Question Set D

CBSE 12th Standard Maths Probability HOT Questions Study Material - QB365

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Questions:

Multiple Choice Question

42 x 1 = 42

If \(P(A \cap B)=\frac{1}{8}\) and \(P(\bar{A})=\frac{3}{4} \text {, then } P\left(\frac{B}{A}\right)\) is equal to

(a)

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

(b)

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

(c)

\(\frac{1}{6}\)

(d)

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

If P(A) = 0.3, P(B) = 0.5 and P(A/B) = 0.4, then P(B/A) is

(a)

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

(b)

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

(c)

\(\frac{3}{5}\)

(d)

none of these

A bag contains 5 red, 6 blue and 4 black balls. Three balls are drawn from the bag. Then the probability that none of them is red, is

(a)

\(\frac{24}{91}\)

(b)

\(\frac{2}{91}\)

(c)

\(\frac{6}{35}\)

(d)

none of these

A pair of dice is thrown and it is known that the second dice always exhibits an odd number. Then the probability that the sum obtained on two dice is 7, is

(a)

\(\frac{1}{6}\)

(b)

\(\frac{5}{6}\)

(c)

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

(d)

none of these

If events A and B are independent, p(A) = 0.35, p(A\(\cup\)B) = 0.60 then P(B) is

(a)

0.25

(b)

(c)

0.95

(d)

none of these

A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then probability that both are dead is

(a)

\(\frac{33}{56}\)

(b)

\(\frac{9}{64}\)

(c)

\(\frac{1}{14}\)

(d)

\(\frac{3}{28}\)

The mean of the number obtained on throwing a die having written 1on three faces, 2 on two faces and 5 on one face is

(a)

(b)

(c)

(d)

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

For the following probability distribution.
\(\begin{array}{c|c|c|c|c|c} \hline \mathbf{X} & -4 & -3 & -2 & -1 & 0 \\ \hline \boldsymbol{P}(\boldsymbol{X}) & 0.1 & 0.2 & 0.3 & 0.2 & 0.2 \\ \hline \end{array}\)
E(X) is equal to

(a)

(b)

-1

(c)

-2

(d)

-1.8

Let {E_l, E_2,..., E₃} be a partition of the sample space S and A be any event associated with S then

(a)

\(P(A)=P\left(E_{1}\right) P\left(A / E_{1}\right)+P\left(E_{2}\right) P\left(A / E_{2}\right)+\ldots +P\left(E_{n}\right) P\left(A / E_{n}\right) \)

(b)

\(P(A)=\sum_{1=1}^{n} P\left(E_{j}\right) P\left(A / E_{j}\right)\)

(c)

Both (a) and (b)

(d)

None of these

10.

Two events A and B are said to be independent, if

(a)

A and B are mutually exclusive

(b)

\(P\left(A^{\prime} \cap B^{\prime}\right)=[1-P(A)][1-P(B)]\)

(c)

\(P(A)=P(B)\)

(d)

P(A) + P(B) = 1

11.

By rule of multiplication of probability \(P(E \cap F)\) is equal to

(a)

P(E)· P(F / E)

(b)

P(F)· P(E / F)

(c)

Both (a) and (b)

(d)

None of these

12.

If \(P(A)=0.4, P(B)=0.8 \text { and } P(B / A)=0.6\) then \(P(A \cup B)\) is equal to

(a)

0.24

(b)

0.3

(c)

0.48

(d)

0.96

13.

If \(P(A)=\frac{3}{10}, P(B)=\frac{2}{5} \text { and } P(A \cup B)=\frac{3}{5}\), then

(a)

\(\frac{1}{4}\)

(b)

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

(c)

\(\frac{5}{12}\)

(d)

\(\frac{7}{12}\)

14.

If \(P(\dot{A})=\frac{4}{5} \text { and } P(A \cap B)=\frac{7}{10}\) , then P(B/ A) is

(a)

\(\frac{1}{10}\)

(b)

\(\frac{1}{8}\)

(c)

\(\frac{7}{8}\)

(d)

\(\frac{17}{20}\)

15.

The vector in the direction of vector \(\hat{i}-2 \hat{j}+2 \hat{k}\) that has magnitude 12 is

(a)

\(\hat{i}-2 \hat{j}+2 \hat{k}\)

(b)

\(\frac{\hat{i}-2 \hat{j}+2 \hat{k}}{3}\)

(c)

\(4(\hat{i}-2 \hat{j}+2 \hat{k})\)

(d)

\(9(\hat{i}-2 \hat{j}+2 \hat{k})\)

16.

