Date: October 2013

If`[ bara bar b barc ] ≠ 0 and barp = [ barb xx barc ]/([ bara bar b barc ]), barq = [ barc xx bara ]/([ bara bar b barc ]), barr = [ bara xx barb ]/[ bara bar b barc ]`

then `bara . barp + barb . barq + barc . barr` is equal to ______.

0

1

2

3

Chapter: [0.07] Vectors

The inverse of the matrix `[[2,0,0],[0,1,0],[0,0,-1]]`is --------

(a) `[[1/2,0,0],[0,1,0],[0,0,-1]]`

(b) `[[-1/2,0,0],[0,-1,0],[0,0,1]]`

(c) `[[-1,0,0],[0,-1/2,0],[0,0,1/2]]`

(d) `1/2[[-1/2,0,0],[0,-1,0],[0,0,-1]]`

Chapter: [0.02] Matrices

Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........

`+-1/sqrt51,+-5/sqrt51,+-1/sqrt51`

`+-5/sqrt51, +-1/sqrt51, +- (-5)/sqrt51`

`+-sqrt5,+-1,+-5`

`+-sqrt51,+-sqrt51+-sqrt51`

Chapter: [0.08] Three Dimensional Geometry

Write truth values of the following statements :`sqrt5` is an irrational number but 3 +`sqrt 5` is a complex number.

True

False

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Write truth values of the following statements: ∃ n ∈ N such that n + 5 > 10.

True

False

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

If `bar c = 3bara- 2bar b ` Prove that `[bar a bar b barc]=0`

Chapter: [0.015] Vectors [0.07] Vectors

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hati + hatj + 2hatk.`

Chapter: [0.1] Plane

The Cartesian equations of line are 3x+1=6y-2=1-z find its equation in vector form.

Chapter: [0.013999999999999999] Pair of Straight Lines [0.09] Line

Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.

Chapter: [0.07] Vectors

Using truth table, prove the following logical equivalence :

(p ∧ q) → r ≡ p → (q → r)

Chapter: [0.01] Mathematical Logic [0.011000000000000001] Mathematical Logic

Find the joint equation of the pair of lines through the origin each of which is making an angle of 30° with the line 3x + 2y - 11 = 0

Chapter: [0.04] Pair of Straight Lines

Show that: `2sin^-1(3/5)=tan^-1(24/7)`

Chapter: [0.03] Trigonometric Functions

Solve the following equations by the method of reduction :

2x-y + z=1, x + 2y +3z = 8, 3x + y-4z=1.

Chapter: [0.012] Matrics [0.02] Matrices

Prove that the volume of a parallelopiped with coterminal edges as ` bara ,bar b , barc `

Hence find the volume of the parallelopiped with coterminal edges `bar i+barj, barj+bark `

Chapter: [0.015] Vectors [0.07] Vectors

Solve the following LPP by using graphical method.

Maximize : Z = 6x + 4y

Subject to x ≤ 2, x + y ≤ 3, -2x + y ≤ 1, x ≥ 0, y ≥ 0.

Also find maximum value of Z.

Chapter: [0.017] Linear Programming [0.11] Linear Programming Problems

In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)

Chapter: [0.03] Trigonometric Functions

If p : It is a day time, q : It is warm, write the compound statements in verbal form

denoted by -

(a) p ∧ ~ q

(b) ~ p → q

(c) q ↔ p

Chapter: [0.01] Mathematical Logic

If the lines `(x-1)/2=(y+1)/3=(z-1)/4 ` and `(x-3)/1=(y-k)/2=z/1` intersect each other then find value of k

Chapter: [0.016] Line and Plane [0.09] Line

Parametric form of the equation of the plane is `bar r=(2hati+hatk)+lambdahati+mu(hat i+2hatj+hatk)` λ and μ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartesian form.

Chapter: [0.1] Plane

If the angle between the lines represented by ax^{2} + 2hxy + by^{2} = 0 is equal to the angle between the lines 2x^{2} - 5xy + 3y^{2} =0,

then show that 100(h^{2} - ab) = (a + b)^{2}

Chapter: [0.08] Three Dimensional Geometry

**Find the general solution of :** sinx · tanx = tanx - sinx + 1

Chapter: [0.03] Trigonometric Functions

The differential equation of the family of curves y=c_{1}e^{x}+c_{2}e^{-x} is......

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

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

(c)`(d^2y)/dx^2+1=0`

(d)`(d^2y)/dx^2-1=0`

Chapter: [0.17] Differential Equation

If X is a random variable with probability mass function

P(x) = kx , x=1,2,3

= 0 , otherwise

then , k=..............

(a) 1/5

(b) 1/4

(c) 1/6

(d) 2/3

Chapter: [0.19] Probability Distribution

If `sec((x+y)/(x-y))=a^2. " then " (d^2y)/dx^2=........`

(a) y

(b) x

(c) y/x

(d) 0

Chapter: [0.13] Differentiation

If `y=sin^-1(3x)+sec^-1(1/(3x)), ` find dy/dx

Chapter: [0.13] Differentiation

Evaluate :`intxlogxdx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If `int_0^h1/(2+8x^2)dx=pi/16 `then find the value of h.

Chapter: [0.15] Integration

The probability that a certain kind of component will survive a check test is 0.5. Find the probability that exactly two of the next four components tested will survive.

Chapter: [0.027999999999999997] Binomial Distribution

Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis

Chapter: [0.16] Applications of Definite Integral

Examine the continuity of the following function at given point:

`f(x)=(logx-log8)/(x-8) , `

` =8, `

Chapter: [0.12] Continuity

If x = Φ(t) differentiable function of ‘ t ' then prove that `int f(x) dx=intf[phi(t)]phi'(t)dt`

Chapter: [0.17] Differential Equation

Solve : 3e^{x} tanydx + (1 +e^{x}) sec^{2} ydy = 0

Also, find the particular solution when x = 0 and y = π.

Chapter: [0.17] Differential Equation

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

Chapter: [0.14] Applications of Derivative

Evaluate : `intlogx/(1+logx)^2dx`

Chapter: [0.15] Integration

If x = f(t), y = g(t) are differentiable functions of parammeter ‘ t ’ then prove that y is a differentiable function of 'x' and hence, find dy/dx if x=a cost, y=a sint

Chapter: [0.13] Differentiation

Show that the function defined by f(x) =|cosx| is continuous function.

Chapter: [0.12] Continuity

Solve the differential equation `dy/dx=(y+sqrt(x^2+y^2))/x`

Chapter: [0.17] Differential Equation

Given X ~ B(n, p). If n = 20, E(X) = 10, find p_{,} Var. (X) and S.D. (X).

Chapter: [0.2] Bernoulli Trials and Binomial Distribution

A bakerman sells 5 types of cakes. Profits due to the sale of each type of cake is respectively Rs. 3, Rs. 2.5, Rs. 2, Rs. 1.5, Rs. 1. The demands for these cakes are 10%, 5%, 25%, 45% and 15% respectively. What is the expected profit per cake?

Chapter: [0.18] Statistics

Verify Lagrange’s mean value theorem for the function f(x)=x+1/x, x ∈ [1, 3]

Chapter: [0.14] Applications of Derivative

Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`

Chapter: [0.023] Indefinite Integration [0.15] Integration

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