Mathematics Test 133

Result

Mathematics Test 133

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Question 1
4 / -1

If then a² + b² is

A
3

B
9

C
16

D
√8

Solution

Taking magnitude and squaring on both sides, we get
Question 2
4 / -1

If g(x) satisfies the conditions of Rolle’s theorem in [1, 2] and g′(x) = f(x), then is equal to

A
1

B
3

C
0

D
2

Solution

As g(x) satisfies the condition of Rolle’s theorem in [1, 2], g(x) is continuous in the interval and g(1) = g(2)
Question 3
4 / -1

If sin then x is equal to

A
1

B
0

C
6/7

D
1/7

Solution
Question 4
4 / -1

The equation of the plane through the line of intersection of planes x + y + z + 3 = 0 and 2x − y + 3z + 1 =0 and parallel to the line is

A
x − 5y + 3z = 7

B
x − 5y + 3z = − 7

C
x + 5y + 3z = 7

D
none of these

Solution

Any plane through the given line is 2x − y + 3z + 1 + λ (x + y + z + 3) = 0 If this plane is parallel to the line then the normal to the plane is also perpendicular to the above line.
∴(l₁l₂ + m₁ m₂ + n₁ n₂ = 0
⇒ (2 + λ) 1 + (λ −1) 2 + (3 + λ) 3 = 0
⇒ λ= -3/2
and the required plane is
2x − y + 3z + 1 + (-3/2) (x + y + z + 3) = 0
⇒ x − 5y + 3z − 7 = 0
or
x − 5y + 3z = 7
Question 5
4 / -1

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is

A
69760

B
30240

C
99784

D
none of these

Solution

The total number of words that can be formed is 10⁵ and number of these words in which no letters are repeated is ¹⁰P₅.
Hence the required number
= 10⁵ − ¹⁰P₅
= 100000 − 10 × 9 × 8 × 7 × 6
= 69760
Question 6
4 / -1

Which of the following is logically equivalent to ∼ (∼p ⇒ q) ?

A
p ∨ q

B
p ∨ ∼ q

C
∼p ∨ q

D
∼p ∧ ∼q

Solution

∼(p ⇒ q) ≡ p ∧ ∼q
∴ ∼(∼p ⇒ q) ≡ ∼p ∧ ∼q
Question 7
4 / -1

If is equal to

A

B

C

D
None of these

Solution
Question 8
4 / -1

If

then f(2x) − f(x) is not divisible by

A
x

B
a

C
2a + 3x

D
x²

Solution

Doing R₃→ R₃−xR₂ and
R₂→R₂− xR₁
we get

So,
f(2x) − f(x)
= a[(a + 2x)² − (a + x)²]
= a (a + 2x − a − x) (a + 2x + a + x)
= ax (2a + 3x)
So, f(2x) - f(x)=ax(2a+3x)
Thus, f(2x) -f(x) is divisible by a, x and (2a+3x).
Question 9
4 / -1

Consider the matrix then B¹⁰²⁷ equals

A

B

C

D

Solution
Question 10
4 / -1

Two vertical poles AL and BM of heights 20 m and 80 m respectively stand apart on a horizontal plane. If A and B be the feet of the poles and AM and BL intersect at P, then the height of P is equal to

A
50 m

B
18 m

C
16 m

D
15 m

Solution