Design of Question Paper Mathematics - Class XII Time : 3 hours

Max. Marks : 100

Weightage of marks over different dimensions of the question paper shall be as follows : A.

Weightage to different topics/content units S.No. 1. 2. 3. 4. 5. 6.

B.

Topics Relations and functions Algebra Calculus Vectors & three-dimensional Geometry Linear programming Probability Total

Marks 10 13 44 17 06 10 100

Weightage to different forms of questions S.No.

Forms of Questions

1. 2. 3.

Very Short Answer questions (VSA) Short answer questions (SA) Long answer questions (LA) Total

Marks for each question 01 04 06

No. of Questions 10 12 07 29

Total Marks 10 48 42 100

C.

Scheme of Options There will be no overall choice. However, an internal choice in any four questions of four marks each and any two questions of six marks each has been provided. D.

Difficulty level of questions S.No. 1. 2. 3.

Estimated difficulty level Easy Average Difficult

Percentage of marks 15 70 15

Based on the above design, separate sample papers along with their blue prints and Marking schemes have been included in this document. About 20% weightage has been assigned to questions testing higher order thinking skills of learners. (1)

(2)

Class XII MATHEMATICS Blue-Print I

Sample Question Paper - I MATHEMATICS Class XII Time : 3 Hours

Max. Marks : 100

General Instructions 1.

All questions are compulsory.

2.

The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each.

3.

All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

4.

There is no overall choice. However, Internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

5.

Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION - A

1.

Which one of the following graphs represent the function of x ? Why ? y

y

x

x

(a)

2.

(b)

What is the principal value of 2π 2π −1 cos −1 cos + sin sin ? 3 3

3.

A matrix A of order 3 × 3 has determinant 5. What is the value of | 3A | ?

4.

For what value of x, the following matrix is singular ?

5 − x 2 (3)

x + 1 4

5.

Find the point on the curve y = x 2 − 2 x + 3 , where the tangent is parallel to x-axis.

6.

What is the angle between vectors a & b with magnitude √3 and 2 respectively ? Given a . b = 3 .

7.

Cartesian equations of a line AB are.

→

→ →

→

2x −1 4 − y z + 1 = = 2 7 2

Write the direction ratios of a line parallel to AB.

∫e

3 log x

(x )dx 4

8.

Write a value of

9.

Write the position vector of a point dividing the line segment joining points A and B with position vectors a & b externally in the ratio

→

→

→

→

1 : 4, where a = 2 iˆ + 3ˆj + 4kˆ and b = − iˆ + ˆj + kˆ

10.

3 − 1 2 1 4 and B = 2 2 If A = 4 1 5 1 3

Write the order of AB and BA. SECTION - B 11.

Show that the function f : R → R defined by f (x ) =

2x −1 , x ∈ R is one-one and onto function. Also find the 3

inverse of the function f. OR Examine which of the following is a binary operation (i)

a*b=

a+b , a, b ∈ N 2

(ii)

a*b =

a+b , a, b ∈ Q 2

for binary operation check the commutative and associative property. 12.

Prove that

63 5 3 tan −1 = sin −1 + cos −1 16 13 5

(4)

13.

Using elementary transformations, find the inverse of

2 − 6 1 − 2 OR Using properties of determinants, prove that

− bc a 2 + ac

b 2 + bc c 2 + bc − ac

a 2 + ab b 2 + ab 14.

c 2 + ac = (ab + bc + ca )3 − ab

Find all the points of discontinuity of the function f defined by x + 2, x ≤1 f (x) = x − 2, 1 < x < 2 0, x≥2

15.

p q If x y = (x + y )

p +q

, prove that

dy y = dx x OR

1 + x2 + 1 − x2 dy −1 , 0 < | x | 0

(1)

6.

0.5

(1)

7.

order 1, degree 2

(½, ½ )

8.

(1)

.9.

(1)

10.

1 – (0.6)3

(1) SECTION - B

11.

(36)

Q. No.

Value Points

Marks

(1)

(1)

(1)

(1)

12. for n = 1

(37)

Q. No.

Value Points

Marks

LHS = aI + bA, RHS = aI + bA ∴ The result is true for n = 1 (1) Let (aI + bA)k = ak I + k a k-1 b A be true = (aI + bA)k . (aI + bA) Now, (aI + bA)k-1 = (kak – 1 bA + akI) (aI + bA) = k ak bA + k ak –1 b2 A2 + ak + 1 I + ak bA (aI + bA)k + 1 = k ak bA+ ak + 1 I + ak bA (a + bA)k + 1 ⇒

=

ak b(k + 1) A + ak + 1 I

= anI + nan –1 bA (aI + bA)n is true v n ∈ N by principle of mathametical induction.

