Category: Maths

Maths

Section – A

Q.1. Find the value of tan^-1 -cot^-1 .

Solution: We have 

Q.2. If the matrix A = is skew-symmetric, find the values of’a’ and ‘b’. 

Solution:

Q.3. Find the magnitude of each of the two vectors  , having the same magnitude such that the angle between them is 60° and their scalar product is 9/2 

Solution: Let be two such vectors. 

Q.4. If a * b denotes the larger of ‘a’ and ‘b’ and if a o b = (a * b) + 3, then write the value of (5) o (10), where * and o are binary operations.** 

 

Section – B

Q.5. Prove that

Solution: R.H.S = sin^-1(3x – 4x³)

Putting x = sin θ in R.H.S, we get

R.H.S = sin^-1(3 sin θ – 4 sin³ θ)

= sin^-1(sin 3θ)

 

Q.6. Given A =   compute A^-1 and show that 2A^-1 = 9I – A. [2]

Solution: 

 

Q.7. Differentiate  with respect to x. 

Solution: 

 

Q.8. The total cost C(x) associated with the pro-duction of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30s + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Solution : Cost function is given as

C(s) = 0.005x³ – 0.02x² + 30s + 5000

Marginal cost (MC) =

= 0.005(3x²) – 0.02 (2x) + 30

= 0.015x² – 0.04s+ 30

When x = 3,MC = 0.015(3)² – 0.04(3) + 30

= 0.135 – 0.12 + 30

= 30.015

 

Q.9. Evaluate, 

Solution: We have, 

 

Q.10. Find the differential equation representing the family of curves y = ae^{bx + 5}, where a and b are arbitrary constants. 

Solution: Given curve is

y = ae^{bx + 5} …(i)

Differentiating (i) w.r.t. s, we get

Q.11. If θ is the angle between two vectors   and , find sin θ.

Solution: 

 

Q.12. A black and a red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Solution : The sample space has 36 Outcomes. Let A be event that the sum of observations is 8.

∴ A = {(2, 6), (3, 5), (5, 3), (4, 4), (6, 2)}

 

Section – C

Q.13. Using properties of determinants, prove that:

Solution:

 

Q.14. If (x² + y²)² = xy, find dy/dx.

Solution: We have,

(x² + y²)² = xy …(i)

OR

Q.14. If x = a(2θ – sin 2θ) and y = a(1 – cos 2θ), find dy/dx when θ =π ⁄3

Solution: Given, x = a (2θ – sin 2θ)  and y = a (1 – cos 2θ)

Differentiating x* and y w.r.t. θ, we get 

 

Q.15. If y = sin(sin x), prove that:

Solution: Given, y = sin (sin x)

Differentiating y w.r.t. x, we get

 = cos (sin x) • cos x

Differentiating again w.r.t. x, we get

 = cos (sin x) (- sin x) + cos x [-sin (sin x)-cos x]

 

Q.16. Find the equations of the tangent and the normal to the curve 16x² + 9y² = 145 at the point (x₁, y₁), where x₁ = 2 and y₁ > 0. 

Solution: Given curve is

16x² + 9y² = 145 ….(i)

Since (x₁, y₁) lies on equation (i),

OR

Q.16. Find the intervals in which the function

(a) strictly increasing,

(b) strictly decreasing.

Solution: Given function is

 

Q.17. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question ? 

Solution: Let the length, breadth and height of the open tank be x, x and y units respectively.

Then, Volume (V) = x²y …(i)

Area is minimum, thus cost is minimum when x = 2y.

i.e., depth of tank is half of the width.

Value : Any relevant value.

 

Q.18. Find

Solution: 

 

Q.19. Find the particular solution of the differential equation e^x tan ydx + (2 – ex) sec² ydy = 0, given that y = 

when x = 0.

Solution: Given differential equation is,

OR

Q.19. Find the particular solution of the differential equation   + 2y tan x = sin x, given than y = 0 when x = .

Solution: Given differential equation is,

Hence, particular solution is

y sec² x = sec x – 2

or y = cos x – 2 cos²x

 

Q.20. Let Find a vector d  which is perpendicular to both .

Solution: Given

Q.21. Find the shortest distance between the lines and .

Solution: Gives lines are,

Q.22. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’, is obtained. If she obtained exactly one “tail7, what is the probability that she threw 3, 4, 5 or 6 with the dice? 

Solution : Let E₁ be the event that girl gets 1 or 2 on the roll and E₂ be the event that girl gets 3, 4, 5, or 6 on the roll of a die.

Let A be the event that she gets exactly one tail. If she tossed coin 3 times and gets exactly one tail then possible outcomes are HTH, HHT, THH.

