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Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 6 Application of Derivatives with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Application of Derivatives Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 6 Application of Derivatives Objective Questions.

Application of Derivatives Class 12 MCQs Questions with Answers

Students are advised to solve the Application of Derivatives Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Application of Derivatives Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Application of Derivatives Class 12 with answers provided with detailed solutions by looking below.

Question 1.
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:
(a) 10π
(b) 12π
(c) 8π
(d) 11π

Answer

Answer: (b) 12π

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Question 2.
The total revenue received from the sale of x units of a product is given by R (x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is:
(a) 116
(b) 96
(c) 90
(d) 126.

Answer

Answer: (d) 126.

---

Question 3.
The interval in which y = x² e^-x is increasing with respect to x is:
(a) (-∞, ∞)
(b) (-2,0)
(c) (2, ∞)
(d) (0, 2).

Answer

Answer: (d) (0, 2).

---

Question 4.
The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is
(a) 3
(b) \(\frac { 1 }{3}\)
(c) -3
(d) –\(\frac { 1 }{3}\)

Answer

Answer: (d) –\(\frac { 1 }{3}\)

---

Question 5.
The line y = x + 1 is a tangent to the curve y² = 4x at the point:
(a) (1, 2)
(b) (2, 1)
(c) (1, -2)
(d) (-1, 2).

Answer

Answer: (a) (1, 2)

---

Question 6.
If f(x) = 3x² + 15x + 5, then the approximate value of f(3.02) is:
(a) 47.66
(b) 57.66
(c) 67.66
(d) 77.66.

Answer

Answer: (d) 77.66.

---

Question 7.
The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is:
(a) 0.06 x³ m³
(b) 0.6 x³ m³
(c) 0.09 x³m³
(d) 0.9 x³ m³

Answer

Answer: (c) 0.09 x³m³

---

Question 8.
The point on the curve x² = 2y, which is nearest to the point (0, 5), is:
(a) (2 √2, 4)
(b) (2 √2, 0)
(c) (0, 0)
(d) (2, 2).

Answer

Answer: (a) (2 √2, 4)

---

Question 9.
For all real values of x, the minimum value of \(\frac { 1-x+x^2 }{1+x+x^2}\) is
(a) 0
(b) 1
(c) 3
(d) \(\frac { 1 }{3}\)

Answer

Answer: (d) \(\frac { 1 }{3}\)

---

Question 10.
The maximum value of [x (x – 1) + 1]^1/3, 0 ≤ x ≤ 1 is
(a) (\(\frac { 1 }{3}\))^{\(\frac { 1 }{3}\)}
(b) \(\frac { 1 }{2}\)
(c) 1
(d) 0

Answer

Answer: (c) 1

---

Question 11.
A cylindrical tank of radius 10 mis being filled with wheat at the rate of 314 cubic m per minute. Then the depth of the wheat is increasing at the rate of:
(a) 1 m/minute
(b) 0 × 1 m/minute
(c) 1 × 1 m/minute
(d) 0 × 5 m/minute.

Answer

Answer: (a) 1 m/minute

---

Question 12.
The slope of the tangent to the curve x = t² + 3t – 8, y = 2 t² – 2t – 5 at the point (2, -1) is:
(a) \(\frac { 22 }{7}\)
(b) \(\frac { 6 }{7}\)
(c) \(\frac { 7 }{6}\)
(d) \(\frac { -6 }{7}\)

Answer

Answer: (b) \(\frac { 6 }{7}\)

---

Question 13.
The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is:
(a) 1
(b) 2
(c) 3
(d) \(\frac { 1 }{2}\)

Answer

Answer: (a) 1

---

Question 14.
The normal at the point (1, 1) on the curve 2y + x² = 3 is
(a) x + y = 0
(b) x – y = 0
(c) x + y + 1 = 0
(d) x – y + 1 = 0.

Answer

Answer: (b) x – y = 0

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Question 15.
The normal to the curve x² = 4y passing through (2, 1) is:
(a) x + y = 3
(b) x – y = 3
(c) x + y = 1
(d) x – y = 1.

Answer

Answer: (a) x + y = 3

---

Question 16.
The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are
(a) (4, ±\(\frac { 8 }{3}\))
(b) (4, –\(\frac { 8 }{3}\))
(c) (4, ±\(\frac { 3 }{8}\))
(d) (±4, \(\frac { 3 }{8}\))

Answer

Answer: (a) (4, ±\(\frac { 8 }{3}\))

---

Question 17.
The abscissa of the point on the curve 3y = 6x – 5x³, the normal at which passes through origin is:
(a) 1
(b) \(\frac { 1 }{3}\)
(c) 2
(d) \(\frac { 1 }{2}\)

Answer

Answer: (a) 1

---

Question 18.
The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2
(a) touch each other
(b) cut at right angle
(c) cut at an angle \(\frac { π }{3}\)
(d) cut at an angle \(\frac { π }{4}\)

Answer

Answer: (b) cut at right angle

---

Question 19.
The tangent to the curve given by:
x = e^t cos t, y = e^t sm t at t = \(\frac { π }{4}\) makes with x-axis an angle:
(a) 0
(b) \(\frac { π }{4}\)
(c) \(\frac { π }{3}\)
(d) \(\frac { π }{2}\)

Answer

Answer: (d) \(\frac { π }{2}\)

---

Question 20.
The equation of the normal to the curve y = sin x at (0, 0) is
(a) x = 0
(b) y = 0
(c) x + y = 0
(d) x – y = 0.

Answer

Answer: (c) x + y = 0

---

Question 21.
The point on the curve y² = x, where the tangent makes an angle of \(\frac { π }{4}\) with x-axis is:
(a) (\(\frac { 1 }{2}\), \(\frac { 1 }{4}\))
(b) (\(\frac { 1 }{4}\), \(\frac { 1 }{2}\))
(c) (4, 2)
(d) (1, 1).

Answer

Answer: (b) (\(\frac { 1 }{4}\), \(\frac { 1 }{2}\))

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Question 22.
Let f: R → R be a positive increasing function with:
\( \lim _{x \rightarrow \infty}\) \(\frac { f(3x) }{f(x)}\) = 1. Then \( \lim _{x \rightarrow \infty}\) \(\frac { f(2x) }{f(x)}\) =
(a) 1
(b) \(\frac { 2 }{3}\)
(c) \(\frac { 3 }{2}\)
(d) 3.

Answer

Answer: (a) 1
Hint:
Since f(x) is a positive increasing function
∴ 0 < f(x) < f(2x) < f(3x)

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Question 23.
The real number k for which the equation 2x³ + 3x + k = 0 has two distinct real roots in [0,1]:
(a) lies between 2 and 3
(b) lies between -1 and 0
(c) does not exist
(d) lies between 1 and 2.

Answer

Answer: (c) does not exist
Hint:
If 2x³ + 3x + k = 0 has two distinct real roots in [0, 1], then f'(x) will change sign.
But f'(x) = cx² + 3 > 0
Hence, no value of k exists.

---

Question 24.
If f and g are differentiable functions on [0, 1] satisfying f(0) = 2 = g(l), g(0) = 0 and f(1) = 6, then for some c ∈ ] 0, 1 [:
(a) 2f'(c) = 3g'(c)
(b) f'(c) = g'(c)
(c) f'(c) = 2g'(c)
(d) 2f'(c) = g'(c).

Answer

Answer: (c) f'(c) = 2g'(c)
Hint:
Let h(x) = f(x) – 2g(x).
∴ h'(x) =f'(x) – 2g'(x).
Here h(0) = h(1) = 2.
By Rolle’s Theorem, h’ (c) = 0
⇒ f'(c) = 2g'(c).

---

Question 25.
Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is:
(a) 25
(b) 30
(c) 12.5
(d) 10.

Answer

Answer: (a) 25
Hint:
Total length = r + r + rθ = 20

∴ r = 5 gives max-area.
Hence, from (2) maximum area,
A =10(5) – 25 = 25.

