+1 Public March 2019 Important One Mark Questions

Download Tamil Nadu 11th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Multiple Choice Question

100 x 1 = 100

The co-ordinates of a point on x + y + 3 = 0 whose distance from x + 2y + 2 = 0 is \(\sqrt 5\), is ______________

(a)

(9, 6)

(b)

(-9, 6)

(c)

(6, -9)

(d)

(-9, -6)

The point (2, 1) and (-3, 5) are on ______________

(a)

Same side of the line 3x - 2y + 1 = 0

(b)

Opposite sides of the line 3x - 2y + 1 = 0

(c)

On the line 3x - 2y + 1 = 0

(d)

On the line x + y = 3

The distance between the parallel lines 3x - 4y + 9 = 0 and 6x - 8y -15 = 0 is ______________

(a)

\(\frac{-33}{10}\)

(b)

\(\frac{10}{33}\)

(c)

\(\frac{33}{10}\)

(d)

\(\frac{33}{20}\)

The equation x²+ kxy + y²- 5x - 7y + 6 = 0 represents a pair of straight lines then k = ______________

(a)

\(\frac{5}{3}\)

(b)

\(\frac{10}{3}\)

(c)

\(\frac{3}{2}\)

(d)

\(\frac{3}{10}\)

The points (a, 0),(0, b) and (1, 1) will be collinear if ______________

(a)

a + b = 1

(b)

a + b = 2

(c)

\(\frac{1}{a}+\frac{1}{b}=1\)

(d)

a + b = 0

The points (k + 1, 1), (2k + 1, 3) and (2k + 2, 2k) are collinear if ______________

(a)

k = -1

(b)

\(k=\frac{1}{2}\)

(c)

k = 3

(d)

k = 2

The value \(\lambda\) for which the equation 12x²- 10xy + 2y²+11x-5y+\(\lambda\) = 0 represent a pair of straight lines is ______________

(a)

\(\lambda\) = 1

(b)

\(\lambda\) = 2

(c)

\(\lambda\) = 3

(d)

\(\lambda\) = 0

Area of triangle ABC is _______________

(a)

\(\frac{1}{2}\)ab cos C

(b)

\(\frac{1}{2}\)ab sin C

(c)

\(\frac{1}{2}\)ab cos B

(d)

\(\frac{1}{2}\)bc sin B

Number of solutions of the equation tan x + sec x = 2 cos x lying in the interval [0, 2π] is _______________

(a)

(b)

(c)

(d)

10.

The value of tan^-1 (1) + cos^-1(\(\frac{-1}{2}\)) + sin^-1(\(\frac{-1}{2}\)) _______________

(a)

\(\frac{\pi}{4}\)

(b)

\(\frac{5\pi}{4}\)

(c)

\(\frac{3\pi}{4}\)

(d)

\(\frac{\pi}{2}\)

11.

If sin θ = sin \(\alpha\), then the angles θ and \(\alpha\) are related by _______________

(a)

\(\theta=n\pi\pm\alpha\)

(b)

\(\theta=2n\pi+(-1)^n\alpha\)

(c)

\(\alpha=n\pi\pm(-1)^n\theta\)

(d)

\(\theta=(2n+1)\pi+\alpha\)

12.

(sec A + tan A-1) (sec A - tan A+1)-2 tan A = _______________

(a)

(b)

(c)

(d)

2 tan A

13.

If A and B are any two finite sets having m and n elements respectively then the cardinality of the power set of A \(\times\) B is ___________

(a)

2^m

(b)

2ⁿ

(c)

(d)

2^mn

14.

If \(f(x)={1-x\over 1+x},(x\neq0)\) then f^-1(x) =

(a)

f(x)

(b)

\(1\over f(x)\)

(c)

-f(x)

(d)

-\(1\over f(x)\)

15.

The function f(x) = log (x + \(\sqrt{x^2+1}\)) is ___________

(a)

an even function

(b)

an odd function

(c)

a periodic function

(d)

neither an even nor an odd function

16.

The domain of the function \(f(x)=\sqrt{ x - 5 }+ \sqrt{6 - x}\) is

(a)

[5, ∞)

(b)

(- ∞, 6)

(c)

[5, 6]

(d)

(-5, ≠6)

17.

The length of perpendicular from the origin to a line is 12 and the line makes an angle of 120° with the positive direction of y-axis. then the equation of line is ______________

(a)

\(x+y\sqrt 3=24\)

(b)

\(x+y=12\sqrt 2\)

(c)

x + y = 24

(d)

\(x+y=12\sqrt 3\)

18.

If ⁿC₁₀ = ⁿC₆, then ⁿC₂ = _________

(a)

(b)

(c)

120

(d)

240

19.

2^1/4 4^1/88^1/16 16^1/32 . . . = ______________

(a)

(b)

(c)

\(\frac{3}{2}\)

(d)

\(\frac{5}{2}\)

20.

