Mathematics Test 124

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Mathematics Test 124

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Weekly Quiz Competition

Question 1
4 / -1

How many terms of the series :  + ..... amounts to  ?

A
8

B
9

C
6

D
10

Solution

Here, the first term a is  and the common ratio r is  .

=  ⇒  ⇒
Question 2
4 / -1

If a,b,c are in harmonic progression, then are in -

A
A.P.

B
G.P.

C
H.P.

D
None of these

Solution
Question 3
4 / -1

The complete set of values of m, for which function ƒ(x) = cos2x + ax + 7 is monotonic, is

A
(–∞, –2] ∪ [2, ∞)

B
[–2, 2]

C
[–1, 1]

D
(–∞, –1] ∪ [1, ∞)

Solution

ƒ'(x) = –2sin2x + a
for ƒ(x) to be monotonic
ƒ'(x) ≥ 0 ∀ x or ƒ'(x) ≤ 0 ∀ x
⇒ m ∈ (–∞, –2] ∪ [2, ∞)
Question 4
4 / -1

Equation of the normal to the curve y = x² which passes through (6, 3) is given by -

A
x + 4y = 18

B
x + 2y = 12

C
x + y = 9

D
x + 3y = 15

Solution

Let the normal from (6, 3) meet the curve y = x² at (t, t²)
∴ Equation of normal is
y – t² = (x – t)
2ty – 2t³ = t – x ...(i)
(i) passes through (6, 3)
⇒ 6t – 2t³ = t – 6 ⇒ 2t³ – 5t – 6 = 0
⇒ (t – 2) (2t² + 4t + 3) = 0 ⇒ t = 2
⇒ Equation is x + 4y = 18
Question 5
4 / -1

Let ƒ(x) = 3x² and g(x) = 2x³ + 1 are two curves and L is the equation of tangent at the point where ƒ(x) and g(x) touch each other

Which of the following options is/are correct ?

A
Equation of L will be 6x – y = 3

B
Equation of normal to g(x) from point (7, 2) is x + 6y = 19

C
Equation of normal to ƒ(x) from point (7, 2) is x + 6y = 19

D
All are correct

Solution

ƒ(x) = 3x² and g(x) = 2x³ + 1
∴ ƒ(x) = g(x) ⇒ 2x³+ 1 – 3x²
= (x – 1)²(2x + 1) = 0
so at (1,3) curves touches each other
∴ equation of tangent L = 6x – y = 3

and equation of normal at (1, 3) is

x + 6y = 19 which is also passing through (7, 2)
Question 6
4 / -1

Let ƒ(x) = 3x² and g(x) = 2x³ + 1 are two curves and L is the equation of tangent at the point where ƒ(x) and g(x) touch each other

Which of the following option(s) is/are correct ?

A
Angle between ƒ(x) and g(x) at point of intersection is tan^–19/7

B
Angle between ƒ(x) and g(x) at point of intersection is sin^–1.

C
Point of intersection of L and g(x) is (–2, –15)

D
All are correct

Solution
Question 7
4 / -1

A geometric series has first term 'a' and common ratio 'r'. The second term of the series is 4 and the sum upto infinite terms of the series is 25

The values of 'a' can be -

A
5

B
15

C
20

D
option 1 and option 3 are correct

Solution
Question 8
4 / -1

A geometric series has first term 'a' and common ratio 'r'. The second term of the series is 4 and the sum upto infinite terms of the series is 25

The smallest value of n for which S_n exceeds 24 is/are -

A
3

B
2

C
15

D
option 1 and option 3 are correct

Solution

⇒ n > 2 ⇒ n ≥ 3

Question 9

4 / -1

List-I		List-II
(P)	ƒ (x) = x⁵ – 5x⁴ + 5x³ – 10 has local maxima and minima at x = ℓ and x = m respectively then 2ℓ +m is	(1)	4
(Q)	Ladder of length 5m leaning against a wall is being pulled along the ground at 2 cm/s. When the foot of the ladder is 4m away from the wall if the top of the ladder slides down on the wall at 8/λ cm/s, then λ =	(2)	5
(R)	If ƒ (x) = x – 2 sinx, 0 ≤ x ≤ 2π is increasing in the interval [aπ, bπ] then a + b is	(3)	3
(S)	If equation of the tangent to the curve y = –e^–x/2, where it crosses the y-axis is , then p – 2q is	(4)	2

P → 2,Q → 3,R → 1,S → 4

P → 2,Q → 3,R → 4,S → 1

P → 2,Q → 4,R → 3,S → 1

P → 2,Q → 4,R → 1,S → 3

Solution

(A)    ƒ '(x) = 5x4 – 20x³ + 15x² = 5x²
(x² – 4x + 3) = 5x² (x – 1)(x – 3)
for maximum & minimum
ƒ '(x) = 0 ⇒ x = 0, 1, 3
ƒ ''(x) = 20x³ – 60x² + 30x = 10x(2x² – 6x + 3)
ƒ ''(0) = 0  Neither point of minimum nor maximum
ƒ ''(1) = –10 point of maximum
ƒ ''(3) = 90 point of minimum
∴   ℓ = 1 & m = 3
2ℓ + m = 5

Question 10
4 / -1

A
P → 1,Q → 3,R → 1,S → 2

B
P → 3,Q → 1,R → 2,S → 4

C
P → 1,Q → 2,R → 3,S → 4

D
P → 2,Q → 1,R → 3,S → 1

Solution

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Re-Attempt Weekly Quiz Competition

Mathematics Test 124

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Mathematics Test 124

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SHARING IS CARING

Question 1

8

9

6

10

Solution

Question 2

A.P.

G.P.

H.P.

None of these

Solution

Question 3

(–∞, –2] ∪ [2, ∞)

[–2, 2]

[–1, 1]

(–∞, –1] ∪ [1, ∞)

Solution

Question 4

x + 4y = 18

x + 2y = 12

x + y = 9

x + 3y = 15

Solution

Question 5

Equation of L will be 6x – y = 3

Equation of normal to g(x) from point (7, 2) is x + 6y = 19

Equation of normal to ƒ(x) from point (7, 2) is x + 6y = 19

All are correct

Solution

Question 6

Angle between ƒ(x) and g(x) at point of intersection is tan–1 9/7

Angle between ƒ(x) and g(x) at point of intersection is sin–1 .

Point of intersection of L and g(x) is (–2, –15)

All are correct

Solution

Question 7

5

15

20

option 1 and option 3 are correct

Solution

Question 8

3

2

15

option 1 and option 3 are correct

Solution

Question 9

P → 2,Q → 3,R → 1,S → 4

P → 2,Q → 3,R → 4,S → 1

P → 2,Q → 4,R → 3,S → 1

P → 2,Q → 4,R → 1,S → 3

Solution

Question 10

P → 1,Q → 3,R → 1,S → 2

P → 3,Q → 1,R → 2,S → 4

P → 1,Q → 2,R → 3,S → 4

P → 2,Q → 1,R → 3,S → 1

Solution

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Angle between ƒ(x) and g(x) at point of intersection is tan^–19/7

Angle between ƒ(x) and g(x) at point of intersection is sin^–1.