---

Question 26.
Let f (x) = x² – \(\frac { 1 }{x^2}\) and g(x) = x – \(\frac { 1 }{x}\), x ∈ R – {-1, 0, 1}. If h(x) = \(\frac { f(x) }{g(x)}\), then the local minimum value of h(x) is:
(a) 3
(b) -3
(c) -2√2
(d) 2√2.

Answer

Answer: (d) 2√2.
Hint:

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Fill in the blanks

Question 1.
Rate of change of the area of a circle with respect to its radius when r = 4 cm is ……………..

Answer

Answer: 8π cm²/m.

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Question 2.
Rate of change of the volume of a ball with respect to its radius is ………………

Answer

Answer: 4πr²

---

Question 3.
The function f(x) = |x| is strictly ……………… in (0, ∞)

Answer

Answer: Increasing.

---

Question 4.
Logarithmic function is strictly ………………. in (0, ∞).

Answer

Answer: Increasing.

---

Question 5.
The value of ‘a’ for which f(x) = sin x – ax + b is decreasing function on R is ……………..

Answer

Answer: ≤ 1.

---

Question 6.
Slope of the tangent to the curve x = at², y = 2t at t = 2 is ……………..

Answer

Answer: \(\frac { 1 }{2}\)

---

Question 7.
If tangent to the curve y² + 3x – 7 = 0 at the point (h, k) is parallel to like x – y – 4, then the value of k is ……………….

Answer

Answer: –\(\frac { 3 }{2}\)

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Question 8.
For the curve y = 5x – 2x³, if x increases at the tangents of 2 units/sec; then at x = 3 the slope of the curve is changes at ……………..

Answer

Answer: decreasing at the rate of 72 units/sec.

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Question 9.
If x > 0, y > 0 and xy = 5, then the minimum value of x + y is ………………

Answer

Answer: 10.

---

Question 10.
Maximum value of:
f(x) = – (x – 1)² + 2 is ………………

Answer

Answer: 2.

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Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 7 Integrals with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Integrals Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 7 Integrals Objective Questions.

Integrals Class 12 MCQs Questions with Answers

Students are advised to solve the Integrals Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Integrals Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Integrals Class 12 with answers provided with detailed solutions by looking below.

Question 1.
The anti-derivative of (√x + \(\frac { 1 }{√x}\)) equals

Answer

Answer: (c) \(\frac { 2 }{3}\) x^{\(\frac { 2 }{3}\)} + 2x^{\(\frac { 1 }{2}\)} + c

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Question 2.
If \(\frac { 1 }{dx}\) (f(x)) = 4x³ – \(\frac { 3 }{x^4}\) such that f(2) = 0 then f(x) is ……………

Answer

Answer: (a) x⁴ + \(\frac { 1 }{x^3}\) – \(\frac { 129 }{8}\)

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Question 3.
∫\(\frac { 10x^9+10^x log_e 10 }{x^{10} + 10^x}\) dx equals
(a) 10^x -x¹⁰ + c
(b) 10^x + x¹⁰ + c
(c) (10^x – x¹⁰)^-1 + c
(d) log (10^x + x¹⁰) + c.

Answer

Answer: (d) log (10^x + x¹⁰) + c.

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Question 4.
∫\(\frac { dx }{sin^2 x cos^2 x}\) equals
(a) tan x + cot x + c
(b) tan x – cot x + c
(c) tan x cot x + c
(d) tan x – cot 2x + c.

Answer

Answer: (b) tan x – cot x + c

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Question 5.
∫\(\frac { sin^2 x – cos ^2 x }{sin^2 x cos^2 x}\) dx is equals to
(a) tan x + cot x + c
(b) tan x + cosec x + c
(c) -tan x + cot x + c
(d) tan x + sec x + c.

Answer

Answer: (a) tan x + cot x + c

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Question 6.
∫\(\frac { e^x(1 + x) }{cos^2(xe^2)}\) dx is equals to
(a) -cot (xe^x) + c
(b) tan (xe^x) + c
(c) tan (e^x) + c
(d) cot (e^x) + c

Answer

Answer: (b) tan (xe^x) + c

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Question 7.
∫\(\frac { dx }{x^2+2x+2}\) equals
(a) x tan^-1 (x + 1) + c
(b) tan^-1 (x + 1) + c
(c) (x + 1) tan^-1 x + c
(d) tan^-1 x + c.

Answer

Answer: (b) tan^-1 (x + 1) + c

---

Question 8.
∫\(\frac { dx }{\sqrt{9-25x^2}}\) equals

Answer

Answer: (b) \(\frac { 1 }{5}\) sin^-1 (\(\frac { 5x }{3}\)) + c

---

Question 9.
∫\(\frac { x dx }{(x-1)(x-2)}\) equals

(d) log |(x – 1) (x – 2)| + c.

Answer

Answer: (b) log |\(\frac { (x-2)^2 }{x-1}\)| + c

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Question 10.
∫\(\frac { dx }{x(x^2+1)}\) equals
(a) log |x| – \(\frac { 1 }{2}\) log (x² + 1) + c
(b) \(\frac { 1 }{2}\) log |x| + \(\frac { 1 }{2}\) log (x² + 1) + c
(c) -log |x| + \(\frac { 1 }{2}\) log (x² + 1) + c
(d) log |x| + log (x² + 1) + c

Answer

Answer: (a) log |x| – \(\frac { 1 }{2}\) log (x² + 1) + c

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Question 11.
∫x² e^x³ dx equals
(a) \(\frac { 1 }{3}\) e^x³ + c
(b) \(\frac { 1 }{3}\) e^x² + c
(c) \(\frac { 1 }{2}\) e^x³ + c
(d) \(\frac { 1 }{2}\) e^x² + c

Answer

Answer: (a) \(\frac { 1 }{3}\) e^x³ + c

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Question 12.
∫e^x sec x (1 + tan x) dx equals
(a) e^x cos x + c
(b) e^x sec x + c
(c) e^x sin x + c
(d) e^x tan x + c.

Answer

Answer: (b) e^x sec x + c

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Question 13.
∫\(\sqrt { 1 + x^2}\) dx is equal to

Answer

Answer:

---

Question 14.
∫\(\sqrt { x^2 – 8x + 7}\) dx is equal to

Answer

Answer:

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Question 15.
\(\int_{1}^{\sqrt{3}}\) \(\frac { dx }{1+x^2}\) equals
(a) \(\frac { π }{3}\)
(b) \(\frac { 2π }{3}\)
(c) \(\frac { π }{6}\)
(d) \(\frac { π }{112}\)

Answer

Answer: (d) \(\frac { π }{112}\)

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Question 16.
\(\int_{1}^{2/3}\) \(\frac { dx }{4+9x^2}\) equals
(a) \(\frac { π }{6}\)
(b) \(\frac { π }{12}\)
(c) \(\frac { π }{24}\)
(d) \(\frac { π }{4}\)

Answer

Answer: (c) \(\frac { π }{24}\)

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Question 17.
The value of the integral \(\int_{1}^{2/3}\) \(\frac { (x-x^3)^{1/3} }{x^4}\) dx is
(a) 6
(b) 0
(c) 3
(d) 4

Answer

Answer: (a) 6

---

Question 18.
If f(x) = \(\int_{0}^{x}\) t sin t dt, then f'(x) is
(a) cos x + x sin x
(b) x sin x
(c) x cos x
(d) sin x + x cos x.

Answer

Answer: (b) x sin x

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Question 19.
The value of
\(\int_{-π/2}^{π/2}\) (x³ + x cos x + tan⁵ x + 1) dx is
(a) 0
(b) 2
(c) π
(d) 1

Answer

Answer: (c) π

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Question 20.
The value of \(\int_{0}^{π/2}\) log (\(\frac { 4+3 sin x }{4+3 cos x}\)) dx is
(a) 2
(b) \(\frac { 3 }{4}\)
(c) 0
(d) -2

Answer

Answer: (c) 0

---

Question 21.
∫\(\frac { dx }{e^x+e{-x}}\) is equal to
(a) tan^-1 (e^x) + c
(b) tan^-1 (e^-x) + c
(c) log (e^x – e^-1) + c
(d) log (e^x + e^-x) + c.

