to hire extra helpers during the Deepawali season, because of a large increase in the volume of mail handling and delivery. Because of the limited office space and the budgetary conditions, the number of temporary helpers must not exceed 10. According to past experience, a man can handle 300 letters and 80 packages per day, on the average, and a woman can handle 400 letters and 50 packets per day. The postmaster believes that the daily volume of extra mail and packages will be no less than 3400 and 680 respectively. A man receives Rs 225 a day and a woman receives Rs 200 a day. How many men and women helpers should be hired to keep the pay-roll at a minimum ? Formulate an LPP and solve it graphically.

Chapter: [0.12] Linear Programming

40% students of a college reside in hostel and the remaining reside outside. At the end of the year, 50% of the hostelers got A grade while from outside students, only 30% got A grade in the examination. At the end of the year, a student of the college was chosen at random and was found to have gotten A grade. What is the probability that the selected student was a hosteler ?

Chapter: [0.13] Probability

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