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QUESTION: 1

Suppose ω is a cube root of unity with ω≠1. Suppose P and Q are the points on the complex plane defined by ω and ω^{2}. If O is the origin, then what is the angle between OP and OQ?

Solution:

P and Q are points on complex plane. Angle between OP and OQ is

QUESTION: 2

Suppose there is a relation * between the positive numbers x and y given by x * y if and only if x ≤ y^{2}. Then which one of the following is correct?

Solution:

x and y are positive numbers.

x ≤ y^{2}

x < x^{2 } positive numbers.

Hence relation is reflexive.

**Transitive -
**

Thus relation is not transitive.

1 ≤ (2)

Hence relation is not symmetric.

Thus x ≤ y

QUESTION: 3

If x^{2}+ px + 4 for all real values of x, then which one of thefollowing is correct?

Solution:

(a) x^{2}-px + 4> 0 real values of x.

If b^{2} - 4ac < 0

⇒p^{2} - 4(1)(4)<0

⇒p^{2} < 16 ⇒|p| < 4

QUESTION: 4

If z=x + iy= , where i = √-1, then what is the fundamental amplitude of

Solution:

z = x + iy

QUESTION: 5

If f(x_{1})- f(x_{2}) is for x_{1}, x_{2} ∈ (-1,1), then what is f(x) equal to?

Solution:

QUESTION: 6

What is the range of the function 2 ,where X ∈**R**?

Solution:

QUESTION: 7

A straight line intersects x and y axes at P and Q respectively If (3,5) is the middle point of PQ, then what is the area ofthe triangle OPQ?

Solution:

As we know that line PQ intersects x-axis andy-axis at Rand Q.

∵ M is the mid point of PQ

⇒ x = 6 and y = 10

Hence area of triangle OPQ

QUESTION: 8

If a circle of radius b units with centre at (0, b) touches the line y = x — a√2 , then what is the value of b?

Solution:

Distance from the centre to the point of line which touches circle is OM = radius

QUESTION: 9

Consider the function f(θ)= 4(sin^{2} θ+ cos^{4} θ)

Q. What is the maximum value of the function f(θ)?

Solution:

f(θ) = 4 (sin^{2}θ + cos^{4} θ)

= 4 (sin^{2 }θ + cos^{2} θ(1- sin^{2} θ))

= 4 (sin^{2} θ + cos^{2} θ - sin^{2} θ cos^{2} θ)

For maximum value of f(θ), sin^{2}2θ should be minimum.

i.e. sin^{2}2θ = 0

f(θ)l_{max}=4(l1-0) = 4

QUESTION: 10

Consider the function f(θ)= 4(sin^{2} θ+ cos^{4} θ)

Q. What is the minimum value of the function f(θ)?

Solution:

f(θ) = 4 (sin^{2}θ + cos^{4} θ)

= 4 (sin^{2 }θ + cos^{2} θ(1- sin^{2} θ))

= 4 (sin^{2} θ + cos^{2} θ - sin^{2} θ cos^{2} θ)

For minimum value ot f(θ), sin^{2}2θ should be maximum i.e. sin^{2}2θ= 1.

QUESTION: 11

Consider the function f(θ)= 4(sin^{2} θ+ cos^{4} θ)

Consider the following statements:

f(θ) = 2 has no solution.

f(θ) = has a solution.

Q. Which of the above statements is/are correct?

Solution:

f(θ) = 4 (sin^{2}θ + cos^{4} θ)

= 4 (sin^{2 }θ + cos^{2} θ(1- sin^{2} θ))

= 4 (sin^{2} θ + cos^{2} θ - sin^{2} θ cos^{2} θ)

Since sin θ cannot have vlaue greater than 1 & less than -1.

Hence f(θ) = 2 has no solution.

QUESTION: 12

Consider the curves

Solution:

Hence f(x ) and g(x) intersects at ( -1 , -2 ) and (2 ,3 ).

QUESTION: 13

What is the area bounded by the curves

Solution:

QUESTION: 14

Consider the function

How many solutions does the function f(x) = 1 have?

Solution:

This is a cubic equation.