                                      = 8/11

 

Q.23. Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Solution: First five positive integers are 1, 2, 3, 4, 5. We select two positive numbers in 5 × 4 = 20 ways. Out of these, two numbers are selected at random. Let X denote larger of the two selected numbers. Then, X can have values 2, 3, 4 or 5.

P(X = 2) – P (larger no. is 2) = {(1, 2) and (2, 1)}

Therefore, mean and variance are 4 and 1 respectively.

 

Q.24. Let A = {x ϵ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b): a, b ϵ A, | a – b | is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class . 

Solution: Given, R = {(a, b): a, b ϵ A, | a – b | is divisible by 4}

Reflexivity: For any a ϵ A

| a – a | = 0, which is divisible by 4

(a, a) ϵ R

So, R is reflexive.

Symmetry: Let (a, b) ϵ R

⇒ | a – b | is divisible by 4

⇒ | b – a | is divisible by 4 [ ∵ | a – b | = | b – a | ]

⇒ (b, a) ϵ R

So, R is symmetric.

Transitive : Let (a, b) ϵ R and (b, c) ϵ R

⇒ | a – b | is divisible by 4<br> ⇒ | a – b | = 4k

∴ a – b = ± 4k, k ϵ Z …(i)

Also, |b – c| is divisible by 4

⇒ |b – c| = 4m

∴ b – c = ±4m, m ϵ Z …(ii)

Adding equations (i) and (ii)

a – b + b – c = ±4 (k + m)

⇒ a – c = ±4 (k + m)

|a – c | is divisible by 4,

⇒ (a, c) ϵ R

So, R is symmetric.

⇒ R is reflexive, symmetric and transitive.

∴ R is an equivalence relation.

Let x be an element of R such that (x, 1) ϵ R

Then | x – 1| is divisible by 4

x – 1 = 0, 4, 8, 12,

⇒ -x = 1, 5, 9 (∵ x ≤ 12)

∴ Set of all elements of A which are related to 1 are {1, 5, 9}.

Equivalence class of 2 i.e.

[2] = {(a, 2): a ϵ A, |a – 2| is divisible by 4}

⇒ | a – 2| = 4k(k is whole number, k ≤ 3)

⇒ a = 2, 6, 10

Therefore, equivalence class [2] is {2, 6, 10]

OR

Q.24. Show that the function f : R → R defined by f(x) = , ∀x ϵ R is neither one-one nor onto. Also,if g: R → R is defined as g(x) = 2x – 1, find fog(x).

Solution: 

Q.25. If A = , find A^-1. Use it to solve the system of equation: 

2x – 3y + 5z = 11

3x + 2y – 4z = -5

x + y – 2z = -3

Solution:

OR

Q.25. Using elementary row transformations, find the inverse of the matrix:

We know that, A = IA

Q.26. Using integration, find the area of the region in the first quadrant enclosed by the X-axis, the line y = x and the circle x² + y² = 32. 

Solution: Given curve is

Q.27. Evalute

Solution:

OR

Q.27. Evaluate  as the limit of the sum.

Solution:

 

Q.28. Find the distance of the point (-1, -5, -10) from the point of intersection of the line 

 and the plane 

Solution:  Equation of line is

 

Q.29. A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘A’ while it takes 6 minutes on the automatic and 3 minutes on the hand- operated machine to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of ₹1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit ? Formulate the above LPP and solve it graphically and find the maximum profit 

Solution: Let the number of packets of screw ‘A’ manufactured in a day be x and that of screw B be y.

Plotting the points on the graph, we get the feasible region OABC as shown (Shaded).

Hence, profit will be maximum if company produces 30 packets of screw A and 20 packets of screw B and maximum profit = ₹ 41.

Previous Paper (SET-1) Click Here!

Maths

Section – A

Q.1. Find | AB |, if A = and B = .

Solution: 

Q.2. Differentiate  , with respect to x.

Solution:

Section – B

Q.6. If A = and A³ = | = 125, the find the value of p.

Solution: 

Q.12. Find the general solution of the differential equation  

Solution: 

Section – C

Q.21. If (a + bx)e^y/x = x, then prove that

Solution: We have

Q.22. The volume of a cube is increasing at the rate of 8 cm³/s. How fast is the surface area increasing when the length of its edge is 12 cm? 

Solution: Let x be the length of side, V be the volume and S be the surface area of cube.

Then, V = x³ and S = 6x², where x is a function of time t

Q.23. Find the cartesian and vector equations of the plane passing through the point A(2, 5, – 3), B(- 2, 3, 5) and C(5, 3, – 3). 

Solution: We know that the general equation of the plane passing through three points (x₁, y₁, z₁) (x₁, y₁, Z₁), (x₃, y₃, z₃)

Section – D

Q.24. Find the point on the curve y² = 4x, which is nearest to the point (2, – 8). 