Answer

Answer: (a) tan^-1 (e^x) + c

---

Question 22.
∫\(\frac { cos 2x }{(sin x + cos x)^2}\) dx is equal to
(a) \(\frac { -1 }{sin x + cos x}\) + c
(b) log |sin x + cos x| + c
(c) log |sin x – cos x| + c
(d) \(\frac { 1 }{(sin x + cos x)^2}\) + c

Answer

Answer: (b) log |sin x + cos x| + c

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Question 23.
If f (a + b – x) = f(x), then \(\int_{a}^{b}\) x f(x) dx is equal to

Answer

Answer: (d) \(\frac { a+b }{2}\) \(\int_{a}^{b}\) f(x) dx

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Question 24.
∫e^x(cos x – sin x)dx is equal to
(a) e^x – cos x + c
(b) e^x sin x + c
(c) -e^x cos x + c
(d) -e^x sin x + c.

Answer

Answer: (a) e^x – cos x + c

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Question 25.
∫\(\frac { dx }{sin^2 x cos^2 x}\) is equal to
(a) tan x + cot x + c
(b) (tan x + cot x)² + c
(c) tan x – cot x + c
(d) (tan x – cot x)² + c.

Answer

Answer: (c) tan x – cot x + c

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Question 26.
If ∫ \(\frac { 3e^x-5e^{-x} }{4r^x+5e^{-x}}\) dx = ax + b log |4e^x + 5e^-x| + c then
(a) a = –\(\frac { 1 }{8}\), b = \(\frac { 7 }{8}\)
(b) a = \(\frac { 1 }{8}\), b = \(\frac { 7 }{8}\)
(c) a = \(\frac { -1 }{8}\), b = –\(\frac { 7 }{8}\)
(d) a = \(\frac { 1 }{8}\), b = –\(\frac { 7 }{8\)

Answer

Answer: (a) a = –\(\frac { 1 }{8}\), b = \(\frac { 7 }{8}\)

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Question 27.
∫tan^-1 √x dx is equal to
(a) (x + 1)tan^-1 √x – √x + c
(b) x tan^-1 √x – √x + c
(c) √x – x tan^-1 √x + c
(d) ^-1x – (x + 1) tan^-1 √x + c

Answer

Answer: (a) (x + 1)tan^-1 √x – √x + c

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Question 28.
∫e^x(\(\frac { 1-x }{(1+x^2)}\))² dx is equal to:

Answer

Answer: (c) \(\frac { e^x }{(1+x^2)^2}\) + c

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Question 29.
\(\int_{a+c}^{b+c}\) f(x)dx is equal to :
(a) \(\int_{c}^{b}\) f(x – c)dx
(b) \(\int_{c}^{b}\) f(x + c)dx
(c) \(\int_{c}^{b}\) f(x)dx
(d) \(\int_{a-c}^{b-c}\)

Answer

Answer: (b) \(\int_{c}^{b}\) f(x + c)dx

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Question 30.
\(\int_{-1}^{1}\) \(\frac { x^3+|x|+1 }{x^2+2|x|+1}\) is equal to
(a) log 2
(b) 2 log 2
(c) \(\frac { 1 }{2}\) log 2
(d) 4 log 2

Answer

Answer: (b) 2 log 2

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Question 31.
\(\int_{c}^{b}\) |x cos πx|dx is equal to
(a) \(\frac { 8 }{π}\)
(b) \(\frac { 4 }{π}\)
(c) \(\frac { 2 }{π}\)
(d) \(\frac { 1 }{π}\)

Answer

Answer: (a) \(\frac { 8 }{π}\)

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Question 32.
If \(\int_{0}^{1}\) \(\frac { e^t }{1+t}\) dt = a, then \(\int_{0}^{1}\) \(\frac { e^t }{(1+t)^2}\)
(a) a – 1 + \(\frac { e }{2}\)
(b) a + 1 – \(\frac { e }{2}\)
(c) a – 1 – \(\frac { e }{2}\)
(d) a + 1 + \(\frac { e }{2}\)

Answer

Answer: (b) a + 1 – \(\frac { e }{2}\)

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Question 33.
If x = \(\int_{0}^{y}\) \(\frac { dt }{\sqrt{1+9t^2}}\) and \(\frac { d^y }{dx^2}\) = ay, then a is equal to
(a) 3
(b) 6
(c) 9
(d) 1.

Answer

Answer: (c) 9

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Question 34.
Let I = \(\int_{0}^{1}\) \(\frac { sin x }{√x}\) dx and J = \(\int_{0}^{1}\) \(\frac { cos x }{√x}\) dx. Then which of the following is true?
(a) I > \(\frac { 2 }{3}\) and J < 2
(b) I > \(\frac { 2 }{3}\) and J > 2
(c) I < \(\frac { 2 }{3}\) and J < 2
(d) I < \(\frac { 2 }{3}\) and J > 2.

Answer

Answer: (c) I < \(\frac { 2 }{3}\) and J < 2
Hint:

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Question 35.
\(\int_{0}^{π}\) [cot x]dx, where [ . ] denotes the greatest integer function, is equal to
(a) \(\frac { π }{2}\)
(b) 2
(c) -1
(d) –\(\frac { π }{2}\)

Answer

Answer: (d) –\(\frac { π }{2}\)
Hint:

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Question 36.
Let p (x) be a function defined on R such that p'(x) = p'(1 – x), for all x ∈ [0, 1], p(0) = 1 and p (1) = 41.
Then \(\int_{0}^{1}\) p(x) dx equals
(a) \(\sqrt { 41}\)
(b) 21
(c) 41
(d) 42

Answer

Answer: (b) 21
Hint:
Here p'(x) – p'(1 – x).
Integrating, p (x) = -p (1 – x) + c ………… (1)
At x = 0, p(0) = -p (1) + c
⇒ 1 = -41 + c ⇒ c = 42.
Putting in (1),
p (x) = -p(1 – x) + 42
∴ \(\int_{0}^{1}\) p(x) dx = –\(\int_{0}^{1}\) p(1 – x)dx + \(\int_{0}^{1}\) 42 dx
⇒ 21 = 42[x]\(_{ 0 }^{1}\)
⇒ 21 = 42
⇒ I = 21.

---

Question 37.
Let I_n = ∫tan” x dx, (n > 1).
If I₄ + I₆ = a tan⁵ x + bx⁵ + c, where c is a constant of integration, then the ordered pair (a, b) is equal to
(a) (\(\frac { 1}{5}\), -1)
(b) (-\(\frac { 1}{5}\), 0)
(c) (-\(\frac { 1}{5}\), 1)
(d) (\(\frac { 1}{5}\), 0)

Answer

Answer: (d) (\(\frac { 1}{5}\), 0)
Hint:
Here I₄ + I₆ = a tan⁵ x + bx⁵ + c
⇒ ∫tan⁴x dx + ∫tan⁶ x dx = a tan⁵ x + bx⁵ + c.
Diff. both sides,
tan⁴ x + tan⁶ x = 5a tan⁴ x sec² x + 5bx⁴
= 5 a tan⁴ x(1 + tan² x) + 5 bx⁴
= 5a tan⁴ x + 5a tan⁶x + 5bx⁴.
Comparing, 1 = 5a and 5b = 0
⇒ a = \(\frac { 1 }{5}\) and b = 0.
Hence, (a, b) = (\(\frac { 1}{5}\), 0)

---

Question 38.
The integral

Answer

Answer: (b) \(\frac {-1}{3(1+tan^3 x)}\) + c
Hint:

---

Question 39.
The value of \(\int_{-π/2}^{π/2}\) \(\frac { sin^2 x }{1 + 2^x}\) dx is
(a) \(\frac { π}{8}\)
(b) \(\frac {π}{2}\)
(c) 4π
(d) \(\frac {π}{4}\)

Answer

Answer: (d) \(\frac {π}{4}\)
Hint:

---

Fill in the blanks

Question 1.
∫(√x + \(\frac {1}{√x}\)) dx (x ≠ 0) = ………………

Answer

Answer: \(\frac { 2 }{3}\) x√x + 2√x

---

Question 2.
∫cot x dx = ………………….