If we put y = then ( 3y + 1) = 0 is a factor o f cubic equation.

QUESTION: 15

Consider the function

How many solutions does the function f(x) = -1 have?

Solution:

Similarly for (f) = -1 we will get 27y^{3} - 27y^{2} - 4 = 0 and after solving it we will find that it has two solutions.

y_{0}=1.1184,-0.05922.

QUESTION: 16

Consider the functions

f(x) = xg(x) and g(x =

Where [•] is the greatest integer function.

What is equal to?

Solution:

As g(x) is a gretest integer function so value of g(x) in integral limit will be

QUESTION: 17

What is equal to ?

Solution:

The value of g(x) in value will be 2 and in range

form(l)

QUESTION: 18

Consider the function f ( x ) = | x - 1 |+ x^{2} , w here x ∈**R** .

Which one of the following statements is correct?

Solution:

f(x) = | x - l | + x^{2} x ∈**R**

f_{1} (x ) = |x - l |, f_{2}(x) = x^{2}

f_{1 }(x) and f_{2}{x) both are continuous.

Hence f(x) is continuous.

f(x) in differentiable at x = 0

f_{1}(x) is not differentiable at x = 1.

Hence(fx) is continuous but not differentiable at x= 1

QUESTION: 19

Consider the function f ( x ) = | x - 1 |+ x^{2} , w here x ∈**R** .

Which one of the following statements is correct?

Solution:

As we know,

f(x) is in quadratic form (parabola). Hence f(x) is decreasing in and increasing

QUESTION: 20

Which one of the following statements is correct?

Solution:

f(x) has local minimum at one point only in (-∞ ,∞ ).

QUESTION: 21

What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates and x = 1 ?

Solution:

Hence area required for given region is

QUESTION: 22

What is the area of the region bounded by x-axis, the curve y = f(x) and the two ordinates x = 1 and

Solution:

Area required for given region is

QUESTION: 23

Given that a_{n} =

Consider the following statements:

1. The sequence {a_{2n}} is in AP with common difference zero.

2. The sequence {a_{2n}+1} is in AP with common difference zero.

Which of the above statements is/are correct?

Solution:

Since it is a definite integral will have a definite value. The sequence {a_{2n}} is in AP with common difference. Statement (1) is correct.

The sequence {a_{2n} + 1} is also in AP with common difference.

Statement (2) is correct.

QUESTION: 24

Given that a_{n} =

What is a_{n-1} - a_{n-4} equal to ?

Solution:

∵ given sequence a_{n} also AP with no difference.

Thus a_{n-1} - a_{n-4} = 0

QUESTION: 25

Consider the equation x + |y| = 2y.

Which of the following statements are **not **correct?

1. y as a function of x is not defined for all real x.

2. y as a function of x is not continuous at x = 0.

3. y as a function of x is differentiable for all x.

Q. Select the correct answer using the code given below.

Solution:

x+ | y |= 2y

x = 2y - |y |

2y-| y | = x

∵ by checking

y as a function of x is continuous at x = 0, but not differentiable at x = 0.

So all of the statements are not correct.

QUESTION: 26

Consider the equation x + |y| = 2y.

What is the derivative of y as a function of x with respect to x for x < 0?

Solution:

Option (d) is correct.

QUESTION: 27

Consider the lines y = 3x, y = 6x and y = 9

What is the area of the triangle formed by these lines?

Solution:

OAB is triangle

QUESTION: 28

Consider the lines y = 3x, y = 6x and y = 9

The centroid of the triangle is at which one of the following points?

Solution:

Coordinates o f O, A, B are (0, 0) respectively.