If a,b, c are in A.P, as well as in G.P then ______________

(a)

a = b ≠ c

(b)

a ≠ b = c

(c)

a ≠b ≠ c

(d)

a = b = c

21.

\(\frac{2}{1!}+\frac{4}{3!}+\frac{6}{5!}+. . .\infty =\) ______________

(a)

(b)

(c)

\(\frac{1}{e}\)

(d)

e²

22.

The coefficient of a⁵ in the expansion of (3a + 5b)⁵ is ______________

(a)

(b)

243

(c)

6750

(d)

9375

23.

If ⁿC_r-1 = 36, ⁿC_r = 84 and ⁿC_r+1 = 126 then r = _________

(a)

(b)

(c)

(d)

24.

Zero of the polynomial p(x) = x² - 4x + 4

(a)

(b)

(c)

-2

(d)

-1

25.

If a and b are roots of x² + x + 1 = 0 then the value of a² + b² = ___________

(a)

(b)

-1

(c)

cannot be determined

(d)

26.

The number of real solutions of the equation |x²| - 3|x| + 2 = 0 is ___________

(a)

(b)

(c)

(d)

27.

The value of \({3^{-3}\times6^4\times 12^{-3}\over 9^{-4}\times 2^{-2}}\) is ___________

(a)

3⁵

(b)

3⁶

(c)

3⁴

(d)

28.

The value of 2 log₁₀3 + log₁₀ 16 - 2 log₁₀ \(6\over 5\) is ___________

(a)

(b)

(c)

(d)

29.

The lines x + 2y - 3 = 0 and 3x - y + 7 = 0 are ______________

(a)

parallel

(b)

neither parallel nor perpendicular

(c)

perpendicular

(d)

parallel as wellas perpendicular

30.

If(1, 3) (2,1) (9, 4) are collinear then a is ______________

(a)

\(\frac{1}{2}\)

(b)

(c)

(d)

-\(\frac{1}{2}\)

31.

The equating straight line with y-intercept -2 and inclination with x-axis is 135° is ______________

(a)

x + y - 2 = 0

(b)

y - x + 2 = 0

(c)

y + x + 2 = 0

(d)

none

32.

\(\sqrt \frac{1-2x}{1+2x}\) is approximately equal to ______________

(a)

1- 2x-x²

(b)

1 + 2x+ x²

(c)

1+ 2x

(d)

1-2x+x²

33.

In the expansion of (2x + 3)⁵ the coefficient of x² is ______________

(a)

720

(b)

1080

(c)

810

(d)

34.

With usual notation C₀ + C₂ + C₄+ ... is ______________

(a)

2^n-1

(b)

2ⁿ

(c)

2ⁿ⁺¹

(d)

2ⁿ⁺²

35.

In \(\triangle\)ABC, \(\hat{C}\) = 90° then a cos A + b cos B is _______________

(a)

2R sin B

(b)

2 sin B

(c)

(d)

2a sin B

36.

If tanx = \({1\over 7}\) ,tan y = \({1\over 3}\) then x + y is ____________

(a)

\({\pi\over 4}\)

(b)

\({\pi\over 3}\)

(c)

\({\pi\over 2}\)

(d)

\({\pi}\)

37.

The number of ways of selecting of 3 poets and 4 scientists such that poets are in even places _________

(a)

(b)

(c)

(d)

144

38.

The number of diagonals of a decagon _________

(a)

(b)

(c)

(d)

39.

Solve \(\sqrt{7+6x-x^2}=x+1\)

(a)

(1, -3)

(b)

(3, -1)

(c)

(1, -1)

(d)

(3, -3)

40.

\(n(p(A))=512,n(p(B))=32,n(A\cup B)=16,\) find \(n(A\cap B) \) ___________

(a)

(b)

(c)

(d)

41.

The value of 2 + 4 + 6 + + 2n is

(a)

\(\frac { n\left( n-1 \right) }{ 2 } \)

(b)

\(\frac { n\left( n+1 \right) }{ 2 } \)

(c)

\(\frac { 2n\left( 2n+1 \right) }{ 2 } \)

(d)

n(n + 1)

42.

If the lines x + q = 0, y - 2 = 0 and 3x + 2y + 5 = 0 are concurrent, then the value of q will be ______________

(a)

(b)

(c)

(d)

43.

Distance between the lines 5x + 3y - 7 = 0 and 15x + 9y + 14 = 0 is ______________

(a)

\(\frac{35}{\sqrt{34}}\)

(b)

\(\frac{1}{3\sqrt{34}}\)

(c)

\(\frac{35}{2\sqrt{34}}\)

(d)

\(\frac{35}{3\sqrt{34}}\)

44.

If the points (a, 0) (0, b) and (x, y) are collinear, then ______________

(a)

\(\frac{x}{a}-\frac{y}{b}=1\)

(b)

\(\frac{x}{a}+\frac{y}{b}=1\)

(c)

\(\frac{x}{a}+\frac{y}{b}=-1\)

(d)

\(\frac{x}{a}+\frac{y}{b}=0\)

45.

