VBQs Differential Equations Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Differential Equations Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

**Differential Equations VBQs Class 12 Mathematics**

**Question. If the normal to the curve y = f(x) at (3, 4) makes an angle 3π/4 p with the positive x-axis, then f ′(3) is equal to**

(a) 1

(b) 3/4

(c) –1

(d) − 3/4

**Answer**

A

**Question. The minimum area of the triangle formed by any tangent to the ellipse x ^{2}/a^{2}+ y^{2}/b^{2 }= 1 with the coordinate axes is**

(a) a

^{2 }+ b

^{2}

(b) (a + b)

^{2}/2

(c) ab

(d) (a − b)

^{2}/2

**Answer**

C

**Question. **

(a) m = 1, n = 0

(b) m = nπ/2 +1

(c) n = mπ/2

(d) m = n = π/2

**Answer**

C

**Question. **

(a) 1

(b) –1

(c) 0

(d) None of these

**Answer**

A

**Question. **

**then a – 2b is equal to**

(a) 1

(b) –1

(c) 0

(d) 2

**Answer**

B

**Question. The solution of the differential equation**

(a) f(x) = y + C

(b) f(x) = y(x + C)

(c) f(x) = x + C

(d) None of the above

**Answer**

B

**Question. The value of f (0) so that (−e ^{x} + 2^{x})/x may be continuous at x = 0 is**

(a) log (1/2)

(b) 0

(c) 4

(d) – 1 + log 2

**Answer**

D

**Question. If (x + y) sin u = x ^{2}y^{2}, then **

**is equal to**(a) sin u

(b) cosec u

(c) 2 tan u

(d) 3 tan u

**Answer**

D

**Question. If there is an error of k% in measuring the edge of a cube, then the percent error in estimating its volume is**

(a) k

(b) 3k

(c) k/3

(d) None of these

**Answer**

B

**Question. The maximum value of 4sin ^{2}x – 12sinx + 7 is**

(a) 25

(b) 4

(c) does not exist

(d) None of these

**Answer**

D

**Question. The value of c from the Lagrange’s mean value theorem for which f (x) = 25 − x ^{2} in [1, 5], is**

(a) 5

(b) 1

(c) 15

(d) None of these

**Answer**

C

**Question. **

**then the value of a and b, if f is continuous at x = 0, are respectively**

(a) 2/3, 3/2

(b) 2/3 , e^{2/3}

(c) 3/2 , e^{3/2}

(d) None of these

**Answer**

B

**Question. The values of constants a and b so that**

(a) a = 0, b = 0

(b) a = 1, b = –1

(c) a = –1, b = 1

(d) a = 2, b = –1

**Answer**

B

**Question. The condition that the line lx + my = 1 may be normal to the curve y ^{2} = 4ax, is**

(a) al

^{3}– 2alm

^{2}= m

^{2}

(b) al

^{2}+ 2alm

^{3}= m

^{2}

(c) al

^{3}+ 2alm

^{2}= m

^{3}

(d) al

^{3}+ 2alm

^{2}= m

^{2}

**Answer**

D

**Question. If x = sec θ – cos θ, y = sec ^{n} θ – cosn θ, then (x^{2} + 4) (dy/dx)^{2 }is equal to**

(a) n

^{2}(y2 – 4)

(b) n

^{2}(4 – y

^{2})

(c) n

^{2}(y

^{2}+ 4)

(d) None of these

**Answer**

C

**Question. **

**Answer**

D

**Question. **

(a) e

(b) e^{2}

(c) e^{3}

(d) e^{5}

**Answer**

C

**Question. If f : R → R is defined by**

**then the value of a so that f is continuous at 0 is**

(a) 2

(b) 1

(c) –1

(d) 0

**Answer**

D

**Question. Find the value of the **

(a) 0

(b) 1

(c) 2

(d) does not exist

**Answer**

D

**Question. If f : R → R be such that f(1) = 3 and f ′(1) = 6.**

(a) 1

(b) e^{1/2}

(c) e^{2}

(d) e^{3 }

**Answer**

C

**Question. The equation of normal to the curve y = (1 + x) ^{y} + sin^{–1}(sin^{2}x) at x = 0 is**

(a) x + y = 1

(b) x – y = 1

(c) x + y = –1

(d) x – y = –1

**Answer**

A

**Question. If y = e ^{a sin–1} x then (1 – x^{2}) y_{n + 2} – (2n + 1)x y_{n + 1} is equal to**

(a) –(n

^{2}+ a

^{2})y

_{n}

(b) (n

^{2}– a

^{2})y

_{n}

(c) (n

^{2}+ a

^{2})y

_{n}

(d) –(n

^{2}– a

^{2})y

_{n }

**Answer**

C

**Question. The function f(x) = x ^{3} + ax^{2} + bx + c, a^{2} ≤ 3b has**

(a) one maximum value

(b) one minimum value

(c) no extreme value

(d) one maximum and one minimum value

**Answer**

C

**Question. If g(x) is a polynomial satisfying g(x) g(y) = g(x) + g(y) + g(xy) – 2 for all real x and y and g(2) = 5, then lim _{x→3} g(x) is**