Answer

Answer: log |sin x| + c

---

Question 3.
∫sec x dx = ………………

Answer

Answer: log |sec x + tan x| + c

---

Question 4.
∫ \(\frac { sin^2 x – cos^2 x }{sin x cos x}\) dx = ………………..

Answer

Answer: log |sec x| – log |sin x| + c

---

Question 5.
∫ \(\frac { x^3+5x^2+4 }{x^2}\) dx = ………………

Answer

Answer: \(\frac { x^2 }{2}\) + 5x – \(\frac { 4 }{x}\) + c

---

Question 6.
∫tan² x dx ………………..

Answer

Answer: tan x – x+ c

---

Question 7.
∫ \(\sqrt { a^2+x^2}\) dx = ………………..

Answer

Answer: \(\frac{x \sqrt{a^{2}+x^{2}}}{2}\) + \(\frac { a^2 }{2}\) log|x + \(\sqrt { a^2+x^2}\)| + c

---

Question 8.
∫ (2 – x) sin x dx = ……………..

Answer

Answer: -2 cos x + x cos x – sin x + c

---

Question 9.
If ∫ e^x(tan x + 1) sec x dx = e^x f(x) + c, then f(x) = ………………

Answer

Answer: sec x

---

Question 10.
∫ \(\frac { 1 }{9x^2-1}\) dx = …………….

Answer

Answer: \(\frac { 1 }{6}\) log |\(\frac { 3x-1 }{3x+1}\)| + c

---

Question 11.
\(\int_{2}^{3}\) 3^x dx = …………………….

Answer

Answer: \(\frac { 18 }{log 3}\)

---

Question 12.
\(\int_{0}^{1}\) \(\frac { dx }{\sqrt{1+x^2}}\) = ……………..

Answer

Answer: log (1 + √2)

---

Question 13.
If \(\int_{0}^{1}\) (3x² + 2x + k) dx = 0, then the value of ‘k’ ………………..

Answer

Answer: -2

---

Question 14.
If f(x) = \(\int_{0}^{x}\) t sin t dt, then the value of f'(x) = ………………….

Answer

Answer: x sin x

---

Question 15.
\(\int_{0}^{1.5}\) [x] dx = ………………. where [x] is greatest integer function.

Answer

Answer: 0.5

---

We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 7 Integrals with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Integrals MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 8 Application of Integrals with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Application of Integrals Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 8 Application of Integrals Objective Questions.

Application of Integrals Class 12 MCQs Questions with Answers

Students are advised to solve the Application of Integrals Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Application of Integrals Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Application of Integrals Class 12 with answers provided with detailed solutions by looking below.

Question 1.
Area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2 is
(a) π
(b) \(\frac { π }{2}\)
(c) \(\frac { π }{3}\)
(d) \(\frac { π }{4}\)

Answer

Answer: (a) π

---

Question 2.
Area of the region bounded by the curve y² = 4x, y-axis and the line y = 3 is
(a) 2
(b) \(\frac { 9 }{4}\)
(c) \(\frac { 9 }{3}\)
(d) \(\frac { 9 }{2}\)

Answer

Answer: (a) 2

---

Question 3.
Smaller area enclosed by the circle x² + y² = 4 and the line x + y = 2 is
(a) 2 (π – 2)
(b) π – 2
(c) 2π – 1
(d) 2 (π + 2).

Answer

Answer: (b) π – 2

---

Question 4.
Area lying between the curves y² = 4x and y = 2 is:
(a) \(\frac { 2 }{3}\)
(b) \(\frac { 1 }{3}\)
(c) \(\frac { 1 }{4}\)
(d) \(\frac { 3 }{4}\)

Answer

Answer: (b) \(\frac { 1 }{3}\)

---

Question 5.
Area bounded by the curve y = x³, the x-axis and the ordinates x = -2 and x = 1 is
(a) -9
(b) –\(\frac { 15 }{4}\)
(c) \(\frac { 15 }{4}\)
(d) \(\frac { 17 }{4}\)

Answer

Answer: (b) –\(\frac { 15 }{4}\)

---

Question 6.
The area bounded by the curve y = x|x|, x-axis and the ordinates x = -1 and x = 1 is given by
(a) 0
(b) –\(\frac { 1 }{3}\)
(c) \(\frac { 2 }{3}\)
(d) \(\frac { 4 }{3}\)

Answer

Answer: (c) \(\frac { 2 }{3}\)

---

Question 7.
The area of the circle x² + y² = 16 exterior to the parabola y² = 6x is
(a) \(\frac { 4 }{3}\) (4π – √3)
(b) \(\frac { 1 }{3}\) (4π + √3)
(c) \(\frac { 2 }{3}\) (8π – √3)
(d) \(\frac { 4 }{3}\) (8π + √3)

Answer

Answer: (c) \(\frac { 2 }{3}\) (8π – √3)

---

Question 8.
The area enclosed by the circle x² + y² = 2 is equal to
(a) 4π sq. units
(b) 2√2 π sq. units
(c) 4π² sq. units
(d) 2π sq. units.

Answer

Answer: (d) 2π sq. units.

---

Question 9.
The area enclosed by the ellipse \(\frac { x^2 }{a^2}\) + \(\frac { y^2 }{b^2}\) = 1 is equal to
(a) π²ab
(b) πab
(c) πa²b
(d) πab².

Answer

Answer: (b) πab

---

Question 10.
The area of the region bounded by the curve y = x² and the line y = 16 is
(a) \(\frac { 32 }{3}\)
(b) \(\frac { 256 }{3}\)
(c) \(\frac { 64 }{3}\)
(d) \(\frac { 128 }{3}\)

Answer

Answer: (b) \(\frac { 256 }{3}\)

---

Question 11.
The area of the region bounded by the y-axis, y = cos x and y = sin x, 0 ≤ x ≤ \(\frac { π }{2}\) is
(a) √2 sq. units
(b) (√2 + 1) sq. units
(c) (√2 – 1) sq. units
(d) (2√2 – 1) sq. units.

Answer

Answer: (c) (√2 – 1) sq. units

---

Question 12.
The area of the region bounded by the curve x² = 4y and the straight line x = 4y – 2 is
(a) \(\frac { 3 }{8}\) sq. units
(b) \(\frac { 5 }{8}\) sq. units
(c) \(\frac { 7 }{8}\) sq. units
(d) \(\frac { 9 }{8}\) sq. units.

Answer

Answer: (d) \(\frac { 9 }{8}\) sq. units.

---

Question 13.
The area (in sq. units) of the region:
{(x, y) : y² ≥ 2x and x² + y2 ≤ 4x, x ≥ 0, y ≥ 0} is
(a) π – \(\frac { 8 }{3}\)
(b) π – \(\frac { 4√2 }{3}\)
(c) \(\frac { π }{2}\) – \(\frac { 2√2 }{3}\)
(d) π – \(\frac { 4 }{3}\)

Answer

Answer: (a) π – \(\frac { 8 }{3}\)
Hint:
y² = 2x is a parabola,
and x² + y² = 4x
⇒ (x – 2)² + y² = 4.
It is a circle having centre (2, 0) and radius 2 units.
∴ Reqd. area = Shaded region

---

Fill in the Blanks

Question 1.
The area of the quadrant of the circle x² + y² = 4 is …………….

Answer

Answer: π sq. units.

---

Question 2.
The area enclosed by the circle x² + y² = a² is ……………….

Answer

Answer: 2 π a² sq. units.

---

Question 3.
The area of the parabola y² = 4ax bounded by the latus-rectum is ………………

Answer

Answer: \(\frac { 8 }{3}\) a² sq. umts.

