ISC (Science) Class 12CISCE
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# Mathematics Set 1 2017-2018 ISC (Science) Class 12 Question Paper Solution

SubjectMathematics
Year2017 - 2018 (March)
Mathematics [Set 1]
Marks: 100Date: 2017-2018 March

1
1.1

The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*4

Concept: Types of Relations
Chapter:  Relations and Functions (Section A)
1.2

if A =((5,a),(b,0)) is symmetric matrix show that a = b

Concept: Symmetric and Skew Symmetric Matrices
Chapter: [2.01] Matrices and Determinants
1.3

Solve 3tan^(-1)x + cot^(-1) x = pi

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter:  Relations and Functions (Section A)
1.4

Without expanding at any stage, find the value of:

|(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|

Concept: Determinant of a Matrix of Order 3 × 3
Chapter: [2.01] Matrices and Determinants
1.5

Find the value of constant ‘k’ so that the function f (x) defined as

f(x) = {((x^2 -2x-3)/(x+1), x != -1),(k, x != -1):}

is continous at x = -1

Concept: Concept of Continuity
Chapter: [3.01] Continuity, Differentiability and Differentiation
1.6

Find the approximate change in the volume ‘V’ of a cube of side x metres caused by decreasing the side by 1%.

Concept: Approximations
Chapter: [3.02] Applications of Derivatives
1.7

Evaluate int(x^3+5x^2 + 4x + 1)/x^2  dx

Concept: Integration as an Inverse Process of Differentiation
Chapter: [3.03] Integrals
1.8

Find the differential equation of the family of concentric circles x^2 + y^2 = a^2

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
1.9

If A and B are events such as that P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/4, then find

1) P(A / B)

2) P(B / A)

Concept: Conditional Probability
Chapter:  Probability (Section A)
1.10

In a race, the probabilities of A and B winning the race are 1/3 and 1/6 respectively. Find the probability of neither of them winning the race.

Concept: Independent Events
Chapter:  Probability (Section A)
2

If the function f(x) = sqrt(2x - 3) is invertible then find its inverse. Hence prove that (fof^(-1))(x) = x

Concept: Types of Functions
Chapter:  Relations and Functions (Section A)
3

if tan^(-1) a + tan^(-1) b + tan^(-1) x = pi, prove that a + b + c = abc

Concept: Inverse Trigonometric Functions - Inverse Trigonometric Functions - Principal Value Branch
Chapter:  Relations and Functions (Section A)
4

Use properties of determinants to solve for x:

|(x+a, b, c),(c, x+b, a),(a,b,x+c)| = 0 and x != 0

Concept: Determinant of a Matrix of Order 3 × 3
Chapter: [2.01] Matrices and Determinants
5 | Attempt Any One
5.1

Show that the function f(x) = {(x^2, x<=1),(1/2, x>1):} is continuous at x = 1 but not differentiable.

Concept: Concept of Continuity
Chapter: [3.01] Continuity, Differentiability and Differentiation
5.2

Verify Rolle’s theorem for the following function:

f(x) = e^(-x) sinx " on"  [0, pi]

Concept: Mean Value Theorem
Chapter: [3.01] Continuity, Differentiability and Differentiation
6

if x = tan(1/a log y), prove that (1+x^2) (d^2y)/(dx^2) + (2x + a) (dy)/(dx) = 0

Concept: Derivatives of Inverse Trigonometric Functions
Chapter: [3.01] Continuity, Differentiability and Differentiation
7

Evaluate int tan^(-1) sqrtx dx

Concept: Integration as an Inverse Process of Differentiation
Chapter: [3.03] Integrals
8 | Attempt Any One
8.1

Find the points on the curve y = 4x^3 - 3x + 5 at which the equation of the tangent is parallel to the x-axis.

Concept: Tangents and Normals
Chapter: [3.02] Applications of Derivatives
8.2

Water is dripping out from a conical funnel of semi-verticle angle pi/4 at the uniform rate of 2 cm^2/secin the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4 cm, find the rate of decrease of the slant height of the water.

