Full Portion - Important One Mark Question Paper

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 locus of a point which is equidistant from (-1, 1) and (4, 2) is ______________

(a)

5x + 3y + 9 = 0

(b)

5x + 3y - 9 = 0

(c)

3x - 5y = 0

(d)

3x + 5y - 9 = 0

Separate equation of lines for a pair of lines whose equation is x²+ xy -12y²= 0 are ______________

(a)

x + 4y = 0 and x + 3y = 0

(b)

2x - 3y = 0 and x - 4y = 0

(c)

x - 6y = 0 and x - 3y = 0

(d)

x + 4y = 0 and x - 3y = 0

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 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

\(\frac{1}{360}\) of a complete rotation clockwise is _______________

(a)

-1°

(b)

-360°

(c)

-90°

(d)

1°

If in a triangle a = 5, b = 4 and cos(A - B) = \(\frac{31}{32}\) then the third side C is equal to _______________

(a)

(b)

(c)

(d)

If cos \(\alpha\) = \(\frac{3}{5}\) and cos \(\beta=\frac{5}{13}\), then _______________

(a)

cos\((\alpha+\beta)=\frac{33}{65}\)

(b)

\(sin(\alpha+\beta)=\frac{56}{65}\)

(c)

\(sin^2\frac{(\alpha-\beta)}{2}=\frac{4}{65}\)

(d)

\(cos(\alpha-\beta)=\frac{66}{65}\)

The value of cos 20°- sin 20° is _______________

(a)

positive

(b)

negative

(c)

(d)

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

(a)

(b)

(c)

(d)

2 tan A

10.

The lines ax + y + 1 = 0, x + by + 1 = 0 and x + y + c = 0(a ≠ b ≠ c ≠ 1) are concurrent, then the value of \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\) ______________

(a)

-1

(b)

(c)

(d)

abc

11.

If A, B and C are three sets and if A ∈ B and B ⊂ C then

(a)

A ⊂ C

(b)

A need not be a subset of C

(c)

A = B

(d)

C ⊂ A

12.

n[P[P[p(Ø)]]] =

(a)

(b)

(c)

(d)

13.

If \(f(x)={1-x\over 1+x}, x≠0\) then \(f[f(x)]+f\left[f\left(1\over x\right)\right]\)

(a)

(b)

> 2

(c)

> 2

(d)

None

14.

The domain of the function \(f(x)=\sqrt{log_{10}{3-x\over x}}\)is

(a)

\((0,{3\over2})\)

(b)

(0, 3)

(c)

\((-\infty, {3\over2}]\)

(d)

\((0, {3\over2}]\)

15.

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

(a)

[5, ∞)

(b)

(- ∞, 6)

(c)

[5, 6]

(d)

(-5, ≠6)

16.

The equation of the straight line which passes through the point (2, 4) and have intercept on the axes equal in magnitude but opposite in sign is ______________

(a)

x - y = 2

(b)

x - y + 2 = 0

(c)

x - y + 1 = 0

(d)

x - y - 1 = 0

17.

The equation of the straight line bisecting the line segment joining the points (2, 4) and (4, 2) and making an angle of 45^o with positive direction of x-axis is ______________

(a)

x + y = 6

(b)

x - y = 0

(c)

x - y = 6

(d)

x + y = 0

18.

If the straight line y = mx + c passes through the point (1, 2) and (-2, 4) then the value of m and c are ______________

(a)

\(\frac{8}{3},\frac{-2}{3}\)

(b)

\(\frac{-2}{3},\frac{8}{3}\)

(c)

\(\frac{2}{3},\frac{-8}{3}\)

(d)

\(\frac{-2}{3},\frac{-8}{3}\)

19.

If ⁿP_t = 720 nCr, then the value of r = _________

(a)

(b)

(c)

(d)

20.

The number of 4 digit numbers, that can be formed by the digits 3, 4, 5, 6, 7, 8, 0 and no digit is being repeated is _________

(a)

720

(b)

840

(c)

280

(d)

560

21.