---

Question 4.
The area bounded by y = x², x = 0, x = 2 and x-axis is ……………….

Answer

Answer: \(\frac { 8 }{3}\) sq. units.

---

Question 5.
The area under the curve y = 2√x between x = 0 and x = 1 is ……………..

Answer

Answer: \(\frac { 4 }{3}\) sq. units.

---

We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 8 Application of Integrals with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Application of Integrals MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Differential Equations Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 9 Differential Equations Objective Questions.

Differential Equations Class 12 MCQs Questions with Answers

Students are advised to solve the Differential Equations Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Differential Equations Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Differential Equations Class 12 with answers provided with detailed solutions by looking below.

Question 1.
The degree of the differential equation:
(\(\frac { d^2y }{dx^2}\))³ + (\(\frac { dy }{dx}\))² + sin (\(\frac { dy }{dx}\)) + 1 = 0 is
(a) 3
(b) 2
(c) 1
(d) not defined.

Answer

Answer: (a) 3

---

Question 2.
The order of the differential equation:
2x² \(\frac { d^2y }{dx^2}\) – 3 \(\frac { dy }{dx}\) + y = 0 is
(a) 2
(b) 1
(c) 0
(d) not defined.

Answer

Answer: (a) 2

---

Question 3.
The number of arbitrary constants in the general solution of a differential equation of fourth order is:
(a) 0
(b) 2
(c) 3
(d) 4.

Answer

Answer: (d) 4.

---

Question 4.
The number of arbitrary constants in the particular solution of a differential equation of third order is:
(a) 3
(b) 2
(c) 1
(d) 0.

Answer

Answer: (d) 0.

---

Question 5.
Which of the following differential equations has y = c₁ e^x+ c₂ e^-x as the general solution?
(a) \(\frac { d^2y }{dx^2}\) + y = 0
(b) \(\frac { d^2y }{dx^2}\) – y = 0
(c) \(\frac { d^2y }{dx^2}\) + 1 = 0
(d) \(\frac { d^2y }{dx^2}\) – 1 = 0

Answer

Answer: (b) \(\frac { d^2y }{dx^2}\) – y = 0

---

Question 6.
Which of the following differential equations has y = x as one of its particular solutions?
(a) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = x
(b) \(\frac { d^2y }{dx^2}\) + x \(\frac { dy }{dx}\) + xy = x
(c) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = 0
(d) \(\frac { d^2y }{dx^2}\) + x \(\frac { dy }{dx}\) + xy = 0

Answer

Answer: (c) \(\frac { d^2y }{dx^2}\) – x² \(\frac { dy }{dx}\) + xy = 0

---

Question 7.
The general solution of the differential equation \(\frac { dy }{dx}\) = e^x+y is
(a) e^x + e^-y = c
(b) e^x + e^y = c
(c) e^-x + e^y = c
(d) e^-x + e^-y = c.

Answer

Answer: (a) e^x + e^-y = c

---

Question 8.
Which of the following differential equations cannot be solved, using variable separable method?
(a) \(\frac { dy }{dx}\) + e^x+y + e^-x+y
(b) (y² – 2xy) dx = (x² – 2xy) dy
(c) xy \(\frac { dy }{dx}\) = 1 + x + y + xy
(d) \(\frac { dy }{dx}\) + y = 2.

Answer

Answer: (b) (y² – 2xy) dx = (x² – 2xy) dy

---

Question 9.
A homogeneous differential equation of the form \(\frac { dy }{dx}\) = h(\(\frac { x }{y}\)) can be solved by making the substitution.
(a) y = vx
(b) v = yx
(c) x = vy
(d) x = v

Answer

Answer: (c) x = vy

---

Question 10.
Which of the following is a homogeneous differential equation?
(a) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
(b) xy dx – (x³ + y²)dy = Q
(c) (x³ + 2y²) dx + 2xy dy = 0
(d) y² dx + (x² – xy – y²)dy = 0.

Answer

Answer: (d) y² dx + (x² – xy – y²)dy = 0.

---

Question 11.
The integrating factor of the differential equation x\(\frac { dy }{dx}\) – y = 2x² is
(a) e^-x
(b) e^-y
(c) \(\frac { 1 }{x}\)
(d) x

Answer

Answer: (c) \(\frac { 1 }{x}\)

---

Question 12.
The integrating factor of the differential equation
(1 – y²) \(\frac { dy }{dx}\) + yx = ay(-1 < y < 1) is
(a) \(\frac { 1 }{y^2-1}\)
(b) \(\frac { 1 }{\sqrt{y^2-1}}\)
(c) \(\frac { 1 }{1-y^2}\)
(d) \(\frac { 1 }{\sqrt{1-y^2}}\)

Answer

Answer: (d) \(\frac { 1 }{\sqrt{1-y^2}}\)

---

Question 13.
The general solution of the differential equation \(\frac { y dx – x dy }{y}\) = 0 is
(a) xy = c
(b) x = cy²
(c) y = cx
(d) y = cx².

Answer

Answer: (c) y = cx

---

Question 14.
The general solution of a differential equation of the type \(\frac { dy }{dx}\) + P₁ x = Q₁ is:
(a) y e^{∫p₁ dy} = ∫(Q₁ e^{∫p₁ dy}) dy + c
(b) y e^{∫p₁ dx} = ∫(Q₁ e^{∫p₁ dx}) dx + c
(c) x e^{∫p₁ dy} = ∫(Q₁ e^{∫p₁ dy}) dy + c
(d) x e^{∫p₁ dx} = ∫(Q₁ e^{∫p₁ dx}) dx + c

Answer

Answer: (c) x e^{∫p₁ dy} = ∫(Q₁ e^{∫p₁ dy}) dy + c

---

Question 15.
The general solution of the differential equation
e^x dy + (y e^x + 2x) dx = 0 is
(a) x e^x + x² = c
(b) x e^y + y² = c
(c) y e^x + x² = c
(d) y e^x + x² = c.

Answer

Answer: (c) y e^x + x² = c

---

Question 16.
The degree of the differential equation representing the family of curves (x – a)² + y² = 16 is
(a) 0
(b) 2
(c) 3
(d) 1.

Answer

Answer: (d) 1.

---

Question 17.
The degree of the differential equation
\(\frac { d^2y }{dx^2}\) + 3(\(\frac { dy }{dx}\))² = x² log (\(\frac { d^2y }{dx^2}\)) is
(a) 1
(b) 2
(c) 3
(d) not defined

Answer

Answer: (d) not defined

---

Question 18.
The order and degree of the differential equation
[1 + (\(\frac { dy }{dx}\))²]² = \(\frac { d^2y }{dx^2}\)
(a) 1, 2
(b) 2, 2
(c) 2, 1
(d) 4, 2.

Answer

Answer: (c) 2, 1

---

Question 19.
The solution of the differential equation:
2x \(\frac { dy }{dx}\) – y = 3 represents a family of:
(a) straight lines
(b) circles
(c) parabolas
(d) ellipses.

Answer

Answer: (c) parabolas

---

Question 20.
A solution of the differential equation:
(\(\frac { dy }{dx}\))² – x \(\frac { dy }{dx}\) + y = 0 is
(a) y = 2
(b) y = 2x
(c) y = 2x – 4
(d) y = 2x² – 4.

Answer

Answer: (c) y = 2x – 4

---

Question 21.
The solution of the differential equation:
x\(\frac { dy }{dx}\) + 2y = x² is
(a) y = \(\frac { x^2+c }{4x^2}\)
(b) y = \(\frac { x^2 }{4}\) + c
(c) y = y = \(\frac { x^4+c }{x^2}\)
(d) y = y = \(\frac { x^4+c }{4x^2}\)

Answer

Answer: (d) y = y = \(\frac { x^4+c }{4x^2}\)

---

Question 22.
The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is
(a) (x – 2)² y’² = 25 – (y – 2)²
(b) (x – 2) y’² = 25 – (y – 2)²
(c) (y – 2) y’² =25 – (y – 2)².
(d) (y – 2)² y’² = 25 – (y – 2)².