QUESTION: 29

Consider the function f(x) = (x - l )^{2} ( x + 1) (x - 2)^{3}

Q. What is the number of points of local minima of the function f(x)?

Solution:

f(x) = (x-l)^{2}(x + l ) (x-2)^{3}

f'(x) = 2(x - l)(x + l)(x - 2)^{3}+ ( x - l)^{2}(x - 2)^{3}+(x - 1)^{2} (x + l)^{3}(x - 2)^{2}

= (x - l)(x -1)^{2} [2(x+1)(x - 2 ) +( x - l) ( x - 2) + 3 ( x - l ) ( x + l)]

f '(x) = ( x - l)(x - 2 )^{2}[2x^{2} - 2x - 4 + x^{2} - 3x + 2 + 3x^{2} - 3]

= (x - l)(x - 2)^{2} [6x^{2} - 5x - 5]

For maxima and minima

f'(x )= 0

(x - l)(x - 2)^{2} [6x^{2} - 5x - 5] = 0

The change in signs of f(x) for dififrent values of x is shown:

∵ Local Minima are

QUESTION: 30

Consider the function f(x) = (x - l )^{2} ( x + 1) (x - 2)^{3}

What is the number of points of local maxima of the function f(x) ?

Solution:

Local Maxima is [x = 1 ]

QUESTION: 31

Let f(x) and g(x) be twice differentiable functions on [0,2] satisfying f"(x) = g"(x), f'(1) = 4, g'(1) = 6,f(2) =3 and g(2) = 9. Then what is f(x) - g(x) at x = 4 equal to ?

Solution:

again integrating equation (1)

Rearranging equation (3) again, we get

QUESTION: 32

Consider the curves y = | x — 1 | and |x| = 2

Q. What is/are the point(s) of intersection of the curves ?

Solution:

and x=2

Hence curves intersect at (-2,3) and (2,1).

QUESTION: 33

Consider the curves y = | x — 1 | and |x| = 2.

Q. What is the area of the region bounded by the curves and x-axis?

Solution:

Bounded region is shaded.

So area of bounded region has two triangles ACB and BDE.

Area of region bounded by curves and x-axis is

QUESTION: 34

Consider the function

Q. What is the value of f (0)?

Solution:

QUESTION: 35

Consider the function

Q. What is the value of p for which f"(0)=0?

Solution:

QUESTION: 36

Consider a tiangle ABC in which

Q. What is the value of sin

Solution:

QUESTION: 37

Consider a tiangle ABC in which

Q. What is the value of

Solution:

As we know that

QUESTION: 38

Given that tan α and tanβ are the roots of the equation x^{2} + bx + c = 0 with b ≠ 0.

Q. What is tan(α + β) equal to?

Solution:

QUESTION: 39

Given that tan α and tanβ are the roots of the equation x^{2} + bx + c = 0 with b ≠ 0.

What is sin(α+ β)sec α see β equal to?

Solution:

sin( α + β) sec α see β

= tan α + tan β

= -b

QUESTION: 40

Consider the two circles (x-l)^{2} + ( y-3)^{2} = r^{2} and x^{2} + y^{2} - 8x + 2y + 8 = 0

Q. What is the distance between the centres of the two circles?

Solution:

Given equation of circles

{h_{1}, k_{1}) = coordinates of centre = (1,3)

x^{2} + y^{2} - 8x + 2y + 8 = 0

(x - 4 )^{2} + (y+1)^{2} =(3)^{2}

(h_{2} , k_{2}) = coordinates o f centre = (4 ,-1)

Distance between centres of two circles

QUESTION: 41

Consider the two circles (x-l)^{2} + ( y-3)^{2} = r^{2} and x^{2} + y^{2} - 8x + 2y + 8 = 0

Q. If the circles intersect at two distinct points, then which one of the following is correct?

Solution:

Given equation of circles

{h_{1}, k_{1}) = coordinates of centre = (1,3)

x^{2} + y^{2} - 8x + 2y + 8 = 0

(x - 4 )^{2} + (y+1)^{2} =(3)^{2}

(h_{2} , k_{2}) = coordinates o f centre = (4 ,-1)

Radius of circle one = r_{1} = r

Radius of circle two = r_{2} = 3

∵ Circle intersects at two points so distance between circle is d < r_{1} + r_{2}

5 < r + 3

r> 2

QUESTION: 42

Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0

Q. What is the equation of the line passing through the point of intersection of the given lines and parallel to x-axis?

Solution:

Equations of lines

x+ y + 1 = 0

3x+2y+ 1=0

Points of intersection (1, -2).