The number of positive integral solution of \(x\times y\times z=30\) is _________

(a)

(b)

(c)

(d)

22.

Each of five questions is a multiple-choice test has 4 possible answers. The number of different sets of possible answers is _________

(a)

4⁵-4

(b)

5⁴-5

(c)

1024

(d)

1023

23.

If x, 2x + 2, 3x + 3 . . . are in G.P, then the 4^th term is ______________

(a)

(b)

-27

(c)

13.5

(d)

-13.5

24.

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

(a)

(b)

(c)

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

(d)

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

25.

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

26.

Sum of the binomial coefficients is ______________

(a)

(b)

n²

(c)

2ⁿ

(d)

n+17

27.

The number of ways of disturbing 7 identical balls in 3 distinct boxes, so that no box is empty is _________

(a)

(b)

(c)

(d)

28.

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

(a)

(b)

(c)

-2

(d)

-1

29.

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

(a)

(b)

(c)

(d)

30.

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

(a)

3⁵

(b)

3⁶

(c)

3⁴

(d)

31.

The locus of a moving point P(a cos³θ, a sin³θ) is ______________

(a)

\({ x }^{ \frac { 2 }{ 3 } }+{ y }^{ \frac { 2 }{ 3 } }={ a }^{ \frac { 2 }{ 3 } }\)

(b)

x²+ y²= a²

(c)

x + y = a

(d)

\({ x }^{ \frac { 3 }{ 2 } }+{ y }^{ \frac { 3 }{ 2 } }={ a }^{ \frac { 3 }{ 2 } }\)

32.

\(\frac{1}{1!}+\frac{1}{3!}+\frac{1}{5!}+...\) is ______________

(a)

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

(b)

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

(c)

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

(d)

none of these

33.

AM, GM, HM denote the Arithmetic mean, Geometric mean and Harmonic mean respectively the relationship between this is ______________

(a)

AM < GM < HM

(b)

AM ≤ GM ≤ HM

(c)

AM>GM>HM

(d)

AM≥GM≥HM

34.

The general solution of cosec_\(\theta\) = -2 is _______________

(a)

\(2n\pi +(-1)^n({\pi\over 6})\)

(b)

\(n\pi +(-1)^n({-\pi\over 6})\)

(c)

\(2n\pi \pm({\pi\over 6})\)

(d)

\(-{\pi\over 6}+n\pi\)

35.

If the arcs of same lengths in two circles sustend central angles 30° and 40° find the ratio of their radii _______________

(a)

3 : 4

(b)

4 : 3

(c)

7 : 12

(d)

none of these

36.

How many words can be formed using all the letters of the word ANAND _________

(a)

(b)

(c)

(d)

37.

The number of parallelogram formed if 5 parallel lines intersect with 4 other paralle llines is _________

(a)

(b)

(c)

(d)

38.

If ⁿP_r = 720, ⁿC_r = 120 then r is _________

(a)

(b)

(c)

(d)

39.

is _________

(a)

\(\lfloor{n}(n+2)\)

(b)

(c)

(d)

none of these

40.

If \(\frac{x}{x^2-5x+6}=\frac{A}{x-2}+\frac{B}{x-3}\) then value of A is _____________

(a)

(b)

(c)

(d)

-2

41.

The value of a when x³- 2x²+ 3x + a is divided by (x - 1), the remainder is 1, is ___________

(a)

-1

(b)

(c)

(d)

-2

42.

The zero of the polynomial function f(x) = 9x²-16 are ____________

(a)

(9, 16)

(b)

(3, 4)

(c)

\((\frac{4}{3},-\frac{4}{3})\)

(d)

\((\frac{3}{4},-\frac{3}{4})\)

43.

Solve 3x² + 5x - 2≤0

(a)

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

(b)

[2,\(\frac{1}{3}\)]

(c)

(-2,\(\frac{1}{3}\))

(d)

(-2,\(\frac{-1}{3}\))

44.

