Mathematics Test 139

Result

Mathematics Test 139

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Question 1
4 / -1

Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r is equal to

A

B

C

D

Solution

From figure it is clear that ΔPRQ and ΔRSP are similar
Question 2
4 / -1

If a³ + b⁶ = 2, then the maximum value of the term independent of x in the expansion of (ax^1/3 + bx^-1/6)⁹ (a > 0, b > 0) is

A
42

B
68

C
148

D
84

Solution
Question 3
4 / -1

The co-ordinates of the point which divides the line segment joining the points (5,4, 2) and (−1,−2, 4) in the ratio 2 : 3 externally is

A
(17, 16,−2)

B
(15, 12,−3)

C
(14, 10,−2)

D
(11, 13,−4)

Solution

Let A (5 , 4 , 2) and B (- 1, - 2, 4) be the given points.
Let P (x , y, z) be the point, which divides the line segment [AB] in the ratio - 2 : 3
Question 4
4 / -1

The area of an expanding rectangle is increasing at the rate of 48 cm²/sec. The length of the rectangle is always equal to the square of the breadth. At the instant when the breadth is 4.5 cm, The length is increasing at the rate of

A
7.11

B
15.08

C
6.11

D
5.11

Solution
Question 5
4 / -1

If ⁿP_r= 1680 and ⁿC_r= 70, then 69 n + r! is equal to

A
128

B
256

C
576

D
625

Solution
Question 6
4 / -1

Consider the straight line ax + by = c, where a, b, c ∈ R⁺this line meets the coordinate axes at A and B respectively. If the area of the ΔOAB, O being origin, is always a constant equal to half, then

A
The three numbers a,c,b are in AP

B
The three numbers a,c,b are in GP

C
The three numbers are a,c,b in HP

D
None of these

Solution

Given line is ax + by = c

⇒ This line cut the coordinate axes as points

Equating this area to half, we get,
c² = ab
Therefore, a, c, b are in GP
Question 7
4 / -1

If for a ΔABC, cot A⋅cot B⋅cot C>0 then the triangle is

A
Right angled

B
Acute angled

C
Obtuse angled

D
All these options are possible

Solution

cot A⋅cot B⋅cot C>0⇒cot A>0, cot B>0, cot C > 0

Because two or more of cot A, cot B, cot C cannot be negative at the same time in a triangle, as no two angle could be more than 90 degree.
Question 8
4 / -1

The equation of the smallest circle passing through the intersection of the line x + y = 1 and the circle x²+ y² = 9 is

A
x²+ y²+ x + y − 8 = 0

B
x²+ y²− x − y − 8 = 0

C
x²+ y²− x + y − 8 = 0

D
None of these

Solution

Any circle passing through the points of intersection of the given line and circle has the equation x² + y² - 9 + λ (x + y - 1) = 0. Its centre

The circle is the smallest if center

Putting this value for λ, the equation of the smallest circle is x² + y² - 9 - (x + y - 1) = 0
⇒ x² + y² - x - y - 8 = 0
Question 9
4 / -1

A
n ⋅ n !

B
n(n + 1) !

C
(n + 1) !

D
None of these

Solution
Question 10
4 / -1

be two perpendicular unit vectors such that

A
1

B
√2

C
1/√2

D
√3

Solution