Mathematics Test

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Mathematics Test - 4

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Question 1
1 / -0.25

The vector equation of the plane in scalar product form is then α = ...

A
A. 2

B
B. 3

C
C. 1

D
D. 0

Solution

Given, equation of the plane

…(i)

Here, plane (i) passing through a (let) and parallel to vector b (let) =

We know that equation of plane passing through a point a and parallel to non-parallel vectors b and c

r·(b x c) = a ·(b x c) = [a b c]

Now,

= 2(0) - 0 + 1(2 - 0) = 2

Therefore α = 2
Question 2
1 / -0.25

The length of the latusrectum of an ellipse is 18/5 and eccentricity is 4/5, then the equation of the ellipse is ...

A
A.

B
B.

C
C.

D
D.

Solution

Given,

2b² /a = 18/5 ⇒ b² /a = 9/5

⇒

⇒ a = 5

∴ Required equation of ellipse,
Question 3
1 / -0.25

“If two triangles are congruent, then their areas are equal" is the given statement then the contrapositive of, toe inverse of the given statement is

A
A. If areas of two triangles are not equal then they are congruent

B
B. If two triangles are not congruent then their areas are equal

C
C. If two triangles are not congruent then their areas are not equal

D
D. If areas of two triangles are equal then they are congruent

Solution

Key Idea: Use p → g then inverse of ft is ~ p → q and contrapositive is ~ q → p.

The inverse of the gi\ren statement: "If two triangles are not congruent, then their areas are not equal”

∴ The contrapositive of the inverse of the given statement: If areas of two triangles are equal, then they are congruent.
Question 4
1 / -0.25

A
A.

B
B.

C
C.

D
D.

Solution

We have,
Question 5
1 / -0.25

The edge of a cube is decreasing at the rate of 0.04 cm/sec. If the edge of the cube is 10 ems, then the rate of decrease of surface area of the cube is...

A
A. 4.8 cm² /sec

B
B. 4.08 cm² /sec

C
C. 48 cm² /sec

D
D. 4.008 cm² /sec

Solution

Let odge of a cube be x cm, then surface area of the cube, A = 6x²

It is given that, dx/dt = -0.04 cm/sec

Now,

= 12x(-0.04)

= -0.48x

when, x = 10, then dA/dt = -0.48 x 10 = -4.8 cm² /sec
Question 6
1 / -0.25

r is the radius of the spherical balloon at time t the surface area of balloon changes at a constant rate K, then......

A
A.

B
B. 8π² = Kt + c

C
C.

D
D. 4π² = kt + c

Solution

According to the question,

(∵ surface area of spherical ballon with radius r is 4π² )

On Integrating both sides, we get

8π∫r dr = K ∫dt

⇒ 4π² = Kt + c
Question 7
1 / -0.25

If ω is a complex cube root of unity and

A
A.

B
B.

C
C.

D
D.

Solution

Given, ω is a complex cube root of unit

∴ ω³ = 1

Given a matrix A can be written as A = IA.

On apply R₁ → ω² R₁ , R₂ → ωR₂ , we get
Question 8
1 / -0.25

= Ax + B log |sin x - cos x| + c then A + B = …...

A
A. 1

B
B. -1

C
C. 0

D
D. -2

Solution

Let sin x - cos x = t

⇒ (cos x + sinx)dx = dt

Here,A=½ , B=½ so, A+B=½+½=1
Question 9
1 / -0.25

a and b are non-collinear vectors. If c = (x - 2) a + b and d = (2x + 1)a - b are collinear vectors, then the value of x = .......

A
A. 1/2

B
B. 1/4

C
C. 1/5

D
D. 1/3

Solution

Given, c = (x - 2)a + b and d = (2x + 1)a - b are collinear

∴ c = λd

⇒ (x - 2) a + b = λ[(2x + 1)a - b]

⇒ (x -2)/(2x + 1) = 1/-1 = λ

⇒ 2x + 1 = -x + 2

⇒ 3x = 1

⇒ x = 1/3
Question 10
1 / -0.25

The slope of normal to the curve x = √t and at t = 4 is ...

A
A. -17/4

B
B. 4/17

C
C. -4/17

D
D. 17/4

Solution

Key Idea: Firstly find then Use the slope of normal

The slope of normal at
Question 11
1 / -0.25

The acute angle between lines x - 3 = 0 and x + y = 19 is...

A
60 º

B
30 º

C
90 º

D
45 º

Solution

Given lines are

x - 3 = 0 ...(i)

Slope of line (i), m₁ = ∞⇒tan θ₁ = ∞⇒θ₁ = 90 ºand x + y = 19

⇒y = - x + 19 ...(ii)

Slope of line (ii), m₂ = tan θ₂ = -1

⇒θ₂ = 135 °

∴Acute angle between given lines is 45 °.
Question 12
1 / -0.25

If the sum of the slopes of the given by x² - 4pxy + 8y² = 0 is three times their product then p = .......

A
3/4

B
1/4

C
4

D
3

Solution

Given, pair of lines, x² - 4pxy + 8y² = 0

Which is in the form of ax² + 2hxy + by² = 0

According to the question,

Sum of the slopes = 3 x product of the slopes

⇒m₁ + m₂ = 3 x (m₁ m₂ )

⇒

⇒4p = 3 ⇒p = 3/4
Question 13
1 / -0.25

The order of the differential equation of all circles which lie in the first quadrant and touch both the axes is...

A
A. two

B
B. three

C
C. one

D
D. four

Solution

Equation of the circle touching both axes is x² + y² - 2ax - 2ay + a² = 0

Here, the number of arbitrary constants is one that is equal to the order of its differential equation.
Question 14
1 / -0.25

The probability that three cards drawn from a pack of 52 cards, all are red is

A
A. 1/17

B
B. 4/17

C
C. 3/17

D
D. 2/17

Solution

Total number of ways that three cards drawn from a pack of 52 cards are ²⁶ C₃ and the number of ways that three red cards drawn are ²⁶ C₃ .

= 2/17
Question 15
1 / -0.25

The domain of the real valued function

A
A. (2,3]

B
B. [2,3)

C
C. (2,3)

D
D. [2,3]

Solution

We have,

Clearly, f(x)will be defined when

3 - x > 0 ⇒ x

and x - 2 ≥ 0 ⇒ x ≥ 2

∴ Domain of f(x) is [2 3).
Question 16
1 / -0.25

lf z = ax + by ; a, b >0 subject to x ≤2, y ≤2, x + y ≥3, x ≥0, y ≥0 has minimum value at (2 , 1) only, then ......

A
a >b

B
a = b

C
a

D
a = 1 + b

Solution

We have, z = ax + by, a, b >0

Subject to constraints x ≤2, y ≤2, x + y ≥3, x , y ≥0

On taking given constraints as equations, we get the following graph

Here. ABCA is the required feasible region whose comer points are A (2, 1), B(1, 2), and C(2, 2).

Since, It is given that z = ax + by ; a , b >0 has minimum value at (2, 1).

∴Value of z at (2, 1)

⇒2a + b

⇒x >b