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1999 Mathematics

1.	If (1-x + x²)ⁿ= a₀+ a₁x + a₂x²+ ...... +a_2nx²ⁿ then a₀+ a₂+ a₄+.... +a_2nequals
	(1) 3ⁿ-1/2 (2) 3ⁿ+1/2 (3) 3ⁿ+1/2 (4) 3ⁿ -1/2

2.	The fourth, seventh and tenth terms of a G.P. are p, q and r respectively then
	(1) p² = q²+ r² (2) p²= q² (3) q²= pr² (4) r²= p²+ q²

3.	cos 1⁰cos 2⁰cos 3⁰ ........... cos 179⁰=
	(1) 1/2 (2) 1 (3) 0 (4) 2

4.	The sum of the slopes of the lines represented by 4x² + 2 hxy - 7y²= 0 is equal to the product of the slopes. Then h is
	(1) -4 (2) 4 (3) -6 (4) -2

5.	If f(9) = 9 and (9) = 4 then
	(1) 3 (2) 4 (3) 1/2 (4) 2

6.	The value of sin² 5⁰ + sin² 10⁰+ sin² 15⁰ +.........+ sin² 85⁰+ sin 90⁰=
	(1) 7 (2) 8 (3) 9 (4) 9 1/2

7.	If the equation x² + y² + 2gx + 2fy + 1 =0 represents a pair of lines then
	(1) f² -g² =1 (2) f² + g² =1 (3) g² -f² =1 (4) f² + g² =1/2

8.	DABC is right angled at C, then tan A + tan B =
	(1) a + b² (2) a²/bc (c) c²/ab (4) b²/ac

9.	If the sum of the distances of a point from two perpendicular lines in a plane is I then its locus is
	(1) Circle (2) square (3) straight line (4) intersecting lines

10.	The value of cot 54⁰/tan 36⁰ + tan 20⁰/ cot 70⁰ =
	(1) 1 (2) 0 (3) 2 (4) 3

11.	The Vectors form a triangle which is
	(1) Equilateral (2) Isosceles (3) Right angled (4) Obtuse angled

12.
	(1) (2) (3) (4)

13.
	(1) (2) (3) (4)

14.	If q is the angle between the vectors
	(1) cot q (2) - cot q (3) tan q (4) -tan q

15.	If A and B are square matrices of order n x n then (A-B)² is equal to
	(1) A² - 2AB + B² (2) A² - B² (3) A² - 2BA + B² (4) A² - AB- BA + B²

16.	Choose the correct answer
	(1) Every scalar matrix is an identity matrix (2) Every identity matrix is a scalar matrix (3) Every diagonal matrix is an identity matrix (4) A square matrix whose each element is 1 is an identity matrix



17.	If f(a) = 2: f' (a) = 1, g(a) =- g'(a) = 2 then
	(1) -5 (2) 1/5 (3) 5 (4) 0

18.	The function f(x) = log_e (1 + ax) - log ( 1-bx)/x is undefined at x = 0. The value which should be assigned to f at x = 0 so that it is continuous at x = 0 is
	(1) a - b (2) a + b/2 (3) a + b (4) log_e(ab)

19.	If f(x) = cos (log x) then f(x) -1/2 [f(y/x) + f(xy)] has the value:
	(1) 0 (2) 1 (3) 1/2 (4) -2

20.	The area of a circle centred at (1,2) and passing through (4, 6) is
	(1) 5psq. units (2) 15psq. units (3) 25psq. units (4) 30psq. units

21.	The eccentricity of the hyperbola �1999/3 ( x² + y²) = 1 is
	(1) 2 (2) 2�2 (3) �3 (4) �2

22.	In the coaxial system of circles x² + y² + 2gx + C = 0 when g is a parameter, if C > 0 then the circles are of.
	(1) non - intersecting type (2) touching type (3) intersecting type (4) orthogonal


23.	If the set A has p electrons, B has q elements, then the number of elements in A x B is
	(1) pq (2) p² (3) p + q (4) P + q + 1

24.	If w is the nth root of unity then I + w + w² + w³ + ..... + w^n-1is
	(1) 0 (2) 1 (3) -1 (4) 2

25.	The contrapositive of (p v q) r is
	(1) p � (q v r) (2) r � (p v q) (3) ~r � ~ (p v p) (4) ~r � (~p � ~q)

26.	For the circuits shown below.
	(1) (p � q) v ( p � ~q) (2) (~p � q) v ( p v ~q) (3) (~p � q) � ( ~q � p) (4) (~p � ~q) � ( q � p)


27.
	(1) X (2) Y (3) � (4) 1

28.	if Z = 1 + i then the multiplicative inverse of Z² is
	(1) 1 - i (2) i/2 (3) -i/2 (4) 2i

29.
	(1) log_e3 (2) log_e2 (3) 0 (4) log_e4

30.	Let f(x) = �₀^x t sin t dt then f' (x) =
	(1) sin x + cos x (2) x sin x (3) x cos x (4) x²/2

31.	The value of �₀^p/2 2^sinx/2^{sin x}+2^cosx dx is
	(1) 2 (2) p (3) p/4 (4) p/2

32.
	(1) 2 (2) log_e2 (3) log_e2/2 (4) 2 log_e2

33.	If f(x) = cos^-1 [1 - (log x)²/1 + (log x)² ] then f' (e) =
	(1) 2/e (2) 1/e (3) 1 (4) does not exist

34.
	(1) p/4 - log 2 (2) p/2 + log 2 (3) p/2 - log 2 (4) p/4 + log 2

35.
	(1) p/ab (2) p/2ab (3) pab (4) p²ab

36.	If x + iy = �a + ib/ c + id then ( x² + y²)² =
	(1) a² - b²/ c² - d² (2) �a² + b²/ c² + d² (3) a² + b²/ c² + d² (4) c² + d²/ a² + b²