Answer

Answer: (d) (y – 2)² y’² = 25 – (y – 2)².
Hint:
The equation of the circle is
(x – c)² + (y – 2)² = 25 …………… (1)
Diff. w.r.t. x,
2(x – c) + 2(y – 2)y’ = 0
⇒ (x – c) = -(y – 2)y’
Putting in (1),
(y – 2)² y’² + (y – 2)² = 25
(y – 2)² y’² = 25 – (y – 2)².

---

Question 23.
The differential equation which represents the family of curves y = e^c₂x, where c₁ and c₂ are arbitrary constants, is:
(a) y” = y’y
(b) yy” = y’
(c) yy” = (y’)²
(d) y’ = y²

Answer

Answer: (d) y’ = y²
Hint:
We have y = c₁ e^c₂x …………… (1)
Diff. w.r.t. x, y’ = c₁c₂ e^c₂x
⇒ y’ = c₂y ………… (2) [Using(1)]
Again diff. w.r.t. x,
y” = c₂y’ …………… (3)
From (2) and (3),
\(\frac { y” }{y’}\) = \(\frac { y’ }{y}\)
⇒ yy” = (y’)²

---

Question 24.
Solution of the differential equation:
cos x dy = y (sin x – y) dx, 0 < x < \(\frac { π }{2}\) is
(a) sec x = (tan x + c)y
(b) y sec x = tan x + c
(c) y tan x = sec x + c
(d) tan x = (sec x + c)y.

Answer

Answer: (a) sec x = (tan x + c)y
Hint:
Here cos x dy =y (sin x – y)dx
⇒ cos x dy – y sin x dx = – y² dx
⇒ d(y cos x) = – y² dx.
Integrating, ∫\(\frac { d(y cos x) }{y^2 cos^2 x}\) = -∫\(\frac { 1 }{cos^2 x}\) dx
⇒ –\(\frac { 1 }{y cos x}\) = -tan x – c
⇒ sec x = (tan x + c)y.

---

Question 25.
At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of worker x is given by:
\(\frac { dP }{dx}\) = 100 – 12√x
If the firm employs 25 more workers, then the new level of production of items is:
(a) 3000
(b) 3500
(c) 4500
(d) 2500.

Answer

Answer: (b) 3500
Hint:
We have: \(\frac { dP }{dx}\) = 100 – 12√x
Integrating,
\(\int_{2000}^{P}\) dP = \(\int_{0}^{25}\) (100 – 12√x)dx
⇒ [P]\(_{2000}^{P}\) = [100x – 12\(\frac { x^{3/2} }{3/2}\)]\(_{0}^{25}\)
⇒ P – 2000 = 100(25)-8(25)^3/2
⇒ P – 2000 = 2500 – 1000
⇒ P = 3500.

---

Fill in the Blanks

Question 1.
The degree of the differential equation:
x²(\(\frac { d^2y }{dx^2}\))³ + y(\(\frac { dy }{dx}\))⁴ + x³ = 0 is …………….

Answer

Answer: 3.

---

Question 2.
The degree and order of the differential equation:
(\(\frac { ds }{dt}\))⁴ + 3s\(\frac { d^2s }{dt^2}\) is ……………. and ………………..

Answer

Answer: 2, 1

---

Question 3.
Differential equation of the family of lines passing through the origin is …………………

Answer

Answer: \(\frac { dy }{dx}\) = \(\frac { y }{x}\).

---

Question 4.
The differential equation of which y = 2 (x² – 1) + ce^-x is a solution is ……………….

Answer

Answer: \(\frac { dy }{dx}\) + 2xy = 4x³

---

Question 5.
General solution of (x² + 1)\(\frac { dy }{dx}\) = 2 is ………………….

Answer

Answer: y = 2 tan^-1 x + c.

---

Question 6.
Solution of \(\frac { dy }{dx}\) = \(\sqrt { 4 – y^2}\) (- 2 < y < 2) is …………….

Answer

Answer: sin^-1 \(\frac { y }{2}\) = x + c.

---

Question 7.
Solution of \(\frac { dy }{dx}\) = \(\frac { y }{x}\) is ……………….

Answer

Answer: y = cx.

---

Question 8.
The differential equation \(\frac { dy }{dx}\) = \(\frac { x-y }{x+y}\) is ………………. equation.

Answer

Answer: homogeneous.

---

Question 9.
The integrating factor of x log x \(\frac { dy }{dx}\) + y = 2 log x is …………………

Answer

Answer: log x.

---

Question 10.
The integrating factor of (\(\frac { e^{-2√x} }{√x}\) – \(\frac { y }{√x}\))\(\frac { dy }{dx}\) = 1 is …………………

Answer

Answer: e^2√x

---

We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 9 Differential Equations with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Differential Equations MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.

Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 10 Vector Algebra with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Vector Algebra Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 10 Vector Algebra Objective Questions.

Vector Algebra Class 12 MCQs Questions with Answers

Students are advised to solve the Vector Algebra Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Vector Algebra Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Vector Algebra Class 12 with answers provided with detailed solutions by looking below.

Question 1.
In ΔABC, which of the following is not true?

(a) \(\vec { AB}\) + \(\vec { BC}\) + \(\vec { CA}\) = \(\vec { 0}\)
(b) \(\vec { AB}\) + \(\vec { BC}\) – \(\vec { AC}\) = \(\vec { 0}\)
(c) \(\vec { AB}\) + \(\vec { BC}\) – \(\vec { CA}\) = \(\vec { 0}\)
(d) \(\vec { AB}\) – \(\vec { CB}\) + \(\vec { CA}\) = \(\vec { 0}\)

Answer

Answer: (c) \(\vec { AB}\) + \(\vec { BC}\) – \(\vec { CA}\) = \(\vec { 0}\)

---

Question 2.
If \(\vec a\) and \(\vec b\) are two collinear vectors, then which of the following are incorrect:
(a) \(\vec b\) = λ\(\vec a\) tor some scalar λ.
(b) \(\vec a\) = ±\(\vec b\)
(c) the respective components of \(\vec a\) and \(\vec b\) are proportional
(d) both the vectors \(\vec a\) and \(\vec b\) have the same direction, but different magnitudes.

Answer

Answer: (d) both the vectors \(\vec a\) and \(\vec b\) have the same direction, but different magnitudes.

---

Question 3.
If a is a non-zero vector of magnitude ‘a’ and λa non-zero scalar, then λ\(\vec a\) is unit vector if:
(a) λ = 1
(b) λ = -1
(c) a = |λ|
(d) a = \(\frac { 1 }{|λ|}\)

Answer

Answer: (d) a = \(\frac { 1 }{|λ|}\)

---

Question 4.
Let λ be any non-zero scalar. Then for what possible values of x, y and z given below, the vectors 2\(\hat i\) – 3\(\hat j\) + 4\(\hat k\) and x\(\hat i\) – y\(\hat j\) + z\(\hat k\) are perpendicular:
(a) x = 2λ. y = λ, z = λ
(b) x = λ, y = 2λ, z = -λ
(c) x = -λ, y = 2λ, z = λ
(d) x = -λ, y = -2λ, z = λ.

Answer

Answer: (c) x = -λ, y = 2λ, z = λ

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Question 5.
Let the vectors \(\vec a\) and \(\vec b\) be such that |\(\vec a\)| = 3 and |\(\vec b\)| = \(\frac { √2 }{3}\), then \(\vec a\) × \(\vec b\) is a unit vector if the angle between \(\vec a\) and \(\vec b\) is:
(a) \(\frac { π }{6}\)
(b) \(\frac { π }{4}\)
(c) \(\frac { π }{3}\)
(d) \(\frac { π }{2}\)

Answer

Answer: (b) \(\frac { π }{4}\)

---

Question 6.
Area of a rectangle having vertices
A(-\(\hat i\) + \(\frac { 1 }{2}\) \(\hat j\) + 4\(\hat k\)),
B(\(\hat i\) + \(\frac { 1 }{2}\) \(\hat j\) + 4\(\hat k\)),
C(\(\hat i\) – \(\frac { 1 }{2}\) \(\hat j\) + 4\(\hat k\)),
D(-\(\hat i\) – \(\frac { 1 }{2}\) \(\hat j\) + 4\(\hat k\)) is
(a) \(\frac { 1 }{2}\) square unit
(b) 1 square unit
(c) 2 square units
(d) 4 square units.