The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is

(a)

(b)

(c)

(d)

17.

A vector of magnitude 14 units, which is parallel to the \(\widehat { i } +2\widehat { j } -3\widehat { k } \) vector

(a)

\(\frac { (\widehat { i } +2\widehat { j } -3\widehat { k } ) }{ 14 } \)

(b)

\(\frac { (\widehat { i } +2\widehat { j } -3\widehat { k } ) }{ \sqrt{14 } }\)

(c)

\(\sqrt { 14 } (\widehat { i } +2\widehat { j } -3\widehat { k } )\)

(d)

14\((\widehat { i } +2\widehat { j } -3\widehat { k } )\)

18.

The unit vector in the direction of \(\overrightarrow { AB } \), where A and B are the points (2, – 3, 7) and (1, 3, – 4) is:

(a)

\(\frac { -\widehat { i } +6\widehat { j } -11\widehat { k } }{ \sqrt { 158 } } \)

(b)

\(\frac { \widehat { i } +6\widehat { j } +11\widehat { k } }{ \sqrt { 158 } } \)

(c)

\(\frac { \widehat { i } -6\widehat { j } +11\widehat { k } }{ \sqrt { 158 } } \)

(d)

\(\frac { -\widehat { i } -11\widehat { k } }{ \sqrt { 122 } } \)

19.

If \(\overrightarrow { a } =\widehat { i } +2\widehat { j } ,\) and \(\overrightarrow { b } =-2\widehat { i } +\widehat { j } \) and \(\overrightarrow { c } =4\widehat { i } +3\widehat { j } \) and \(\overrightarrow { c } =x\overrightarrow { a } +y\overrightarrow { b } \), then the value of scalars x and y are:

(a)

x = 1 and y = -2

(b)

x = -2 and y = 1

(c)

x = 2 and y = -1

(d)

x = 2 and y = 1

20.

If \(\overrightarrow { a } =2\widehat { i } +3\widehat { j } -6\widehat { k } \) and \(\overrightarrow { b } =6\widehat { i } -2\widehat { j } +3\widehat { k } \), then

(a)

\(|\overrightarrow { a } |=|\overrightarrow { b } |\)

(b)

\(\overrightarrow { a } +\overrightarrow { b } =0\)

(c)

\(\overrightarrow { a } =\overrightarrow { b } \)

(d)

\(2\overrightarrow { a } =\overrightarrow { b } \)

21.

If the magnitude of the position vector \(\overrightarrow { a } =x\widehat { i } +2\widehat { j } -2x\widehat { k } \) is 7, the value of x is:

(a)

±1

(b)

±5

(c)

±3

(d)

±2

22.

If \(\overrightarrow { b } =\lambda \overrightarrow { a } \), the vectors a and b are ______ .

(a)

coinitial

(b)

free vector

(c)

zero vector

(d)

collinear

23.

Two or more vectors having the same initial point are called

(a)

co-terminus vectors

(b)

zero vectors

(c)

co-initial vectors

(d)

unit vectors

24.

For what values of x and y, the vectors \(\overrightarrow { a } =3\widehat { i } +y\widehat { j } -\widehat 3{ k } \) and \(\overrightarrow { b } =2x\widehat { i } +2x\widehat { i } -3\widehat { k } \) are equal?

(a)

\(x=3,y=\frac { 3 }{ 2 } \)

(b)

x = 3, y = 6

(c)

\(x=\frac{3}{2},y=3\)

(d)

x = 6, y = 3

25.

If a, b, c and d are the position vectors of the points A, B, C and D such that a + c = b + d, then ABCD is a

(a)

Trapezium

(b)

Rectangle

(c)

Square

(d)

Parallelogram

26.

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

(a)

10^–1

(b)

\({ \left( \frac { 1 }{ 2 } \right) }^{ 5 }\)

(c)

\({ \left( \frac { 9 }{ 2 } \right) }^{ 5 }\)

(d)

\(\frac { 9 }{ 10 } \)

27.

Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. Then the value of E(X) is

(a)

\(\frac{37}{221}\)

(b)

\(\frac{5}{13}\)

(c)

\(\frac{1}{13}\)

(d)

\(\frac{2}{13}\)

28.

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

(a)

(b)

(c)

(d)

\(\frac83\)

29.

The probability distribution of the discrete variable X is given as:

X	2	3	4	5
P(X)	\(\frac{5}{k}\)	\(\frac{7}{k}\)	\(\frac{9}{5}\)	\(\frac{11}{k}\)

The value of k is

(a)

(b)

(c)

(d)

30.