The value of x so that 2 is the slope of the line through (2, 5) and (x, 3) is ______________

(a)

-1

(b)

(c)

(d)

46.

The series 1+4x+8x²+ \(\frac { 32 }{ 3 } { x }^{ 3 }+.....+\infty \ is\) ______________

(a)

e^x

(b)

e^4x

(c)

e^2x

(d)

e^8x

47.

If a vertex of a square is at the origin and its one side lies along the line 4x + 3y - 20 = 0, then the area of the square is

(a)

20 sq. units

(b)

16 sq. units

(c)

25 sq. units

(d)

4 sq.units

48.

For all n \(\in \) N, 3\(\times\) 5²ⁿ⁺¹+2³ⁿ⁺¹ is divisible by _________

(a)

(b)

(c)

(d)

49.

The first and last term of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be ______________

(a)

(b)

(c)

(d)

50.

The Co-efficient of x^-17 in \({ \left( { x }^{ 4 }-\frac { 1 }{ { x }^{ 3 } } \right) }^{ 15 }\)is _____________

(a)

1365

(b)

-1365

(c)

3003

(d)

-3003

51.

Among the players 5 are bowlers. In how many ways a team of 11 may be formed with atleast 4 bowlers?

(a)

265

(b)

263

(c)

264

(d)

275

52.

5c₁+ 5c₂+ 5c₃+ 5c₄+ 5c₅ is equal to _________

(a)

(b)

(c)

(d)

53.

The number of ways to average the letters of the word CHEESE are _________

(a)

120

(b)

240

(c)

720

(d)

54.

The value of \(\frac { 1 }{ 2! } +\frac { 1 }{ 4! } +\frac { 1 }{ 6! } +....is\)

(a)

\(\frac { { e }^{ 2 }+1 }{ 2e } \)

(b)

\(\frac { { (e+1) }^{ 2 } }{ 2e } \)

(c)

\(\frac { { (e-1) }^{ 2 } }{ 2e } \)

(d)

\(\frac { { e }^{ 2 }+1 }{ 2e } \)

55.

The coefficient of x⁵ in the series e^-2x is

(a)

\(\frac { 2 }{ 3 } \)

(b)

\(\frac { 2 }{ 3 } \)

(c)

\(\frac { -4 }{ 15 } \)

(d)

\(\frac { 4 }{ 15 } \)

56.

The slope of the line which makes an angle 45^o with the line 3x- y = -5 are:

(a)

1, -1

(b)

\(\frac{1}{2},-2\)

(c)

\(1,\frac{1}{2}\)

(d)

\(2,-\frac{1}{2}\)

57.

If P_r stands for ^r P_r then the sum of the series 1+ P₁ + 2P₂ + 3P₃ +...+ nP_n is

(a)

P_n+1

(b)

P_n+1-1

(c)

P_n-1+1

(d)

⁽ⁿ⁺¹⁾P_(n-1)

58.

The number of 10 digit number that can be written by using the digits 2 and 3 is

(a)

¹⁰C₂+⁹C₂

(b)

2¹⁰

(c)

2¹⁰-2

(d)

10!

59.

In ²ⁿC₃ : ⁿC₃= 11 : 1 then n is

(a)

(b)

(c)

(d)

60.

In a plane there are 10 points are there out of which 4 points are collinear, then the number of triangles formed is

(a)

110

(b)

¹⁰C₃

(c)

120

(d)

116

61.

If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

(a)

(b)

(c)

10!

(d)

2¹⁰

62.

Number of sides of a polygon having 44 diagonals is

(a)

(b)

(c)

(d)

63.

The value of the series\(\frac { 1 }{ 2 } +\frac { 7 }{ 4 } +\frac { 13 }{ 8 } +\frac { 19 }{ 16 } +\).....is

(a)

(b)

(c)

(d)

64.

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

(a)

(b)

(c)

(d)

65.

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

(a)

(b)

(c)

(d)

66.

The number of five digit telephone numbers having at least one of their digits repeated is

(a)

90000

(b)

10000

(c)

30240

(d)

69760

67.

The n^thterm of the sequence 1, 2, 4, 7, 11,... is

(a)

n³+ 3n²+ 2n

(b)

n³- 3n²+ 3n

(c)

\(\frac { n(n+1)(n+2) }{ 3 } \)

(d)

\(\frac { { n }^{ 2 }-n+2 }{ 2 } \)

68.

The remainder when 38¹⁵ is divided by 13 is

(a)

(b)

(c)

(d)

69.

\((\sqrt { 5 } -2)(\sqrt { 5 } +2)\) is equal to ___________

(a)

(b)

(c)

(d)

70.

If |x + 3| ≥ 10 then ___________

(a)

x ∊ (-13, 7]

(b)

x ∊ [-13, 7)

(c)

x ∊ (-∞, -13] \(\cup\) [7, ∞)

(d)

x ∊ (-∞, -13] \(\cup\) [7, ∞)

71.

