Subject

Mathematics

Class

JEE Class 12

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 Multiple Choice QuestionsMultiple Choice Questions

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21.

Suppose the cube x3– px + q has three distinct real roots where p > 0 and q > 0. Then which one of the following holds?

  • The cubic has minima at square root of straight p over 3 end root and maxima at –square root of straight p over 3 end root

  • The cubic has minima at –square root of straight p over 3 end root and maxima atsquare root of straight p over 3 end root

  • The cubic has minima at both square root of straight p over 3 end root and- square root of straight p over 3 end root

  • The cubic has minima at both square root of straight p over 3 end root and- square root of straight p over 3 end root


A.

The cubic has minima at square root of straight p over 3 end root and maxima at –square root of straight p over 3 end root

Let f(x) = x3– px + q
Now for maxima/minima
f′(x) = 0
⇒ 3x2– p = 0
⇒ x2
= p/3
therefore space straight x space equals plus-or-minus square root of straight p over 3 end root

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22.

The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is 

  • (x – 2)y′2 = 25 – (y – 2)2 

  • (y – 2)y′2 = 25 – (y – 2)2

  • (y – 2)2y′2= 25 – (y – 2)

  • (y – 2)2y′2= 25 – (y – 2)

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23.

Let A be a square matrix all of whose entries are integers. Then which one of the following is true?

  • If det A = ± 1, then A–1 exists but all its entries are not necessarily integers

  • If detA ≠ ± 1, then A–1 exists and all its entries are non-integers

  • If detA = ± 1, then A–1 exists and all its entries are integers

  • If detA = ± 1, then A–1 exists and all its entries are integers

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24.

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay. Then a2 + b2 + c2 + 2abc is equal to

  • 2

  • -1

  • 0

  • 0

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25.

It is given that the events A and B are such that straight P left parenthesis straight A right parenthesis space equals space 1 fourth comma space straight P space open parentheses straight A over straight B close parentheses space equals space 1 half space and space straight P space open parentheses straight B over straight A close parentheses space equals space 2 over 3 then P(B) is 


  • 1/6

  • 1/3

  • 2/3

  • 2/3

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26.

The vector straight a with rightwards arrow on top space equals space straight alpha space straight i with hat on top space plus 2 straight j with hat on top space plus straight beta straight k with hat on top lies in the plane of the vectors straight b with rightwards arrow on top space equals straight i with hat on top space plus straight j with hat on top space and space straight c with rightwards arrow on top space equals space straight j with hat on top space plus straight k with hat on top and bisects the angle between straight b with rightwards arrow on top space and space straight c with rightwards arrow on top.Then which one of the following gives possible values of α and β?

  • α = 2, β = 2

  • α = 1, β = 2

  • α = 2, β = 1

  • α = 2, β = 1

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27.

The non−zero verctors straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space and space straight c with rightwards arrow on topare related by straight a with rightwards arrow on top space equals space 8 straight b with rightwards arrow on top space and space straight c with rightwards arrow on top space equals space minus space 7 straight b with rightwards arrow on top.Then the angle between straight a with rightwards arrow on top space and space straight c with rightwards arrow on top is

  • 0

  • π/4

  • π/2

  • π/2

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28.

The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz−plane at the point open parentheses 0 space comma 17 over 2 comma space fraction numerator negative 13 over denominator 2 end fraction close parentheses. Then

  • a = 2, b = 8

  • a = 4, b = 6

  • a = 6, b = 4

  • a = 6, b = 4

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29.

If the straight lines fraction numerator straight x minus 1 over denominator straight k end fraction space equals fraction numerator straight y minus 2 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 3 end fraction space and space fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y minus 3 over denominator straight k end fraction space equals fraction numerator straight z minus 1 over denominator 2 end fraction intersect at a point, then the integer k is equal to

  • -5

  • 2

  • 5

  • 5

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30.

In a shop, there are five types of ice-creams available. A child buys six ice-creams.
Statement -1: The number of different ways the child can buy the six ice-creams is 10C5.
Statement -2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row.

  • Statement −1 is false, Statement −2 is true

  • Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

  • Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

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