Date: March 2019

If f : R → R, f(x) = x^{3 } and g: R → R , g(x) = 2x^{2 }+ 1, and R is the set of real numbers, then find fog(x) and gof (x)

Chapter: [0.01] Relations and Functions (Section A)

Solve: sin(2 tan^{ -1 }x)=1

Chapter: [0.01] Relations and Functions (Section A)

Using determinants, find the values of k, if the area of triangle with vertices (–2, 0), (0, 4) and (0, k) is 4 square units.

Chapter: [0.021] Matrices and Determinants

Show that (A + A') is symmetric matrix, if `A = ((2,4),(3,5))`

Chapter: [0.021] Matrices and Determinants

`f(x)=(x^2-9)/(x - 3)` is not defined at x = 3. what value should be assigned to f(3) for continuity of f(x) at = 3?

Chapter: [0.031] Continuity, Differentiability and Differentiation

Prove that the function `f(x) = x^3- 6x^2 + 12x+5` is increasing on R.

Chapter: [0.032] Applications of Derivatives

Evaluate : `int sec^2x/(cosec^2x)dx`

Chapter: [0.033] Integrals

Using L’Hospital Rule, evaluate: `lim_(x->0) (8^x - 4^x)/(4x

)`

Chapter: [0.033] Integrals

Two balls are drawn from an urn containing 3 white, 5 red and 2 black balls, one by one without replacement. What is the probability that at least one ball is red?

Chapter: [0.04] Probability (Section A)

If events A and B are independent, such that `P(A)= 3/5`, `P(B)=2/3` 'find P(A ∪ B).

Chapter: [0.04] Probability (Section A)

If f A→ A and A=R - `{8/5}` , show that the function `f (x) = (8x + 3)/(5x - 8)` is one-one onto. Hence,find `f^-1`.

Chapter: [0.01] Relations and Functions (Section A)

Solve for x:

`tan^-1 [(x-1),(x-2)] + tan^-1 [(x+1),(x+2)] = x/4`

Chapter: [0.01] Relations and Functions (Section A)

if sec^{-1} x = cosec^{-1 }v. show that `1/x^2 + 1/y^2 = 1`

Chapter: [0.01] Relations and Functions (Section A)

Using propertiesof determinants prove that:

`|(x , x(x^2), x+1), (y, y(y^2 + 1), y+1),( z, z(z^2 + 1) , z+1) | = (x-y) (y - z)(z - x)(x + y+ z)`

Chapter: [0.021] Matrices and Determinants

Show that the function `f(x) = |x-4|, x ∈ R` is continuous, but not diffrent at x = 4.

Chapter: [0.031] Continuity, Differentiability and Differentiation

**Verify the Lagrange’s mean value theorem for the function: **`f(x)=x + 1/x ` in the interval [1, 3]

Chapter: [0.031] Continuity, Differentiability and Differentiation

IF `y = e^(sin-1x) and z =e^(-cos-1x),` prove that `dy/dz = e^x//2`

Chapter: [0.031] Continuity, Differentiability and Differentiation

A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

Chapter: [0.032] Applications of Derivatives

Solve the differential equation `dy/dx = (x + y+2)/(2(x+y)-1)`

Chapter: [0.01] Relations and Functions (Section A)

Evaluate: `int_-6^3 |x+3|dx`

Chapter: [0.01] Relations and Functions (Section A)

Solve the differential equation `dy/dx = (x + y+2)/(2(x+y)-1)`

Chapter: [0.01] Relations and Functions (Section A)

Bag A contains 4 white balls and 3 black balls. While Bag B contains 3 white balls and 5 black balls. Two balls are drawn from Bag A and placed in Bag B. Then, what is the probability of drawing a white ball from Bag B?

Chapter: [0.04] Probability (Section A)

Solve the following system of linear equation using matrix method:

`1/x + 1/y +1/z = 9`

`2/x + 5/y+7/z = 52`

`2/x+1/y-1/z=0`

Chapter: [0.021] Matrices and Determinants

The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.