Three balls are drawn from a bag containing 2 red and 5 black balls, if the random variable X represents the number of red balls drawn, then X can take values

(a)

0, 1, 2

(b)

0, 1, 2, 3

(c)

(d)

1, 2

31.

Two dice are thrown once. If it is known that the sum of the numbers on the dice was less than 6 the probability of getting a sum 3 is

(a)

\(\frac{1}{18}\)

(b)

\(\frac{5}{18}\)

(c)

\(\frac15\)

(d)

\(\frac25\)

32.

Let A and B be two given events such that P(A) = 0.6, P(B) = 0.2 and P(A/B) = 0.5. Then P(A’/B’) is

(a)

\(\frac { 1 }{ 10 } \)

(b)

\(\frac { 3 }{ 10 } \)

(c)

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

(d)

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

33.

If vectors \(\widehat { i } +\widehat { j } +3\widehat { k } \), \(2\widehat { i } +\widehat { j } -\lambda \widehat { k } \), \(5\widehat { i } +2\widehat { j } +3\widehat { k } \) are coplanar, then value or λ is

(a)

(b)

(c)

-3

(d)

34.

If \(\left| \vec { a } \right| =8,\) \(\left| \vec { b } \right| =3\) and \(\left| \vec { a.b } \right| =12\sqrt { 3 } \) then the value of \(\left| \vec { a } \times \vec { b } \right| \) is

(a)

(b)

\(12\sqrt { 3 } \)

(c)

(d)

\(4\sqrt { 3 } \)

35.

The area of a parrallelgram whose one diagonal is \(2\widehat { i } +\widehat { j } -2\widehat { k } \) and one side is \(3\widehat { i } +\widehat { j } -\widehat { k } \) is

(a)

\(\widehat { i } -4\widehat { j } -\widehat { k } \)

(b)

\(3\sqrt { 2 } \) sq unts

(c)

\(6\sqrt { 2 } \) sq units

(d)

6 sq units

36.

If for non zero vectors \(\vec { a } \) and \(\vec { b } \), \(\vec { a } \) x \(\vec { b } \) is a unit vector and |\(\vec { a } \)| = |\(\vec { b } \)| = \(\sqrt2\), then angle θ between vectors \(\vec { a } \) and \(\vec { b } \) is

(a)

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

(b)

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

(c)

\(\frac { \pi }{ 6 } \)

(d)

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

37.

If \(\vec { a } \) and \(\vec { b } \) are unit vectors, then what is the angle between \(\vec { a } \) and \(\vec { b } \) for \(\sqrt { 3 } \vec { a } -\vec { b } \) to be a unit vector?

(a)

30°

(b)

45°

(c)

60°

(d)

90°

38.

If |\(\vec { a } \)| = 4 and -3 ≤ λ ≤ 2 then the range of |λ\(\vec { a } \)| is

(a)

[0, 8]

(b)

[-12, 8]

(c)

[0, 12]

(d)

[8, 12]

39.

A vector in the direction of vector \(\widehat { i } -2\widehat { j } +\widehat { k } \) that has magnitude 15 is

(a)

\(\frac { \widehat { i } -2\widehat { j } +2\widehat { k } }{ 3 } \)

(b)

\(15\widehat { i } -30\widehat { j } +30\widehat { k } \)

(c)

\(\widehat { i } -2\widehat { j } +15\widehat { k } \)

(d)

\(5\widehat { i } -10\widehat { j } +10\widehat { k } \)

40.

The position vector of a point which divides the join of points with position vectors \(\vec { a } +\vec { b } \) and \(2\vec { a } -\vec { b } \) in the ratio 1:2 internally is

(a)

\(\frac { 3\vec { a } +a\vec { b } }{ 3 } \)

(b)

\(\vec { a } \)

(c)

\(\frac { 5\vec { a } -\vec { b } }{ 3 } \)

(d)

\(\frac { 4\vec { a } +\vec { b } }{ 3 } \)

41.

A vector equally inclined to axes is

(a)

\(\widehat { i } +\widehat { j } +\widehat { k } \)

(b)

\(\widehat { i } -\widehat { j } +\widehat { k } \)

(c)

\(\widehat { i } -\widehat { j } -\widehat { k } \)

(d)

\(-\widehat { i } +\widehat { j } -\widehat { k } \)

42.

Let the vectors \(\vec{a} \text { and } \vec{b} \text { be such that }|a| \overrightarrow{=} 3 \text { and } \overrightarrow{|b|}=\frac{\sqrt{2}}{3} \text { then } \vec{a} \times \vec{b}\) is a unit is a vector, if the angle between is:

(a)

\(\frac { \pi }{ 6 } \)

(b)

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

(c)

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

(d)

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

Assertion and reason

10 x 1 = 10

43.

