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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.
Engineering
113 Views   |

The system of linear equation  $x+y+z=2,\, \, 2x+y-z=3, \, \, 3x+2y+kz=4$

has a unique solution if

• Option 1)

$K\neq0$

• Option 2)

$-1

• Option 3)

$-2

• Option 4)

$K= 0$

As learnt in Solution of a system of equations -  satisfy the system of linear equations   - wherein    For unique solution  Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
125 Views   |

The system of equation$kx+y+z=1,x+ky+z=k$  and $x+y+kz=k^{^{2}}$ have resolution if k equals

• Option 1)

0

• Option 2)

1

• Option 3)

-1

• Option 4)

-2

As learnt in Solution of a system of equations -  satisfy the system of linear equations   - wherein     Determinant is    on solving,  For      It has a solution when  and  are all zero for      Option 1) 0 This option is incorrect. Option 2) 1 This option is incorrect. Option 3) -1 This option is incorrect. Option 4) -2 This option is correct.
Engineering
105 Views   |

The solution set of $\frac{a^{2} -3a+4}{a+1} > 1, a \epsilon R$ is

• Option 1)

$\left ( 3, \infty \right )$

• Option 2)

$\left ( -1, 1 \right ) U \left ( 3, \infty \right )$

• Option 3)

$\left [ -1, 1 \right ] U \left [ 3, \infty \right )$

• Option 4)

None.

As learnt in concept Range - The range of the relation R is the set of all second elements of the ordered pairs in a relation R. - wherein eg. R={(a,b),(c,d)}. Then Range is {b,d}   and wavy curves,    Option 1) This option is incorrect. Option 2) This option is correct. Option 3) This option is incorrect. Option 4) None.  This option is incorrect.
Engineering
193 Views   |

The sum of an infinite geometric series of real number is 14, and the sum of the cube of the terms of this series is 392 then the first term of the series is

• Option 1)

-14

• Option 2)

10

• Option 3)

7

• Option 4)

-5

As learnt in Sum of infinite terms of a GP - - wherein first term common ratio       Also if series is    Divide  and  For      Option 1) -14 This option is incorrect. Option 2) 10 This option is incorrect. Option 3) 7 This option is correct. Option 4) -5 This option is incorrect.
Engineering
108 Views   |

The root of the question is?

$\begin{bmatrix} x &1&2 \end{bmatrix}$$\begin{bmatrix} 0 &1&1 \\1&0&1\\ 1&1&0\end{bmatrix}$$\begin{bmatrix} x \\-1\\1 \end{bmatrix}= 0$

• Option 1)

$\frac{1}{3}$

• Option 2)

$\frac{-1}{3}$

• Option 3)

0

• Option 4)

1

As learnt in

Multiplication of matrices -

-

$\begin{bmatrix} x & 1 &2 \end{bmatrix} \begin{bmatrix} 0 &1 &1 \\ 1 &0 & 1\\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x\\-1 \\1 \end{bmatrix}=0$

$\begin{bmatrix} 3 & x+2 & x+1 \end{bmatrix}\begin{bmatrix} x\\-1 \\1 \end{bmatrix}=0$

$3x-x-2+x+1=0$

$x=\frac{1}{3}$

Option 1)

$\frac{1}{3}$

This option is correct.

Option 2)

$\frac{-1}{3}$

This option is incorrect.

Option 3)

0

This option is incorrect.

Option 4)

1

This option is incorrect.

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Engineering
114 Views   |

The real part of  $[1+\cos \left( \frac{\pi}{5} \right )+i \sin \left( \frac{\pi}{5} \right )]^{-1}$   is:

• Option 1)

1

• Option 2)

$\frac{1}{2}$

• Option 3)

$\frac{1}{2} \cos \left ( \frac{\pi}{10} \right )$

• Option 4)

$\frac{1}{2} \cos \left ( \frac{\pi}{5} \right )$

As learnt in

Definition of Argument/Amplitude of z in Complex Numbers -

$\theta =tan^{-1}|\frac{y}{x}|, z\neq 0$

$\boldsymbol{\theta,\pi-\theta,-\pi+\theta,-\theta}$ are Principal Argument if z lies in first, second, third or fourth quadrant respectively.

