Available Questions Page 11 of 14
Standalone Questions
#653
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Let \(f^{\prime}(x)=3(x^{2}+2x)-\frac{4}{x^{3}}+5,\) and \(f(1)=0\). Then, \(f(x)\) is:
(A) \(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x+11\)
(B) \(x^{3}+3x^{2}+\frac{2}{x^{2}}+5x-11\)
(C) \(x^{3}+3x^{2}-\frac{2}{x^{2}}+5x-11\)
(D) \(x^{3}-3x^{2}-\frac{2}{x^{2}}+5x-11\)
Key: B
Sol:
Sol:
#652
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The order and degree of the following differential equation are, respectively: \(-\frac{d^{4}y}{dx^{4}}+2e^{dy/dx}+y^{2}=0\)
(A) 4, 1
(B) 4, not defined
(C) 1, 1
(D) 4, 1
Key: B
Sol:
Sol:
#651
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The solution for the differential equation \(\log(\frac{dy}{dx})=3x+4y\) is:
(A) \(3e^{4y}+4e^{-3x}+C=0\)
(B) \(e^{3x+4y}+C=0\)
(C) \(3e^{-3y}+4e^{4x}+12C=0\)
(D) \(3e^{-4y}+4e^{3x}+12C=0\)
Key: D
Sol:
Sol:
#650
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The integrating factor of the differential equation \((1-x^{2})\frac{dy}{dx}+xy=ax,\) \(-1 \lt x\lt 1\) is:
(A) \(\frac{1}{x^{2}-1}\)
(B) \(\frac{1}{\sqrt{x^{2}-1}}\)
(C) \(\frac{1}{1-x^{2}}\)
(D) \(\frac{1}{\sqrt{1-x^{2}}}\)
Key: D
Sol:
Sol:
#649
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The order and degree of the differential equation \([1+(\frac{dy}{dx})^{2}]^{3}=\frac{d^{2}y}{dx^{2}}\) respectively are:
(A) 1, 2
(B) 2, 3
(C) 2, 1
(D) 2, 6
Key: C
Sol:
Sol:
#648
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The differential equation \(\frac{dy}{dx}=F(x,y)\) will not be a homogeneous differential equation, if \(F(x,y)\) is :
(A) \(\cos~x-\sin(\frac{y}{x})\)
(B) \(\frac{y}{x}\)
(C) \(\frac{x^{2}+y^{2}}{xy}\)
(D) \(\cos^{2}(\frac{x}{y})\)
Key: A
Sol:
Sol:
#647
Mathematics
Differential Equations
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The degree of the differential equation \((y^{\prime\prime})^{2}+(y^{\prime})^{3}=x~\sin(y^{\prime})\) is:
(A) 1
(B) 2
(C) 3
(D) not defined
Key: D
Sol:
Sol:
#646
Mathematics
Differential Equations
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(x~\log~x\frac{dy}{dx}+y=2~\log~x\) is an example of a :
(A) variable separable differential equation.
(B) homogeneous differential equation.
(C) first order linear differential equation.
(D) differential equation whose degree is not defined.
Key: C
Sol:
Sol:
#645
Mathematics
Differential Equations
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The general solution of the differential equation \(x~dy+y~dx=0\) is:
(A) \(xy=c\)
(B) \(x+y=c\)
(C) \(x^{2}+y^{2}=c^{2}\)
(D) \(log~y=log~x+c\)
Key: A
Sol:
Sol:
#644
Mathematics
Differential Equations
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The integrating factor of the differential equation \((x+2y^{2})\frac{dy}{dx}=y(y>0)\) is:
(A) \(\frac{1}{x}\)
(B) x
(C) y
(D) \(\frac{1}{y}\)
Key: D
Sol:
Sol:
#643
Mathematics
Differential Equations
MCQ_SINGLE
ANALYZE
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The order of the differential equation \(\frac{d^{4}y}{dx^{4}}-sin(\frac{d^{2}y}{dx^{2}})=5\) is:
(A) 4
(B) 3
(C) 2
(D) not defined
Key: D
Sol:
Sol:
#642
Mathematics
Applications of Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The area of the shaded region (figure) represented by the curves \(y=x^{2}\), \(0\le x\le2\) and y-axis is given by(A) \(\int_{0}^{2} x^2 dx\)
(B) \(\int_{0}^{2} \sqrt{y} dy\)
(C) \(\int_{0}^{4} x^2 dx\)
(D) \(\int_{0}^{4} \sqrt{y} dy\)
Key: D
Sol:
Sol:
\(\int_{0}^{4} \sqrt{y} dy\)
#641
Mathematics
Applications of Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The area of the region enclosed by the curve \(y=\sqrt{x}\) and the lines \(x=0\) and \(x=4\) and x-axis is:
(A) \(\frac{16}{9}\) sq. units
(B) \(\frac{32}{9}\) sq. units
(C) \(\frac{16}{3}\) sq. units
(D) \(\frac{32}{3}\) sq. units
Key: C
Sol:
Sol:
The area (\(A\)) of the region enclosed by the curve \(y = f(x)\), the \(x\)-axis, and the vertical lines \(x=a\) and \(x=b\) is given by the definite integral:\[A = \int_{a}^{b} f(x) dx\]
\[A = \int_{0}^{4} x^{1/2} dx\]
\[A = \frac{2}{3} \left[ x^{3/2} \right]_{0}^{4}\]
\[A = \frac{2}{3} \left[ (4)^{3/2} - (0)^{3/2} \right]\]
\[A = \frac{16}{3}\]
#639
Mathematics
Applications of Integrals
MCQ_SINGLE
APPLY
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
Area of the region bounded by curve \(y^{2}=4x\) and the X-axis between \(x=0\) and \(x=1\) is:
(A) \(\frac{2}{3}\)
(B) \(\frac{8}{3}\)
(C) 3
(D) \(\frac{4}{3}\)
Key:
Sol:
Sol:
The region is bounded by the parabola $y^2=4x$, the X-axis ($y=0$), $x=0$, and $x=1$.