If - 3x + 17 < -13 then ___________

(a)

x ∈ (10, ∞)

(b)

x ∈ [10, ∞)

(c)

x ∈ (-∞, 10]

(d)

x ∈ [10, 10)

72.

\(\frac { cos3x }{ 2cos2x-1 } \) is _______________

(a)

cos x

(b)

sin x

(c)

tan x

(d)

cot x

73.

If tan x = \(\frac { -1 }{ \sqrt { 5 } } \) and x lies in the IV quadrant, then the value of cos x is ___________

(a)

\(\sqrt { \frac { 5 }{ 6 } } \)

(b)

\(\frac { 2 }{ \sqrt { 6 } } \)

(c)

\(\frac { 1 }{ 2 } \)

(d)

\(\frac { 1 }{ \sqrt { 6 } } \)

74.

If the angles of a triangle are in A.P., then the measure of one of the angles in radians is ___________

(a)

\(\frac { \pi }{ 6 } \)

(b)

\(\frac { \pi }{ 3 } \)

(c)

\(\frac { \pi }{ 2 } \)

(d)

\(\frac { 2\pi }{ 3 } \)

75.

The value of log₃ 11.log₁₁ 13.log₁₃ 15.log₁₅ 27.log₂₇ 81 is

(a)

(b)

(c)

(d)

76.

The number of roots of (x + 3)⁴+ (x + 5)⁴= 16 is

(a)

(b)

(c)

(d)

77.

If a and b are the real roots of the equation x²- kx + c = 0, then the distance between the points (a, 0) and (b, 0) is

(a)

\(\sqrt { { k }^{ 2 }-4c } \)

(b)

\(\sqrt { { 4k }^{ 2 }-c } \)

(c)

\(\sqrt { 4c-{ k }^{ 2 } } \)

(d)

\(\sqrt { k-8c } \)

78.

If 8 and 2 are the roots of x²+ ax + c = 0 and 3, 3 are the roots of x²+ dx + b = 0; then the roots of the equation x²+ ax + b = 0 are

(a)

1, 2

(b)

-1, 1

(c)

9, 1

(d)

-1, 2

79.

The equation whose roots are numerically equal but opposite in sign to the roots 3x²- 5x -7 = 0 is

(a)

3x²- 5x - 7 = 0

(b)

3x²+ 5x - 7 = 0

(c)

3x²- 5x + 7 = 0

(d)

3x²+ x - 7

80.

If a and b are the roots of the equation x²- kx + 16 = 0 and a²+ b²= 32, then the value of k is

(a)

(b)

-8

(c)

-8, 8

(d)

81.

If 3 is the logarithm of 343, then the base is

(a)

(b)

(c)

(d)

82.

Given that x, y and b are real numbers x < y, b > 0, then

(a)

xb < yb

(b)

xb > yb

(c)

xb ≤ yb

(d)

\(\frac { x }{ b } \ge \frac { y }{ b } \)

83.

If |x+2| \(\le\) 9, then x belongs to

(a)

\((-\infty ,-7)\)

(b)

[-11, 7]

(c)

\((-\infty ,-7)\cup (11,\infty)\)

(d)

(-11, 7)

84.

If f(ፀ) = |sin ፀ| + |cos ፀ|, ፀ\(\in \)R, then f(ፀ) is in the interval

(a)

[0, 2]

(b)

[1,\(\sqrt { 2 } \)]

(c)

[1, 2]

(d)

[0, 1]

85.

cos 2ፀ cos 2ф + sin²(ፀ - ф) - sin²(ፀ + ф) is equal to

(a)

sin2(ፀ+\(\phi \))

(b)

cos2(ፀ+\(\phi \))

(c)

sin2(ፀ-\(\phi \))

(d)

cos2(ፀ-\(\phi \))

86.

Let f_k(x) = \(\frac { 1 }{ k } \)[sin^kx + cos^kx] where x\(\in \)R and k ≥ 1. Then f₄(x) - f₆(x) =

(a)

\(\frac { 1 }{ 4 } \)

(b)

\(\frac { 1 }{ 12 } \)

(c)

\(\frac { 1 }{ 6 } \)

(d)

\(\frac { 1 }{ 3 } \)

87.

If tan40⁰= λ, then \(\frac { tan{ 140 }^{ 0 }-tan{ 130 }^{ 0 } }{ 1+tan{ 140 }^{ 0 }.tan{ 130 }^{ 0 } } \) =

(a)

\(\frac { 1-\lambda ^{ 2 } }{ \lambda } \)

(b)

\(\frac { 1+{ \lambda }^{ 2 } }{ \lambda } \)

(c)

\(\frac { 1+{ \lambda }^{ 2 } }{ 2\lambda } \)

(d)

\(\frac { 1-{ \lambda }^{ 2 } }{ 2\lambda } \)

88.