- wherein

$\left[ 1+\cos \frac {\pi}{5}+ i \sin \frac{\pi}{5} \right ]^{-1}$

$\left[ 2\cos^{2} \frac {\pi}{10}+ 2i \sin \frac {\pi}{10} \cos \frac {\pi}{10} \right ]^{-1}$

$\frac{1}{2 \cos \frac{\pi}{10}}\times \left[ \cos \frac{\pi}{10} - i \sin \frac {\pi}{10} \right ]$

Option 1)

1

This option is incorrect.

Option 2)

$\frac{1}{2}$

This option is correct.

Option 3)

$\frac{1}{2} \cos \left ( \frac{\pi}{10} \right )$

This option is incorrect.

Option 4)

$\frac{1}{2} \cos \left ( \frac{\pi}{5} \right )$

This option is incorrect.

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Engineering
116 Views   |

The Range of the function

$f \left ( m \right ) = n^{2}+\frac{1}{n^{2}+1}$

• Option 1)

$\left [ 1,\infty \right )$

• Option 2)

$\left [ 2,\infty \right )$

• Option 3)

$\left [ \frac{3}{2},\infty \right )$

• Option 4)

None

as learnt in Relation between AM, GM and HM of two positive numbers - - wherein Inequality of the three given means.      We will use    Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) None This option is incorrect.
Engineering
135 Views   |

The quadratic equation $x^{2}+bx+c=0$ with real coefficients $b$ and $c$, has a complex root$\sqrt 3 - i$ , then which of the following represents the value of $c-ib$ ?

• Option 1)

$4-2\sqrt3 i$

• Option 2)

$4+2\sqrt3 i$

• Option 3)

$2\sqrt3 +4i$

• Option 4)

$2\sqrt3 -4i$

As learnt in To form a Quadratic Equation given the roots - - wherein S = Sum of roots P = Product of roots         ;     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
119 Views   |

The order of $\begin{bmatrix} x&y&z \end{bmatrix} \begin{bmatrix} a&h&g\\ h&b&f\\g&f&c\end{bmatrix} \begin{bmatrix} x \\ y\\z\end{bmatrix}\, \, \, \, is$

• Option 1)

$3\times 1$

• Option 2)

$1\times1$

• Option 3)

$7\times3$

• Option 4)

$3\times3$

As learnt in concept Multiplication of matrices - -    Order, is  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
88 Views   |

The order of $\begin{bmatrix} x&y&z \end{bmatrix} \begin{bmatrix} a&h&g\\ h&b&f\\g&f&c\end{bmatrix} \begin{bmatrix} x \\ y\\z\end{bmatrix}$

• Option 1)

$3\times 1$

• Option 2)

$1\times 1$

• Option 3)

$7\times 3$

• Option 4)

$3\times 3$

As learnt in concept Multiplication of matrices - -    Order is = Option 1) This is incorrect option Option 2) This is correct option Option 3) This is incorrect option Option 4) This is incorrect option
Engineering
137 Views   |

The number of ways in which 12 balls can be divided between two friends, one receiving 8 and the other 4, is:

• Option 1)

$\frac{12!}{8!4!}$

• Option 2)

$\frac{12!2!}{8!4!}$

• Option 3)

$\frac{12!}{8!4!2!}$

• Option 4)

None of these

As learnt in concept Rule for Division into Groups - The number of ways in which (m + n) different things can be divided into two groups which contain m and n things respectively is . -    grouping ways are now they can be done in any way So, Option 1) This is incorrect option Option 2) This is correct option Option 3) This is incorrect option Option 4) None of these This is incorrect option
Engineering
114 Views   |

The number of terms in the expansion of $(a+b+c)^{n}$ is

• Option 1)

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

• Option 2)

$n+3$

• Option 3)

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

• Option 4)

None of these

As learnt in concept 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 .     The general term is Sum of powers =n x+y+z=n Thus no of solutions = Option 1) This is correct option Option 2) This is incorrect option Option 3) This is incorrect option Option 4) None...
Engineering
622 Views   |

The number of selections of four letters from the letters of the word ASSASSINATION is

• Option 1)

72

• Option 2)

71

• Option 3)

66

• Option 4)

52

As learnt in concept 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      The Number of ways of Arrangement of objects - The number of ways of n different objects taken all at a time  - wherein Where 0! = 1    Case I All letters different Case II Only 2 letters are same Case III 2...
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