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Mathematics (Solution)

This document contains 12 multiple choice or short answer mathematics questions. Some of the questions involve trigonometric identities, solving inequalities, finding common roots of equations, and evaluating expressions. The questions cover topics like trigonometry, algebra, and functions. The full worked out solutions and step-by-step explanations are provided for each question.

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18 views5 pages

Mathematics (Solution)

This document contains 12 multiple choice or short answer mathematics questions. Some of the questions involve trigonometric identities, solving inequalities, finding common roots of equations, and evaluating expressions. The questions cover topics like trigonometry, algebra, and functions. The full worked out solutions and step-by-step explanations are provided for each question.

Uploaded by

qtwxcttj9k
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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MATHEMATICS

Q. 1 Multiple Choice Question


(i) Ans.: (a)
sin 15 + cos 105
= sin 15 + cos (90 + 15)
= sin 15  sin 15
=0

(ii) tan (A + B) =

= =
tan (A + B) =  1

A+B=

A+B=
(iii) Ans.: (a)
cos 1 cos 2 cos 3…… cos (100)
= cos 1  cos 2 cos 3….(0) 90 …cos 100
=0
(iv) Ans.: (d)
3x  18 < 4 + x
2x < 22
x < 11

(v) Ans.: (d)


|x|<a
a<x<a
  10 < x < 10

(vi) Ans.: (c)


Domain :
3  2x > 0 4x + 1 > 0

x< x>  (2)


log5 (3  2x)  log5 (4x + 1)
 3  2x  4x +1
2  6x
1
 x   (3)
From (1), (2), (3)

x
(vii) Ans.: (c)
x2  7x + 10 = 0

sum of roots = =

(viii) Ans.: (a)


1 + x + x2 = 0

Product of roots = = = 1
(ix) Ans.: (c)
a1 = 1 b1 = 5 c1 = 10
a1 = 3 b1 = 15 c2 = c
Both roots are common


 c = 30

10. Ans.: (a)


D = b2  4ac
= ( 10)2  4  1  20
= 100  80
D = 20
 Real and distinct

Q. 2 1. (i)
1 + tan2  = sec2 

1+


16x2  9y2 = 144

(ii)
1 + cot2  = cosec2 

2
1+
16x2  9y2 = 576
2. L.H.S. = 1 + tan2 A + 1 + cot2 A
= sec2 A + cosec2 A

=
= R.H.S.

3. sin 3x sin x + sin2 x + sin2 x + cos 3x cos x  cos2 x


= (cos 3x cos x + sin 3x sin x) + sin2 x  (0)2 x
= cos (3x  x)  (cos2 x  sin2 x)
= (0) 2x  cos 2x
=0

4.

=
= + 1 = R.H.S.

5. A + B + C = 

tan

= cot

 + tan = 1  tan

3
 tan + tan

6.  2 < 2x  1 < 2
 1 < 2x < 3

x

7. x  1 > 5 or  (x  1) > 5
x>6 x1<5
x  (6, ) x<4
x  (,  4)
 x  ( ,  4)  (6, )

8.


+   + +
1 2 4


9. =
 =
 4x + 12x  4 = 3x2 + 9x + 6
2

 x2 + 3x  10 = 0
(x + 5) (x  2) = 0
x=5 or x = 2

10. Let  be common root


a  3 = 0
2 + a  15 = 0
  +
 2a + 12 = 0
a = 6

=

4
2  a  3 = 0

 3=0

9=0

 a2 =
a2 = 4
a=2

11. ||x1|1|1
1|x1|11
0|x1|2
 | x  1 | 
 x  1 < 2
1x3
 x  [ 1, 3]

12. Solve for x :


(| x |  1) (| x |  2) < 0
Case (i) : | x |  1 < 0 and | x |  2 > 0
|x|<1 |x|>2
Which is not possible
Case (ii)
|x|1>0 and | x |  2 < 0
|x|>1 |x|<2
x  ( ( ,  1)  (1,  x < 2
x  ( ( 2, 2)

2 1 1 2

 x  ( 2,  1)  (1, 2)

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