(½)

(2)

(½)

OR

c1 → c1 + c2 + c3

(1)

(1) R2 → R2 – R1, R3 → R3 – R1

(1)

(1) 13.

(38)

Q. No.

Value Points

Marks

(1)

⇒

(1½)

(½)

(1) 14. Let P (x1, y1) be the point on the curve y = 4 x 3 − 3x + 5 where the tangent is perpendicular to the line 9 y + x + 3 = 0 ⇒

(½)

⇒

(½) Slope of a line perpendicular to

is 9

∴ ⇒ ⇒

(1) Two coresponding points on the curve

are (1, 6) and (–1, 4).

(½)

Therefore, equations of tangents are (1½) 15.

Let

f(x) = | x + 2 |

⇒

(½)

When x > –2 or x < –2, f(x) being a polynomial function is continuous. We check continuity at x = –2

(½)

(39)

Q. No.

Value Points

Also ⇒ ⇒ Now, LHD

Marks

f(-2) = 0 f(x) is continuous at x = –2 (1) f(x) is continuous v x ∈ R We check differentiability at x = –2 at x = –2

(½)

(½)

(½) ⇒

f(x) is not differentiable at x = –2

(½) (½)

16.

(1)

(1)

(1)

(½)

(40)

Q. No.

Value Points

Marks

(½)

17.

(1)

(1½)

(½)

=

(½) OR

(1)

(½)

(1)

(41)

Q. No.

Value Points

Marks

(1)

(½)

18.

(1)

area

=

1 2

→ → → → b − a × c − a

(½)

(1) If A, B, C are collinear then area ∆ ABC = 0

(½)

⇒

⇒

(½)

⇒

(½) OR

We have,

(1)

Since

∴ A, B, C form a triangle.

(½)

Also,

∴ BC ⊥ CA ⇒

Max. Marks : 100

Weightage of marks over different dimensions of the question paper shall be as follows : A.

Weightage to different topics/content units S.No. 1. 2. 3. 4. 5. 6.

B.

Topics Relations and functions Algebra Calculus Vectors & three-dimensional Geometry Linear programming Probability Total

Marks 10 13 44 17 06 10 100

Weightage to different forms of questions S.No.

Forms of Questions

1. 2. 3.

Very Short Answer questions (VSA) Short answer questions (SA) Long answer questions (LA) Total

Marks for each question 01 04 06

No. of Questions 10 12 07 29

Total Marks 10 48 42 100

C.

Scheme of Options There will be no overall choice. However, an internal choice in any four questions of four marks each and any two questions of six marks each has been provided. D.

Difficulty level of questions S.No. 1. 2. 3.

Estimated difficulty level Easy Average Difficult

Percentage of marks 15 70 15

Based on the above design, separate sample papers along with their blue prints and Marking schemes have been included in this document. About 20% weightage has been assigned to questions testing higher order thinking skills of learners. (1)

(2)

Class XII MATHEMATICS Blue-Print I

Sample Question Paper - I MATHEMATICS Class XII Time : 3 Hours

Max. Marks : 100

General Instructions 1.

All questions are compulsory.

2.

The question paper consist of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, section B comprises of 12 questions of four marks each and section C comprises of 07 questions of six marks each.

3.

All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.

4.

There is no overall choice. However, Internal choice has been provided in 04 questions of four marks each and 02 questions of six marks each. You have to attempt only one of the alternatives in all such questions.

5.

Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION - A

1.

Which one of the following graphs represent the function of x ? Why ? y

y

x

x

(a)

2.

(b)

What is the principal value of 2π 2π −1 cos −1 cos + sin sin ? 3 3

3.

A matrix A of order 3 × 3 has determinant 5. What is the value of | 3A | ?

4.

For what value of x, the following matrix is singular ?

5 − x 2 (3)

x + 1 4

5.

Find the point on the curve y = x 2 − 2 x + 3 , where the tangent is parallel to x-axis.

6.

What is the angle between vectors a & b with magnitude √3 and 2 respectively ? Given a . b = 3 .

7.

Cartesian equations of a line AB are.