If \(\alpha\) and \(\beta\) are the roots of 2x²+ 4x + 5 = 0 the equation where roots are 2\(\alpha\) and 2\(\beta\) is ___________

(a)

4x²+ 4x + 5 = 0

(b)

2x² + 4x + 50 = 0

(c)

x²+ 4x + 5 = 0

(d)

x²+ 4x + 10 = 0

45.

If \(f:[-2,2]\rightarrow A\) is given by f(x) = 3³ then f is onto, if A is ___________

(a)

[3, 3]

(b)

(3, 3)

(c)

[-24,24]

(d)

(-24, 24)

46.

If a is the arithmetic mean and g is the geometric mean of two numbers, then

(a)

a \(\le \) g

(b)

a \(\ge\) g

(c)

a = g

(d)

a > g

47.

If ⁿC₁₀ > ⁿC_r for all possible r, then a value of n is

(a)

(b)

(c)

(d)

48.

The coefficient of x⁶ in (2 + 2x)¹⁰ is

(a)

¹⁰C₆

(b)

2⁶

(c)

¹⁰C₆2⁶

(d)

¹⁰C₆2¹⁰

49.

If 7x² - 8xy +A = 0 represents a pair of perpendicular lines, the A is ______________

(a)

(b)

-7

(c)

-8

(d)

50.

The figure formed by the lines ax ± by ± c = 0 is a ______________

(a)

rectangle

(b)

square

(c)

rhombus

(d)

none of these

51.

Slope of x-axis or a line parallel to x-axis is ______________

(a)

(b)

positive

(c)

negative

(d)

infinity

52.

The Co-efficient of x³ in \(\sqrt { \frac { 1-x }{ 1+x } } ,\left| x \right| <1\ is\ \)______________

(a)

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

(b)

\(\frac { 3 }{ 8 } \)

(c)

\(\frac { -3 }{ 8 } \)

(d)

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

53.

The area of the triangle formed by the lines x² - 4y² = 0 and x = a is

(a)

2a²

(b)

\(\frac{\sqrt3}{2}a^2\)

(c)

\(\frac12a^2\)

(d)

\(\frac{2}{\sqrt3}a^2\)

54.

The image of the point (2, 3) in the line y = -x is

(a)

(-3, -2)

(b)

(-3, 2)

(c)

(-2, -3)

(d)

(3, 2)

55.

The nth term of a G.P is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is ______________

(a)

(b)

(c)

(d)

none of these

56.

If p(n):49ⁿ + 16ⁿ +\(\lambda \) is divisible by 64 for n \(\in \) N is true, then the least negative integral value of \(\lambda \) is _________

(a)

-3

(b)

-2

(c)

-1

(d)

-4

57.

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

(a)

(b)

(c)

(d)

58.

The term without x in \({ \left( 2x-\frac { 1 }{ 2{ x }^{ 2 } } \right) }^{ 12 }\) is ______________

(a)

495

(b)

-495

(c)

-7920

(d)

7920

59.

The number of different signals which can be given from 6 flags of different colours taking one or more at a time is _________

(a)

1958

(b)

1956

(c)

(d)

60.

The intercepts of the perpendicular bisector of the line segment joining (1, 2) and (3, 4) with coordinate axes are

(a)

5, -5

(b)

5, 5

(c)

5, 3

(d)

5, -4

61.

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)

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

62.

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)

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

63.

If the point (8, -5) lies on the locus \(\frac{x^2}{16}-\frac{y^2}{25}=k\), then the value of k is

(a)

(b)

(c)

(d)

64.

The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

(a)

⁵²C₅

(b)

⁴⁸C₅

(c)

⁵²C₅ + ⁴⁸C₅

(d)

⁵²C₅ - ⁴⁸C₅

65.

In ²ⁿC₃ : ⁿC₃= 11 : 1 then n is

(a)

(b)

(c)

(d)

66.

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

67.

Number of sides of a polygon having 44 diagonals is

(a)

(b)

(c)

(d)

68.