Equation ofx-axis

y=0

Line parallel to x axis is

y = k

If this line passes through (1, -2) then

⇒ y = - 2

⇒ y + 2 = 0

Equation of line passing through (1, -2) and parallel to x-axis is

y + 2 = 0

QUESTION: 43

Consider the two lines x + y + 1 = 0 and 3x + 2y + 1 = 0

Q. What is the equation of the line passing through the point of intersection of the given lines and parallel to y-axis?

Solution:

Equations of lines

x+ y + 1 = 0

3x+2y+ 1=0

Points of intersection (1, -2).

Equation of y-axis

x = 0

Equation of line parallel to x -axis is

x = k

If this line passes through (1,-2 then)

x = 1

Hence equation of line which passes through point of intersection of given line (1, -2) and parallel to y-axis

x = 1

⇒ x-1= 0

QUESTION: 44

Consider the equation

k sinx + cos 2x = 2k - 7

If the equation possesses solution, then what is the minimum value of k?

Solution:

K sin x + cos 2x = 2 K - 7

K sin x + (1 -2 sin^{2} x ) = (2K - 7)

2 sin^{2} x - K sin x + (2k - 8) = 0

This is a quadratic equation in sin x.

For minimum value of k

sin x = -l

Squaring both sides, we get

K^{2} - 16 K+ 64 = K^{2} + 16 + 8 K

24K=48

K=2

QUESTION: 45

Consider the equation

k sinx + cos 2x = 2k - 7

If the equation possesses solution, then what is the maximum value of k?

Solution:

For maximum value of K

sin x = 1

QUESTION: 46

Consider the function f (x) where [•] denotes the greatest integer function.

Q. What is equal to?

Solution:

QUESTION: 47

Consider the function f (x) where [•] denotes the greatest integer function.

Q. What is equal to?

Solution:

QUESTION: 48

Let z_{1,} z_{2} and z_{3} be non-zero com plex num bers satisfying , where i =√- 1 .

Q. What is Z_{1} + z_{2} + z_{3} equal to?

Solution:

Given

Let us suppose that z - x + iy

x^{2} - y^{2} +2xyi = ix+y

Comparing real and imaginary part of both sides

x^{2} - y^{2} = y and 2 xy = x.

Taking 2xy=x

(2y - l)x = 0

ifx = 0

Since given numbers are non zero complex numbers.

So, z_{1} - 0 + ( - 1 )i = - i

QUESTION: 49

Let z_{1,} z_{2} and z_{3} be non-zero com plex num bers satisfying , where i =√- 1 .

Consider the following statements:

1. z_{1}z_{2}z_{3} is purely imaginary.

2. z_{1}z_{2} + z_{2}z_{3} + z_{3}z_{1} is purely real.

Q. Which of the above statements is/are correct?

Solution:

Hence z_{1}z_{2} + z_{2}z_{3} + z_{3}Z_{1} = 0 is purely real.

Hence both statements are correct.

QUESTION: 50

Given that log_{x} y, log_{z} x, log_{y} z are in GP, xyz = 64 and x^{3}, y^{3}, z^{3 }are inA.P.

Q. Which one of the following is correct ?

x,y and z are

Solution:

Thus x, y z are in A.P. and G.P. both.

QUESTION: 51

Given that log_{x} y, log_{z} x, log_{y} z are in GP, xyz = 64 and x^{3}, y^{3}, z^{3 }are inA.P.

Q. Which one of the following is correct?

xy, yz and zx are

Solution:

Similarly xy,yz, zx are also in A.P. and G.P. both.

QUESTION: 52

Let z be a complex number satisfying

Q. What is |z| equal to?

Solution:

⇒|z-4| = |z-8|

Let z = x + iy

| x + iy - 4 | = | x + iy -8 |

Squaring both sides, we get

[ ( x - 4 )^{2} + y^{2}] = [(x-8)^{2} + y^{2}]

(x-4)^{2} =(x-8)^{2}

x^{2} + 16 - 8x = x^{2} + 64 - 1 6x

8x = 48 ⇒ x= 6

Squaring both sides, we get

4(x^{2} + y^{2}) = 9 [ (x - 2)^{2} + y^{2}]

⇒ 4 X^{2} + 4 y^{2} = 9x^{2} + 36 - 36 x + 9 y^{2}

⇒ 5x^{2} +5y^{2} -36x+36 = 0

as we know x = 6

5(6)^{2} + 5y^{2} - 36 x 6 + 36 = 0

⇒ 5y^{2} = 0 ⇒ y = 0.