If cos 28⁰+ sin 28⁰= k³, then cos 17⁰ is equal to

(a)

\(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

(b)

-\(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

(c)

±\(\frac { { k }^{ 3 } }{ \sqrt { 2 } } \)

(d)

-\(\frac { { k }^{ 3 } }{ \sqrt { 3 } } \)

89.

\(\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } } \)=

(a)

\(\sqrt{2}\)

(b)

\(\sqrt{3}\)

(c)

(d)

90.

For non-empty sets A and B, if A ⊂ B then (A \(\times\)B) ⋂ (B \(\times\)A) is equal to

(a)

A ⋂ B

(b)

A \(\times\)A

(c)

B \(\times\)B

(d)

none of these.

91.

Let f : R➝R be given by f(x) = x + \(\sqrt { { x }^{ 2 } } \) is __________

(a)

injective

(b)

Surjective

(c)

bijective

(d)

none of these

92.

If f(x) = 2x - 3 and g(x) = x²+ x - 2 then gof(x) is __________

(a)

2(2x²- 5x + 2)

(b)

(2x²- 5x - 2)

(c)

2(2x²+ 5x + 2)

(d)

2x²+ 5x - 2

93.

The number of students who take both the subjects Mathematics and Chemistry is 70. This represents 10% of the enrollment in Mathematics and 14% of the enrollment in Chemistry. The number of students take at least one of these two subjects, is

(a)

1120

(b)

1130

(c)

1100

(d)

insufficient data

94.

Let A and B be subsets of the universal set N, the set of natural numbers. Then A^'∪[(A⋂B)∪B^'] is

(a)

(b)

(c)

(d)

95.

If A = {(x,y) : y = sin x, x ∈ R} and B = {(x,y) : y = cos x, x ∈ R} then A∩B contains

(a)

no element

(b)

infinitely many elements

(c)

only one element

(d)

cannot be determined

96.

If A = {(x,y) : y = e^x, x∈R} and B = {(x,y) : y = e^-x, x ∈ R} then n(A∩B) is

(a)

Infinity

(b)

(c)

(d)

97.

Given A = {5,6,7,8}. Which one of the following is incorrect?

(a)

Ø ⊆ A

(b)

A⊆A

(c)

{7,8,9}⊆A

(d)

{5}⊆A

98.

The function f:R➝R be defined by f(x) = sinx + cosx is

(a)

an odd function

(b)

neither an odd function nor an even function

(c)

an even function

(d)

both odd function and even function

99.

Let f:R➝R be defined by f(x) = 1 - |x|. Then the range of f is

(a)

(b)

(1,∞)

(c)

(-1,∞)

(d)

(-∞,1]

100.

The number of constant functions from a set containing m elements to a set containing n elements is

(a)

(b)

(c)

(d)

m+n

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Answers Key:

Multiple Choice Question

100 x 1 = 100

(b)
(-9, 6)

(b)
Opposite sides of the line 3x - 2y + 1 = 0

(b)
\(\frac{10}{3}\)

(d)
k = 2

(b)
\(\lambda\) = 2

(b)
\(\frac{1}{2}\)ab sin C

10.

11.

12.

(a)
0

13.

(d)
2^mn

14.

(a)
f(x)

15.

(b)
an odd function

16.

17.

(a)
\(x+y\sqrt 3=24\)

18.

19.

(b)
2

20.

(d)
a = b = c

21.

(a)
e

22.

(b)
243

23.

24.

(b)
2

25.

(a)
1

26.

(d)
4

27.

(b)
3⁶

28.

29.

(b)
neither parallel nor perpendicular

30.

(a)
\(\frac{1}{2}\)

31.

32.

(d)
1-2x+x²

33.

(b)
1080

34.

(a)
2^n-1

35.

(d)
2a sin B

36.

(a)
\({\pi\over 4}\)

37.

(d)
144

38.

39.

(b)
(3, -1)

40.

(a)
2

41.

(d)
n(n + 1)

42.

43.

44.

(b)
\(\frac{x}{a}+\frac{y}{b}=1\)

45.

(b)
1

46.

(b)
e^4x

47.

Perpendicular distance from origin to the line is
4x + 3y - 20 = 0 is
\(\left(\frac{-20}{\sqrt{16+9}}\right)=\frac{20}{\sqrt{25}}=\frac{20}{5}=4 \text { units }\)
Area of the square = 4 \(\times\) 4 = 16 sq. units.

48.

(b)
17

49.

(b)
6

50.

(b)
-1365

51.

52.

(b)
31

53.

(a)
120

54.

\(e=1+\frac{1}{\lfloor 1}+\frac{1}{\lfloor 2}+\frac{1}{\lfloor 3}+\frac{1}{\lfloor 4}+\ldots \ldots\)
\(e^{-1}=1-\frac{1}{\lfloor 1}+\frac{1}{\lfloor 2}-\frac{1}{\lfloor 3}+\frac{1}{4}-\ldots\)
\(\frac{e+e^{-1}}{2}=1+\frac{1}{\lfloor 2}+\frac{1}{\lfloor 4}+\ldots\)
\(\frac{1}{\lfloor 2}+\frac{1}{\lfloor 4}+\ldots . \quad=\frac{e+e^{-1}}{2}-1\)
\(=\frac{\mathrm{e}-2+\mathrm{e}^{-1}}{2} \)
\(=\frac{(\mathrm{e}-1)^{2}}{2 \mathrm{e}} \)

55.