The sum up to n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\).....is

(a)

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

(b)

2n(n+1)

(c)

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

(d)

69.

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

(a)

30⁴\(\times\) 29²

(b)

30³\(\times\) 29³

(c)

30²\(\times\) 29⁴

(d)

30\(\times\)29⁵

70.

If S_n denotes the sum of n terms of an AP whose common difference is d, the value of S_n- 2S_n-1+ S_n-2 is

(a)

(b)

(c)

(d)

d²

71.

The sequence \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } }, \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } },...... \)form an

(a)

(b)

(c)

(d)

AGP

72.

The factors of the polynomial \(6\sqrt { { 3x }^{ 2 } } -47x+5\sqrt { 3 } \) are ___________

(a)

\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

(b)

\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x+1)\)

(c)

\((2x+5\sqrt { 3 } )(3\sqrt { 3 } x+1)\)

(d)

\((2x+5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

73.

The value of log₁₀⁸+ log₁₀⁵- log₁₀⁴= ___________

(a)

\({ log }_{ 10 }^{ 9 }\)

(b)

\({ log }_{ 10 }^{ 36 }\)

(c)

(d)

-1

74.

If the roots of x²-bx + c = 0 are two consecutive integer,then b²- 4c is ___________

(a)

(b)

(c)

(d)

none of these

75.

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

(a)

(b)

(c)

(d)

76.

The rationalising factor of \(\frac { 5 }{ \sqrt [ 3 ]{ 3 } } \) is

(a)

\(\sqrt [ 3 ]{ 6 } \)

(b)

\(\sqrt [ 3 ]{ 3 } \)

(c)

\(\sqrt [ 3 ]{ 9 } \)

(d)

\(\sqrt [ 3 ]{ 27 } \)

77.

If x < 7, then ___________

(a)

-x < -7

(b)

- x ≤ -7

(c)

-x > -7

(d)

-x ≥ -7

78.

If cos x = \(\frac { -1 }{ 2 } \) \(0 < x < 2\pi\) and, then the solutions are _______________

(a)

x = \(\frac { \pi }{ 3 } ,\frac { 4\pi }{ 3 } \)

(b)

x = \(\frac { 2\pi }{ 3 } ,\frac { 4\pi }{ 3 } \)

(c)

x = \(\frac { 2\pi }{ 3 } ,\frac { 7\pi }{ 6 } \)

(d)

x = \(\frac { 2\pi }{ 3 } ,\frac { 5\pi }{ 3 } \)

79.

If tan A = \(\frac { a }{ a+1 } \) and B = \(\frac { 1 }{ 2a+1 } \) then the value of A + B is ___________

(a)

(b)

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

(c)

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

(d)

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

80.

The angle between the minute and hour hands of a clock at 8.30 is ___________

(a)

80⁰

(b)

75⁰

(c)

60⁰

(d)

105⁰

81.

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

(a)

(b)

(c)

(d)

82.

If \(\frac { 1-2x }{ 3+2x-{ x }^{ 2 } } =\frac { A }{ 3-x } +\frac { B }{ x+1 } \), then the value of A + B is

(a)

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

(b)

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

(c)

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

(d)

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

83.

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)

84.

The solution 5x-1<24 and 5x+1 > -24 is

(a)

(4,5)

(b)

(-5,-4)

(c)

(-5,5)

(d)

(-5,4)

85.

If sin α + cos α = b, then sin 2α is equal to

(a)

b²- 1, if b ≤\(\sqrt { 2 } \)

(b)

b²- 1, if b >\(\sqrt { 2 } \)

(c)

b²-1, if b ≥ 1

(d)

b²- 1, if b ≥\(\sqrt { 2 } \)

86.

\(\frac { cos6x+6cos4x+15cos2x+10 }{ cos5x+5cos3x+10cosx } \) is equal to

(a)

cos 2x

(b)

cos x

(c)

cos 3x

(d)

2 cos x

87.

