JEE Advanced Maths Mock Test – 1
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Jee Advanced Maths Mock Test -1
Total Questions: 20
Total Marks: 80
Duration: 48 Minutes
- Correct Answer : 4 Marks
- Wrong Answer: -1 Mark
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Question 1 of 20
1. Question
Evaluate $\int_0^\infty \frac{e^{-x} \sin x}{x} \, dx$.
Correct
Use the Laplace transform technique.
Incorrect
Use the Laplace transform technique.
Unattempted
Use the Laplace transform technique.
-
Question 2 of 20
2. Question
Solve $\int \frac{\ln x}{1+x^2} \, dx$.
Correct
Use integration by parts with substitution $u = \ln x$.
Incorrect
Use integration by parts with substitution $u = \ln x$.
Unattempted
Use integration by parts with substitution $u = \ln x$.
-
Question 3 of 20
3. Question
Find the eccentricity of the ellipse $\frac{x^2}{9} + \frac{y^2}{16} = 1$.
Correct
Use the formula $e = \sqrt{1 – _x000c_rac{b^2}{a^2}}$, where $a > b$.
Incorrect
Use the formula $e = \sqrt{1 – _x000c_rac{b^2}{a^2}}$, where $a > b$.
Unattempted
Use the formula $e = \sqrt{1 – _x000c_rac{b^2}{a^2}}$, where $a > b$.
-
Question 4 of 20
4. Question
Find the radius of convergence of the series $\sum_{n=1}^\infty \frac{x^n}{n}$.
Correct
Use the ratio test $\lim_{n o \infty} \left| _x000c_rac{a_{n+1}}{a_n}
ight|$.Incorrect
Use the ratio test $\lim_{n o \infty} \left| _x000c_rac{a_{n+1}}{a_n}
ight|$.Unattempted
Use the ratio test $\lim_{n o \infty} \left| _x000c_rac{a_{n+1}}{a_n}
ight|$. -
Question 5 of 20
5. Question
Evaluate $\int_0^\pi x^2 \sin x \, dx$.
Correct
Use integration by parts twice and solve.
Incorrect
Use integration by parts twice and solve.
Unattempted
Use integration by parts twice and solve.
-
Question 6 of 20
6. Question
Find the equation of the normal to the hyperbola $\frac{x^2}{9} – \frac{y^2}{16} = 1$ at the point $(3, 2)$.
Correct
Use the derivative of the hyperbola equation and normal slope formula.
Incorrect
Use the derivative of the hyperbola equation and normal slope formula.
Unattempted
Use the derivative of the hyperbola equation and normal slope formula.
-
Question 7 of 20
7. Question
Solve $\int_0^\infty e^{-x} x^2 \, dx$.
Correct
Use the Gamma function $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1} \, dx$.
Incorrect
Use the Gamma function $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1} \, dx$.
Unattempted
Use the Gamma function $\Gamma(n) = \int_0^\infty e^{-x} x^{n-1} \, dx$.
-
Question 8 of 20
8. Question
Evaluate $\lim_{x \to 0} \frac{\cos(x) – e^{-x^2}}{x^2}$.
Correct
Expand $\cos(x)$ and $e^{-x^2}$ using Taylor series and simplify.
Incorrect
Expand $\cos(x)$ and $e^{-x^2}$ using Taylor series and simplify.
Unattempted
Expand $\cos(x)$ and $e^{-x^2}$ using Taylor series and simplify.
-
Question 9 of 20
9. Question
If $\int_0^\infty e^{-x} \sin(ax) \, dx = \frac{a}{1+a^2}$, find the value of $a$.
Correct
Use Laplace transform properties to simplify.
Incorrect
Use Laplace transform properties to simplify.
Unattempted
Use Laplace transform properties to simplify.
-
Question 10 of 20
10. Question
Solve $\int \frac{1}{(1+x^2)^2} \, dx$.
Correct
Use substitution $x = an u$ and trigonometric identities.
Incorrect
Use substitution $x = an u$ and trigonometric identities.
Unattempted
Use substitution $x = an u$ and trigonometric identities.