Solution: Given curve is of the form, y² = Ax and let p(x, y) is a point on the curve which is nearest to the point (2, – 8).

Q.25. Find as the limit of sums  .

Solution: We have

OR

Q.24. Using integration, find the area of the triangular region whose sides have the equation y = 2x + 1, y = 3x + 1 and x = 4.

Solution: The equations of sides of triangle are

y = 2x + 1, …(i)

y = 3x + 1 …(ii)

and x = 4 …(iii)

The equation y = 2x + 1 meets x and y axes at  and (0, 1). By joining these two points we obtain the graph of x + 2y = 2. Similarly, graphs of other equations are drawn.

Solving equation (i), (ii) and (iii) in pairs, we obtain the coordinates of vertices of ∆ABC are A(0, 1), B (4, 13) and C (4, 9).

Then, area of ∆ABC = Area (OLBAO) – Area (OLCAO)

Maths

SET-III

Section – A

Q.1.Find the differenital equation representing the family of curves y = ae^2x + 5, where a is an arbitrary constant. 

Solution:

Given, y = ae^2x + 5

On differentiating w.r.t. x, we get

 

Q.2. If y = cos√3x, then find ^dy/_dx. 

Solution:

Section – B

Q.5. Show that the points  ,  and   are collinear. 

Solution:

Q.6. Find:

Solution:

 

Section – C

Q.13. Solve for x: tan^-1 + tan^-1(x – 1) = tan^-1(⁸/₃₁).

Solution:

 

Q.14. If x = ae^t (sin t + cost t) and y = ae^t (sin t – cos t), then prove that 

^dy/_dx= ^x+y/_x-y.

Solution:

                              ^dy/_dx= ^x+y/_x-y                  Hence proved.

 OR

Q.14. Differentiate x^sinx + (sinx)^cosx with respect to x.

 Solution:

 

Q.15. Find:

Solution:

 

Section – D

Q.24. 

Show that for the matrix A = , A³ – 6A² + 5A + 111 = 0. Hence, find A ^-1.

Solution:

OR

Q.24. Using matrix method, solve the following system of equations:

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

Solution:

The given equations are

3x – 2y + 3z = 8

2x + y – z = 1

4x – 3y + 2z = 4

These equations can be written in the form AX = B, where

 

Q.26. A bag contains 5 red and 4 black balls, a second bag contains 3 red and 6 black balls. One of the two bags is selected at random and two balls are drawn at random (without replacement) both of which are found to be red. Find the probability that the balls are drawn from the second bag. 

Solution:

 Let E₁ be the event of choosing the bag I, E₂ be the event of choosing the bag II and A be the event of drawing a red ball.

 Then,

Maths

SET-I

Section – A

Q.1. If A is a square matrix satisfying A’ A = I, write the value of |A|.

Solution: Given, A’ A= I

 

Q.2. If y = x | x |, find for x &lt; 0. 

Solution: If y = x | x |

 

Q.3. Find the order and degree (if defined) of the differential equation 

Solution:

Order of this equation is 2.

Degree of this equation is not defined.

 

Q.4. Find the direction cosines of a line which makes equal angles with the coordinate axes. 

Solution : Let the direction cosines of the line make an angle with each of the coordinate axes and direction cosines be l, m and n.

OR

Q.4 A line passes through the point with position vector  and is in the direction of the vector  . Find the equation of the line in cartesian form.

Solution: The line passes through a point (2, -1, 4) and has direction ratios proportional to (1, 1, – 2).

Cartesian equation of the line

Section – B

Q.5. Examine whether the operation * defined on R, the set of all real numbers, by   is a binary operation or not, and if it is a  binary operation, find out whether it is associative or not.** 

Q.6. If A = , show that (A – 2I)(A – 3I) = 0. 

Solution: Given,

 

Q.7. Find

Solution:

 

Q.8. Find 

Solution:

OR

Q.8. Find

Solution:

 

Q.9. Find the differential equation of the family of curves y = Ac^2x + Be^-2x, where A and B are arbitrary constants. 

Solution: Given, y = Ae^2x + Be^-2x         ………(i)

On differentiating equation (i) w.r.t. x, we get

 

Q.10. If  and , find the angle between and .

Solution: Given,

OR

Q.10. Find the volume of a cuboid whose edges are given by  and .

Solution: If a, b, c are edges of a cuboid.

 

Q.11. If P(not A) = 0·7, P(B) = 0·7 and P(B/A) = 0·5, then find P(A∆B).

Solution:

 

Q.12. A coin is tossed 5 times. What is the probability of getting (i) 3 heads, (ii) at most 3 heads? 

Solution:

OR

Q.12. Find the probability distribution of X, the number of heads in a simultaneous toss of two coins.

