Date & Time: 19th March 2017, 12:30 pm

Duration: 3h

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^{–1}) = (det A)^{k}

Chapter: [2.02] Matrices

Determine the value of the constant 'k' so that function f(x) `{((kx)/|x|, ","if x < 0),(3"," , if x >= 0):}` is continuous at x = 0

Chapter: [3.01] Continuity and Differentiability

Evaluate : `int_2^3 3^x dx`

Chapter: [3.05] Integrals

If a line makes angles 90° and 60° respectively with the positive directions of *x* and *y* axes, find the angle which it makes with the positive direction of *z*-axis.

Chapter: [1.01] Inverse Trigonometric Functions

Show that all the diagonal elements of a skew symmetric matrix are zero.

Chapter: [2.02] Matrices

Find `(dy)/(dx)` at x = 1, y = `pi/4` if `sin^2 y + cos xy = K`

Chapter: [3.04] Differential Equations

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm

Chapter: [3.02] Applications of Derivatives

Show that the function f(x) = 4x^{3} - 18x^{2} + 27x - 7 is always increasing on R.

Chapter: [3.02] Applications of Derivatives

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Chapter: [4.02] Vectors

Chapter: [6.01] Probability

A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most 24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is Rs 100 and that on a bracelet is Rs 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit?

It is being given that at least one of each must be produced.

Chapter: [5.01] Linear Programming

Find `int dx/(x^2 + 4x + 8)`

Chapter: [3.05] Integrals

Prove that `tan {pi/4 + 1/2 cos^(-1) a/b} + tan {pi/4 - 1/2 cos^(-1) a/b} = (2b)/a`

Chapter: [1.01] Inverse Trigonometric Functions

Using properties of determinants, prove that `|(x,x+y,x+2y),(x+2y, x,x+y),(x+y, x+2y, x)| = 9y^2(x + y)`

Chapter: [2.01] Determinants

Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O

Chapter: [2.02] Matrices

Differentiate the function with respect to *x*.

`(sin x)^x + sin^(-1) sqrtx`

Chapter: [3.01] Continuity and Differentiability

if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`

Chapter: [3.01] Continuity and Differentiability

Find `int (2x)/(x^2 + 1)(x^2 + 2)^2 dx`

Chapter: [3.05] Integrals

Evaluate `int_0^pi (x sin x)/(1 + cos^2 x) dx`

Chapter: [3.05] Integrals

Evaluate `int_0^(3/2) |x sin pix|dx`

Chapter: [3.05] Integrals

Prove that x^{2} – y^{2} = c(x^{2} + y^{2})^{2} is the general solution of the differential equation (x^{3} – 3xy^{2})dx = (y^{3} – 3x^{2}y)dy, where C is parameter

Chapter: [3.04] Differential Equations

Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then

1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar

2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` coplanar

Chapter: [4.02] Vectors

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `veca + vecb+ vecc` is equally inclined to `veca, vecb` and `vecc`.

Chapter: [4.02] Vectors

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, find the angle which `veca + vecb + vecc`make with `veca or vecb or vecc`

Chapter: [4.02] Vectors

The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that `Sigmap_i x_i^2 = 2Sigmap_ix_i`. Find the value of p

Chapter: [6.01] Probability

Often it is taken that a truthful person commands, more respect in the society. A man is known to speak the truth 4 out of 5 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Do you also agree that the value of truthfulness leads to more respect in the society?

Chapter: [6.01] Probability

Solve the following L.P.P. graphically:

Minimise Z = 5x + 10y

Subject to x + 2y ≤ 120

Constraints x + y ≥ 60

x – 2y ≥ 0 and x, y ≥ 0

Chapter: [5.01] Linear Programming

Use product `[(1,-1,2),(0,2,-3),(3,-2,4)][(-2,0,1),(9,2,-3),(6,1,-2)]` to solve the system of equations x + 3z = 9, −x + 2y − 2z = 4, 2x − 3y + 4z = −3

Chapter: [2.02] Matrices

Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`

Hence Find

1) `f^(-1)(10)`

2) y if `f^(-1) (y) = 4/3`

where R_{+} is the set of all non-negative real numbers.

Chapter: [1.02] Relations and Functions

Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule *a* * *b*= *a* − *b* + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.

Chapter: [1.02] Relations and Functions

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Chapter: [3.02] Applications of Derivatives

Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).

Chapter: [3.03] Applications of the Integrals

Find the area bounded by the circle *x*^{2} + y^{2} = 16 and the line `sqrt3 y = x` in the first quadrant, using integration.

Chapter: [3.03] Applications of the Integrals

Solve the differential equation `x dy/dx + y = x cos x + sin x`, given that y = 1 when `x = pi/2`

Chapter: [3.04] Differential Equations

Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line *x* − 1 = 2*y* − 4 = 3*z* − 12.

Chapter: [4.01] Three - Dimensional Geometry

Find the vector and Cartesian equations of a line passing through (1, 2, –4) and perpendicular to the two lines `(x - 8)/3 = (y + 19)/(-16) = (z - 10)/7` and `(x - 15)/3 = (y - 29)/8 = (z - 5)/(-5)`

Chapter: [4.01] Three - Dimensional Geometry

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