37.	The matrix is known as
	(1) Symmetric matrix (2) Diagonal matrix (3) Upper triangular matrix (4) Skew symmetric matrix


38.	The number of improper subgroups of G = { I, -I, i, -i} w.r.t. multiplication is
	(1) 1 (2) 2 (3) 3 (4) 4

39.	In the group G = { 0, 1, 2, 3, 4, 5 } under addition modulo 6, the value of {3 x 5^-1}^-1is
	(1) 3 (2) 5 (3) 4 (4) 2

40.	An ellipse has its centre at (1, -1) and semi major axis = 8, which passes through the point (1, 3) Then the equation of the ellipse is
	(1) (x + I)²/64 + (y + I)²/16 =1 (2) (x - I)²/64 + (y + I)²/ 16 =1 (3) (x - I)²/16 + (y + I)²/ 64 =1 (4) (x + I)²/64 + (y - I)²/ 16 =1


41.	In the multiplicative group of 2 x 2 matrices of the form a � o and a � R the inverse is
	(1) (2) (3) (4) does not exist

42.	The circle x² + y² - 8x + 4y + 4 = 0 touches
	(1) x - axis (2) y- axis (3) both x and y axes (4) does not touch the axes

43.	Focus of the parabola (y- 2)²= 20 ( x + 3 ) is
	(1) (2, 2) (2) (-3, 2) (3) (3, -2) (4) (2, -3)

44.	The locus of the centre of a circle which touches externally the given two circles is
	(1) Circle (2) Parabola (3) Ellipse (4) Hyperbola

45.	The line p = x cos α + y sin α becomes tangent to x²/a² - y²/b² =1 if
	(1) p = a cos α -b sin α (2) p² = a² cos α -b² sin α (3) p² = a² cos α + b² sin² α (4) p² = a² cos² α -b² sin² α


46.	Equation of the normal to the hyperbola x²/a² - y²/b² =1 at the point (a sec q) is
	(1) ax/ sec q - by/tan q = a²- b² (2) ax/ sec q + by/tan q = a²+ b² (3) ax/ sec q + by/tan q = a²- b² (4) ax/ sec q - by/tan q = a- b


47.	If tan^-1 (x) + 2 cot^-1 (x) = 2p/3 then x =
	(1) �3 (2) �2 (3) �3 -1/ �3 + 1 (4) 3

48.	The angle between the curve y² = 4x and x² + y²= 5 at (1, 2) is
	(1) p/2 (2) p/4 (3) tan^-1 (3) (4) tan^-1 (2)

49.	For the curve yⁿ = a^n-1 x the subnormal at the point is constant. The value of n must be
	(1) 0 (2) 1 (3) 2 (4) 3

50.	The maximum value of the function f(x) = x^1/xis
	(1) e (2) e^1/e (3) 1/e (4) 2/e

51.	� 1/sin x + cos x dx is
	(1) 1/�2 log (x/2 + p/8) +C (2) log tan (x/2 + p/8) +C (3) 1/2 log tan ( x/2 + p/8) +C (4) 1/�2 log tan (x + p/4) +C


52.	� e^x (I + tan x + tan² x) dx =
	(1) e^xtan x + c (2) e^xsec x + c (3) e^xsin x + c (4) e^xcos x + c

53.	�^p/2₀ a sin x + b cos x/ sin x + cos x dx =
	(1) p/4 (2) (a + b) p/2 (3) (a + b)p (4) (a + b)p/4

54.	�^p/2₀ log (tan x) dx =
	(1) p/4 (2) p/2 (3) 0 (4) 1

55.	The area enclosed between the parabolas y²= 4x and x²= 4y is
	(1) 1/16 sq. units (2) 16/3 sq. units (3) 14/3 sq. units (4) 3/4 sq. units

56.	Solution of the differential equation tan y sec² x dx + tan x sec² y dy =0 is
	(1) tan x + tan y = k (2) tan x - tan y = k (3) tan x /tan y = k (4) tan x tan y = k

57.	The area bounded by the curve y = log_e x, the X axis, and the straight line x = e is
	(1) e. sq. units (2) 1. sq. unit (3) 1- 1/e. sq. units (4) 1+ 1/e. sq. units

58.	Let f be a polynomial. Then the second derivative of f(e^x) is
	(1) f" (e^x) e^x.f'(e^x) (2) f"(e^x) e^2x+ f"(e^x)e^x (3) f"(e^x) (4) f" (e^x) e^2x+ f'(e^x)e^x


59.	If m and n are the order and degree of the differential equation
	(1) m = 3 , n = 3 (2) m = 3 , n = 2 (3) m = 3 , n = 5 (4) m = 3 , n = 1

60.	The value of is
	(1) 0 (2) a + b + c (3) 4 abc (4) abc