If cos pፀ + cos qፀ = 0 and if p ≠ q, then ፀ is equal to (n is any integer)

(a)

\(\frac { \pi (3n+1) }{ p-q } \)

(b)

\(\frac { \pi (2n+1) }{ p\pm q } \)

(c)

\(\frac { \pi (n\pm 1) }{ p\pm q } \)

(d)

\(\frac { \pi (n+2) }{ p+q } \)

88.

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)

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

89.

If \(\pi <2\theta <\frac { 3\pi }{ 2 } \), then \(\sqrt { 2+\sqrt { 2+2cos4\theta } } \) equals to

(a)

-2 cosፀ

(b)

-2 sinፀ

(c)

2 cosፀ

(d)

2 sinፀ

90.

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

(a)

\(\sqrt{2}\)

(b)

\(\sqrt{3}\)

(c)

(d)

91.

The range of the function \({1\over 1-2sinx}\) is

(a)

\((-∞,-1)\cup\left( {1\over 3},\infty\right)\)

(b)

\(\left( -1,{1\over 3}\right)\)

(c)

\(\left[ -1,{1\over 3}\right]\)

(d)

\((-∞,-1]\cup [\frac { 1 }{ 3 } ,∞)\)

92.

Let X = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (1, 3), (2, 2), (3, 3), (2, 1), (3, 1), (1, 4),(4, 1)}. Then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

93.

Let R be the universal relation on a set X with more than one element. Then R is

(a)

not reflexive

(b)

not symmetric

(c)

transitive

(d)

none of the above

94.

The number of relations on a set containing 3 elements is

(a)

(b)

(c)

512

(d)

1024

95.

If n(A) = 2 and n(B ∪ C) = 3, then n[(A \(\times\) B) ∪ (A \(\times\) C)] is

(a)

2³

(b)

3²

(c)

(d)

96.

Let R be the relation over the set of all straight lines in a plane such that l₁Rl₂ ⇔ l₁丄l₂ . Then R is __________

(a)

symmetric

(b)

reflexive

(c)

transitive

(d)

an equivalent relation

97.

If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by "x is greater than y". The range of R is __________

(a)

{1, 4, 6, 9}

(b)

{4, 6, 9}

(c)

{1}

(d)

None of these

98.

If A⊆B, then A\B is ________

(a)

(b)

(c)

(d)

\(\frac{B}{A}\)

99.

The function f:R➝R is defined by f(x)=\(\frac { \left( { x }^{ 2 }+cosx \right) \left( 1+{ x }^{ 4 } \right) }{ \left( x-sinx \right) \left( 2x-{ x }^{ 3 } \right) } +{ e }^{ -\left| x \right| }\) 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.

100.

If the function f:[-3,3]➝S defined by f(x) = x² is onto, then S is

(a)

[-9,9]

(b)

(c)

[-3,3]

(d)

[0,9]

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Answers Key:

Multiple Choice Question

100 x 1 = 100

(b)
5x + 3y - 9 = 0

(d)
x + 4y = 0 and x - 3y = 0

(b)
\(\lambda\) = 2

(a)
-1°

(b)
6

(b)
\(sin(\alpha+\beta)=\frac{56}{65}\)

(a)
positive

(a)
0

10.

(a)
-1

11.

(b)
A need not be a subset of C

12.

13.

(b)
> 2

14.

(d)
\((0, {3\over2}]\)

15.

16.

(b)
x - y + 2 = 0

17.

(b)
x - y = 0

18.

(b)
\(\frac{-2}{3},\frac{8}{3}\)

19.

(a)
6

20.

(a)
720

21.

(d)
27

22.

23.

(a)
27

24.

(b)
2

25.

(d)
a = b = c

26.

27.

(d)
15

28.

(b)
2

29.

(d)
4

30.

(b)
3⁶

31.

(a)
\({ x }^{ \frac { 2 }{ 3 } }+{ y }^{ \frac { 2 }{ 3 } }={ a }^{ \frac { 2 }{ 3 } }\)

32.

33.

(d)
AM≥GM≥HM

34.

