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.

1. Mathematics:
F.M: 40
Group A
Circle or tick the correct answer of the following questions. [ 1 × 10=10]
1. The domain of the function f ( x) = x + 1 is :
(a) x ≥ −1 (b) x ≤ −1 (c) x =-1 (d) x ≥ 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 ∫ 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 ∪ B) is equals to :
(a) 6 (b) 8 (c) 4 (d) 16
−1 −1 2π
9. If sin x + sin y = , then the value of cos −1 x + cos −1 y is equals to :
3
π π
(a) π (b) (c) (d) 3π
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 × 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 α and β are the roots of the equation x 2 − x − 6 = 0 , then the equation whose roots are α +
β
1
and β + is:
α
(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 = λ x + 3 are concurrent , then the value of λ 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:
π π π π
(a) (b) (c) (d)
3 4 6 2
dx
7. The value of the integral ∫ 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 → 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 × 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 ∫ a−x
dx .