Chapter: [0.032] Applications of Derivatives

Find the point on the straight line 2x+3y = 6, which is closest to the origin.

Chapter: [0.032] Applications of Derivatives

Evaluate: `int_0^x (xtan x)/(sec x + tan x) dx`

Chapter: [0.033] Integrals

Given three identical Boxes A, B and C, Box A contains 2 gold and 1 silver coin, Box B contains 1 gold and 2 silver coins and Box C contains 3 silver coins. A person choose a Box at random and takes out a coin. If the coin drawn is of silver, find the probability that it has been drawn from the Box which has the remaining two coins also of silver.

Chapter: [0.033] Integrals

Determine the binomial distribution where mean is 9 and standard deviation is `3/2` Also, find the probability of obtaining at most one success.

Chapter: [0.04] Probability (Section A)

If \[\vec{a}\] and \[\vec{b}\] are perpendicular vectors, \[\left| \vec{a} + \vec{b} \right| = 13\] and \[\left| \vec{a} \right| = 5\] find the value of \[\left| \vec{b} \right|\]

Chapter: [0.05] Vectors (Section B)

Find the length of the perpendicular from origin to the plane `vecr. (3i - 4j-12hatk)+39 = 0`

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Find the angle between the two lines `2x = 3y = -z and 6x =-y = -4z`

Chapter: [0.06] Three - Dimensional Geometry (Section B)

(a) If `veca = hati - 2j + 3veck , vecb = 2hati + 3hatj - 5hatk,` prove that `veca and vecaxxvecb` are perpendicular.

Chapter: [0.05] Vectors (Section B)

Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Find the equation of the plane passing through the interesection of the planes 2x + 2y -3z -7 =0 and 2x +2y - 3z -7=0 such that the intercepts made by the resulting plane on the x - axis and the z - axis are equal.

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines

Chapter: [0.06] Three - Dimensional Geometry (Section B)

Draw a rough sketch and find the area bounded by the curve x^{2} = y and x + y = 2.

Chapter: [0.07] Application of Integrals (Section B)

A company produces a commodity with Rs. 24,000 as fixed cost. The variable cost estimated to be 25% of the total revenue received on selling the products, is at the rate of Rs. 8 per unit. Find the break-even point.

Chapter: [0.08] Application of Calculus (Section C)

The total cost function for a production is given by `C(x) = 3/4 x^2 - 7x + 27`

Find the number of units produced for which M.C. = A.C

(M.C. = Marginal Cost and A. C. = Average Cost.)

Chapter: [0.08] Application of Calculus (Section C)

If `vecx = 18, vecy=100,σ_{y }=20` , bary` and correlation coefficient xyr 0.8, find the regression equation of y on x.

Chapter: [0.09] Linear Regression (Section C)

The following results were obtained with respect to two variables x and y.

∑ x = 15 , ∑y = 25, ∑xy = 83, ∑xy = 55, ∑y^{2 }=135 and n =5

(i) Find the regression coefficient xy b .

(ii) Find the regression equation of x on y.

Chapter: [0.09] Linear Regression (Section C)

Find the equation of the regression line of y on x, if the observations (x, y) are as follows :

(1,4),(2,8),(3,2),(4,12),(5,10),(6,14),(7,16),(8,6),(9,18)

Also, find the estimated value of y when x = 14.

Chapter: [0.09] Linear Regression (Section C)

The cost function of a product is given by C(x) =`x^3/3 - 45x^2 - 900x + 36` where x is the number of units produced. How many units should be produced to minimise the marginal cost?

Chapter: [0.08] Application of Calculus (Section C)

The marginal cost function of x units of a product is given by 2MC= 3x^{2 }-10x +3x^{2 }The cost of producing one unit is Rs. 7. Find the cost function and average cost function.

Chapter: [0.08] Application of Calculus (Section C)

A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for Rs. 48 per unit and product B is sold for Rs. 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income? Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph.

Chapter: [0.1] Linear Programming (Section C)

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