(b)
\(n\pi +(-1)^n({-\pi\over 6})\)

35.

(b)
4 : 3

36.

(a)
30

37.

(d)
60

38.

39.

(a)
\(\lfloor{n}(n+2)\)

40.

(d)
-2

41.

(a)
-1

42.

43.

(d)
(-2,\(\frac{-1}{3}\))

44.

(d)
x²+ 4x + 10 = 0

45.

46.

\(\mathrm{AM} \geq \mathrm{GM} \)
\(\Rightarrow a \geq g \)

47.

\({ }^{20} \mathrm{C}_{10}>{ }^{20} \mathrm{C}_{\mathrm{r}} \text { for all possible value of } \mathrm{r} \text {. }\)

48.

\((2+2 x)^{10} \text { Term containing } x^{6} \text { is }\)
\({ }^{10} \mathrm{C}_{6}(2)^{10-6}(2 x)^{6}={ }^{10} \mathrm{C}_{6} 2^{4} 2^{6} x^{6}\)
\(\text { Coefficient of } x^{6} \text { is }{ }^{10} \mathrm{C}_{6} 2^{10}\)

49.

(b)
-7

50.

51.

(a)
0

52.

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

53.

\(x-2 y =0 \)
\(x+2 y =0 \)
\(x =a \)
\(\text {The points are }(0,0),\left(a, \frac{a}{2}\right),\left(a, \frac{-a}{2}\right)\)
\(\text {Area }=-\frac{1}{2} \text { (base } \times \text { height } \text { ) }\)
\(=\frac{1}{2}(a \cdot a)\)
\(\text {Area }=\frac{1}{2} a^{2}\)

54.

The required point is (-3, -2)

55.

(a)
1

56.

57.

(b)
17

58.

(d)
7920

59.

(b)
1956

60.

Equation of line joining (1, 2) and (3, 4) is
\(\frac{y-2}{4-2} =\frac{x-1}{3-1} \)
\(x-y+1 =0 \)
Any line perpendicular to this x + y + k = 0
This passes through midpoint of (1, 2) and (3, 4)
That is (2, 3)
k = -5, x+y-5 = 0
x intercept is 5, y intercept is 5.

61.

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

62.

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

63.

\(\frac{\left(8^{2}\right)}{16}-\frac{(-5)^{2}}{25} =\mathrm{k} \)
\(\frac{64}{16}-\frac{25}{25} =\mathrm{k} \)
\(\mathrm{k} =4-1=3 \)

64.

\({ }^{52} \mathrm{C}_{5} \text { includes all possibilities (zero king, }1 \text { king, } 2 \text { kings, } 3 \text { kings, } 4 \text { kings }){ }^{48} C_{5} \text { has no kings}\)
\(\text { Réquired possibilities }{ }^{52} \mathrm{C}_{5}-{ }^{48} \mathrm{C}_{5}\)

65.

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

66.

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

67.

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

68.

\(\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32} \ldots . . =\sqrt{2}+2 \sqrt{2}+3 \sqrt{2}+4 \sqrt{2} . \)
\(=\sqrt{2}[1+2+3+\ldots .] \)
\(S_{n} =\frac{\sqrt{2}[n(n+1)]}{2} \)
\(=\frac{n(n+1)}{\sqrt{2}} \)

69.

First in Maths = 30 ways
Second in Maths - 29 ways
Similarly for other subjects
30 \(\times\) 29 \(\times\) 30 \(\times\) 29 \(\times\) 30 \(\times\)30 = 30⁴ \(\times\)29²

70.