Hence x = 6 and y = 0.

⇒ z = 6

|z| = 6

QUESTION: 53

Let z be a complex number satisfying

Q. What is equal to?

Solution:

QUESTION: 54

A function f(x) is defined as follows:

Consider the following statements:

1. The function f(x) is continuous at x = 0.

2. The function f(x) is continuous at x =

Q. Which of the above statements is/are correct?

Solution:

Given

For continuity,

f(0)=L.HL. = RHL

Hence function is continuous at x = 0.

Statement (1) is correct.

Hence fimction is continuous at

Statement (2) is correct.

QUESTION: 55

A function f(x) is defined as follows:

Consider the following statements:

1. The function f(x) is differentiable at x = 0. 71

2. The function f(x) is differentiable at x =

Q. Which of the above statements is/are correct ?

Solution:

For differentiability,

L.HD. = RH.D.

Thus at x= 0

**L.HD.≠ RHD.**

So at x = 0 function is not differentiable. Statement (1) is not correct.

Hence function is not differentiable at x =

Statement (2) is not correct.

QUESTION: 56

Let α and β (α < β ) be th e roots of the equatio n x^{2} + bx + c = 0, where b > 0 and c < 0.

Consider the following:

1. β < - α

2. P < | a |

Q. Which of the above is/are correct?

Solution:

Given quadratic equation,

x^{2 }+ bx + c = 0 and roots are α and β.

where a < p. Hence roots of given quadratic equation are

QUESTION: 57

Consider the following:

1. α + β + αβ > 0

2. α^{2} β + β^{2}α > 0

Which of the above is/are correct?

Solution:

Sum of roots = α + β = -b

Multiplication of roots = αβ = c

Hence

QUESTION: 58

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the value o f AC^{2} - BD^{2} ?

Solution:

Suppose Mid point of AC and BD is M (a, b).

QUESTION: 59

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the point of intersection of the diagonals?

Solution:

Point of intersection (a, b) is

QUESTION: 60

Consider a parallelogram whose vertices are A (1,2), B (4, y), C (x, 6) and D (3,5) taken in order.

Q. What is the area of the parallelogram?

Solution:

Area of parallelogram=2 area of Δ ADB

QUESTION: 61

Let f : **R** → **R** be a function such that

f(x ) = x^{3} + x^{2} f '(1) + xf "(2)+ f "'(3)

for x ∈ **R**

What is f(1) equal to?

Solution:

QUESTION: 62

Let f : **R** → **R** be a function such that

f(x ) = x^{3} + x^{2} f '(1) + xf "(2)+ f "'(3)

for x ∈ **R**

Q. What is f '(1) equal to?

Solution:

f'(l)=-5

QUESTION: 63

Let f : **R** → **R** be a function such that

f(x ) = x^{3} + x^{2} f '(1) + xf "(2)+ f "'(3)

for x ∈ **R**

Q. What is f'""(10) equal to?

Solution:

f "'(10)=6

QUESTION: 64

Consider the following:

1. f(2 ) = f(1) - f(0)

2. f "(2) - 2f '(1) = 12

Q. Which of the above is/are correct?

Solution:

QUESTION: 65

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. What are the direction ratios of the line of intersection of the given planes?

Solution:

Hence direction ratios ofthe line of intersection of given plane <2,-5,-3 >

QUESTION: 66

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. What is the equation of the plane P?

Solution:

QUESTION: 67

A plane P passes through the line of intersection of the planes 2x - y + 3z = 2, x + y - z = 1 and the point (1 ,0 ,1 ).

Q. If the plane P touches the sphere x^{2} + y^{2} + z^{2} = r^{2}, then what is r equal to?

Solution:

Plane P touches the sphere x^{2} + y^{2} + z^{2} = r^{2} then r=Distane between centre of sphere (0,0,0) to plance P.

