# Mathematics Foreign Set 1 2013-2014 CBSE (Science) Class 12 Question Paper Solution

Mathematics [Foreign Set 1]
Date: March 2014

[1]1

Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.

Concept: Types of Relations
Chapter: [1.02] Relations and Functions
[1]2

Write the value of $\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)$ .

Concept: Inverse Trigonometric Functions (Simplification and Examples)
Chapter: [1.01] Inverse Trigonometric Functions
[1]3

Use elementary column operations  $C_2 \to C_2 - 2 C_1$ in the matrix equation $\begin{pmatrix}4 & 2 \\ 3 & 3\end{pmatrix} = \begin{pmatrix}1 & 2 \\ 0 & 3\end{pmatrix}\begin{pmatrix}2 & 0 \\ 1 & 1\end{pmatrix}$ .

Concept: Elementary Operation (Transformation) of a Matrix
Chapter: [2.02] Matrices
[1]4

If $\begin{pmatrix}a + 4 & 3b \\ 8 & - 6\end{pmatrix} = \begin{pmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{pmatrix},$ ,write the value of a − 2b.

Concept: Operations on Matrices - Properties of Matrix Addition
Chapter: [2.02] Matrices
[1]5

If A is a 3 × 3 matrix |3A| = k|A|, then write the value of k.

Concept: Order of a Matrix
Chapter: [2.02] Matrices
[1]6

Evaluate : $\int\frac{dx}{\sin^2 x \cos^2 x}$ .

Concept: Definite Integrals Problems
Chapter: [3.05] Integrals
[1]7

Evaluate : $\int\limits_0^\frac{\pi}{4} \tan x dx$ .

Concept: Properties of Indefinite Integral
Chapter: [3.05] Integrals
[1]8

Write the projection of $\hat{i} + \hat{j} + \hat{k}$ along the vector $\hat{j}$

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [4.02] Vectors
[1]9

Find a vector in the direction of vector $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ which has magnitude 21 units.

Concept: Magnitude and Direction of a Vector
Chapter: [4.02] Vectors
[1]10

Find the angle between the lines

$\vec{r} = \left( 2 \hat{i} - 5 \hat{j} + \hat{k} \right) + \lambda\left( 3 \hat{i} + 2 \hat{j} + 6 \hat{k} \right)$ and $\vec{r} = 7 \hat{i} - 6 \hat{k} + \mu\left( \hat{i} + 2 \hat{j} + 2 \hat{k} \right)$

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
[4]11

Let f : W → W be defined as f(x) = x − 1 if x is odd and f(x) = x + 1 if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers.

Concept: Composition of Functions and Invertible Function
Chapter: [1.02] Relations and Functions
[4]12 | Attempt any one of the following
[4]12.1

Solve for x : $\cos \left( \tan^{- 1} x \right) = \sin \left( \cot^{- 1} \frac{3}{4} \right)$ .

Concept: Properties of Inverse Trigonometric Functions
Chapter: [1.01] Inverse Trigonometric Functions
[4]12.2

Prove that:
cot−1 7 + cot​−1 8 + cot​−1 18 = cot​−1 3 .

Concept: Basic Concepts of Trigonometric Functions
Chapter: [1.01] Inverse Trigonometric Functions
[4]13

Using properties of determinants, prove that $\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)$ .

Concept: Properties of Determinants
Chapter: [2.01] Determinants
[4]14

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that y^2 (d^2y)/(dx^2)-xdy/dx+y=0

Concept: Second Order Derivative
Chapter: [3.01] Continuity and Differentiability
[4]15

If xmyn = (x + y)m+n, prove that $\frac{dy}{dx} = \frac{y}{x} .$

Concept: Basic Concepts of Differential Equation
Chapter: [3.04] Differential Equations
[4]16 | Attempt any one of the following.
[4]16.1

Find the approximate value of f(3.02), up to 2 places of decimal, where f(x) = 3x2 + 5x + 3.

Concept: Approximations
Chapter: [3.02] Applications of Derivatives
[4]16.2

Find the intervals in which the function $f(x) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51$ is

(a) strictly increasing
(b) strictly decreasing

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
[4]17 | Attempt any one of the following.
[4]17.1

Evaluate : $\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx$ .

