Classes of Mathematics

Date : 02-11-2023 STD 12 Science Mathematics Total Marks : 30

* Choose The Right Answer From The Given Options.[1 Marks Each] [30]
1. Choose the correct answer from the given four options.
^ ^ ^
The vectors from origin to the points A and B are a⃗ = 2 i − 3 j + 2k and
⃗ ^ ^ ^
b = 2 i + 3 j + k, respectively, then the area of the triangle OAB is:
a. 340
−−
b. √25
−−−
c. √229
−−−
d.
1
√229
2

2. If A(6, 3, 2), B(5, 1, 4), C(3, −4, 7), D(0, 2, 5) are four points, then projection
of CD on AB is:
a.
13
−
7

b.
13

c.
−

−
7

13
11
d.
7
21
−
13

3. If a⃗, b⃗ are the vectors forming consecutive sides of a regular hexagon ABCDEF, then
34

the vector representing side CD is,

a. a⃗ + b
⃗
90

b. a⃗ − b
⃗

c. ⃗
b − a⃗
99

d. ⃗
−( a⃗ + b)

4. −
−→
find the coordinate of the tip of the position vector which is equivalent to AB where the
coordinates of A and B are (-1, 3) and (-2, 1) respectively:
a. (+1, +2)
b. (+1, -2)
c. (-1, +2)
d. (-1, -2)
5. If a⃗ and b⃗ are two unit vectors inclined at an angle θ , such that ∣∣a⃗ + b⃗∣∣ < 1, then:
a.
π
θ <
3

b.
2π
θ >
3

c.
π 2π
< θ <
3 3

d.
2π
< θ < π
3

– –
6. If in a
△ABC, A = (0, 0), B = (3, 3√3), C = (−3√3, 3) , then the vecctor of
–
magnitude 2√2 units directed along AO, where O is the circumcenter of △ABC is,
– –
a. ^ ^
(1 − √3) i + (1 + √3) j

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 – –
b. ^ ^
(1 + √3) i + (1 − √3) j
– –
c. ^ ^
(1 + √3) i + (√3 − 1) j

d. None of these

7. → →
If a is vector of magnitude x, m is non-zero scalar and m a is a unit vector then x in
terms of m is:
a. m = x

b. x =∣ m∣

c.
1
x =
∣m∣

d.
m
x =
2

8. If a⃗ ^ ^ ^ ⃗ ^ ^ ^
= 2 i − 3 j + 5k, b = 3 i − 4 j + 5k and c ⃗ ^ ^ ^
= 5 i − 3 j − 2k, then the volume of
the parallelopiped with contermious edges a⃗ + b⃗, b⃗ + c,⃗ c ⃗ + a⃗ is:
a. 2
b. 1
c. -1
d. 0
9. If a⃗ be the position vector whose tip is (5, -3) find the coordinates of a point B such that
→
AB = a⃗
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the coordinates of A being (4, -1):
a. (9, -4)
21
b. (-9, -4)
c. (9, 4)
d.
34

None of these

10. The position vectors of P and Q are respectively a and b.If R is a point on PQ, PQ such
that PR = 5PQ, then the position vector of R is:
90

a. 5b − 4a
b. 5b + 4a
c. 4b − 5a
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d. 4b + 5a

11. Which of the following represents coinitial vector:

a. c, d
b. m, b
c. b, d
d. Both (a) and (b)

12. If u, v, w are non-coplanar vector and p, q are real numbers, then the equality
[3u pv pw] - [pv w qw] - [2w qv qu] = 0 holds for:
a. Exactly two values of (p, q)
b. More than but not all values of (p, q)
c. All values of (p, q)
d. Exactly one values of (p, q)

13. Which of the following is not a vector quantity:

a. Speed
b. Density
c. Force
d. Velocity

14.

