Mathematics
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The document contains a mathematics exam with three groups of questions testing different concepts: Group A contains 10 multiple choice questions covering domains of functions, trigonometric functions, derivatives, integrals, determinants, and properties related to maxima and minima of functions. Group B contains another 10 multiple choice questions testing concepts like distance between parallel lines, matrix operations, complex numbers, solving equations, properties of concurrent lines, integrals involving logarithms, and solving inequalities. Group C contains 2 problems to be solved in detail, the first finding the length of a perpendicular from a point to a line, and the second evaluating a definite integral.
![Mathematics:
F.M: 40
Group A
Circle or tick the correct answer of the following questions. [ 1 ¡Á 10=10]
1. The domain of the function f ( x) = x + 1 is :
(a) x ¡Ý ?1 (b) x ¡Ü ?1 (c) x =-1 (d) x ¡Ý 0 .
2
2. If tan x = , then the value of sinx is equals to :
3
2 3
(a) 13 (b) 13 (c) (d) .
13 13
3. If f ( x) = x x , then its differential coefficient is :
(a) x x ?1 (b) x x (c) x x (1 + log x) (d) ( x x + log x) .
dx
4. The value of the integral ¡Ò 2
x +x
is equals to :
(a) log( x 2 + x) + C (b) [log x][log( x + 1)] + C
? x ?
(c) log(2 x + 1) + C (d) log ? ?+C
? x +1 ?
1 a b+c
5. The value of the determinant 1 b c + a is equals to
1 c a+b
(a) 0 (b) a + b + c (c) 1 (d) abc
6. The direction cosines of the line perpendicular to the plane x ? 2 y + 2 z = 1 is:
1 ?1 1 ?2 2
(a) 1, -2, 2 (b) 1, , (c) 3, -2, 2 (d) , ,
2 2 3 3 3
7. The function y = f (x) has maximum value at x = a if
dy d2y dy d2y
(a) = 0 and >0 (b) = 0 and <0
dx dx 2 dx dx 2
dy d2y dy d2y
(c) > 0 and <0 (d) < 0 and =0
dx dx 2 dx dx 2
8. If A = {1,2,3,4} and B = { 3,4,5,6} , then n( A ¡È B) is equals to :
(a) 6 (b) 8 (c) 4 (d) 16
?1 ?1 2¦Ð
9. If sin x + sin y = , then the value of cos ?1 x + cos ?1 y is equals to :
3
¦Ð ¦Ð
(a) ¦Ð (b) (c) (d) 3¦Ð
2 3
10. In a triangle ABC, the value of a (b cos C ? c cos B) is equals to :
(a) b 2 ? c 2 (b) b 2 + c 2 (c) c 2 ? b 2 (d) b 2 c 2
Group B
Circle or tick the correct answer of the following questions. [ 2 ¡Á 10 = 20 ]
1. The distance between the parallel lines y = 2 x + 4 and 6 x ? 3 y = 5 is
17 17
(a) 17 (b) (c) (d) does not exists.
45 45](https://image.slidesharecdn.com/mathematics-101003064510-phpapp02/85/Mathematics-1-320.jpg)
![?1 2?
2. If A = ? , then the value of A 2 ? 2 A ? 5 I is equals to
? 0 1??
?? 6 0 ? ?6 0 ? ?0 0 ? ?6 0 ?
(a) ? ? (b) ?0 ? 6? (c) ? ? (d) ? ?
? 0 ? 6? ? ? ?0 0 ? ?0 6 ?
3. The value of the complex number (1 + i ) 8 is equals to
(a) 2 (b) 2 (c) 81 (d) 16
1
4. If ¦Á and ¦Â are the roots of the equation x 2 ? x ? 6 = 0 , then the equation whose roots are ¦Á +
¦Â
1
and ¦Â + is:
¦Á
(a) x 2 + x ? 6 = 0 (b) 6 x 2 ? 5 x ? 6 = 0 (c) 6 x 2 ? 5 x ? 52 = 0 (d)
6 x 2 ? 5 x ? 25 = 0
5. If the lines y = x + 1 , y = 2 x + 2 and y = ¦Ë x + 3 are concurrent , then the value of ¦Ë is
1
(a) 3 (b) 2 (c) (d) 1
2
6. The angle between the planes x + 2 y + z + 7 = 0 and 2 x + y ? z + 13 = 0 is:
¦Ð ¦Ð ¦Ð ¦Ð
(a) (b) (c) (d)
3 4 6 2
dx
7. The value of the integral ¡Ò x[log x] 2 is equals to:
1 1 1
(a) 2 log[log( x)] + C (b) +C (c) +C (d) - +C
log x x log x
x y
8. If the area of the triangle formed by coordinate axes and the line + = 1 is equals to 1, then the
2 k
value of ¡® k¡¯ is :
(a) 2 (b) -1 (c) 1 (d) -2
lim ? a+x ? a?
9. The value of x ¡ú 0 ? ? , is equals to :
?
? x ?
?