Since the parabola is symmetric about the X-axis, the total area can be found by calculating the area of the upper half (where $y = 2\sqrt{x}$) and multiplying by 2.
1. From $y^2=4x$, we get $y = \pm \sqrt{4x} = \pm 2\sqrt{x}$.
2. The area $A$ is given by the integral of $y$ with respect to $x$ from $x=0$ to $x=1$. Due to symmetry, we calculate the area for $y=2\sqrt{x}$ and multiply by 2.
$$A = 2 \int_{0}^{1} 2\sqrt{x} \, dx$$
3. Evaluate the integral:
$$A = 4 \int_{0}^{1} x^{1/2} \, dx$$
$$A = 4 \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{1}$$
$$A = 4 \left[ \frac{2}{3} x^{3/2} \right]_{0}^{1}$$
4. Apply the limits of integration:
$$A = \frac{8}{3} \left( (1)^{3/2} - (0)^{3/2} \right)$$
$$A = \frac{8}{3} (1 - 0)$$
$$A = \frac{8}{3}$$
The final answer is $\boxed{\frac{8}{3}}$.
#638
Mathematics
Applications of Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The area of the shaded region bounded by the curve \(y^2=x,x=4\) and the \(x-\)axis is given by (A) \( \int_{0}^{4} x \, dx\)
(B) \( \int_{0}^{2} y^2 \, dy\)
(C) \( 2 \int_{0}^{4} \sqrt{x} \, dx\)
(D) \( \int_{0}^{4} \sqrt{x} \, dx\)
Key: D
Sol:
Sol:
The curve \( y^2 = x \) is symmetric about the x-axis.
The shaded area is the area above the x-axis:
Answer:
\( \int_0^4 \sqrt{x} \, dx \)
\( \int_0^4 \sqrt{x} \, dx \)
#637
Mathematics
Applications of Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The area of the region bounded by the curve \(y^{2}=x\) between \(x=0\) and \(x=1\) is:
(A) \(\frac{3}{2}\) sq units
(B) \(\frac{2}{3}\) sq units
(C) 3 sq units
(D) \(\frac{4}{3}\) sq units
Key: B
Sol:
Sol:
**Correct Option if MCQ:** B
**Reasoning:**
* \(y = \sqrt{x}\)
* Area = \(\int_{0}^{1} \sqrt{x} dx\)
* Area = \([\frac{2}{3}x^{\frac{3}{2}}]_{0}^{1} = \frac{2}{3}\)
#636
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
If \(f(2a-x)=f(x)\), then \(\int_{0}^{2a}f(x)dx\) is
(A) \(\int_{0}^{2a}f(\frac{x}{2})dx\)
(B) \(\int_{0}^{a}f(x)dx\)
(C) \(2\int_{a}^{0}f(x)dx\)
(D) \(2\int_{0}^{a}f(x)dx\)
Key:
Sol:
Sol:
**Correct Option if MCQ:** D
**Reasoning:**
* Let \(I = \int_{0}^{2a}f(x)dx\).
* Using the property \(\int_{0}^{na}f(x)dx = n\int_{0}^{a}f(x)dx\) for \(f(x)=f(2a-x)\).
* Thus, \(I = 2\int_{0}^{a}f(x)dx\).
#635
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
The value of \(\int_{0}^{1}\frac{dx}{e^{x}+e^{-x}}\) is:
(A) \(-\frac{\pi}{4}\)
(B) \(\frac{\pi}{4}\)
(C) \(\tan^{-1}e-\frac{\pi}{4}\)
(D) \(\tan^{-1}e\)
Key: C
Sol:
Sol:
\[\frac{1}{e^{x}+e^{-x}} = \frac{1}{e^{x} + \frac{1}{e^{x}}} = \frac{1}{\frac{e^{2x} + 1}{e^x}} = \frac{e^x}{e^{2x} + 1}\]
The integral becomes:
\[I = \int_{0}^{1}\frac{e^x}{e^{2x} + 1}dx\]Now use Substitution
Let \(u = e^x\). Then, \(du = e^x dx\).
Change the limits of integration:
- Lower limit (\(x=0\)): \(u_1 = e^0 = 1\)
- Upper limit (\(x=1\)): \(u_2 = e^1 = e\)
The integral becomes:
\[I = \int_{1}^{e}\frac{du}{u^{2} + 1}\]Now evaluate and Apply Limits
\[I = \left[ \tan ^{-1}(u) \right]_{1}^{e}\] \[=\tan ^{-1}(e) - \frac{\pi}{4}\]
#634
Mathematics
Definite Integrals
MCQ_SINGLE
APPLY
2025
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{0}^{\pi/2}\cos x\cdot e^{\sin x}dx\) is equal to:
(A) 0
(B) \(1-e\)
(C) \(e-1\)
(D) e
Key:
Sol:
Sol:
#633
Mathematics
Definite Integrals
MCQ_SINGLE
UNDERSTAND
2024
AISSCE(Board Exam)
KNOWLEDGE
1 Marks
\(\int_{a}^{b}f(x)dx\) is equal to:
(A) \(\int_{a}^{b}f(a-x)dx\)
(B) \(\int_{a}^{b}f(a+b-x)dx\)
(C) \(\int_{a}^{b}f(x-(a+b))dx\)
(D) \(\int_{a}^{b}f((a-x)+(b-x))dx\)
Key:
Sol:
Sol: