Date & Time: 28th February 2015, 11:00 am

Duration: 3h

if `A=[[2,0,0],[0,2,0],[0,0,2]]` then A6= ......................

6A

12A

16A

32A

Chapter: [0.02] Matrices

The principal solution of `cos^-1(-1/2)` is :

`pi/3`

`pi/6`

`(2pi)/3`

`(3pi)/2`

Chapter: [0.03] Trigonometric Functions

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then __________

fg = ch

gh = cf

Jh = cg

hf= - eg

Chapter: [0.04] Pair of Straight Lines

**Write the converse and contrapositive of the statement -**

“If two triangles are congruent, then their areas are equal.”

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

Find ‘k' if the sum of slopes of lines represented by equation x^{2}+ kxy - 3y^{2} = 0 is twice their product.

Chapter: [0.04] Pair of Straight Lines

Find the angle between the planes `bar r.(2bar i+barj-bark)=3 and bar r.(hati+2hatj+hatk)=1`

Chapter: [0.1] Plane

The Cartesian equations of line are 3x -1 = 6y + 2 = 1 - z. Find the vector equation of line.

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

If `bara=bari+2barj, barb=-2bari+barj,barc=4bari+3barj`, find x and y such that `barc=xbara+ybarb`

Chapter: [0.07] Vectors

If A, B, C, D are (1, 1, 1), (2, I, 3), (3, 2, 2), (3, 3, 4) respectively, then find the volume of parallelopiped with AB, AC and AD as the concurrent edges.

Chapter: [0.015] Vectors [0.07] Vectors

Discuss the statement pattern, using truth table : ~(~p ∧ ~q) v q

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

If point C `(barc)` divides the segment joining the points A(`bara`) and B(`barb`) internally in the ratio m : n, then prove that `barc=(mbarb+nbara)/(m+n)`

Chapter: [0.015] Vectors [0.07] Vectors

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1

Chapter: [0.08] Three Dimensional Geometry

In any ΔABC if a^{2} , b^{2} , c^{2} are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.

Chapter: [0.013000000000000001] Trigonometric Functions [0.03] Trigonometric Functions

The sum of three numbers is 6. When second number is subtracted from thrice the sum of first and third number, we get number 10. Four times the sum of third number is subtracted from five times the sum of first and second number, the result is 3. Using above information, find these three numbers by matrix method.

Chapter: [0.02] Matrices

If θ is the acute angle between the lines represented by equation ax^{2} + 2hxy + by^{2} = 0 then prove that `tantheta=|(2sqrt(h^2-ab))/(a+b)|, a+b!=0`

Chapter: [0.04] Pair of Straight Lines

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

Construct the switching circuit for the following statement : [p v (~ p ∧ q)] v [(- q ∧ r) v ~ p]

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

Find the general solution of : cos x - sin x = 1.

Chapter: [0.03] Trigonometric Functions

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).

Chapter: [0.1] Plane

A diet of a sick person must contain at least 48 units of vitamin A and 64 units of vitamin B. Two foods F_{ 1} and F_{2} are available. Food F_{1} costs Rs. 6 per unit and food F_{2} costs Rs. 10 per unit. One unit of food F_{1} contains 6 units of vitamin A and 7 units of vitamin B. One unit of food F_{2} contains 8 units of vitamin A and 12 units of vitamin B.Find the minimum cost for the diet that consists of mixture of these two foods and also meeting the minimal nutritional requirements.

Chapter: [0.11] Linear Programming Problems

A random variable X has the following probability distribution:

then E(X)=....................

0.8

0.9

0.7

1.1

Chapter: [0.027000000000000003] Probability Distributions [0.19] Probability Distribution

If `int_0^alpha3x^2dx=8` then the value of α is :

(a) 0

(b) -2

(c) 2

(d) ±2

Chapter: [0.15] Integration

The differential equation of `y=c/x+c^2` is :

(a)`x^4(dy/dx)^2-xdy/dx=y`

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

(c)`x^3(dy/dx)^2+xdy/dx=y`

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

Chapter: [0.17] Differential Equation

Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`

Chapter: [0.15] Integration

If `y=sqrt(sinx+sqrt(sinx+sqrt(sinx+..... oo))),` then show that `dy/dx=cosx/(2y-1)`

Chapter: [0.17] Differential Equation

Evaluate :`int_0^(pi/2)1/(1+cosx)dx`

Chapter: [0.15] Integration

If y=e^{ax },show that `xdy/dx=ylogy`

Chapter: [0.021] Differentiation [0.13] Differentiation

A fair coin is tossed five times. Find the probability that it shows exactly three times head.

Chapter: [0.19] Probability Distribution

Integrate : sec^{3} x w. r. t. x.

Chapter: [0.023] Indefinite Integration [0.15] Integration

If y = (tan^{-1} x)^{2}, show that

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

Chapter: [0.17] Differential Equation

If `f(x)=[tan(pi/4+x)]^(1/x), `

= k ,for x=0

is continuous at x=0 , find k.

Chapter: [0.12] Continuity

Find the co-ordinates of the points on the curve y=x-(4/x) where the tangents are parallel to the line y=2x

Chapter: [0.06] Conics

Prove that `int sqrt(x^2-a^2)dx=x/2sqrt(x^2-a^2)-a^2/2log|x+sqrt(x^2-a^2)|+c`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`

Chapter: [0.15] Integration

Find a and b, so that the function f(x) defined by

f(x)=-2sin x, for -π≤ x ≤ -π/2

=a sin x+b, for -π/2≤ x ≤ π/2

=cos x, for π/2≤ x ≤ π

is continuous on [- π, π]

Chapter: [0.12] Continuity

If `log_10((x^3-y^3)/(x^3+y^3))=2 "then show that" dy/dx = [-99x^2]/[101y^2]`

Chapter: [0.13] Differentiation

Let the p. m. f. (probability mass function) of random variable x be

`p(x)=(4/x)(5/9)^x(4/9)^(4-x), x=0, 1, 2, 3, 4`

=0 otherwise

find E(x) and var (x)

Chapter: [0.19] Probability Distribution

Examine the maxima and minima of the function f(x) = 2x^{3} - 21x^{2} + 36x - 20 . Also, find the maximum and minimum values of f(x).

Chapter: [0.022000000000000002] Applications of Derivatives [0.14] Applications of Derivative

Solve the differential equation (x^{2} + y^{2})dx- 2xydy = 0

Chapter: [0.026000000000000002] Differential Equations [0.17] Differential Equation

Given the p. d. f. (probability density function) of a continuous random variable x as :

`f(x)=x^2/3, -1`

= 0 , otherwise

Determine the c. d. f. (cumulative distribution function) of x and hence find P(x < 1), P(x ≤ -2), P(x > 0), P(1 < x < 2)

Chapter: [0.19] Probability Distribution

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