(a) 9

(b) 10

(c) 25

(d) 20

**Answer**

B

**Question. The function f(x) = x ^{2} e^{–2x}, x > 0. Then the maximum value of f(x) is**

(a) 1/e

(b) 1/2e

(c) 1/e

^{2}

(d) 4/e

^{4 }

**Answer**

C

**Question. Let f ′(x) be differentiable ∀ x. If f(1) = –2 and f ′(x) ≥ 2 ∀ x ∈[1, 6], then**

(a) f(6) < 8

(b) f(6) ≥ 8

(c) f(6) ≥ 5

(d) f(6) ≤ 5

**Answer**

B

**Question. The angle between the tangents at those points on the curve x = t ^{2} + 1 and y = t^{2} – t – 6 where it meets x-axis is**

(a) ± tan

^{−1}(4/29)

(b) ± tan

^{−1}(5/49)

(c) ± tan

^{−1}(10/49)

(d) ± tan

^{−1}(8/29)

**Answer**

C

**Question. Let [·] denote the greatest integer function and f (x) = [tan ^{2}x]. Then**

(a) lim

_{x→0}f (x) does not exist

(b) f (x) is continuous at x = 0

(c) f (x) is not differentiable at x = 0

(d) f (x) = 1

**Answer**

B

**Question. The rate of change of the surface area of a sphere of radius r, when the radius is increasing at the rate of 2 cm s ^{–1} is proportional to**

(a) 1/r

(b) 1/r

^{2}

(c) r

(d) r

^{2 }

**Answer**

C

**Question. **

(a) 0

(b) tan t

(c) 1

(d) sin t cos t

**Answer**

C

**Question. **

(a) point of minima

(b) point of maxima

(c) point of discontinuity

(d) None of the above

**Answer**

A

**Question. **

(a) 0

(b) 1

(c) –1

(d) e

**Answer**

B

**Question. The minimum value of x/log x is**

(a) e

(b) 1/e

(c) e^{2}

(d) e^{3 }

**Answer**

A

**Question. A spherical balloon is expanding. If the radius is increasing at the rate of 2 centimetres per minute, the rate at which the volume increases (in cubic centimetres per minute) when the radius is 5 centimetres is **

(a) 10π

(b) 100π

(c) 200π

(d) 50π

**Answer**

C

**Question. If y(x) is the solution of the differential equation (x+2) dy/dx = x ^{2} + 4x – 9, x ≠ -2 and y(0) = 0, then y(-4) is equal to :**

(a) 0

(b) 2

(c) 1

(d) –1

## Answer

A

**Question. 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 workers x is given by dP/dx = 100 – 12√x . If the firm employs 25 more workers, then the new level of production of items is**

(a) 2500

(b) 3000

(c) 3500

(d) 4500

## Answer

C

**Question. The population p(t) at time t of a certain mouse species satisfies the differential equation dp(t)/dt = 0.5 p(t) – 450. If p (0) = 850, then the time at which the population becomes zero is :**

(a) 2ln 18

(b) ln 9

(c) (1/2) ln 18

(d) ln 18

## Answer

A

**Question. The solution of the differential equaiton dy/dx + (y/2)sec x = tanx/2y, where 0 ≤ x < π/2, and y(0) = 1, is given by :**

(image72)

## Answer

D

**Question. Let y(x) be the solution of the differential equation (x log x) dy/dx + y = 2x log x, (x ≥ 1). Then y(e) is equal to:**

(a) 2

(b) 2e

(c) e

(d) 0

## Answer

A

**Question. Let y(x) be a solution of (image 47) If y(0) = 2, then y(π/2) equals**

(a) 5/2

(b) 2

(c) 7/2

(d) 3

## Answer

C

**Question. The curve that passes through the point (2, 3), and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact is given by :**

(a) 2y – 3x = 0

(b) y = 6/x

(c) x^{2} + y^{2} =13

(d) (x/2)^{2} + (y/3)^{2} = 2

## Answer

B

**Question. The solution of the differential equation, dy/dx = (x – y) ^{2}, when y (1) = 1, is**

(image 31)

## Answer

B

**Question. (image) 32)**

(a) 1/3 + e^{6}

(b) 1/3

(c) − 4/3

(d) 1/3 + e^{3}

## Answer

A

**Question. The solution of the differential equation dy/dy = (x + y)/x satisfying the condition y(1) =1 is**

(a) y = ln x + x

(b) y = x ln x + x^{2}

(c) y = xe^{(x – 1)}

(d) y = x ln x + x

## Answer

D

**Question. If the differential equation representing the family of all circles touching x-axis at the origin is (x ^{2} − y^{2}) dy/dx = g(x)y, then g(x) equals:**

(a) (1/2)x

(b) 2x

^{2}

(c) 2x

(d) (1/2)x

^{2}

## Answer

C

**Question. Statement-1:** The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P.**Statement-2:** The system of parabolas y^{2} = 4ax satisfies a differential equation of degree 1 and order 1.

