Date & Time: 21st March 2019, 10:30 am

Duration: 2h30m

- This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
- All questions in Section A are to be answered in one word, one sentence, or as per the exact requirement of the question.
- There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C, and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions.

If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.

Chapter: [2.02] Matrices

If f(x) = x + 1, find `d/dx (fof) (x)`

Chapter: [3.01] Continuity and Differentiability

Find the order and the degree of the differential equation `x^2 (d^2y)/(dx^2) = { 1 + (dy/dx)^2}^4`

Chapter: [3.04] Differential Equations

If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.

Chapter: [4.01] Three - Dimensional Geometry

Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.

Chapter: [4.01] Three - Dimensional Geometry

Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?

Chapter: [1.02] Relations and Functions

Find a matrix A such that 2A − 3B + 5C = 0, where B =`[(-2, 2, 0), (3, 1, 4)] and "C" = [(2, 0, -2),(7, 1, 6)]`.

Chapter: [2.02] Matrices

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Chapter: [3.05] Integrals

Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`

Chapter: [3.05] Integrals

Find: `int sin^-1 (2x) dx.`

Chapter: [3.05] Integrals

Form the differential equation representing the family of curves y = e^{2}x (a + bx), where 'a' and 'b' are arbitrary constants.

Chapter: [3.04] Differential Equations

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is `sqrt(3)`.

Chapter: [4.02] Vectors

if `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`

Chapter: [2.02] Matrices

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

Chapter: [6.01] Probability

A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of

(i) 5 successes?

(ii) at least 5 successes?

(iii) at most 5 successes?

Chapter: [6.01] Probability

The random variable X has probability distribution P(X) of the following form, where k is some number:

`P(X = x) {(k, if x =0),(2k, if x =1),(3k, if x = 2),(0, "otherwise"):}`

Determine the value of 'k'.

Chapter: [6.01] Probability

Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.

Chapter: [1.02] Relations and Functions

Prove that the function f : N → N, defined by f(x) = x^{2} + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Chapter: [3.02] Applications of Derivatives

Solve: tan^{-1} 4 x + tan^{-1} 6x `= π/(4)`.

Chapter: [1.01] Inverse Trigonometric Functions

Using properties of determinants, prove that

`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`

Chapter: [2.01] Determinants

If log `("x"^2 + "y"^2) = 2 tan^-1 ("y"/"x"), "show that" (d"y")/(d"x") = ("x" + "y")/("x" - "y")`

Chapter: [3.01] Continuity and Differentiability

If x^{y }- y^{x }= a^{b}, find `(dy)/(dx)`.

Chapter: [3.01] Continuity and Differentiability

If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`

Chapter: [3.01] Continuity and Differentiability

Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.

Chapter: [3.02] Applications of Derivatives

Find: `int (3x +5)/(x^2+3x-18)dx.`

Chapter: [3.05] Integrals

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Chapter: [3.05] Integrals

Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.

Chapter: [3.04] Differential Equations

Solve the differential equation: `(1 + x^2) dy/dx + 2xy - 4x^2 = 0,` subject to the initial condition y(0) = 0.

Chapter: [3.04] Differential Equations

if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.

Chapter: [4.02] Vectors

Find the value of λ, so that the lines `(1-"x")/(3) = (7"y" -14)/(λ) = (z -3)/(2) and (7 -7"x")/(3λ) = ("y" - 5)/(1) = (6 -z)/(5)` are at right angles. Also, find whether the lines are intersecting or not.

Chapter: [4.01] Three - Dimensional Geometry

If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A^{-1}. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

Chapter: [2.01] Determinants

Find the inverse of the following matrix using elementary operations.

`"A" = [(1,2,-2), (-1,3,0),(0,-2,1)]`

Chapter: [2.02] Matrices

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m^{3}. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

Chapter: [3.02] Applications of Derivatives

Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2).

Chapter: [2.01] Determinants

Find the area of the region lying above x-axis and included between the circle x^{2} + y^{2} = 8x nd inside of the parabola y^{2} = 4x.

Chapter: [3.03] Applications of the Integrals

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

Chapter: [4.01] Three - Dimensional Geometry

Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

Chapter: [4.01] Three - Dimensional Geometry

A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that was produced by A?

Chapter: [6.01] Probability

A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours of work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer's profit on an item of model A is ₹ 15 and on an item of model B is ₹ 10. How many items of each model should be made per day in order to maximize daily profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Chapter: [5.01] Linear Programming

#### Other Solutions

#### Submit Question Paper

Help us maintain new question papers on Shaalaa.com, so we can continue to help studentsonly jpg, png and pdf files

## CBSE previous year question papers Class 12 Mathematics with solutions 2018 - 2019

Previous year Question paper for CBSE Class 12 Maths-2019 is solved by experts. Solved question papers gives you the chance to check yourself after your mock test.

By referring the question paper Solutions for Mathematics, you can scale your preparation level and work on your weak areas. It will also help the candidates in developing the time-management skills. Practice makes perfect, and there is no better way to practice than to attempt previous year question paper solutions of CBSE Class 12.

How CBSE Class 12 Question Paper solutions Help Students ?

• Question paper solutions for Mathematics will helps students to prepare for exam.

• Question paper with answer will boost students confidence in exam time and also give you an idea About the important questions and topics to be prepared for the board exam.

• For finding solution of question papers no need to refer so multiple sources like textbook or guides.