# Mathematics Delhi Set 3 2013-2014 CBSE (Arts) Class 12 Question Paper Solution

Mathematics [Delhi Set 3]
Date: March 2014

[1] 1

if 2[[3,4],[5,x]]+[[1,y],[0,1]]=[[7,0],[10,5]] , find (xy).

Concept: Equality of Matrices
Chapter: [0.03] Matrices
[1] 2

Solve the following matrix equation for x: [x 1] [[1,0],[−2,0]]=0

Concept: Operations on Matrices - Addition of Matrices
Chapter: [0.03] Matrices
[1] 3

If $\begin{vmatrix}2x & 5 \\ 8 & x\end{vmatrix} = \begin{vmatrix}6 & - 2 \\ 7 & 3\end{vmatrix}$ , write the value of x.

Concept: Applications of Determinants and Matrices
Chapter: [0.04] Determinants
[1] 4

Write the antiderivative of (3sqrtx+1/sqrtx).

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.07] Integrals
[1] 5

If sin (sin^(−1)(1/5)+cos^(−1) x)=1, then find the value of x.

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[1] 6

Let * be a binary operation, on the set of all non-zero real numbers, given by $a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}$

Write the value of x given by 2 * (x * 5) = 10.

Concept: Concept of Binary Operations
Chapter: [0.01] Relations and Functions
[1] 7

Find the projection of the vector hati+3hatj+7hatk  on the vector 2hati-3hatj+6hatk

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[1] 8

Write the vector equation of the plane, passing through the point (a, b, c) and parallel to the plane vec r.(hati+hatj+hatk)=2

Concept: Vector and Cartesian Equation of a Plane
Chapter: [0.11] Three - Dimensional Geometry
[1] 9

Evaluate each of the following integral:

$\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx$

Concept: Definite Integrals Problems
Chapter: [0.07] Integrals
[1] 10

Write a unit vector in the direction of the sum of the vectors $\overrightarrow{a} = 2 \hat{i} + 2 \hat{j} - 5 \hat{k}$ and $\overrightarrow{b} = 2 \hat{i} + \hat{j} - 7 \hat{k}$.

Concept: Vectors and Their Types
Chapter: [0.1] Vectors
[4] 11 | Attempt any one of the following
[4] 11.1

Prove that, for any three vectors $\vec{a} , \vec{b} , \vec{c}$ $\left[ \vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a} \right] = 2 \left[ \vec{a} , \vec{b} , \vec{c} \right]$.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[4] 11.2

Vectors veca,vecb and vecc  are such that veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7  Find the angle between veca and vecb

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [0.1] Vectors
[4] 12

Solve the following differential equation: (x^2-1)dy/dx+2xy=2/(x^2-1)

Concept: Solutions of Linear Differential Equation
Chapter: [0.09] Differential Equations
[4] 13 | Attempt any one of the follwoing
[4] 13.1

Evaluate : ∫(sin^6x+cos^6x)/(sin^2x.cos^2x)dx

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.07] Integrals
[4] 13.2

Evaluate : int(x-3)sqrt(x^2+3x-18)  dx

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
[4] 14 | Attempt any one of the follwoing
[4] 14.1

Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is

(a) strictly increasing

(b) strictly decreasing

Concept: Increasing and Decreasing Functions
Chapter: [0.06] Applications of Derivatives
[4] 14.2

Find the equations of the tangent and normal to the curve x = a sin3θ and y = a cos3θ at θ=π/4.

Concept: Tangents and Normals
Chapter: [0.06] Applications of Derivatives
[4] 15

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].

Concept: Types of Relations
Chapter: [0.01] Relations and Functions
[4] 16 | Attempt any one of the following
[4] 16.1

Prove that cot^(-1)((sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx)))=x/2;x in (0,pi/4)

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 16.2

Prove that

$2 \tan^{- 1} \left( \frac{1}{5} \right) + \sec^{- 1} \left( \frac{5\sqrt{2}}{7} \right) + 2 \tan^{- 1} \left( \frac{1}{8} \right) = \frac{\pi}{4}$ .