-
Question 11 of 20
11. Question
Find the point on the parabola $y^2 = 8x$ closest to the point $(2, 0)$.
Correct
Use the distance formula and minimize using differentiation.
Incorrect
Use the distance formula and minimize using differentiation.
Unattempted
Use the distance formula and minimize using differentiation.
-
Question 12 of 20
12. Question
Find the angle between the planes $2x – y + 2z = 5$ and $x + y + 2z = 3$.
Correct
Use the formula $\cos heta = _x000c_rac{|_x000b_ec{n_1} \cdot _x000b_ec{n_2}|}{|_x000b_ec{n_1}||_x000b_ec{n_2}|}$.
Incorrect
Use the formula $\cos heta = _x000c_rac{|_x000b_ec{n_1} \cdot _x000b_ec{n_2}|}{|_x000b_ec{n_1}||_x000b_ec{n_2}|}$.
Unattempted
Use the formula $\cos heta = _x000c_rac{|_x000b_ec{n_1} \cdot _x000b_ec{n_2}|}{|_x000b_ec{n_1}||_x000b_ec{n_2}|}$.
-
Question 13 of 20
13. Question
Solve $\frac{dy}{dx} = \frac{x + y}{x – y}$.
Correct
Use the substitution $v = _x000c_rac{y}{x}$.
Incorrect
Use the substitution $v = _x000c_rac{y}{x}$.
Unattempted
Use the substitution $v = _x000c_rac{y}{x}$.
-
Question 14 of 20
14. Question
Evaluate $\int_0^1 \frac{x^4 – x^2 + 1}{x^2 + 1} \, dx$.
Correct
Simplify the integrand by dividing $x^4 – x^2 + 1$ by $x^2 + 1$.
Incorrect
Simplify the integrand by dividing $x^4 – x^2 + 1$ by $x^2 + 1$.
Unattempted
Simplify the integrand by dividing $x^4 – x^2 + 1$ by $x^2 + 1$.
-
Question 15 of 20
15. Question
Find the shortest distance between the point $(1, 2, 3)$ and the plane $x + y – z = 1$.
Correct
Use the formula $D = _x000c_rac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$.
Incorrect
Use the formula $D = _x000c_rac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$.
Unattempted
Use the formula $D = _x000c_rac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$.
-
Question 16 of 20
16. Question
Solve $\int_0^\infty \frac{1}{(1+x^4)} \, dx$.
Correct
Use substitution $x^2 = t$.
Incorrect
Use substitution $x^2 = t$.
Unattempted
Use substitution $x^2 = t$.
-
Question 17 of 20
17. Question
Find the volume of the region enclosed by $z = 4 – x^2 – y^2$ and $z = 0$.
Correct
Use double integration $\int \int (4 – x^2 – y^2) \, dx \, dy$.
Incorrect
Use double integration $\int \int (4 – x^2 – y^2) \, dx \, dy$.
Unattempted
Use double integration $\int \int (4 – x^2 – y^2) \, dx \, dy$.
-
Question 18 of 20
18. Question
Solve the differential equation $\frac{dy}{dx} + y\cot x = 1$, where $x \in (0, \pi)$.
Correct
Use the integrating factor $e^{\int \cot x \, dx} = \sin x$.
Incorrect
Use the integrating factor $e^{\int \cot x \, dx} = \sin x$.
Unattempted
Use the integrating factor $e^{\int \cot x \, dx} = \sin x$.
-
Question 19 of 20
19. Question
Evaluate $\lim_{x \to \infty} \frac{\sqrt{x^2 + x} – \sqrt{x^2 – x}}{x}$.
Correct
Multiply numerator and denominator by the conjugate.
Incorrect
Multiply numerator and denominator by the conjugate.
Unattempted
Multiply numerator and denominator by the conjugate.
-
Question 20 of 20
20. Question
Find the sum of the infinite series $\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$.
Correct
Use the series summation property for alternating series.
Incorrect
Use the series summation property for alternating series.
Unattempted
Use the series summation property for alternating series.