Date: March 2015

If `veca=2hati+hatj+3hatk and vecb=3hati+5hatj-2hatk` ,then find ` |veca xx vecb|`

Chapter: [4.02] Vectors

Find the angle between the vectors `hati-hatj and hatj-hatk`

Chapter: [4.02] Vectors

Find the distance of a point (2, 5, −3) from the plane `vec r.(6hati-3hatj+2 hatk)=4`

Chapter: [4.01] Three - Dimensional Geometry

Write the element a_{12} of the matrix A = [a_{ij}]_{2 × 2}, whose elements aij are given by a_{ij} = e^{2ix }sin jx.

Chapter: [2.02] Matrices

Find the differential equation of the family of lines passing through the origin.

Chapter: [3.04] Differential Equations

Find the integrating factor for the following differential equation:`x logx dy/dx+y=2log x`

Chapter: [3.04] Differential Equations

If `A=[[1,2,2],[2,1,2],[2,2,1]]` ,then show that `A^2-4A-5I=0` and hence find A^{-1}.

Chapter: [2.02] Matrices

If `A=|[2,0,-1],[5,1,0],[0,1,3]|` , then find A^{-1} using elementary row operations

Chapter: [2.02] Matrices

Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`

Chapter: [2.01] Determinants

Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`

Chapter: [3.05] Integrals

Evaluate `int_(-1)^2(e^3x+7x-5)dx` as a limit of sums

Chapter: [3.05] Integrals

Evaluate:

`int x^2/(x^4+x^2-2)dx`

Chapter: [3.05] Integrals

In a set of 10 coins, 2 coins are with heads on both the sides. A coin is selected at random from this set and tossed five times. If all the five times, the result was heads, find the probability that the selected coin had heads on both the sides.

Chapter: [6.01] Probability

How many times must a fair coin be tossed so that the probability of getting at least one head is more than 80%?

Chapter: [6.01] Probability

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

Chapter: [4.02] Vectors

A line passing through the point A with position vector `veca=4hati+2hatj+2hatk` is parallel to the vector `vecb=2hati+3hatj+6hatk` . Find the length of the perpendicular drawn on this line from a point P with vector `vecr_1=hati+2hatj+3hatk`

Chapter: [4.02] Vectors

Solve the following for x:

`sin^(-1)(1-x)-2sin^-1 x=pi/2`

Chapter: [1.01] Inverse Trigonometric Functions

Show that:

`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`

Chapter: [1.01] Inverse Trigonometric Functions

If y = e^{ax}. cos bx, then prove that

`(d^2y)/(dx^2)-2ady/dx+(a^2+b^2)y=0`

Chapter: [3.01] Continuity and Differentiability

if x^{x}+x^{y}+y^{x}=a^{b}, then find `dy/dx`.

Chapter: [3.01] Continuity and Differentiability

If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `

Chapter: [3.01] Continuity and Differentiability

Evaluate:

`int((x+3)e^x)/((x+5)^3)dx`

Chapter: [3.05] Integrals

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-helds fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:

School/Article | School X | School Y | School z |

Hand-held fans | 30 | 40 | 35 |

Mats | 12 | 15 | 20 |

toys | 70 | 55 | 75 |

Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation.

Chapter: [2.02] Matrices

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: [1.02] Relations and Functions

Let f : W → W be defined as

`f(n)={(n-1, " if n is odd"),(n+1, "if n is even") :}`

Show that f is invertible a nd find the inverse of f. Here, W is the set of all whole

numbers.

Chapter: [1.02] Relations and Functions

Sketch the region bounded by the curves `y=sqrt(5-x^2)` and y=|x-1| and find its area using integration.

Chapter: [3.03] Applications of the Integrals

Find the particular solution of the differential equation x^{2}dy = (2xy + y^{2}) dx, given that y = 1 when x = 1.

Chapter: [3.05] Integrals

Find the particular solution of the differential equation `(1+x^2)dy/dx=(e^(mtan^-1 x)-y)` , give that y=1 when x=0.

Chapter: [3.04] Differential Equations

Find the absolute maximum and absolute minimum values of the function f given by f(x)=sin^{2}x-cosx,x ∈ (0,π)

Chapter: [3.02] Applications of Derivatives

Show that lines:

`vecr=hati+hatj+hatk+lambda(hati-hat+hatk)`

`vecr=4hatj+2hatk+mu(2hati-hatj+3hatk)` are coplanar

Also, find the equation of the plane containing these lines.

Chapter: [4.01] Three - Dimensional Geometry

Minimum and maximum z = 5x + 2y subject to the following constraints:

x-2y ≤ 2

3x+2y ≤ 12

-3x+2y ≤ 3

x ≥ 0,y ≥ 0

Chapter: [5.01] Linear Programming

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

Chapter: [6.01] Probability

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