Date & Time: 2nd March 2019, 11:00 am

Duration: 3h

The principal solutions of cot x = -`sqrt3` are .................

`pi/6 ,(5pi)/6`

`(5pi)/6 , (7pi)/6`

`(5pi)/6,(11pi)/6`

`pi/6,(11pi)/6`

Chapter: [0.03] Trigonometric Functions

The acute angle between the two planes x+y+2z = 3 and 3x -2y +2z = 7 ________.

`"sin"^-1( 5/sqrt102)`

`"cos"^-1(5/sqrt102)`

`"sin"^-1(15/sqrt102)`

`"cos"^-1(15/sqrt102)`

Chapter: [0.1] Plane

The direction ratios of the line which is perpendicular to the lines with direction ratios –1, 2, 2 and 0, 2, 1 are _______.

–2, –1, –2

2, 1, 2

2, –1, –2

–2, 1, –2

Chapter: [0.08] Three Dimensional Geometry

If f(x) = `(1+2"x")^(1/"x)"` for x≠0 is continuous at x = 0 then f(0) = _______.

e

e^{2}

0

2

Chapter: [0.12] Continuity

`int "dx"/(9"x"^2 + 1)= ______. `

`1/3 "tan"^-1(2"x") +"c"`

`1/3 "tan"^-1"x" +"c"`

`1/3 "tan"^-1(3"x") +"c"`

`1/3 "tan"^-1(6"x") +"c"`

Chapter: [0.023] Indefinite Integration [0.15] Integration

If y = ae^{5x} + be^{-5x} then the differential equation is _________.

`("d"^2"y")/"dx"^2 = 25"y"`

`("d"^2"y")/"dx"^2 = -25"y"`

`("d"^2"y")/"dx"^2 = -5"y"`

`("d"^2"y")/"dx"^2 = 5"y"`

Chapter: [0.17] Differential Equation

Write the truth values of the following statements:

(1) 2 is a rational number and`sqrt2` is an irrational number. .

(2) 2 + 3 = 5 or `sqrt 2 +sqrt3 = sqrt5`

Chapter: [0.01] Mathematical Logic

Find the volume of the parallelopiped, if the coterminus edges are given by the vectors `2hat"i" + 5hat"j" -4 hat"k", 5hat"i" +7hat"j"+5 hat "k" , 4hat"i" +5hat"j" - 2 hat"k"`.

Chapter: [0.015] Vectors [0.07] Vectors

Find the value of p, if the vectors `hat"i" - 2hat"j" + hat"k", 2hat"i" -5hat"j"+"p" hat "k" , 5hat"i" -9hat"j" + 4 hat"k"` are coplanar.

Chapter: [0.015] Vectors [0.07] Vectors

Show that the points A (-7 , 4 , -2),B (-2 , 1 , 0)and C (3 ,-2 ,2) are collinear.

Chapter: [0.07] Vectors

Write the equation of the plane 3x + 4y -2z = 5 in the vector form.

Chapter: [0.1] Plane

If `"y" ="x"^"x" , "find" "dy"/"dx"`.

Chapter: [0.12] Continuity [0.13] Differentiation

Find the equation of tangent to the curve y = x^{2} +4x + 1 at (-1 , -2).

Chapter: [0.14] Applications of Derivative

Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`

Chapter: [0.15] Integration

In , ΔABC prove that

`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`

Chapter: [0.03] Trigonometric Functions

Show that `"sin"^-1(5/13) + "cos"^-1(3/5) = "tan"^-1(63/16)`

Chapter: [0.03] Trigonometric Functions

If `"A" (bar"a")` and `"B" (bar"b")` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that

`bar "r" = ("m"bar"b" + "n"bar"a")/("m" + "n") `

Chapter: [0.015] Vectors [0.07] Vectors

The equation of a line is 2x -2 = 3y +1 = 6z -2 find the direction ratios and also find the vector equation of the line.

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

Discuss the continuity of the function

f(x) = `("log"(2+"x") - "log"(2-"x"))/("tan""x")` , for x ≠ 0

= 1 for x = 0 at the point x =0

Chapter: [0.12] Continuity

x | 0 | 1 | 2 | 3 | 4 |

P(X = x) | 0.45 | 0.35 | 0.15 | 0.03 | 0.02 |

Find the variance of X.

Chapter: [0.19] Probability Distribution

For the following probability density function (p. d. f) of X, find : (i) P ( X<1), (ii) P |x| < 1

if `f(x) ="x"^2/18 , -3 < x < 3`

= 0, otherwise

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

Given is X ~ B (n,p)

If E(X) = 6,Var(X) = 4.2 , find n and p.

Chapter: [0.027999999999999997] Binomial Distribution [0.2] Bernoulli Trials and Binomial Distribution

Find the symbolic form of the given switching circuit. Construct its switching table and interpret yout result. diagram

Chapter: [0.01] Mathematical Logic

If three numbers are added, their sum is 2. If two times the second number is subtracted from the sum of first and third numbers we get 8 and if three times the first number is added to the sum of second and third numbers we get 4. Find the numbers using matrices.

Chapter: [0.012] Matrics [0.02] Matrices

In ,Δ ABC with usual notations prove that

b^{2} = c^{2} +a^{2} - 2 ca cos B

Chapter: [0.03] Trigonometric Functions

In , ΔABC with usual notations prove that

(a-b)^{2} cos^{2} `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`

Chapter: [0.03] Trigonometric Functions

Find 'p' and 'q' if the equation px^{2} - 8xy +3y^{2} +14 x +2y +q = 0 represents a pair of perpendicular lines.

Chapter: [0.04] Pair of Straight Lines

Maximize: z = 3x + 5y Subject to

x +4y ≤ 24 3x + y ≤ 21

x + y ≤ 9 x ≥ 0 , y ≥0

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

If X = f(t) and Y = g(t) Are Differentiable Functions of t , then prove that y is a differentiable function of x and

`"dy"/"dx" =("dy"/"dt")/("dx"/"dt" ) , "where" "dx"/"dt" ≠ 0`

Hence find `"dy"/"dx"` if x = a cos^{2} t and y = a sin^{2} t.

Chapter: [0.13] Differentiation

f(x) = (x-1)(x-2)(x-3) , x ε[0,4], find if 'c' LMVT can be applied

Chapter: [0.14] Applications of Derivative

A rod of 108 meters long is bent to form a rectangle. Find its dimensions if the area is maximum. Let x be the length and y be the breadth of the rectangle.

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

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

Chapter: [0.023] Indefinite Integration [0.15] Integration

Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`

Chapter: [0.023] Indefinite Integration [0.15] Integration

Solve the differential equation:

`"dy"/"dx" + "y" "sec" "x" = "tan""x"`

Chapter: [0.13] Differentiation

Solve the differential equation:

`("x" + "y") "dy"/"dx" = 1`

Chapter: [0.13] Differentiation

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