\(\mathrm{S}_{\mathrm{n}} =\frac{\mathrm{n}}{2}[2 a+(\mathrm{n}-1) \mathrm{d}] \)
\(\mathrm{S}_{\mathrm{n}-1} =\frac{\mathrm{n}-1}{2}[2 a+(\mathrm{n}-2) \mathrm{d}] \)
\(\mathrm{S}_{\mathrm{n}-2} =\frac{\mathrm{n}-2}{2}[2 a+(\mathrm{n}-3) \mathrm{d}] \)
\(\mathrm{S}_{\mathrm{n}}-2 \mathrm{~S}_{\mathrm{n}-1}+\mathrm{S}_{\mathrm{n}-2} =2 a\left[\frac{\mathrm{n}-2(\mathrm{n}-1)}{2}+\frac{(\mathrm{n}-2)}{2}\right]+\frac{\mathrm{d}}{2}[\mathrm{n}(\mathrm{n}-1)-2(\mathrm{n}-1)(\mathrm{n}-2)+(\mathrm{n}-2)(\mathrm{n}-3)] \)
\(=2 a\left[\frac{\mathrm{n}-2 \mathrm{n}+2+\mathrm{n}-2}{2}\right]+\frac{\mathrm{d}}{2}\left[\mathrm{n}^{2}-\mathrm{n}-2 \mathrm{n}^{2}+6 \mathrm{n}-4+\mathrm{n}^{2}-5 \mathrm{n}+6\right] \)
\(= 0+\frac{\mathrm{d}}{2}(2)=\mathrm{d} \)

71.

\(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}+\sqrt{2}}, \frac{1}{\sqrt{3}+2 \sqrt{2}}, \ldots \ldots \text { form } a \text { HP. }\)

72.

(a)
\((2x-5\sqrt { 3 } )(3\sqrt { 3 } x-1)\)

73.

74.

(b)
1

75.

(a)
1

76.

77.

78.

(b)
x = \(\frac { 2\pi }{ 3 } ,\frac { 4\pi }{ 3 } \)

79.

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

80.

(b)
75⁰

81.

(a)
4

82.

\(\frac{1-2 x}{3+2 x-x^{2}}= \frac{\mathrm{A}}{3-x}+\frac{\mathrm{B}}{x+1} \)
\( 1-2 x=\mathrm{A}(x+1)+\mathrm{B}(3-x) \)
\( \text { Put } x=3,-5=\mathrm{A}(4) \Rightarrow \mathrm{A}=\frac{-5}{4} \)
\( \text { Put } x=-1,3=\mathrm{B}(4) \Rightarrow \mathrm{B}=\frac{3}{4} \)
\(\mathrm{~A}+\mathrm{B}= \frac{-5}{4}+\frac{3}{4}=\frac{-2}{4}=\frac{-1}{2} \)

83.

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

84.

\(5 x-1 <24 \ \ \ 5 x+1 >-24 \)
\(5 x <25 \ \ 5 x >-25\)
\(x <5 \ x >-5 \)
\(x \in(-5,5)\)

85.

\(\sin \alpha+\cos \alpha=b \Rightarrow(\sin \alpha+\cos \alpha)^{2} =b^{2} \)
\(\sin ^{2} \alpha+\cos ^{2} \alpha+2 \sin \alpha \cos \alpha =b^{2} \)
\(\sin 2 \alpha =b^{2}-1 \)
Since \(-1<\sin 2 \alpha \leq 1 \)
\(-1 \leq \mathrm{b}^{2}-1 \leq 1 \)
\(\mathrm{~b}^{2}-1 \leq 1 \)
\(\mathrm{~b}^{2} \leq 2 \)
\(\text { This is possible if } \mathrm{b} \leq \sqrt{2}\)
\(\sin 2 \alpha=b^{2}-1, \text { if } b \leq \sqrt{2}\)

86.

\(\text { Numerator }=\cos 6 x+6 \cos 4 x+15 \cos 2 x+10\)
\(=(\cos 6 x+\cos 4 x)+5(\cos 4 x+\cos 2 x)+ 10(\cos 2 x+1) \)
\(=2 \cos 5 x \cos x+5(2 \cos 3 x \cdot \cos x)+10\left(2 \cos ^{2} x\right) \)
\(=2 \cos x[\cos 5 x+5 \cos 3 x+10 \cos x] \)
\(\therefore \frac{\mathrm{Nr}}{\mathrm{Dr}} =\frac{2 \cos x(\cos 5 x+5 \cos 3 x+10 \cos x)}{\cos 5 x+5 \cos 3 x+10 \cos x} \)
\(=2 \cos x \)

87.