\(\mathrm{e}^{-2 x}=1-\frac{2 x}{1 !}+\frac{(2 x)^{2}}{2 !}-\frac{(2 x)^{3}}{3 !}+\frac{(2 x)^{4}}{4 !}-\frac{(2 x)^{5}}{5 !}+\ldots\)
\(\text { Coefficient of } x^{5} \text { is } \frac{-2^{5}}{5 !}=\frac{-32}{120}=\frac{-4}{15}\)

56.

\(\text { Slope of } 3 x-y+5=0 \text { is } \frac{-3}{-1}=3=m_{1}\)
Let m₂ be the slope of the second line
\(\text {Given } \tan \theta=\tan 45^{\circ}=1 \Rightarrow \frac{m_{1}-m_{2}}{1+m_{1} m_{2}}=\pm 1\)
\(\frac{3-m_{2}}{1+3 m_{2}}=1 \quad \frac{m_{2}-3}{1+3 m_{2}}=1\)
\(3-m_{2}=1+3 m_{2} \quad m_{2}-3=1+3 m_{2}\)
\(2=4 m_{2} \quad-2 m_{2}=4\)
\(\mathrm{m}_{2}=\frac{1}{2} \quad \mathrm{~m}_{2}=-2\)
\(\left(\frac{1}{2},-2\right)\)

57.

\(1+1 \mid 1+2\lfloor 2+3\lfloor 3+\ldots \ldots . n\lfloor n \quad=\lfloor n+1\)
\(\text { Let } \mathrm{n}=1, \quad \text { L.H.S }=1+1=2\)
\(\text { R.H.S }=\lfloor 2=2\)
It is true for n = 1, In fact it is true for n = 0 also let us assume that it is true for n=k
\(1+1 \mid 1+2\lfloor 2+3\lfloor 3+\ldots \ldots . n\lfloor n=\lfloor k+1\)
\(1+1 \mid 1+2\lfloor 2+3\lfloor 3+\ldots \ldots k \mid k+(k+1)\lfloor k+1\)
\(=\lfloor k+1+(k+1)\lfloor k+1=\lfloor k+1[1+k+1]\)
\(\lfloor k+1 (k+ 2)\)
\(\lfloor k+2\)
It is true for (k + 1)
Also by mathematical induction, it is true for all value of \(n \geq 0, n \in Z\)

58.

Number of 10 digit number that can be written by using the digits 2 and 3 is 2¹⁰

59.

\(\frac{{ }^{2 n} C_{3}}{{ }^{n} C_{3}} =\frac{11}{1} \)
\(\frac{(2 n)(2 n-1)(2 n-2)}{n(n-1)(n-2)} =\frac{11}{1} \)
\(\frac{2 n(2 n-1) 2(n-1)}{n(n-1)(n-2)} =11 \)
\(4(2 n-1) =11(n-2) \)
\(8 n-4 =11 n-22 \)
\(18 =3 n \)
\(n = 6\)

60.

\(\text { Number of triangles }={ }^{10} \mathrm{C}_{3}-{ }^{4} \mathrm{C}_{3}\)
\(=\frac{10 \times 9 \times 8}{1 \times 2 \times 3}-4 \)
\(=120-4 =116 \)

61.

\(\text { Number of points of intersection }={ }^{10} \mathrm{C}_{2}\)
\(=\frac{10 \times 9}{1 \times 2}=45\)

62.

\(\text { Number of diagonals }={ }^{n} C_{2}-n\)
\(\frac{n(n-1)}{2}-n=44 \)
\(n^{2}-n-2 n=88 \)
\(n^{2}-3 n-88=0 \)
\((n-11)(n+8)=0 \)
\(n=11(\text { or }) n=-8 \)

63.

\(\mathrm{a} =1, \quad \mathrm{~d}=6, \quad \mathrm{r}=\frac{1}{2} \)
\(\mathrm{~S}_{\infty} =\frac{a}{1-\mathrm{r}}+\frac{\mathrm{dr}}{(1+\mathrm{r})^{2}} \)
\(=\frac{1}{1-\frac{1}{2}}+\frac{6 \times \frac{1}{2}}{\left(\frac{1}{2}\right)^{2}} \)
\(=2+(3 \times 4)=14 \)

64.

Number of parallelograms
\(={ }^{4} C_{2} \times{ }^{3} C_{2}=6 \times 3=18\)

65.

\(\text { No. of lines }{ }^{10} \mathrm{C}_{2}-{ }^{4} \mathrm{C}_{2}+1=45-6+1=40\)

66.