QUESTION: 68

Consider th e function f (x ) = | x^{2} - 5x + 6 |

Q. What is f '(4) equal to?

Solution:

QUESTION: 69

Consider th e function f (x ) = | x^{2} - 5x + 6 |

Q. What is f"(2.5) equal to?

Solution:

f"(2.5) =-2

QUESTION: 70

Let f(x) be the greatest integer function and g(x) be the modulus function.

Q. What is equal to?

Solution:

f(x) → greatest integer function

f(x)=[x]

g(x) → modulus fuction

g(x)=| x |

QUESTION: 71

Let f(x) be the greatest integer function and g(x) be the modulus function.

Q. What is equal to?

Solution:

f(x) → greatest integer function

f(x)=[x]

g(x) → modulus fuction

g(x)=| x |

QUESTION: 72

Consider a circle passing through the origin and the points (a, b) and (-b, -a).

Q. On which line does the centre of the circle lie?

Solution:

Suppose; x^{2 }+ y^{2} + 2gx + 2fy + c = 0 is the eq. of the circle.

Since; it passes through

x + y =0 is the line which passes through (f, -f)

QUESTION: 73

Consider a circle passing through the origin and the points (a, b) and (-b, -a).

Q. What is the sum of the squares of the intercepts cut off by the circle on the axes?

Solution:

The two intercepts are : -2g & -2f

∵ from eq (l) &( 2) we get;

is sum of squares of intercepts

QUESTION: 74

Let be two unit vectors and 0 be the angle between them.

Q. What is cos equal to?

Solution:

QUESTION: 75

Let be two unit vectors and 0 be the angle between them.

What is sin equal to?

Solution:

QUESTION: 76

Consider the following statements:

1. There exists

2. sin^{-1 }

Q. Which of the above statements is/are correct ?

Solution:

Hence, statement (2) is correct.

QUESTION: 77

Consider the following statements:

1.

2. There exist x ,y ∈[-l, 1], where x ≠ y such that

Which of the above statements is/are correct ?

Solution:

Statement-1

Statement (1) is wrong.

Statement 2,

Only when x = y

Here x ≠ y .

Statement (2) is also wrong.

QUESTION: 78

What are the order and degree respectively of the differential equation whose solution is y = cx + c^{2} - 3c^{3/2} + 2, where c is a parameter?

Solution:

Given:

Solution of differential equation is

y - cx + c^{2 }- 3c^{3/2} + 2 ...........(1)

To find order and degree of differential equation, we will find differential equation first.

Now differentiating equation (1) w.r.t. x and putting value of c to remove it, we get

Hence order of differential equation is 1 and degree is 4.

QUESTION: 79

What is

equal to, where [•] is the greatest integer function?

Solution:

QUESTION: 80

If then what is equal to?

Solution:

QUESTION: 81

If , where a ≠ 0 then what is equal to ?

Solution:

QUESTION: 82

What is equal to ?

Solution:

QUESTION: 83

If A is a square matrix, then what is adj(A^{-1}) - (adj A)^{-1} equal to?

Solution:

QUESTION: 84

What is the binary equivalent ofthe decimal number 0.3125?

Solution:

QUESTION: 85

Let R be a relation on the set N of natural numbers defined by 'nRM n is a factor of m'. Then which one of the following is correct?

Solution:

QUESTION: 86

What is equal to ?

Solution:

QUESTION: 87

What is the number of natural numbers less than or equal to 1000 which are neither divisible by 10 nor 15 nor 25?

Solution:

Let A, B & C be the sets of numbers divisible by 10,15 & 25 respectively

No. divisible by 10 = 100 = n(A)

No. divisible by 15 = 66 = n (B)

No. divisible by 25 = 40 = n (C)

No. divisible by (10 & 15) = 33 = n(A B)

No. divisible by (15 & 25) = 13 = n (B C)

No. divisible by (25 & 10) = 20 = n (A C)

No. divisible by (10,15 & 2 5 ) = 6 = n ( A B C )

No. divisible by 10,15 and 25 = n ( A B C )

= 100 + 66 + 4 0 -3 3 -1 3 -2 0 + 6=146

Thus, no. which are neither divisible by 10 nor 15 nor

25=1000-146 = 854

QUESTION: 88

(a, 2b) is the mid-point of the line segment joining the points

(10, -6) and (k, 4). If a - 2b = 7, then what is the value of k?

