B. R.
BIRLA PUBLIC SCHOOL, JODHPUR
Terminal Examination 2020
Class: XII Subject: Mathematics
Time allowed: 3 hours Maximum Marks: 80
General Instructions
(a) All questions are compulsory.
(b) The question paper consists of 36 questions divided into 4 Sections A, B, C and D.
(c) Section A comprises of 20 Objective Type Questions carrying 01 mark each,
(d) Section B comprises of 6 Questions carrying 02 marks each,
(e) Section C comprises of 6 Questions carrying 04 marks each,
(f) Section D comprises of 4 Questions carrying 06 marks each,
(g) There is no overall choice in the question paper. However, an internal choice has been
provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks
and 2 questions of six marks. Only one of the choices in such questions have to be
attempted.
(h) Use of calculator is not permitted.
SECTION A
(Question no. 1 to 20 are of Objective Type carrying 01 Mark each)
1. If R be a relation in A= {1, 2, 3, 4} given by R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1),
(3, 1), (1, 3)}, then R is
(a) Symmetric only (b) Reflexive only
(c) An equivalence (d) Transitive Only
2
2. The function f : R R given by f(x) x is
(a) one – one onto (b) many – one onto
(c) many– one into (d) one – one into
1 1
3. The value of tan 3 sec (2) is equal to
2
(a) (b) (c) (d)
3 3 3
1
4. The value of tan(cos x) =
1 x2 x 1 x2
(a) (b) (c) (d) 1 x 2
x 1 x2 x
1 / 3 2 3 6
5. If A ,B and AB I , then x =
0 2x 3 0 1
(a) –1 (b) 1 (c) 0 (d) 2
6. If the order of a square matrices A and B are 3, and |A| = 2 and |B|= – 1, then
|2AB|is
(a) – 4 (b) 32 (c) – 32 (d) – 16
OR
3 5
If A , then A2 5A
4 2
(a) I (b) 14 I (c) 0 (d) None of these
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7. If the order of matrices A, B, C and D are p k , 2 n , 3 m and 4 k respectively then
the values of k and m for which ACB is defined are
(a) k = 2, m = 2 (b) k = 3, m = 4 (c) k = 3, m = 2 (d) k = 2, m = 4
d 1 cos x
8. tan
dx 1 sin x
1 1
(a) (b) (c) 1 (d) 1
2 2
9. If f( x) mx c, f(0) f'(0) 1 then f( 2)
(a) 1 (b) 2 (c) 3 (d) – 3
10. The slope of tangent to the curve x t 3t 8 , y 2t 2t 5 at the point (2, –1) is
2 2
22 6
(a) (b) (c) – 6 (d) None of these
7 7
Fill in the blanks in question no. 11 to 15.
11. If the function f : R+ → S be defined by f (x) = x 4 is onto function , then S = _______
2
12. The slope of the normal to the curve y x 3 2x at the point (1, 3) is _______
1 3 2
13. If the matrix 2 4
8 is singular, then _____________
3 5 10
sin x
cos x , if x 0
14. If f( x) x is continuous at x = 0, then the value of k is _________
k , if x 0
x
15. The simplest form of tan 1 is _________
2
1 x
OR
We can express sin x as tan ________
1 1
Answer the question no. 16 to 20 in short.
16. Write the ordered pairs which must be included in the relation is R defined as
R = { (1, 1), (1, 2), (3, 3), (2, 1), (2, 3), (3, 2)} to make it a transitive and reflexive relation
but not symmetric.
OR
Show with examples that the relation A in the set R of real numbers, defined as
A = {(a, b) : a ≤ b2 } is neither symmetric nor transitive.
17. If A and B are symmetric matrices then show that AB – BA is a skew symmetric matrix.
0 1 0
18. If A and B , then find the value of for which A 2 B .
1 1 5 1
5
19. Find the value of tan 1 tan .
3
1 kx 1 kx
, for 1 x 0
20. If f ( x ) x , is continuous at x 0 , then find k.
2x 2 3x 2 , for 0 x 1
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SECTION B
(Question 21 to 26 are carrying 02 Marks)
21. Show that relation R defined on set Z, given by R = {(a, b) : a > b} is neither reflexive
nor symmetric but transitive.
3 3
22. Evaluate sin sin 1 cos 1
7 49
4 3 3
23. If adjoint of the matrix N 1
0 1 is equal to kN, then find k.
4 4 3
OR
3 1 5 1
If . X 2 3 , then the matrix X.
4 1
24. Find interval in which the given function f(x) 2x3 9x 2 12x 1 is strictly
decreasing.
1 0 1 1
25. If A B and A 2 B 0 1 , then find the matrix A
1 1
dy
26. If x a cos3 , y a sin3 , then find at
dx 4
OR
dy
If y = tan x tan x tan x ... , prove that (2y - 1) = sec2x.
dx
SECTION C
(Question 27 to 32 are carrying 04 Marks)
1 1 2 1 3
27. Prove that tan tan 1 cos 1 .
4 9 2 5
OR
1
1 x2 1 x2
Express tan in the simplest form.
2 2
1 x 1 x
1 2 2
28. If A = 2 1 2 , find k so that A 2 4A kI 0
2 2 1
3 1
29. If P
2 2 , A 1 1 and Q PAP where P is transpose of P, then find
0 1
1 3
2 2
(i) PP
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(ii) P Q 4 P
1 cos 8x
, when x 0
bx 2
30. If f ( x ) a , when x 0 , is continuous at x 0 , then find values of 'a' and
x
, when x 0
(16 x ) 4
‘b’.
dy
31. If (1 x6 ) (1 y6 ) a3 ( x3 y3 ) , then find
dx
OR
2
n d y dy
If y x 1 x 2 , then (1 x 2 ) 2 x n2y
dx dx
2
32. Find the points on the curve y = x 3x 4 at which the tangents passes through the
origin.
SECTION D
(Question 33 to 36 are carrying 06 Marks each)
33. Answer the following
7 1 x4
(i) Function f is from R – to R – defined f(x) = and show that it is a
5 5 5x 7
bijective function.
(ii) Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither
one- one nor onto, where [x] denotes the greatest integer less than or equal to x.
34. Using matrix method solve the system of equations 3x y z 1 0 , x 2 y 3z 4 and
x yz3 0
d 2 y sec 3
35. If x a(cos sin ), y a(sin cos ); 0 , prove that
2 dx 2 a
OR
dy y x x
Find at x 1 & y 1 , if x y x
dx
36. Prove that the altitude of the largest cone that can be inscribed in a sphere of radius R
4R
is of that of the sphere, and also find the maximum volume of cone.
3
OR
Prove that of all the rectangles that can be inscribed in a circle of radius R, square has
the maximum area. Also find the maximum area of rectangle.
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