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

Duration: 3h

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: [0.02] Inverse Trigonometric Functions

Evaluate : `int_2^3 3^x dx`

Chapter: [0.07] Integrals

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

Chapter: [0.03] 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: [0.05] Continuity and Differentiability

Chapter: [0.13] 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: [0.12] Linear Programming

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

Chapter: [0.07] Integrals

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

Chapter: [0.03] Matrices

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

Chapter: [0.09] Differential Equations

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

Chapter: [0.06] 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: [0.1] Vectors

For the curve y = 5x – 2x^{3}, if x increases at the rate of 2 units/sec, then find the rate of change of the slope of the curve when x = 3

Chapter: [0.09] Differential Equations

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

Chapter: [0.07] Integrals

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

Chapter: [0.07] 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: [0.09] 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: [0.1] Vectors

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: [0.13] Probability

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

Chapter: [0.02] 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: [0.04] 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: [0.03] Matrices

Differentiate the function with respect to *x*.

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

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.05] Continuity and Differentiability

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

Chapter: [0.13] Probability

Using vectors find the area of triangle ABC with vertices A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1).

Chapter: [0.1] Vectors

Solve the following L.P.P. graphically Maximise Z = 4x + y

Subject to following constraints x + y ≤ 50

3x + y ≤ 90,

x ≥ 10

x, y ≥ 0

Chapter: [0.12] Linear Programming

Find `int (2x)/((x^2 + 1)(x^4 + 4))`dx

Chapter: [0.07] Integrals

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

Chapter: [0.08] 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: [0.08] 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: [0.09] 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: [0.11] 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: [0.11] Three - Dimensional Geometry

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: [0.01] 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: [0.01] Relations and Functions

A metal box with a square base and vertical sides is to contain 1024 cm^{3}. The material for the top and bottom costs Rs 5 per cm2 and the material for the sides costs Rs 2.50 per cm2. Find the least cost of the box

Chapter: [0.06] Applications of Derivatives

if A = `((2,3,10),(4,-6,5),(6,9,-20))`, Find `A^(-1)`. Using `A^(-1)` Solve the system of equation `2/x + 3/y +10/z = 2`; `4/x - 6/y + 5/z = 5`; `6/x + 9/y - 20/z = -4`

Chapter: [0.04] Determinants

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