1 1
(a) 0 (b) (c) (d) Does not exists.
a 2 a
10. The value of 2 x + 5 < 3 , is same as
(a) ?3 < x < 3 (b) ?4 < x < ?1 (c) 1 < x < 4 (d) ?5 < x < 3
Group C
Find the solution of the following questions. Show in detail in extra page if necessary.[ 2 ¡Á 5 =10 ]
1. Find the length of perpendicular drawn from the point (1,2) upon the line passing through the
origin and the point of intersection of the lines 2 x ? 3 y + 14 = 0 and 5 x + 4 y ? 7 = 0 .
a+x
2. Evaluate the integral ¡Ò a?x
dx .](https://image.slidesharecdn.com/mathematics-101003064510-phpapp02/85/Mathematics-2-320.jpg)
Mathematics
- 1. Mathematics: F.M: 40 Group A Circle or tick the correct answer of the following questions. [ 1 ¡Á 10=10] 1. The domain of the function f ( x) = x + 1 is : (a) x ¡Ý ?1 (b) x ¡Ü ?1 (c) x =-1 (d) x ¡Ý 0 . 2 2. If tan x = , then the value of sinx is equals to : 3 2 3 (a) 13 (b) 13 (c) (d) . 13 13 3. If f ( x) = x x , then its differential coefficient is : (a) x x ?1 (b) x x (c) x x (1 + log x) (d) ( x x + log x) . dx 4. The value of the integral ¡Ò 2 x +x is equals to : (a) log( x 2 + x) + C (b) [log x][log( x + 1)] + C ? x ? (c) log(2 x + 1) + C (d) log ? ?+C ? x +1 ? 1 a b+c 5. The value of the determinant 1 b c + a is equals to 1 c a+b (a) 0 (b) a + b + c (c) 1 (d) abc 6. The direction cosines of the line perpendicular to the plane x ? 2 y + 2 z = 1 is: 1 ?1 1 ?2 2 (a) 1, -2, 2 (b) 1, , (c) 3, -2, 2 (d) , , 2 2 3 3 3 7. The function y = f (x) has maximum value at x = a if dy d2y dy d2y (a) = 0 and >0 (b) = 0 and <0 dx dx 2 dx dx 2 dy d2y dy d2y (c) > 0 and <0 (d) < 0 and =0 dx dx 2 dx dx 2 8. If A = {1,2,3,4} and B = { 3,4,5,6} , then n( A ¡È B) is equals to : (a) 6 (b) 8 (c) 4 (d) 16 ?1 ?1 2¦Ð 9. If sin x + sin y = , then the value of cos ?1 x + cos ?1 y is equals to : 3 ¦Ð ¦Ð (a) ¦Ð (b) (c) (d) 3¦Ð 2 3 10. In a triangle ABC, the value of a (b cos C ? c cos B) is equals to : (a) b 2 ? c 2 (b) b 2 + c 2 (c) c 2 ? b 2 (d) b 2 c 2 Group B Circle or tick the correct answer of the following questions. [ 2 ¡Á 10 = 20 ] 1. The distance between the parallel lines y = 2 x + 4 and 6 x ? 3 y = 5 is 17 17 (a) 17 (b) (c) (d) does not exists. 45 45
- 2. ?1 2? 2. If A = ? , then the value of A 2 ? 2 A ? 5 I is equals to ? 0 1?? ?? 6 0 ? ?6 0 ? ?0 0 ? ?6 0 ? (a) ? ? (b) ?0 ? 6? (c) ? ? (d) ? ? ? 0 ? 6? ? ? ?0 0 ? ?0 6 ? 3. The value of the complex number (1 + i ) 8 is equals to (a) 2 (b) 2 (c) 81 (d) 16 1 4. If ¦Á and ¦Â are the roots of the equation x 2 ? x ? 6 = 0 , then the equation whose roots are ¦Á + ¦Â 1 and ¦Â + is: ¦Á (a) x 2 + x ? 6 = 0 (b) 6 x 2 ? 5 x ? 6 = 0 (c) 6 x 2 ? 5 x ? 52 = 0 (d) 6 x 2 ? 5 x ? 25 = 0 5. If the lines y = x + 1 , y = 2 x + 2 and y = ¦Ë x + 3 are concurrent , then the value of ¦Ë is 1 (a) 3 (b) 2 (c) (d) 1 2 6. The angle between the planes x + 2 y + z + 7 = 0 and 2 x + y ? z + 13 = 0 is: ¦Ð ¦Ð ¦Ð ¦Ð (a) (b) (c) (d) 3 4 6 2 dx 7. The value of the integral ¡Ò x[log x] 2 is equals to: 1 1 1 (a) 2 log[log( x)] + C (b) +C (c) +C (d) - +C log x x log x x y 8. If the area of the triangle formed by coordinate axes and the line + = 1 is equals to 1, then the 2 k value of ¡® k¡¯ is : (a) 2 (b) -1 (c) 1 (d) -2 lim ? a+x ? a? 9. The value of x ¡ú 0 ? ? , is equals to : ? ? x ? ? 1 1 (a) 0 (b) (c) (d) Does not exists. a 2 a 10. The value of 2 x + 5 < 3 , is same as (a) ?3 < x < 3 (b) ?4 < x < ?1 (c) 1 < x < 4 (d) ?5 < x < 3 Group C Find the solution of the following questions. Show in detail in extra page if necessary.[ 2 ¡Á 5 =10 ] 1. Find the length of perpendicular drawn from the point (1,2) upon the line passing through the origin and the point of intersection of the lines 2 x ? 3 y + 14 = 0 and 5 x + 4 y ? 7 = 0 . a+x 2. Evaluate the integral ¡Ò a?x dx .