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 Let a⃗ and b⃗ be two unit vectors and a be the angle between them. Then, a⃗ + b⃗ is a unit
vector if:
a. a =
π

b.
π
a =
3

c.
2π
a =
3

d.
π
a =
2

15. Choose the correct answer from the given four options.
The vectors λ^ ^ ^
i + j + 2k,
^ ^ ^
i + λj − k and 2^ ^ ^
i − j + λk are coplanar if:

a. λ = −2

b. λ = 0

c. λ = 1

d. λ = −1

16. If θ is the angle between any two vectors a⃗ and b⃗, then ∣∣a⃗. b⃗∣∣ = ∣
⃗
when θ is equal
∣a⃗ × b∣ ∣
to:
a. 0

b.
π

c.
4
π

2
11
d. π
21
17. −
−→ −
−→
Forces 3OA , 5OB act along OA and OB. If their resultant passes through C on AB,
then,
34

a. C is a mid-point of AB.
b. C divides AB in the ratio 2 : 1
c. 3AC = 5CB
90

d. 2AC = 3CB

18. If a⃗, b⃗, c ⃗ and d⃗ are the position vector of points A, B, C, D such that no three of them are
99

⃗ ⃗
collinear and a⃗ + c ⃗ = b + d, then ABCD is a,
a. Rhombus.
b. Rectangle.
c. Square.
d. Parallelogram.

19. −
−→ −→ −→
If ABCDEF is a regular hexagon, then AD + EB + FC equals,
−
−→
a. 2AB

b. 0
⃗

−
−→
c. 3AB

−
−→
d. 4AB

20. Which of the below given is a vector quantity:

a. 8kg
b. 4 seconds
c. 6 Newton

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 d. 90cm3

21. For any three vectors a⃗, b⃗, c ⃗ the expression ( a⃗ − b⃗). {( b⃗ − c)
⃗ × ( c ⃗ − a⃗)} equals:

a. ⃗ ⃗
[a⃗bc]

b. ⃗
2[a⃗bc]⃗

c. ⃗ ⃗
[a⃗bc]

d. None of these
22. Which of the following holds true for a vector quantity:
a. It has only magnitude
b. It has only direction
c. A vector has both direction and magnitude
d. A vector can never be negative

23. Choose the correct answer from the given four options.
The number of vectors of unit length perpendicular to the vectors
^
^ ^
a⃗ = 2 i + j + 2k and b⃗ ^ ^
= j +k is:
a. One.
b. Two.
c.
d.
Three.
Infinite.
11
21
24. The position vectors of the points A, B, C are ^ ^ ^ ^ ^ ^
2 i + j − k, 3 i − 2 j + k and
^ ^ ^
i + 4 j − 3k respectively. These points,
34

a. Form an isosceles triangle.

b. Form a right triangle.
c. Are collinear.
90

d. Form a scalene triangle.

25. ^ ^ ^ ^ ^ ^ ^ ^ ^
The value of i ⋅ ( j × k) + j ⋅ ( i × k) + k ⋅ ( i × j ) is
99

A. 0
B. -1
C. 1
D. 3

26. If ∣∣a⃗ × b⃗∣∣ = 4, ∣

⃗ 2
then |a⃗| ⃗
2
∣a⃗. b∣ ∣ = 2, ∣
∣b∣ ∣ =

a. 6
b. 2
c. 20
d. 8

27. Let a⃗, b⃗, c ⃗ be three unit vectors, such that ∣∣a⃗ + b⃗ + c∣∣⃗ = 1 and a⃗ is perpendicular to
⃗
b. If c ⃗ makes angle α and β with a⃗ and b⃗ respectively, then cos α + cos β =

a.
3
−
2

b.
3

c. 1

d. −1

28. If ∣a × b ∣= 4 and ∣a.b ∣= 2 then ∣a 2

∣ ∣ b∣
2
is equal to:

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 a. 4
b. 6
c. 20
d. 2

29. Choose the correct answer from the given four options.
Assume that in a family, each child is equally likely to be a boy or a girl. A family with
three children is chosen at random. The probability that the eldest child is a girl given
that the family has at least one girl is:
a. 1

b.
1

c.
2

d. 4

30. If θ is the angle between the vectors 2^ ^ ^ ^ ^ ^

i − 2 j + 4k and 3 i + j + 2k, then sin θ =

a.
2

b. 2

√7

√2
c.

d. √
7
−
−
2

7
11
21
----- -----
34
90
99

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