Solution:

M = mid point of line segment PQ

Put the values of a & ab in eq (1), we get

QUESTION: 89

Consider the following statements:

1. If ABC is an equilateral triangle, then 3tan( A+B) tan C =1.

2. If ABC is a triangle in which A= 78°, B =66°, then

3. If ABC is any triangle, then

Q. Which of the above statements is/are correct?

Solution:

∵ ABC is an equilateral triangle.

∴A = B = C=60°

L.H.S. = 3 tan (A + B) tan C

= 3 tan 120° tan 60°

= 3(-√3)(√3)

=-9 ≠ 1

Hence statement (1) is incorrect.

**Statement-2**

ABC is a triangle such that A=78° and B = 66°

C = 180 - (78 + 66) = 180 -144 = 36°

Hence statement (2) is correct.

**Statement (3)
**

We can see that statement (3) is not correct. Hence only 2nd statement is correct.

QUESTION: 90

if and then what is equal to ?

Solution:

QUESTION: 91

What is the mean deviation from the mean of the numbers 10,9,21,16,24?

Solution:

Given numbers-10,9,21,16,24

QUESTION: 92

Three dice are thrown simultaneously. What is the probability that the sum on the three faces is at least 5?

Solution:

As we know that 3 dice are thrown. We want prob. of sum on three faces at least 5 i.e. some may be 5 or more. We will find prob. of sum on three faces not 5 or less, i.e. sum on faces is 3 and 4 (1,2 is not possible because of 3 dice).

No. of ways for sum on faces not 5 or more = 4

[(1, 1, 1) , ( 1, 2, 1) , ( 1, 1, 2),(2, 1, 1)]

Total out comes = 216

Prob. of not 5 or more

Prob. of sum on three faces at least 5

QUESTION: 93

Two independent events A and B have P(A) and P(B) What is the probability that exactly one of the two events A or B occurs?

Solution:

A and B are independent.

We want to find probability that exactly one of the two events^ or B occurs i.e. when^4 occurs B does not and vice-versa.

Lets take desired prob. is P.

QUESTION: 94

A coin is tossed three times. What is the probability of getting head and tail alternately?

Solution:

Coin is tossed three tim es i.e. total outcomes = 2^{3} = 8 [(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H),

(T, H, T), (T, T, H), (T, T, T)]

Alternate head and tail are coming two times only.

Thus prob. of getting head and tail alternately

QUESTION: 95

If the total number of observations is 20 and then what is the variance of the distribution?

Solution:

Total no. of observation (n) = 20

QUESTION: 96

A card is drawn from a wel-shuffled deck of 52 cards. What is the probability that it is queen of spade?

Solution:

Prob. of getting queen of spade

QUESTION: 97

If two dice are thrown, then what is the probability that the sum on the two faces is greater than or equal to 4?

Solution:

Since two dice are thrown so number of outcomes are 36.

No. of ways when sum on two faces less than 4 = 3.

[(1, 1), ( 1, 2), (2, 1)]

Hence prob of getting sum on two faces less than 4

Thus required prob. that sum on the two faces is greater

than or equal to 4

QUESTION: 98

A certain type of missile hits the target with probability p = 0.3. What is the least number of missiles should be fired so that there is at least an 80% probability that the target is hit?

Solution:

Probability of hittiy the forget = 0.3

If'n' is the no. of times that the Missile is fired.

Probability of hitting at least once = 1-[1-0.3]^{n} = 0.8

0.7^{n}=0.2

n log 0.7 = log 0.2

⇒n=4.512

for n = 4 ; p < 0 . 8

taken = 5

QUESTION: 99

For two mutually exclusive events A and B, P(A) = 0.2 and = 0.3. What is equal to?

Solution:

Events A and B are mutually exclusive.

QUESTION: 100

What is the probability of 5 Sundays in the month of December?

Solution:

In month of December 31 days i.e. (28 + 3) days.