\(\cos p \theta+\cos q \theta=0 \)
\(2 \cos \left(\frac{p+q}{2}\right) \theta \cdot \cos \left(\frac{p-q}{2}\right) \theta=0 \)
\(\Rightarrow 2 \cos \left(\frac{p+q}{2}\right) \theta=0 \)
\(\cos \left(\frac{\mathrm{p}-\mathrm{q}}{2}\right) \theta=0\)
\(\text { Principal angle is } \pi / 2\)
\(\therefore\left(\frac{p+q}{2}\right) \theta =n \pi \pm \pi / 2 \)
\(\Rightarrow(p+q) \cdot \theta =2 n \pi \pm \pi \)
\(\theta =\frac{\pi(2 n \pm 1)}{p+q} \)
\(\text { Similarly, } \theta=\frac{\pi(2 n \pm 1)}{p-q}\)

88.

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

89.

\(\sqrt{2+2 \cos 4 \theta} =\sqrt{2+2\left(2 \cos ^{2} 2 \theta-1\right)} \)
\(=\sqrt{4 \cos ^{2} 2 \theta} \)
\(=2 \cos 2 \theta \)
\(\sqrt{2+\sqrt{2+2 \cos 4 \theta}} =\sqrt{2+2 \cos 2 \theta} \)
\(=\sqrt{2+2\left(2 \cos ^{2} \theta-1\right)} \)
\(=\sqrt{4 \cos ^{2} \theta} \)
\(=\pm 2 \cos \theta \)
\(\Rightarrow \pi<2 \theta<\frac{3 \pi}{2} \)
\(\Rightarrow \frac{\pi}{2}<\theta<\frac{3 \pi}{4} \text { in II quadrant. } \)
\(\therefore \sqrt{2+\sqrt{2+2 \cos 4 \theta}}=-2 \cos \theta\)

90.

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

91.

\(-1 \leq \sin x \leq 1 \)
\(-2 \leq 2 \sin x \leq 2 \)
\(2 \geq-2 \sin x \geq-2 \)
\(-2 \leq-2 \sin x \leq 2 \)
Adding 1,
\(1-2 \leq 1-2 \sin x \leq 1+2 \)
\(-1 \leq 1-2 \sin x \leq 3 \)
\(-1 \geq \frac{1}{1-2 \sin x} \geq \frac{1}{3} \)
\(\frac{1}{3} \leq \frac{1}{1-2 \sin x} \leq-1 \)
\(\therefore \text { Range is }(-\infty,-1] \cup\left[\frac{1}{3}, \infty\right)\)

92.

\(\text { If } 4 \in X \text { then }(4,4) \notin \mathrm{R}\)
R is not reflexive
Symmetric can be easily checked

93.

Let X = (a,b,c)
Then R = Universal relation
= {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}.
It is transitive

94.

95.

\(n[(A \times B) \cup(A \times C)]=n(A) \times n(B \cup C)\)
= 2 \(\times\) 3 = 6

96.

(a)
symmetric

97.

98.

99.

\(f(x) =\frac{\left(x^{2}+\cos x\right)\left(1+x^{4}\right)}{(x-\sin x)\left(2 x-x^{3}\right)}+e^{-|x|} \)
\(f(-x) =\frac{\left[(-x)^{2}+\cos (-x)\right]\left[1+(-x)^{4}\right]}{[-x-\sin (-x)]\left[-2 x-\left(-x^{3}\right)\right]}+e^{+x \mid} \)
\(=\frac{\left(x^{2}+\cos x\right)\left(1+x^{4}\right)}{(x-\sin x)\left(2 x-x^{3}\right)}+e^{-|x|} \)
\(=f(x) \)
Hence f(x) is an even function

100.

f(0) = 0, f(-3) = 9 and f(3) = 9
.'. S is [0,9]