(a) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1.

(b) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.

(c) Statement-1 is true; Statement-2 is false.

(d) Statement-1 is false; Statement-2 is true.

## Answer

B

**Question. The solution of the equation (d ^{2}y)/dx^{2} = e^{−2x}**

(image 54)

## Answer

B

**Question. Let y = y(x) be the solution of the differential equation, (2 + sin x)/(y+1) · dy/dx = −cos x, y > 0, y(0) = 1 If y(π) = a and dy/dx at x = π is b, then the ordered pair (a, b) is equal to :**

(a) (2, 3/2)

(b) (1, –1)

(c) (1, 1)

(d) (2, 1)

## Answer

C

**Question. If a curve y = ƒ(x), passing through the point (1, 2), is the solution of the differential equation, 2x ^{2}dy = (2xy + y^{2})dx, then ƒ(1/2) is equal to :**

(a) 1/(1 + log

_{e}2)

(b) 1/(1 − log

_{e}2)

(c) 1 + log

_{e}2

(d) −1/(1 + log

_{e}2)

## Answer

A

**Question. Let y = y(x) be the solution of the differential equation cos x(dy/dx) + 2ysin x = sin2x, x ∈ (0, π/2). If y(π / 3) = 0, then y(π / 4) is equal to :**

(a) 2 – √2

(b) 2 + √2

(c) √2 – 2

(d) 1/√2 – 1

## Answer

C

**Question. Let y = y(x) be the solution curve of the differential equation, (y ^{2} – x) dy/dx = 1, satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is:**

(a) 2 – e

(b) – e

(c) 2

(d) 2 + e

## Answer

A

**Question. Consider the differential equation, y ^{2}dx + (x − 1/y)dy = 0. If value of y is 1 when x = 1, then the value of x for which y = 2, is :**

(a) 5/2 + 1/√e

(b) 3/2 – 1/√e

(c) 1/2 + 1/√e

(d) 3/2 – √e

## Answer

B

**Question. Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equation dV(t)/dt = −K(T −t), where k > 0 is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is**

(a) I – KT^{2}/2

(b) I – ( K(T–t)^{2 })/ 2

(c) e^{–kT}

(d) T^{2} – 1/K

## Answer

A

**Question. If dy/dx = y + 3 > 0 and y (0) = 2, then y (ln 2) is equal to :**

(a) 5

(b) 13

(c) – 2

(d) 7

## Answer

D

**Question. The solution of the differential equation x(dy/dx) + 2y = x ^{2} (x ≠ 0) with y(1) = 1, is:**

(image 62)

## Answer

C

**Question. Solution of the differential equation ydx + (x + x ^{2}y)dy = 0 is**

(a) log y = Cx

(b) – 1/xy + log y = C

(c) 1/xy + log y = C

(d) – 1/xy = C

## Answer

B

**Question. The solution of the differential equation (image 82)**

(image 82)

## Answer

C

**Question. Let y = y(x) be the solution of the differential equation, (image 63) such that y(0) = 0. If √a y(1) = π/32, then the value of ‘a’ is :**

(a) 1/4

(b) 1/2

(c) 1

(d) 1/16

## Answer

D

**Question. If dy/dx = xy/(x ^{2}+y^{2}) ; y(1) = 1; then a value of x satisfying y(x) = e is:**

(a) (1/2)√3e

(b) e/√2

(c) √2e

(d) √3e

## Answer

D

**Question. Let f(x) = (sin(tan ^{–1} x) + sin(cot^{–1} x))^{2} – 1, |x| > 1. If dy/dx = (1/2) d/dx (sin^{–1} (f (x))) and y (√3) π/6 = , then y (-√3) is equal to:**

(a) 2π/3

(b) − π/6

(c) 5π/6

(d) π/3

## Answer

B

**Question. Let y = y(x) be the solution of the differential equation, x(dy/dx) + y = x log _{e}x, (x>1). If 2y(2) = log_{e} 4 –1, then y(e) is equal to :**

(a) − e/2

(b) − e

^{2}/2

(c) e/4

(d) e

^{2}/4

## Answer

C

**Question. The differential equation which represents the family of curves y = c _{1}e^{c2x} , where c_{1}, and c_{2} are arbitrary constants, is**

