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Engineering
139 Views   |

Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of AB having 3 or more elements is :

• Option 1)

211

• Option 2)

256

• Option 3)

220

• Option 4)

219

n(A) = 4, n(B) = 2

$n(A\times B)=8$

Number of sbsets having atlest 3 elements

$=2^{8}-\left(1+^{8}C_{1}+^{8}C_{2} \right )=219$

Option 1)

211

Incorrect

Option 2)

256

Incorrect

Option 3)

220

Incorrect

Option 4)

219

Correct

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Engineering
440 Views   |

The length of a focal chord of the parabola y2 = 4ax at a distance b from the vertex is C. Then

• Option 1)

2a2 = bc

• Option 2)

a3 = b2c

• Option 3)

ac = b2

• Option 4)

b2c = 4a3

Find intersections A(x1, y1), B(x2, y2) of P & L Eliminate y from (1) & (2): m2x2 - 2am x + m2 a2 = 4ax  m2 x2-2a(m + 2) x + m2 a2 = 0                     ..............(3) x1, x2 are the roots, x1 + x2 = 2a (m+2)/m2 ;; x1 x2 = a2         .........(4) Eliminate x from (1) & (2): y = m(y2/4a) - ma my2 - 4a y-4a2 m = 0                        ..........(5) y1, y2 are the roots, y1 + y2 = 4a/m   ...
Engineering
136 Views   |

The inverse of the function   $y=\left [ 1-\left ( x-3 \right )^{4} \right ]^{\frac{1}{7}}$  is

• Option 1)

$3+\left ( 1-x^{7} \right )^{\frac{1}{4}}$

• Option 2)

$3-\left ( 1+x^{7} \right )^{\frac{1}{4}}$

• Option 3)

$3-\left ( 1-x^{7} \right )^{\frac{1}{4}}$

• Option 4)

None of these

As we learnt in  Co - Domain of function - All possible outcomes for the function f(x) is known as co - domain unless not specified in question. -     Range of function - All possible values of  domain   is known as Range  -     Option 1) Option is correct Option 2) Option is incorrect Option 3) Option is incorrect Option 4) None of these Option is incorrect
Engineering
118 Views   |

The radius of the circle passing through the points (1,2) (5,2)and (5,-2) is

• Option 1)

$5\sqrt{2}$

• Option 2)

$2\sqrt{5}$

• Option 3)

$3\sqrt{2}$

• Option 4)

$2\sqrt{2}$

As we learnt in Equation of a circle - - wherein Circle with centre and radius .    On substituting the  value of  (x, y) as (1, 2), we get              ....................(1)  On substituting the  value of  (x, y) as (5, 2), we get         .........................(2) On substituting the  value of  (x, y) as (5, -2), we get        .......................(3) On subtracting  Eq. (1) - (2), we...
Engineering
104 Views   |

The radius of the circle passing through the points (1,2),(5,2) and (5,-2) is

• Option 1)

$5\sqrt{2}$

• Option 2)

$2\sqrt{5}$

• Option 3)

$3\sqrt{2}$

• Option 4)

$2\sqrt{2}$

As we learnt in

Equation of a circle -

$\left ( x-h \right )^{2}+\left ( y-k \right )^{2}= r^{2}$

- wherein

Circle with centre $\left ( h,k \right )$ and radius $r$.

On substituting the  value of  (x, y) as (1, 2), we get

$(1-h)^{2}+(2-k)^{2}=r^{2}$            ....................(1)

On substituting the  value of  (x, y) as (5, 2), we get

$(5-h)^{2}+(2-k)^{2}=r^{2}$        .........................(2)

On substituting the  value of  (x, y) as (5, -2), we get

$(5-h)^{2}+(-2-k)^{2}=r^{2}$       .......................(3)

On subtracting  Eq. (1) - (2), we get

h = 3

On subtacting Eq. (2) - Eq. (3), we get

k = 0

$\therefore r = 2\sqrt{2}$

On

Option 1)

$5\sqrt{2}$

Incorrect

Option 2)

$2\sqrt{5}$

Incorrect

Option 3)

$3\sqrt{2}$

Incorrect

Option 4)

$2\sqrt{2}$

Correct

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Engineering
102 Views   |

$\lim_{x \to b}\frac{\sqrt{x-a}-\sqrt{b-a}}{x^{2}-b^{2}}$

• Option 1)

$\frac{1}{4b\sqrt{a-b}}$

• Option 2)

$\frac{1}{4b\sqrt{b-a}}$

• Option 3)

$\frac{1}{4a\sqrt{a-b}}$

• Option 4)

$\frac{1}{b\sqrt{b-a}}$

As we learnt in Method of Rationalisation - Rationalisation method is used when we have RADICAL SIGNS in an expression.(like  1/2,  1/3 etc) and there exists a negative sign between two terms of an algebraic expression. - wherein                 Option 1) Incorrect Option 2) Correct Option 3) Incorrect Option 4) Incorrect
Engineering
116 Views   |

In how many differnt ways can 3 different rings be worn in 5 fingers of a hand?

• Option 1)

3! x 5!

• Option 2)

35

• Option 3)

210

• Option 4)

180

As learnt in Number of Permutations without repetition - Arrange n objects taken r at a time equivalent to filling r places from n things.   - wherein Where      First ring can be worn in 7 ways Second in 6 ways Third in 5 ways Option 1) 3! x 5! This option is incorrect. Option 2) 35 This option is incorrect. Option 3) 210 This option is correct. Option 4) 180 This option is incorrect.
Engineering
115 Views   |

if a, b and c from G.P with common ration r ,the sum of the y cordinates of the points of intersection of the line ax+by+c=0 and the curve $x+2y^{2}=0$ is

• Option 1)

$-\frac{r}{4}$

• Option 2)

$-\frac{r}{2}$

• Option 3)

$\frac{r}{2}$

• Option 4)

$\frac{r}{4}$

As learnt in

General term of a GP -

$T_{n}= ar^{n-1}$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

And,

$ax+by+c=0$

$b=ar, c=ar^{2}$

$x+ry+r^{2}=0$

Also,

$x+2y^{2}=0$

$2y^{2}-ry-r^{2}=0$

$Sum= \frac{r}{2}$

Option 1)

$-\frac{r}{4}$

This option is incorrect.

Option 2)

$-\frac{r}{2}$

This option is incorrect.

Option 3)

$\frac{r}{2}$

This option is correct.

Option 4)

$\frac{r}{4}$

This option is incorrect.

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Engineering
95 Views   |

$If \begin{vmatrix} y+z & x & x \\ y & z+x &y \\ z & z & x+y\end{vmatrix} =K(xyz)$

Then K is equal to

• Option 1)

4

• Option 2)

-4

• Option 3)

0

• Option 4)

None

As learnt in concept Value of determinants of order 3 - -       Option 1) 4 This option is correct. Option 2) -4 This option is incorrect. Option 3) 0 This option is incorrect. Option 4) None This option is incorrect.
Engineering
118 Views   |

With the help of matrices , the solution of the equations  $3x+y+2z=3,\, \, 2x-3y-z=-3\, \, x+2y+z=4, is\, \, given\, \, by$

• Option 1)

x=1, y=2,z=-1

• Option 2)

x=-1,y=2,z=1

• Option 3)

x=6,y=-2,z=-1

• Option 4)

x=-1,y=-2,z=1

As learnt in Cramer's rule for solving system of linear equations - When   and  , then  the system of equations has infinite solutions. - wherein and   are obtained by replacing column 1,2,3 of  by   column     We should solve such questions analytically by cross-checking options because of the time constraint. , satisfy all these equations Option 1) x=1, y=2,z=-1 This option is...
Engineering
146 Views   |

what is the 2015th term of a sequence of natural numbers written in accending order, that does not have any perfect squares.

• Option 1)

2058

• Option 2)

2059

• Option 3)

2060

• Option 4)

2062

As learnt in Sequence - Arragement of real numbers specified in a definite order, by some assigned law. - wherein Notations - or     We get,   Thus total terms are   i.e., 2060 Option 1) 2058 This option is incorrect. Option 2) 2059 This option is incorrect. Option 3) 2060 This option is correct. Option 4) 2062 This option is incorrect.
Engineering
187 Views   |

Three pairs of dice are rolled simultaneously. In how many different ways can the sum of the numbers appearing on the top faces of the 3 dices be 9?

• Option 1)

18

• Option 2)

25

• Option 3)

22

• Option 4)

27

As learnt in concept Sum Rule of Association - Let A can occurs in m ways and B can occurs in n ways and both cannot occur simultaneously. Then A or B can occurs in (m + n) ways. - wherein A or B means at least one of them.     Sum=9 can be written in several ways. (i)    6+2+1    6 ways (ii)    4+3+2    6 ways (iii)    3+3+3    1 way (iv)    2+2+5    3 ways (v)    1+3+5    6 ways (vi)   ...
Engineering
122 Views   |

There are two parallel lines with 6 points on one line and 8 points on the other. How many quadrilaterals can be formed by joining points on the two lines?