Answer

Answer: (c) 2 square units

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Question 7.
If θ is the angle between two vectors \(\vec a\), \(\vec b\), then \(\vec a\).\(\vec b\) ≥ 0 only when
(a) 0 < θ < \(\frac { π }{2}\)
(b) 0 ≤ θ ≤ \(\frac { π }{2}\)
(c) 0 < θ < π
(d) 0 ≤ θ ≤ π

Answer

Answer: (b) 0 ≤ θ ≤ \(\frac { π }{2}\)

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Question 8.
Let \(\vec a\) and \(\vec b\) be two unit vectors and 6 is the angle between them. Then \(\vec a\) + \(\vec b\) is a unit vector if:
(a) θ = \(\frac { π }{4}\)
(b) θ = \(\frac { π }{3}\)
(c) θ = \(\frac { π }{2}\)
(d) θ = \(\frac { 2π }{3}\)

Answer

Answer: (d) θ = \(\frac { 2π }{3}\)

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Question 9.
If {\(\hat i\), \(\hat j\), \(\hat k\)} are the usual three perpendicular unit vectors, then the value of:
\(\hat i\).(\(\hat j\) × \(\hat k\)) + \(\hat j\).(\(\hat i\) × \(\hat k\)) + \(\hat k\).(\(\hat i\) × \(\hat j\)) is
(a) 0
(b) -1
(c) 1
(d) 3

Answer

Answer: (d) 3

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Question 10.
If θ is the angle between two vectors \(\vec a\) and \(\vec b\), then |\(\vec a\).\(\vec b\)| = |\(\vec a\) × \(\vec b\)| when θ is equal to:
(a) 0
(b) \(\frac { π }{4}\)
(c) \(\frac { π }{2}\)
(d) π

Answer

Answer: (b) \(\frac { π }{4}\)

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Question 11.
The area of the triangle whose adjacent sides are
\(\vec a\) = 3\(\hat i\) + \(\hat j\) + 4\(\hat k\) and \(\vec b\) = \(\hat i\) – \(\hat j\) + \(\hat k\) is
(a) \(\frac { 1 }{2}\) \(\sqrt{ 42 }\)
(b) 42
(c) \(\sqrt{ 42 }\)
(d) \(\sqrt{ 21 }\)

Answer

Answer: (a) \(\frac { 1 }{2}\) \(\sqrt{ 42 }\)

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Question 12.
The magnitude of the vector 6\(\hat i\) + 2\(\hat j\) + 3\(\hat k\) is
(a) 5
(b) 7
(c) 12
(d) 1.

Answer

Answer: (b) 7

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Question 13.
The vector with initial point P (2, -3, 5) and terminal point Q (3, -4, 7) is
(a) \(\hat i\) – \(\hat j\) + 2\(\hat k\)
(b) 5\(\hat i\) – 7\(\hat j\) + 12\(\hat k\)
(c) –\(\hat i\) + \(\hat j\) – 2\(\hat k\)
(d) None of these.

Answer

Answer: (a) \(\hat i\) – \(\hat j\) + 2\(\hat k\)

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Question 14.
The angle between the vectors \(\hat i\) – \(\hat j\) and \(\hat j\) – \(\hat k\) is
(a) \(\frac { π }{3}\)
(b) \(\frac { 2π }{3}\)
(c) –\(\frac { π }{3}\)
(d) \(\frac { 5π }{6}\)

Answer

Answer: (b) \(\frac { 2π }{3}\)

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Question 15.
The value of ‘λ’ for which the two vectors:
2\(\hat i\) – \(\hat j\) + 2\(\hat k\) and 3\(\hat i\) + λ\(\hat j\) + \(\hat k\) are perpendicular is
(a) 2
(b) 4
(c) 6
(d) 8.

Answer

Answer: (d) 8.

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Question 16.
If |\(\vec a\)| = 8, |\(\vec b\)| = 3 and |\(\vec a\) × \(\vec b\)|= 12, then value of \(\vec a\).\(\vec b\) is
(a) 6√3
(b) 8√3
(c) 12√3
(d) None of these.

Answer

Answer: (c) 12√3

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Question 17.
The non-zero vectors \(\vec a\), \(\vec b\) and \(\vec c\) are related by \(\vec a\) = 8\(\vec b\) and \(\vec c\) = -7\(\vec b\). Then the angle between \(\vec a\) and \(\vec c\) is
(a) π
(b) 0
(c) \(\frac { π }{4}\)
(d) \(\frac { π }{2}\)

Answer

Answer: (a) π
Hint:
\(\vec a\) = 8\(\vec b\) and \(\vec c\) = -7\(\vec b\)
Clearly \(\vec a\) and \(\vec b\) are parallel and \(\vec b\) and \(\vec c\) are anti-parallel.
∴ \(\vec a\) and \(\vec c\) are anti-parallel.
Hence, angle between \(\vec a\) and \(\vec c\) is π.

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Question 18.
If the vectors \(\vec a\) = \(\hat i\) – \(\hat j\) + 2\(\hat k\), \(\vec b\) = 2\(\hat i\) + 4\(\hat j\) + \(\hat k\) and \(\vec c\) = λ\(\hat i\) + \(\hat j\) + µ\(\hat k\) are mutually orthogonal, then (λ, µ) =
(a) (-3, 2)
(b) (2, -3)
(c) (-2, 3)
(d)(3, -2).

Answer

Answer: (a) (-3, 2)
Hint:
\(\vec a\), \(\vec b\) and \(\vec c\) are mutually orthogonal
⇒ \(\vec b\).\(\vec c\) = 0 and \(\vec a\).\(\vec c\) = 0
⇒ (2\(\hat i\) + 4\(\hat j\) + \(\hat k\)). (λ\(\hat i\) + \(\hat j\) + µ\(\hat i\)) = 0
⇒ 2λ + 4 + µ = 0
⇒ 2λ + µ = -4 …………(1)
and (\(\hat i\) – \(\hat j\) + 2\(\hat k\)).(λ\(\hat i\) + \(\hat j\) + µ\(\hat k\)) = 0
⇒ λ – 1 + 2µ = 0
⇒ λ + 2µ = 1 ………….. (2)
Solving (1) and (2),
λ = -3 and µ = 2.

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Question 19.
If (2\(\hat i\) + 6\(\hat j\) + 27\(\hat k\)) × (\(\hat i\) + p\(\hat j\) + q\(\hat k\)) = \(\vec 0\), then the values ofp and q are?
(a) p = 6, q = 27
(b) p = 3, q = \(\frac { 27 }{2}\)
(c) p = 6, q = \(\frac { 27 }{2}\)
(d) p = 3, q = 27.

Answer

Answer: (b) p = 3, q = \(\frac { 27 }{2}\)
Hint:
(2\(\hat i\) + 6\(\hat j\) + 27\(\hat k\)) × (\(\hat i\) + p\(\hat j\) + q\(\hat k\))
\(\left[\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
2 & 6 & 27 \\
1 & p & q
\end{array}\right]\)
By the question,
\(\hat i\) (6q – 27p) –\(\hat j\) (2q – 27) +\(\hat k\) (2p – 6) = \(\vec 0\)
⇒ 6q – 27p = 0 ⇒ 2q – 9p = 0
2q – 27 = 0 ⇒ q = \(\frac { 27 }{2}\)
and 2p – 6 = 0 ⇒ p = 3.
Hence, p = 3 and q = \(\frac { 27 }{2}\).