Two coins are tossed once.
Assertion (A) If E : tail appears on one coin and F : one coin shows head, then P(E/F) is 1.
Reason (R) If E : no tail appears and F : no head appears, then P(E/F) is 0.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

44.

Assertion (A) If P(A) = \(\frac{3}{5}\)and P(B)=\(\frac{1}{5}\), then P(A\(\cap\)B), if A and B are independent events, is \(\frac{3}{25}\).
Reason (R) Two cards are drawn at random and without replacement from a pack of 52 playing cards. Then, the probability that both the cards are 25 black, is \(\frac{25}{102}\).
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

45.

Assertion: The projection of the vector a = 2\(\hat{i}+3\hat{j}+2\hat{k}\)on the vector \(\vec{b}=\hat{i}+2\hat{j}+\hat{k}\) is \(\frac{5}{3}\sqrt{6}\)
Reason: The projection of vector a on vector b is \(\frac{1}{|a|}\)(a.b).
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

46.

Assertion: The adjacent sides of a parallelogram are along \(\vec{a}=\hat{i}+2\hat{j}\) and \(\vec{b}=2\hat{i}+\hat{j}\). The angle between the diagonal is 150°.
Reason: Two vectors are perpendicular to each other if their dot product is zero.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

47.

Assertion: If the point \(\vec{p}=(\vec{a}+\vec{b}-\vec{c}),\vec{Q}=(2\vec{a}+\vec{b})\) and \(\vec{R}=(\vec{b}+t\vec{c})\) are collinear, where \(\vec{a},\vec{b},\vec{c}\) are three non-coplanar vectors, then the value of t is -2.
Reason: If P, Q, R are collinear, then \(\overline{PQ}||\overline{PR} or \overline{PQ}=\lambda \overline{PR},\lambda \in R\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

48.

Assertion: \(\bar{a}\) = i + pj + 2k and \(\bar{b}\) = 2i + 3j + qk are parallel vectors if p = \(\frac{3}{2}\), q = 4
Reason: If \(\vec{a}\)= a₁ \(\hat{i}\)+a₂ \(\hat{j}\) + a₃ \(\hat{k}\) and \(\vec{b}\) = b₁\(\hat{i}\)+b₂\(\hat{j}\)+b₃ \(\hat{k}\) are parallel \(\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}=\frac{a_{3}}{b_{3}}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

49.

Assertion: In \(\Delta\)ABC, \(\overline{AB}+\overline{BC}+\overline{CA}=0.\)
Reason: If \(\overline{OA}\) = \(\overline{a}\), \(\overline{OB}\) = \(\overline{b}\), then \(\overline{AB}\) = \(\overline{a}\)+ \(\overline{b}\) (triangle law of addition)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

50.

Consider the following statements
Assertion: Let A and B be two independent events. Then P(A\(\cap\)B) = P(A) + P (B)
Reason: Three events A, B, and C are said to be independent, if P(A\(\cap\)B\(\cap\)C) = P(A)P(B)P(C).
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

51.

Consider the two events E and F which are associated with the sample space of a random experiment.
Assertion: P(E/F)=\(\frac{n(E\cap F)}{n(F)}\)
Reason: P(E/F) = \(\frac{P(E\cap F)}{P(F)}\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

52.

Assertion: Consider the experiment of drawing a card from a deck of 52 playing cards, in which the elementary events are assumed to be equally likely.
If E and F denote the events the card drawn is a spade and the card drawn is an ace respectively. then P(E / F) = \(\frac{1}{4}\)and P(F / E) = \(\frac{1}{13}\)
Reason: E and F are two events such that the probability of occurrence of one of them is not affected by occurrence of the other. Such events are called independent events.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

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Answers Key:

Multiple Choice Question

42 x 1 = 42

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

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

(a)
\(\frac{24}{91}\)

(a)
\(\frac{1}{6}\)

(d)
none of these

(d)
\(\frac{3}{28}\)

(b)
2

(d)
-1.8

10.

(b)
\(P\left(A^{\prime} \cap B^{\prime}\right)=[1-P(A)][1-P(B)]\)

11.

12.

(d)
0.96

13.

(d)
\(\frac{7}{12}\)

14.

15.

16.

(b)
7

17.

18.

(a)
\(\frac { -\widehat { i } +6\widehat { j } -11\widehat { k } }{ \sqrt { 158 } } \)

19.