Concept: Fundamental Theorem of Calculus
Chapter: [3.05] Integrals
[4]17.2

Evaluate : $\int(3x - 2) \sqrt{x^2 + x + 1}dx$ .

Concept: Properties of Definite Integrals
Chapter: [3.05] Integrals
[4]18

Solve the differential equation (x2 − yx2) dy + (y2 + x2y2) dx = 0, given that y = 1, when x = 1.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
[4]19

Solve the differential equation $\frac{dy}{dx}$ + y cot x = 2 cos x, given that y = 0 when x = $\frac{\pi}{2}$ .

Concept: Methods of Solving First Order, First Degree Differential Equations - Linear Differential Equations
Chapter: [3.04] Differential Equations
[4]20 | Attempt any one of the following.
[4]20.1

Show that the vectors $\vec{a,} \vec{b,} \vec{c}$ are coplanar if and only if $\vec{a} + \vec{b}$, $\vec{b} + \vec{c}$ and $\vec{c} + \vec{a}$ are coplanar.

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter: [4.02] Vectors
[4]20.2

Find a unit vector perpendicular to both the vectors $\vec{a} + \vec{b} \text { and } \vec{a} - \vec{b}$ ,where $\vec{a} = \hat{i}+ \hat{j} + \hat{k} , \vec{b} =\hat {i} + 2 \hat{j} + 3 \hat{k}$.

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter: [4.02] Vectors
[4]21

Find the shortest distance between the following pairs of lines whose vector are: $\overrightarrow{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( 2 \hat{i} - \hat{j} + \hat{k} \right) \text{ and } , \overrightarrow{r} = 2 \hat{i} + \hat{j} - \hat{k} + \mu\left( 3 \hat{i} - 5 \hat{j} + 2 \hat{k} \right)$

Concept: Equation of a Line in Space
Chapter: [4.01] Three - Dimensional Geometry
[4]22

Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution.

Concept: Conditional Probability
Chapter: [6.01] Probability
[6]23

Two schools P and Q want to award their selected students on the values of tolerance, kindness and leadership. School P wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively, with a total award money of Rs 2,200. School Q wants to spend Rs 3,100 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each value is Rs 1,200, using matrices, find the award money for each value.

Concept: Invertible Matrices
Chapter: [2.02] Matrices
[6]24

Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.

Concept: Maxima and Minima
Chapter: [3.02] Applications of Derivatives
[6]25

Evaluate : $\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x}dx$ .

Concept: Properties of Indefinite Integral
Chapter: [3.05] Integrals
[6]26

Find the area of the smaller region bounded by the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the line $\frac{x}{3} + \frac{y}{2} = 1 .$

Concept: Area Under Simple Curves
Chapter: [3.03] Applications of the Integrals
[6]27 | Attempt any one of the following.
[6]27.1

Find the equation of the plane that contains the point (1, –1, 2) and is perpendicular to both the planes 2x + 3y – 2z = 5 and x + 2y – 3z = 8. Hence, find the distance of point P (–2, 5, 5) from the plane obtained

Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
Chapter: [4.01] Three - Dimensional Geometry
[6]27.2

Find the distance of the point P (–1, –5, –10) from the point of intersection of the line joining the points A (2, –1, 2) and B (5, 3, 4) with the plane x – y + z = 5.

Concept: Plane - Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
Chapter: [4.01] Three - Dimensional Geometry
[6]28

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 25 and that from a shade is Rs 15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit? Formulate an LPP and solve it graphically.

Concept: Different Types of Linear Programming Problems
Chapter: [5.01] Linear Programming
[6]29 | Attempt any one of the following.
[6]29.1

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probabilities of an accident for them are 0.01, 0.03 and 0.15, respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver or a car driver?

Concept: Baye'S Theorem
Chapter: [6.01] Probability
[6]29.2

Five cards are drawn one by one, with replacement, from a well-shuffled deck of 52 cards. Find the probability that
(i) all the five cards diamonds
(ii) only 3 cards are diamonds
(iii) none is a diamond

Concept: Bernoulli Trials and Binomial Distribution
Chapter: [6.01] Probability

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