Solution: If we toss two coins simultaneously then sample space is given by (HH, HT, TH, IT)

Then probability distribution is,

 

Section – C

Q.13. Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive. 

Solution: Here, R = {(a, b): b = a +1}

 ∴ R = {(a, a + 1): a, a + 1 ϵ (1, 2, 3, 4, 5 ,6)}

 ⇒ R = {(1, 2,) (2, 3), (3, 4), (4, 5), (5, 6)}

(i) R is not reflexive as {a, a} ∉ R∀a

(ii) R is not symmetric as (1, 2) ϵ R but (2, 1) ∉ R

(iii) R is not transitive as (1, 2) ϵ R, (2, 3) ϵ R but (1, 3) ϵ R

OR

Q.13. Let f: N → Y be a function defined as

f(x) = 4x + 3,

where Y = {y ϵ N : y = 4x + 3, for some x ϵ N}.

Show that f is invertible. Find its inverse.

Solution: Consider an arbitrary element of Y. By the definition of y, y = 4x + 3, for some x in the domain N.

Q.14. Find the value

Solution:

Q.15. Using properties of determinants, show that 

Solution :

Q.16. If = 0 and x ≠ y, prove that 

Solution:

Solution:

Q.17. If If (x – a)² + (y – b)² = c², for some c > 0, prove that  is a constant independent of a and b. 

Solution:

Q.18. Find the equation of the normal to the curve x² = 4y which passes through the point (-1, 4). 

Solution: Suppose the normal at P(x₁, y₁) on the parabola x 1 = 4y passes through (-1, 4)

Since, P(x₁, y₁) lies on x₁ = 4y

∴ = 4y₁

The equation of curve is x² = 4y.

Differentiating with respect to x, we have

2x = 4

 

Q.19. Find

Solution:

 

Q.20. Prove that  and hence evaluate .

Solution:

Q.21. Solve the differential equation: 

Solution:

 

Q.22. The scalar product of the vector    with a unit vector along the sum of the vectors  and  is equal to 1. Find the value of λ and hence find the unit vector along  .

Solution:

 

Q.23. If the lines  and  are perpendicular, find the value of λ. Hence find whether the lines are intersecting or not.

Solution: 

Section – D

Q.24. If A =, Find A^-1_.

Hence solve the system of equations

x + 3y + 4z = 8

2x + y + 2z = 5

and 5x + y + z = 7

Solution :

OR

Q.24. Find the inverse of the following matrix, using elementary transformation:

A =  

Solution:

Q.25. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is . Also find the maximum volume.

Solution: Let, ‘x’ be the diameter of the base of the cylinder and let ‘h’ be the height of the cylinder.

Q.26. Using method of integration, find the area of the triangle whose vertices are (1, 0), (2, 2) and (3, 1).

Solution: A(1, 0), B(2, 2) and C(3, 1)

OR

Q.26. Using method of integration, find the area of the region enclosed between two circles x² + y² = 4 and (x – 2)² + y² = 4.

Solution: Equations of the given circles are,

x² + y² = 4 …(i)

(x – 2)² + y² = 4 …(ii)

Equation (i) is a circle with centre O at the origin and radius 2. Equation (ii) is a circle with centre C (2, 0) and radius 2.

Solving equation (i) and (ii) we have

(x – 2)² + y² = x² + y²

or x² – 4x + 4 + y² = x² + y² or x = 1 which gives y = ±

Thus, the points of intersection of the given circles are A (1, 3) and A (1, 3)

Q.27. Find the vector and cartesian equations of the plane passing through the points having position vectors and . Write the equation of a plane passing through a point (2, 3, 7) and parallel to the plane obtained above. Hence, find the distance between the two parallel planes.

Solution: Let A, B, C be the points with position

OR

Q.27. Find the equation of the line passing through (2, -1, 2) and (5, 3, 4) and of the plane passing through (2, 0, 3), (1, 1, 5) and (3, 2, 4). Also, find their point of intersection.

Solution: We know that the equation of a line passing through points (x₁, y₁, z₁,) and (x₂, y₂, z₂) ) is given by

 

Q.28. There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time and the third is an unbiased coin. One of the three coins is chosen at random and tossed. If it shows heads, what is the probability that it is the two-headed coin?

Solution: Given, there are three coins.

Let,

E₁ = coin is two headed

E₂ = biased coin

E₃ = unbiased coin

A = shows only head

 

Q.29. A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3g of silver and 1 g of gold while that of type B requires 1 g of silver and 2g of gold. The company can use at most 9 g of silver and 8 of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, find the number of units of each type that the company should produce to maximize profit. Formulate the above LPP and solve it graphically and also find the maximum profit.

Solution: There are two types of goods, A and B and let units of type A be x and units of type B be y. 

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