Concept: Properties of Inverse Trigonometric Functions
Chapter: [0.02] Inverse Trigonometric Functions
[4] 17

If y = xx, prove that $\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .$

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[4] 18

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls? Give that
(i) the youngest is a girl.
(ii) at least one is a girl.

Concept: Conditional Probability
Chapter: [0.13] Probability
[4] 19

Using properties of determinants, prove the following:

$\begin{vmatrix}x^2 + 1 & xy & xz \\ xy & y^2 + 1 & yz \\ xz & yz & z^2 + 1\end{vmatrix} = 1 + x^2 + y^2 + z^2$ .
Concept: Properties of Determinants
Chapter: [0.04] Determinants
[4] 20

Differentiate$\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)$ with respect to $\sin^{-1} \left( \frac{2x}{1 + x^2} \right)$, If $- 1 < x < 1, x \neq 0 .$ ?

Concept: Simple Problems on Applications of Derivatives
Chapter: [0.06] Applications of Derivatives
[4] 21

Find the particular solution of the differential equation $\frac{dy}{dx} = \frac{x\left( 2 \log x + 1 \right)}{\sin y + y \cos y}$ given that

$y = \frac{\pi}{2}$ when x = 1.
Concept: General and Particular Solutions of a Differential Equation
Chapter: [0.09] Differential Equations
[4] 22

Prove that the line $\vec{r} = \left( \hat{i }+ \hat{j }- \hat{k} \right) + \lambda\left( 3 \hat{i} - \hat{j} \right) \text{ and } \vec{r} = \left( 4 \hat{i} - \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{k} \right)$ intersect and find their point of intersection.

Concept: Equation of a Line in Space
Chapter: [0.11] Three - Dimensional Geometry
[6] 23

A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs 5,760 to invest and has space for at most 20 items for storage. An electronic sewing machine cost him Rs 360 and a manually operated sewing machine Rs 240. He can sell an electronic sewing machine at a profit of Rs 22 and a manually operated sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as a LPP and solve it graphically.

Concept: Graphical Method of Solving Linear Programming Problems
Chapter: [0.12] Linear Programming
[6] 24 | Attempt any one of the following
[6] 24.1

A card from a pack of 52 playing cards is lost. From the remaining cards of the pack three cards are drawn at random (without replacement) and are found to be all spades. Find the probability of the lost card being a spade.

Concept: Independent Events
Chapter: [0.13] Probability
[6] 24.2

From a lot of 15 bulbs which include 5 defectives, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence find the mean of the distribution.

Concept: Random Variables and Its Probability Distributions
Chapter: [0.13] Probability
[6] 25

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.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter: [0.08] Applications of the Integrals
[6] 26 | Attempt any one of the following
[6] 26.1

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).

Concept: Distance of a Point from a Plane
Chapter: [0.11] Three - Dimensional Geometry
[6] 26.2

Find the distance of the point (−1, −5, −10) from the point of intersection of the line vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk)  and the plane vec r (hati-hatj+hatk)=5

Concept: Three - Dimensional Geometry Examples and Solutions
Chapter: [0.11] Three - Dimensional Geometry
[6] 27

Two schools P and Q want to award their selected students on the values of discipline, politeness and punctuality. The school P wants to award Rs x each, Rs y each and Rs z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs 1,000. School Q wants to spend Rs 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs 600, using matrices, find the award money for each value.
Apart from the above three values, suggest one more value for awards.

Concept: Invertible Matrices
Chapter: [0.03] Matrices
[6] 28

Evaluate: $\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x}dx$ .

Concept: Evaluation of Definite Integrals by Substitution
Chapter: [0.07] Integrals
[6] 29

Of all the closed right circular cylindrical cans of volume 128π cm3, find the dimensions of the can which has minimum surface area.

Concept: Graph of Maxima and Minima
Chapter: [0.06] Applications of Derivatives

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