The number of five digit telephone numbers which can be formed using the digits 0,1,2.... 9 is 10⁵.
The number of 5 digit numbers which has none of their digit repeated is ¹⁰P₅, = 30240
The required number of telephone. number is 10⁵ =- 30240 = 69,760

67.

\(n^{\text {th }} \text { term is } \frac{n^{2}-n+2}{2}\)

68.

\((38)^{5} =(39-1)^{15} \)
\(=(39)^{15}-{ }^{15} C_{1}(39)^{14}+\ldots+{ }^{15} C_{14}(39)-1 \)
The remainder will be 12 because all the terms except the last is divisible by 39 and so by 13, -1 remains

69.

(a)
1

70.

(d)
x ∊ (-∞, -13] \(\cup\) [7, ∞)

71.

(a)
x ∈ (10, ∞)

72.

(a)
cos x

73.

(a)
\(\sqrt { \frac { 5 }{ 6 } } \)

74.

(b)
\(\frac { \pi }{ 3 } \)

75.

The value of
\(\log _{3} 11 \log _{11} 13 \log _{13} 15 \log _{15} 27 \log _{27} 81=\log _{3} 81 \)
\(=\log _{3} 3^{4}=4 \log _{3} 3=4 \)

76.

(a)
4

77.

\(x^{2}-\mathrm{k} x+\mathrm{c}=0\)
a and b are the roots
\(\therefore a+b=k, a b=c\)
To find
\(\sqrt{(a-b)^{2}+0^{2}}=a-b=\sqrt{(a+b)^{2}-4 a b}=\sqrt{k^{2}-4 c}\)

78.

\(x^{2}+a x+c=0 \)
\(x^{2}+d x+b=0 \)
\(8 \& 2 \text { are the roots }\)\(\text { 3. } 3 \text { are the roots }\)
\(\therefore a=-10 ; c=16 \quad d=-6, \quad b=9\)
\(x^{2}+a x+b =0 \)
\(x^{2}-10 x+9 =0 \)
\(\Rightarrow(x-1)(x-9) =0 \)
\(\therefore x =1 \text { (or) } 9 \)

79.

\(3 x^{2}-5 x-7=0\)
\(\text { Let the roots be } \alpha, \beta\)
\(\Rightarrow \operatorname{Sum}: \alpha+\beta=\frac{5}{3}\)
\(\text { Product : } \alpha \beta=\frac{-7}{3}\)
\(\text { Now take the roots are }-\alpha,-\beta\)
\(\Rightarrow \text { Sum : }-\alpha-\beta \quad-(\alpha+\beta)=-\frac{5}{3}\)
\(\text { Product : }(-\alpha)(-\beta)=\alpha \beta=\frac{-7}{3}\)
\(\text { The reouired equation }\)
\(x^{2}+\frac{5}{3} x-\frac{7}{3}=0 \)
\(3 x^{2}+5 x-7=0 \)

80.

\(x^{2}-k x+16=0, a^{} \& b \text { are roots }\)
\(a+b =k ; a b=16 \)
\(a^{2}+b^{2} =(a+b)^{2}-2 a b \)
\(32 =k^{2}-32 \)
\(k^{2} =64 \)
\(k =\pm 8 \Rightarrow k=-8,8 \)

81.

(b)
7

82.

(a)
xb < yb

83.

\(|x+2| \leq 9 \)
\( \Rightarrow-9 \leq x+2 \leq 9 \)
\( \Rightarrow-11 \leq x \leq 7 \)
\(x \text { belongs to }[-11,7]\)

84.

\(\text { It lies in } \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}} \text { and } 0+1\)
\(=\sqrt{2} \text { and } 1=[1, \sqrt{2}]\)

85.

\(\cos 2 \theta \cos 2 \phi+\sin ^{2}(\theta-\phi)-\sin ^{2}(\theta+\phi) \)
\(=\cos 2 \theta \cos 2 \phi+\frac{1}{2}-\frac{1}{2} \cos (2 \theta-2 \phi)-\frac{1}{2}+\frac{1}{2} \cos (2 \theta+2 \phi) \)
\(=\cos 2 \theta \cos 2 \phi+\frac{1}{2}[\cos (2 \theta+2 \phi)-\cos (2 \theta-2 \phi)] \)
\(=\cos 2 \theta \cos 2 \phi-\sin 2 \theta \cos 2 \phi \)
\(=\cos (2 \theta+2 \phi) \)
\(=\cos 2(\theta+\phi) \)

86.

