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

Mathematics [Delhi Set 3]
Date: March 2014

 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
 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
 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
 4

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

Concept: Integration as an Inverse Process of Differentiation
Chapter: [0.07] Integrals
 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
 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
 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
 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
 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
 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
 11 | Attempt any one of the following
 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
 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
 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
 13 | Attempt any one of the follwoing
 13.1

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

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

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

Concept: Methods of Integration: Integration by Substitution
Chapter: [0.07] Integrals
 14 | Attempt any one of the follwoing
 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
 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
 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
 16 | Attempt any one of the following
 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
 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
 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
 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
 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
 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
 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
 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
 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
 24 | Attempt any one of the following
 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
 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
 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
 26 | Attempt any one of the following
 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
 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
 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
 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
 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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