• Option 1)

210

• Option 2)

205

• Option 3)

418

• Option 4)

420

as learnt in Rule of Geometrical Permutations - There are n points in a plane such that no three of them are in the same straight line then the number of lines that can be formed by joining is/are  and number of triangle is/are . -     To form a quadrilateral we need two points from each line.  We get,  Option 1) 210 This option is incorrect. Option 2) 205 This option is incorrect. Option...
Engineering
108 Views   |

There are three papers of 100 marks each in an examination. Then the number of ways a student gets 150 marks such that he gets at least 60% in two papers.

• Option 1)

3C2 x 32C2

• Option 2)

4C3 x 32C2

• Option 3)

4Cx 36C2

• Option 4)

4Cx 32C3

As learnt in Theorem of Combinations - The number of combinations of n distinct objects taken r at a time when any object may be repeated any number of times is . - wherein Coefficient of  in .     150 marks Thus,  We get    solutions and the two papers can be chosen in  ways.   Option 1) 3C2 x 32C2 This option is correct. Option 2) 4C3 x 32C2 This option is incorrect. Option 3) 4C3 x...
Engineering
102 Views   |

The value of a for which one of the roots of $x^{2}-3x+2a=0$ is double of one of the roots of $x^{2}-x+a=0$ is

• Option 1)

0, 2

• Option 2)

0, -2

• Option 3)

2, -2

• Option 4)

None of these

As learnt in Roots of Quadratic Equation with real Coefficients - are roots if is satisfied by   - wherein       and,  Let first root be  , second root = Thus  i.e.,  and,  Thus,  Option 1) 0, 2 This option is incorrect. Option 2) 0, -2 This option is correct. Option 3) 2, -2 This option is incorrect. Option 4) None of these This option is incorrect.
Engineering
109 Views   |

The value of $b$ for which the sum of the squares of the roots of the equation $x^{2}-(b-2)x-b-1=0$ assumes the least value is

• Option 1)

0

• Option 2)

1

• Option 3)

2

• Option 4)

3

As learnt in concept Roots of Quadratic Equation with real Coefficients - are roots if is satisfied by   - wherein      And,   Sum of Roots in Quadratic Equation - - wherein are root of quadratic equation    And,   Product of Roots in Quadratic Equation - - wherein are roots of quadratic equation:     Minimum value at b=1 Option 1) 0 This option is incorrect. Option...
Engineering
112 Views   |

The value of $m, \: log_{e}(m-3)<1$ is

• Option 1)

(0,3)

• Option 2)

(0,e)

• Option 3)

(0,e+3)

• Option 4)

(3,3+e)

As learnt in concept Logarithmic Function - - wherein     Also,  Thus,    Option 1) (0,3) This option is incorrect. Option 2) (0,e) This option is incorrect. Option 3) (0,e+3) This option is incorrect. Option 4) (3,3+e) This option is correct.
Engineering
112 Views   |

The value of $\lambda\, \, and \, \, \mu$  for which the system of equations $x+y+z=6$, $x+2y+3z=10$  and  $x+2y+\lambda z=\mu$  have no solution are

• Option 1)

$\lambda = 3,\mu =10$

• Option 2)

$\lambda = 3,\mu \ne 10$

• Option 3)

$\lambda \neq 3,\mu =10$

• Option 4)

$\lambda \neq 3,\mu \ne 10$

As learnt in Inconsistent system of linear equation - If the system of equations has no solutions -   For no solution    Also,   Option 1) This option is incorrect. Option 2) This option is correct. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
81 Views   |

The total number of ways of selecting six coins out of 20 one rupee coins, 10 fifty paise coins and 7 twenty-five paise coins is:

• Option 1)

28

• Option 2)

56

• Option 3)

37C6

• Option 4)

None of these

As learnt in Theorem of Combination - Each of the different groups or selection which can be made by taking r things from n things is called a combination. - wherein Where     Total coins=37 Number of selections=37C6   Option 1) 28 This option is incorrect. Option 2) 56 This option is incorrect. Option 3) 37C6 This option is correct. Option 4) None of these This option is incorrect.
Engineering
384 Views   |

The total number of ways of arranging the letters AAAA BBB CC DE F in a row such that letter C are separated from one another is:

• Option 1)

2772000

• Option 2)

1386000

• Option 3)

4158000

• Option 4)

None of these

As learnt in concept Conditions for Circular Permutation - The number of circular permutations of n distinct things = (n - 1)! -    AAAABBBCCDEF Ways= Total - Two C's are together Option 1) 2772000 This option is incorrect. Option 2) 1386000 This option is correct. Option 3) 4158000 This option is incorrect. Option 4) None of these This option is incorrect.
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