20.

(a)
\(|\overrightarrow { a } |=|\overrightarrow { b } |\)

21.

22.

(d)
collinear

23.

24.

(b)
x = 3, y = 6

25.

(d)
Parallelogram

26.

27.

(d)
\(\frac{2}{13}\)

28.

(b)
2

29.

As ΣP(X) = 1
\(\Rightarrow \frac { 5 }{ k } +\frac { 7 }{ k } +\frac { 9 }{ k } +\frac { 11 }{ k } =1\)
\(\Rightarrow k=32\)

30.

As there are 2 red balls, so maximum ’ red balls can be 2.

31.

As favourable cases for sum less than 6 are 10 and favourable for a total of 3 is 2.

32.

As, \(P(A/B)=\frac { P(A\cap B) }{ P(B) } \)
\(\Rightarrow P(A\cap B)=0.5\times 0.2=0.1\)
\(P(A'/B')=\frac { P(A'\cap B') }{ P(B') } \)
\(=\frac { 1-P(A\cup B) }{ 1-P(B) } =\frac { 3 }{ 8 } \)

33.

As \(\left| \begin{matrix} 1 & 1 & -3 \\ 2 & 1 & -\lambda \\ 5 & 2 & 3 \end{matrix} \right| =0\)
⇒ 1(3 + 2λ) - 1(6 + 5λ) - 3(-1) = 0
⇒ 3 + 2λ - 6 - 5λ + 3 = 0
⇒ 3λ = 0
⇒ λ = 0

34.

As \({ \left| \vec { a } \times \vec { b } \right| }^{ 2 }+({ \vec { a } .\vec { b } ) }^{ 2 }=|{ \vec { a } | }^{ 2 }|{ \vec { b } | }^{ 2 }\)
\(\Rightarrow { \left| \vec { a } \times \vec { b } \right| }^{ 2 }\) = 64 x 9 - 144 x 3
= 576 - 432 = 144
\(\Rightarrow { \left| \vec { a } \times \vec { b } \right| }^{ 2 }\) = 12

35.

As area of parallelogram
= \(\left| \begin{matrix} \widehat { i } & \widehat { j } & \widehat { k } \\ 2 & 1 & -2 \\ 3 & 1 & -1 \end{matrix} \right| \)
= \(\left| \widehat { i } -4\widehat { j } -\widehat { k } \right| \)
= \(\sqrt { 1+16+1 } \)
= \(3\sqrt { 2 } \) sq units

36.

As sin \(\theta \) = \(\frac { |\vec { a } \times \vec { b } | }{ |\vec { b } ||\vec { b } | } \)
\(=\frac { 1 }{ \sqrt { 2 } .\sqrt { 2 } } =\frac { 1 }{ 2 } \)
\(\Rightarrow \theta =\frac { \pi }{ 6 } \)

37.

As \({ \left| \sqrt { 3 } \vec { a } -\vec { b } \right| }^{ 2 }=({ \sqrt { 3 } \vec { a } -\vec { b } ) }^{ 2 }\)
\(=3\vec { { a }^{ 2 } } +\vec { { b }^{ 2 } } -2\sqrt { 3 } \vec { a } .\vec { b } \)
\(1=3+1-2\sqrt { 3 } \vec { a } .\vec { b } \)
\(\Rightarrow \vec { a } .\vec { b } =\frac { \sqrt { 3 } }{ 2 } \)
\(\therefore cos\theta =\frac { \vec { a } .\vec { b } }{ |\vec { b } ||\vec { b } | } =\frac { \sqrt { 3 } }{ 2 } \)\(\Rightarrow \theta ={ 30 }^{ 0 }\)

38.

As |λ\(\vec { a } \)| = |λ| |\(\vec { a } \)| = 4|λ|
Also -3 ≤ λ ≤ 2 ⇒ 0 ≤ |λ| ≤ 3
⇒ 0 ≤ 4 |λ| ≤ 12

39.

As vector = 15\(\left( \frac { \widehat { i } -2\widehat { j } +2\widehat { k } }{ \sqrt { 1+4+4 } } \right) \)
= \(5\widehat { i } -10\widehat { j } +10\widehat { k } \)

40.

As position vector = \(\frac { 2(\vec { a } +\vec { b } )+1(2\vec { a } -\vec { b } ) }{ 1+2 } \) = \(\frac { 4\vec { a } +\vec { b } }{ 3 } \)

41.

As direction ratios are 1, 1, 1 and direction cosines \(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } } \)
⇒ cos α = cos β = cos γ
⇒ α = β = γ

42.