Concept: Increasing and Decreasing Functions
Chapter: [3.02] Applications of Derivatives
9 | Attempt Any One
9.1

Solve sin x dy/dx - y = sin x.tan  x/2

Concept: Solutions of Linear Differential Equation
Chapter: [3.04] Differential Equations
9.2

The population of a town grows at the rate of 10% per year. Using differential equation, find how long will it take for the population to grow 4 times.

Concept: General and Particular Solutions of a Differential Equation
Chapter: [3.04] Differential Equations
10 | Attempt Any One
10.1

Using matrices, solve the following system of equations :

2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

Concept: Minors and Co-factors
Chapter: [2.01] Matrices and Determinants
10.2

Using elementary transformation, find the inverse of the matrix :

[(0,1,2),(1,2,3),(3,1,1)]

Concept: Matrices - Proof of the Uniqueness of Inverse
Chapter: [2.01] Matrices and Determinants
11

A speaks the truth in 60% of the cases, while B is 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

Concept: Independent Events
Chapter:  Probability (Section A)
12

A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height

Concept: Maximum and Minimum Values of a Function in a Closed Interval
Chapter: [3.02] Applications of Derivatives
13 | Attempt Any One
13.1

Evaluate int (x-1)/(sqrt(x^2 - x)) dx

Concept: Methods of Integration - Integration by Substitution
Chapter: [3.03] Integrals
13.2

Evaluate int_0^(pi/2) cos^2x/(1+ sinx cosx) dx

Concept: Properties of Definite Integrals
Chapter: [3.03] Integrals
14

From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample.

If the sample is drawn without replacement, find :

1) The probability distribution of X

2) Mean of X

3) Variance of X

Concept: Probability Distribution Function
Chapter:  Probability (Section A)
15
15.1

Find lambda if the scalar projection of vec a = lambda hat i + hat j + 4 hat k on vec b = 2hati + 6hatj + 3hatk is 4 units

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Chapter:  Vectors (Section B)
15.2

The Cartesian equation of the line is 2x - 3 = 3y + 1 = 5 - 6z. Find the vector equation of a line passing through (7, –5, 0) and parallel to the given line.

Concept: Vector and Cartesian Equation of a Plane
Chapter:  Three - Dimensional Geometry (Section B)
15.3

Find the equation of the plane through the intersection of the planes vecr.(hati + 3hatj - hatk) = 9 and vecr.(2hati - hatj + hatj) = 3 and passing through the origin.

Concept: Plane - Equation of Plane Passing Through the Intersection of Two Given Planes
Chapter:  Three - Dimensional Geometry (Section B)
16 | Attempt Any One
16.1

If A, B, C are three non- collinear points with position vectors vec a, vec b, vec c, respectively, then show that the length of the perpendicular from Con AB is |(vec a xx vec b)+(vec b xx vec c) + (vec b xx  vec a)|/|(vec b -  vec a)|

Concept: Product of Two Vectors - Vector (Or Cross) Product of Two Vectors
Chapter:  Vectors (Section B)
16.2

Show that the four points A, B, C and D with position vectors 4hati + 5hatj + hatk, -hatj-hatk, 3hati + 9hatj + 4hatk and 4(-hati + hatj + hatk) respectively are coplanar

Concept: Scalar Triple Product of Vectors
Chapter:  Vectors (Section B)
17 | Attempt Any One
17.1

Draw a rough sketch of the curve and find the area of the region bounded by curve y2 = 8x and the line x =2.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter:  Application of Integrals (Section B)
17.2

Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.

Concept: Area of the Region Bounded by a Curve and a Line
Chapter:  Application of Integrals (Section B)
18

Find the image of a point having the position vector: 3hati - 2hatj + hat k in the plane vec r.(3hati - hat j + 4hatk) = 2

Concept: Vector and Cartesian Equation of a Plane
Chapter:  Three - Dimensional Geometry (Section B)

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