\(\mathrm{f}_{4}(x)-\mathrm{f}_{6}(x)=\frac{1}{4}\left[\sin ^{4} x+\cos ^{4} x\right]-\frac{1}{6}\left[\sin ^{6} x+\cos ^{6} x\right] \)
\(=\frac{1}{4}\left[\left(\sin ^{2} x+\cos ^{2} x\right)^{2}-2 \sin ^{2} x \cos ^{2} x\right]-\frac{1}{6}\left[\left(\sin ^{2} x+\cos ^{2} x\right)^{3}\right. \)
\(\left.=-3 \sin ^{2} x \cos ^{2} x+\left(\sin ^{2} x+\cos ^{2} x\right)\right] \)
\(=\frac{1}{4}\left(1-2 \sin ^{2} x \cos ^{2} x\right)-\frac{1}{6}\left(1-3 \sin ^{2} x \cos ^{2} x\right) \)
\(=\frac{1}{4}-\frac{1}{2} \sin ^{2} x \cos ^{2} x-\frac{1}{6}+\frac{1}{2} \sin ^{2} x \cos ^{2} x \)
\(=\frac{1}{4}-\frac{1}{6} \)
\(=\frac{3-2}{12}=\frac{1}{12} \)

87.

\(\frac{\tan 140^{\circ}-\tan 130^{\circ}}{1+\tan 140^{\circ} \tan 130^{\circ}} =\tan \left(140^{\circ}-130^{\circ}\right) \)
\(=\tan 10^{\circ} \ldots \ldots(1) \)
\(\tan 40^{\circ} =\lambda \)
\(\tan 80^{\circ}=\frac{2 \tan 40^{\circ}}{1-\tan ^{2} 40^{\circ}} =\frac{2 \lambda}{1-\lambda^{2}} \)
\(\tan \left(90^{\circ}-10^{\circ}\right) =\frac{2 \lambda}{1-\lambda^{2}} \)
\(\cot 10^{\circ} =\frac{2 \lambda}{1-\lambda^{2}} \)
\(\Rightarrow \tan 10^{\circ} =\frac{1-\lambda^{2}}{2 \lambda} \)
\(\frac{\tan 140^{\circ}-\tan 130^{\circ}}{1+\tan 140^{\circ} \tan 130^{\circ}} \)
\(=\frac{1-\lambda^{2}}{2 \lambda}\)

88.

\(\cos 28^{\circ}+\sin 28^{\circ} =\mathrm{k}^{3} \)
\(\cos 28^{\circ}+\sin \left(90^{\circ}-62^{\circ}\right) =\mathrm{k}^{3} \)
\(\cos 28^{\circ}+\cos 62^{\circ} =\mathrm{k}^{3} \)
\(2 \cos 45^{\circ} \cos 17^{\circ} =\mathrm{k}^{3} \)
\(2 \frac{1}{\sqrt{2}} \cos 17^{\circ} =\mathrm{k}^{3} \)
\(\cos 17^{\circ} =\frac{\mathrm{k}^{3}}{\sqrt{2}} \)

89.

\(x =\frac{1}{\cos 80^{\circ}}-\frac{\sqrt{3}}{\sin 80^{\circ}} \)
\(=\frac{\sin 80^{\circ}-\sqrt{3} \cos 80^{\circ}}{\sin 80^{\circ} \cos 80^{\circ}} \)
\(\frac{x}{2} =\frac{\frac{1}{2} \sin 80^{\circ}-\frac{\sqrt{3}}{2} \cos 80^{\circ}}{\frac{1}{2} \sin 80^{\circ} \cos 80^{\circ}} \)
\(=\frac{\sin 80^{\circ} \cos 60^{\circ}-\cos 80^{\circ} \sin 60^{\circ}}{\frac{1}{2} \sin 80^{\circ} \cos 80^{\circ}} \)
\(=\frac{\sin 20^{\circ}}{\frac{1}{2} \sin 80^{\circ} \cos 80^{\circ}} \)
\(=\frac{\sin 160^{\circ}}{\frac{1}{2} \sin 80^{\circ} \cos 80^{\circ}} \)
\(=\frac{2 \sin 80^{\circ} \cos 80^{\circ}}{\frac{1}{2} \sin 80^{\circ} \cos 80^{\circ}}=4 \)

90.

LetA = (a, b),B = (a,b,c)
A \(\times\)B = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c)}
B \(\times\)A = {(a, a),(a, b), (b, a), (b, b), (c, a),(c, b)}
(A \(\times\)B) ⋂ (B \(\times\)A) = {(a, a), (a,b), (b, a): (b, b)}
= A \(\times\)A

91.

(d)
none of these

92.

(a)
2(2x²- 5x + 2)

93.

\(M \cap C=70 \text { which is } 10 \% \text { of } M \text { and } 14 \% \text { of } C\)
\(M =700, C=500\)
\(M \cup C=700+500-70=1130\)

94.

95.

96.

\(n(A \cap B)=1\)

97.

98.

\(f(x) =\sin x+\cos x \)
\(f(-x) =\sin (-x)+\cos (-x) \)
\(=-\sin x+\cos x \)
\(-f(-x) =\sin x-\cos x \)
\(f(x) \neq-f(-x) \)
f(x) is neither odd function nor even function.

99.

\(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R} \text { is defined by }\)
\(\mathrm{f}(x)=1-|x|\)
\(\text { The range is }(-\infty, 1] \text { as } f(-\infty)=-\infty\)
\(f(0)=1\)
\(f(\infty) =-\infty\)

100.