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Question 20.
If the vectors \(\bar { AB }\) = 3\(\hat i\) + 4\(\hat k\) and \(\bar { AC }\) = 5\(\hat i\) – 2\(\hat j\) + 4\(\hat k\) are the sides ofa triangle ABC, then the length of the median through A is
(a) \(\sqrt {72}\)
(b) \(\sqrt {33}\)
(c) \(\sqrt {45}\)
(d) \(\sqrt {18}\)

Answer

Answer: (b) \(\sqrt {33}\)
Hint:

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Question 21.
If [\(\vec a\) × \(\vec b\) \(\vec b\) × \(\vec c\) \(\vec c\) × \(\vec a\)] = λ [\(\vec a\) \(\vec b\) \(\vec c\)]², then λ is equal to
(a) 3
(b) 0
(c) 1
(d) 2.

Answer

Answer: (c) 1
Hint:
As usual, we will have:
[\(\vec a\) × \(\vec b\) \(\vec b\) × \(\vec c\) \(\vec c\) × \(\vec a\)] = [\(\vec a\) \(\vec b\) \(\vec c\)]²
Given:
[\(\vec a\) × \(\vec b\) \(\vec b\) × \(\vec c\) \(\vec c\) × \(\vec a\)] = [\(\vec a\) \(\vec b\) \(\vec c\)]²
Hence, λ = 1.

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Question 22.
Let \(\vec a\), \(\vec b\) and \(\vec c\) be three unit vectors such that:
\(\vec a\) × (\(\vec b\) × \(\vec c\)) = \(\frac { √3 }{2}\) (\(\vec b\) + \(\vec c\))
If \(\vec b\) is not parallel to \(\vec c\), then the angle between \(\vec a\) and \(\vec b\) is:
(a) \(\frac { π }{2}\)
(b) \(\frac { 2π }{3}\)
(c) \(\frac { 5π }{6}\)
(d) \(\frac { 3π }{4}\)

Answer

Answer: (c) \(\frac { 5π }{6}\)
Hint:

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Fill in the blanks

Question 1.
The magnitude of projection of (2\(\hat i\) – \(\hat j\) + \(\hat k\))
on (\(\hat i\) – 2\(\hat j\) + 2\(\hat k\)) is ……………….

Answer

Answer: 2 units.

---

Question 2.
Vector of magnitude 5 units and in the direction opposite to 2\(\hat i\) + 3\(\hat j\) – 6\(\hat k\) is ……………..

Answer

Answer: \(\frac { 5 }{7}\) (-2\(\hat i\) – 3\(\hat j\) + 6\(\hat k\))

---

Question 3.
The sum of the vectors
\(\vec a\) = \(\hat i\) – 2\(\hat j\) + \(\hat k\), \(\vec b\) = -2\(\hat i\) + 4\(\hat j\) + 5\(\hat k\) and \(\vec c\) = \(\hat i\) – 6\(\hat j\) – 7\(\hat k\) is ……………….

Answer

Answer: -4\(\hat i\) – \(\hat k\)

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Question 4.
The value of ‘a’ when the vectors:
2\(\hat i\) – 3\(\hat j\) + 4\(\hat k\) and a\(\hat i\) + b\(\hat j\) – 8\(\hat k\) are collinear is ……………….

Answer

Answer: -4

---

Question 5.
If \(\vec a\) = 2\(\hat i\) + \(\hat j\) – 2\(\hat k\), then |\(\vec a\)| = ……………….

Answer

Answer: 3.

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Question 6.
If \(\vec a\) is a unit vector and (\(\vec x\) – \(\vec a\)).(\(\vec x\) + \(\vec a\)) = 8, then |\(\vec x\)| = …………….

Answer

Answer: 3.

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Question 7.
(\(\hat i\) × \(\hat j\)).\(\hat k\) + \(\hat i\).\(\hat j\) = ……………..

Answer

Answer: 1.

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Question 8.
The value of ‘λ’ of (2\(\hat i\) + 6\(\hat j\) + 14\(\hat k\)) × (\(\hat i\) – λ\(\hat j\) + 7\(\hat k\)) = \(\vec 0\) is ……………….

Answer

Answer: -3

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Question 9.
If any two vectors \(\vec a\), \(\vec b\), \(\vec c\) are parallel, then [\(\vec a\). \(\vec b\). \(\vec c\)] = …………………

Answer

Answer: 0.

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Question 10.
The value of ‘λ’ such that the vectors:
3\(\hat i\) + \(\hat j\) + 5\(\hat k\), \(\hat i\) + 2\(\hat j\) – 3\(\hat k\) and 2\(\hat i\) – \(\hat j\) + \(\hat k\) are coplanar is ………………..

Answer

Answer: -4.

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Students can access the NCERT MCQ Questions for Class 12 Maths Chapter 12 Linear Programming with Answers Pdf free download aids in your exam preparation and you can get a good hold of the chapter. Use MCQ Questions for Class 12 Maths with Answers during preparation and score maximum marks in the exam. Students can download the Linear Programming Class 12 MCQs Questions with Answers from here and test their problem-solving skills. Clear all the fundamentals and prepare thoroughly for the exam taking help from Class 12 Maths Chapter 12 Linear Programming Objective Questions.

Linear Programming Class 12 MCQs Questions with Answers

Students are advised to solve the Linear Programming Multiple Choice Questions of Class 12 Maths to know different concepts. Practicing the MCQ Questions on Linear Programming Class 12 with answers will boost your confidence thereby helping you score well in the exam.

Explore numerous MCQ Questions of Linear Programming Class 12 with answers provided with detailed solutions by looking below.

Question 1.
The point which does not lie in the half plane 2x + 3y -12 < 0 is
(a) (1, 2)
(b) (2, 1)
(c) (2, 3)
(d) (-3, 2).

Answer

Answer: (c) (2, 3)
Hint:
Putting (2, 3) in 2x + 3y – 12.
Which becomes:
2(2)+ 3(3) – 12 = 4 + 9 – 12 = 1 > 0 and not ≤ 0.

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Question 2.
The corner points of the feasible region determined by the following system of linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5).
Let Z = px + qy, where p, q > 0. Conditions on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is
(a) p = 3q
(b) p = 2q
(c) p = q
(d) q = 3p.

Answer

Answer: (d) q = 3p.

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Question 3.
The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is
(a) p = q
(b) p = 2q
(c) q = 2p
(d) q = 3p.

Answer

Answer: (d) q = 3p.

---

Question 4.
The feasible solution for a LPP is shown in the following figure. Let Z = 3x – 4y be the objective function.

Minimum of Z occurs at:
(a) (0, 0)
(b) (0, 8)
(c) (5, 0)
(d) (4, 10).

Answer

Answer: (b) (0, 8)

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Question 5.
The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q> 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is Maximum of Z occurs at:
(a) (5, 0)
(b) (6, 5)
(c) (6, 8)
(d) (4, 10).

Answer

Answer: (a) (5, 0)

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Fill in the Blanks

Question 1.
Maximum of Z = x + 2y subject to
x + y ≥ 5, x ≥ 0, y ≥ 0 is …………….. at …………….

Answer

Answer: 10, (0,5).

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Question 2.
Minimum of Z = x + y subject to:
2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0 is ………………. at …………….

Answer

Answer: 3, (0.3).

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Question 3.
Maximum and Minimum of Z = 3x + 4 v ≤ 80 subject to 3x + 4v ≤ 80, x + 3y ≤ 30, x ≥ 0, y ≥ 0 are …………….. and …………….

Answer

Answer: 21600, 0.

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We believe the knowledge shared regarding NCERT MCQ Questions for Class 12 Maths Chapter 12 Linear Programming with Answers Pdf free download has been useful to the possible extent. If you have any other queries regarding CBSE Class 12 Maths Linear Programming MCQs Multiple Choice Questions with Answers, feel free to reach us via the comment section and we will guide you with the possible solution.