(a) y” = y’y

(b) yy” = y’

(c) yy” = (y’)

^{2}

(d) y’ = y

^{2}

## Answer

C

**Question. The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is**

(a) (x – 2)y’^{2} = 25 –(y – 2)^{2}

(b) (y – 2)y’^{2} = 25 –(y – 2)^{2}

(c) (y – 2)^{2}y’^{2} = 25 –(y – 2)^{2}

(d) (x – 2)^{2}y’^{2} = 25 –(y – 2)^{2}

## Answer

C

**Question. If a curve passes through the point (1, – 2) and has slope of the tangent at any point (x, y) on it as (x ^{2} – 2y)/x, then the curve also passes through the point :**

(a) (3, 0)

(b) ( √3,0)

(c) (–1, 2)

(d) (– √2,1)

## Answer

B

**Question. If y(x) is the solution of the differential equation dy/dx + ( (2x+1)/x )y = e ^{–2x}, x > 0, where y(1) = (1/2)e^{–2}, then**

(a) y (log

_{e}2) = log

_{e}4

(b) y (log

_{e}2) = (log

_{e}2)/4

(c) y(x) is decreasing in (1/2, 1)

(d) y(x) is decreasing in (0, 1)

## Answer

C

**Question. The curve satisfying the differential equation, ydx–(x + 3y ^{2}) dy = 0 and passing through the point (1, 1), also passes through the point :**

(a) (1/4, − 1/2)

(b) (− 1/3, 1/3)

(c) (1/3, − 1/3)

(d) (1/4, 1/2)

## Answer

B

**Question. If dy /dx + y tan x = sin 2x and y(0) = 1, then y(π) is equal to:**

(a) 1

(b) – 1

(c) – 5

(d) 5

## Answer

C

**Question. If y = y(x) is the solution of the differential equation, e ^{y} = e^{x} such that y(0) = 0, then y(l) is equal to:**

(a) l + log

_{e}2

(b) 2 + log

_{e}2

(c) 2e

(d) log

_{e}2

## Answer

A

**Question. The general solution of the differential equation (y ^{2} – x^{3}) dx – xydy = 0 (x ≠ 0) is :**

(where c is a constant of integration)

(a) y

^{2}– 2x

^{2}+ cx3 = 0

(b) y

^{2}+ 2×3 + cx

^{2}= 0

(c) y

^{2}+ 2x

^{2}+ cx3 = 0

(d) y

^{2}– 2×3 + cx

^{2}= 0

## Answer

B

**Question. The general solution of the differential equation, sin 2x(dy/dx − √tanx) − y = 0, is :**

(a) y √(tan x) = x + c

(b) y √(cot x) = tan x + c

(c) y √(tan x) = cot x + c

(d) y √(cot x) = x + c

## Answer

D

**Question. The general solution of the differential equation dy/dx + (2/x)y = x ^{2} is**

(a) y = cx

^{–3}– x

^{2}/4

(b) y = cx

^{3}– x

^{2}/4

(c) y = cx

^{2}– x

^{3}/5

(d) y = cx

^{–2}– x

^{3}/5

## Answer

D

**Question. The degree and order of the differential equation of the family of all parabolas whose axis is x – axis, are respectively.**

(a) 2, 3

(b) 2, 1

(c) 1, 2

(d) 3, 2

## Answer

C

**Question. The order and degree of the differential equation (image 13)**

(a) (1, 2/3)

(b) (3, 1)

(c) (3, 3)

(d) (1, 2)

## Answer

C

**Question. If a curve passes through the point (2, 7/2) and has slope (1 − 1/x ^{2}) at any point (x, y) on it, then the ordinate of the point on the curve whose abscissa is – 2 is :**

(a) – 3/2

(b) 3/2

(c) 5/2

(d) – 5/2

## Answer

A

**Question. Consider the differential equation : **

(image 45)**Statement-1:** The substitution z = y2 transforms the above equation into a first order homogenous differential equation.**Statement-2:** The solution of this differential equation is (image 45)

(a) Both statements are false.

(b) Statement-1 is true and statement-2 is false.

(c) Statement-1 is false and statement-2 is true.

(d) Both statements are true.

## Answer

D

**Question. Let y = y (x) be the solution of the differential equation, xy’ – y = x ^{2} (x cos x + sin x), x > 0. If y(π) = π, then y”(π/2) + y(π/2) is equal to :**

(a) 2 + π/2

(b) 1 + π/2 + π

^{2}/4

(c) 2 + π/2 + π

^{2}/4

(d) 1 + π/2

## Answer

A

**Question. If for x ≥ 0, y = y(x) is the solution of the differential equation, (x + 1)dy = ((x + 1) ^{2} + y – 3)dx, y(2) = 0, then y(3) is equal to _____.**

## Answer

3