In 28 days will get 4 Sundays.

If we get any Sunday in first 3 days of December than only we can get 5 Sundays in month.

n (5th Sunday) = 3 [4 weeks + 3 days]

n(5) = 7

Hence prob. of 5 Sundays in month of December =

QUESTION: 101

If m is the geometric mean of

then what is the value of m?

Solution:

Three terms are

QUESTION: 102

A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?

Solution:

Probability that the randomly selected point is at least one inch from the edge of the rectangle

QUESTION: 103

The mean of the series x_{1,}x_{2}, . . . , x_{n} is If x2 is replaced by λ , th en what is the new mean?

Solution:

Mean of series (x_{1}, x_{2}, x_{3}.....x_{n})

Now we will replace x_{2} by λ so no. of elements in series will not change.

New series will include λ and exclude x_{2} Hence new series sum :

QUESTION: 104

For the data

3, 5, 1, 6, 5, 9, 5, 2, 8, 6

the mean, median and mode are x, y and z respectively. Which one of the following is correct?

Solution:

Given data 3,5,1,6,5,9,5,2,8,6 and mean, median and mode are x, y, z respectively.

Rearranging data

1,2,3,5,5,5,6,6,8,9

Mode (z) = most frequently occuring value = 5 Hence x=y=z.

QUESTION: 105

Consider the following statements in respect of a histogram:

1. The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequnecy polygon and the x-axis.

2. When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency.

Q. Which of the above statements is/are correct?

Solution:

Statement (1) is correct because total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and x-axis. Statement (2) is also correct.

QUESTION: 106

A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

Solution:

Let x denote number of tails. Then, X is a binomial variate with parameters:

QUESTION: 107

What is the number of ways in which 3 holiday travel tickets are to be given to 10 employees of an organization, if each employee is eligible for any one or more of the tickets?

Solution:

No. of ways in which 3 holiday travel tickets are to be given to 10 employees = 10^{3} = 1000

QUESTION: 108

If one root of the equation (1 - m) x^{2} +1 x +1 = 0 is double the other and 1 is real, then what is the greatest value of m?

Solution:

Given equation is

Roots are α, β.

∵ One root is double the other.

β = 2α

Sum of roots = α + β

QUESTION: 109

What is the number of four-digit decimal numbers (<1) in which no digit is repeated?

Solution:

Let the given 4 digit decimal number is

Places after decimal can be filled in the following ways:

Total number of ways = 7 x 8 x 9 x 9 = 4536

QUESTION: 110

What is a vector of unit length orthogonal to both the vectors

Solution:

Vector of unit length orthogonal to both the vectors

QUESTION: 111

If are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

Solution:

Position vectors o f vertices A, B and C are

∵ triangle is equilateral.

∴ Centroid and orthocenter will coincide. Centroid = orthocenter position vector

∵ given in question orthocenter is at origin.

Hence

QUESTION: 112

What is the area of the parallelogram having diagonals

Solution:

QUESTION: 113

Consider the following in respect of the matrix

Q. Which of the above is/are correct?

Solution:

QUESTION: 114

Which of the following determinants have value ‘zero’?

Q. Select the correct answer using the code given below,

Solution:

two columns are same so value of determinant is zero.

∵ diagonal is zero so value of determinant is zero.

QUESTION: 115

What is the acute angle between the lines represented by the equations

Solution:

QUESTION: 116

The system of linear equations kx + y + z =1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions?

Solution:

Linear equations

Linear equantion will have unique solution when A^{-1}x exist:

QUESTION: 117

What is the number of different messages that can be represented by three 0’s and two 1 ’s?

Solution:

Number of different messages that can be represented by three 0's and two l's is 10.

Option (a) is correct.

QUESTION: 118

If log_{a}(ab) = x, then what is log_{b}(ab) equal to?

Solution:

QUESTION: 119

then what is equal to?

Solution:

QUESTION: 120

Suppose ω_{1}, and ω_{2} are two distinct cube roots of unity different from 1. Then what equal to?

Solution:

Cube root of unity are

w_{1} and w_{2} are two distinct cube roots of unity different from 1.

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