→

→ →

→

2x −1 4 − y z + 1 = = 2 7 2

Write the direction ratios of a line parallel to AB.

∫e

3 log x

(x )dx 4

8.

Write a value of

9.

Write the position vector of a point dividing the line segment joining points A and B with position vectors a & b externally in the ratio

→

→

→

→

1 : 4, where a = 2 iˆ + 3ˆj + 4kˆ and b = − iˆ + ˆj + kˆ

10.

3 − 1 2 1 4 and B = 2 2 If A = 4 1 5 1 3

Write the order of AB and BA. SECTION - B 11.

Show that the function f : R → R defined by f (x ) =

2x −1 , x ∈ R is one-one and onto function. Also find the 3

inverse of the function f. OR Examine which of the following is a binary operation (i)

a*b=

a+b , a, b ∈ N 2

(ii)

a*b =

a+b , a, b ∈ Q 2

for binary operation check the commutative and associative property. 12.

Prove that

63 5 3 tan −1 = sin −1 + cos −1 16 13 5

(4)

13.

Using elementary transformations, find the inverse of

2 − 6 1 − 2 OR Using properties of determinants, prove that

− bc a 2 + ac

b 2 + bc c 2 + bc − ac

a 2 + ab b 2 + ab 14.

c 2 + ac = (ab + bc + ca )3 − ab

Find all the points of discontinuity of the function f defined by x + 2, x ≤1 f (x) = x − 2, 1 < x < 2 0, x≥2

15.

p q If x y = (x + y )

p +q

, prove that

dy y = dx x OR

1 + x2 + 1 − x2 dy −1 , 0 < | x | 0

(1)

6.

0.5

(1)

7.

order 1, degree 2

(½, ½ )

8.

(1)

.9.

(1)

10.

1 – (0.6)3

(1) SECTION - B

11.

(36)

Q. No.

Value Points

Marks

(1)

(1)

(1)

(1)

12. for n = 1

(37)

Q. No.

Value Points

Marks

LHS = aI + bA, RHS = aI + bA ∴ The result is true for n = 1 (1) Let (aI + bA)k = ak I + k a k-1 b A be true = (aI + bA)k . (aI + bA) Now, (aI + bA)k-1 = (kak – 1 bA + akI) (aI + bA) = k ak bA + k ak –1 b2 A2 + ak + 1 I + ak bA (aI + bA)k + 1 = k ak bA+ ak + 1 I + ak bA (a + bA)k + 1 ⇒

=

ak b(k + 1) A + ak + 1 I

= anI + nan –1 bA (aI + bA)n is true v n ∈ N by principle of mathametical induction.

(½)

(2)

(½)

OR

c1 → c1 + c2 + c3

(1)

(1) R2 → R2 – R1, R3 → R3 – R1

(1)

(1) 13.

(38)

Q. No.

Value Points

Marks

(1)

⇒

(1½)

(½)

(1) 14. Let P (x1, y1) be the point on the curve y = 4 x 3 − 3x + 5 where the tangent is perpendicular to the line 9 y + x + 3 = 0 ⇒

(½)

⇒

(½) Slope of a line perpendicular to

is 9

∴ ⇒ ⇒

(1) Two coresponding points on the curve

are (1, 6) and (–1, 4).

(½)

Therefore, equations of tangents are (1½) 15.

Let

f(x) = | x + 2 |

⇒

(½)

When x > –2 or x < –2, f(x) being a polynomial function is continuous. We check continuity at x = –2

(½)

(39)

Q. No.

Value Points

Also ⇒ ⇒ Now, LHD

Marks

f(-2) = 0 f(x) is continuous at x = –2 (1) f(x) is continuous v x ∈ R We check differentiability at x = –2 at x = –2

(½)

(½)

(½) ⇒

f(x) is not differentiable at x = –2

(½) (½)

16.

(1)

(1)

(1)

(½)

(40)

Q. No.

Value Points

Marks

(½)

17.

(1)

(1½)

(½)

=

(½) OR

(1)

(½)

(1)

(41)

Q. No.

Value Points

Marks

(1)

(½)

18.

(1)

area

=

1 2

→ → → → b − a × c − a

(½)

(1) If A, B, C are collinear then area ∆ ABC = 0

(½)

⇒

⇒

(½)

⇒

(½) OR

We have,

(1)

Since

∴ A, B, C form a triangle.

(½)

Also,

∴ BC ⊥ CA ⇒