Mathematics Set 1 2018-2019 ISC (Commerce) Class 12 Question Paper Solution

If f : R → R, f(x) = x³  and g: R → R , g(x) =  2x² + 1, and R is the set of real numbers, then find fog(x) and gof (x)

Solve:  sin(2 tan ^-1 x)=1

 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. 

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

`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?

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

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

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

 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?

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

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`.

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

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

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)`

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

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

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

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?

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

 Evaluate: `int_-6^3 |x+3|dx`

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

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?

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`

 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. 

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

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

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. 

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

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 \[\left| \vec{b} \right|\]

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

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

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

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

 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. 

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

Draw a rough sketch and find the area bounded by the curve x² = y and x + y = 2.

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. 

 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.) 

If `vecx = 18, vecy=100,σ<sub>y</sub> =20` , bary` and correlation coefficient xyr 0.8,  find the regression equation of y on x. 

The following results were obtained with respect to two variables x and y. 
∑ x = 15 , ∑y = 25, ∑xy = 83, ∑xy = 55, ∑y² =135 and n =5
(i) Find the regression coefficient xy b . 
(ii) Find the regression equation of x on y. 

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.

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?

The marginal cost function of x units of a product is given by 2MC= 3x² -10x +3x² The cost of producing one unit is Rs. 7. Find the cost function and average cost function. 

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.