Math Practice MCQs

Mathematics - Trigonometry

1. If in a triangle $ABC$, $\frac{2\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c} = \frac{a}{bc} + \frac{b}{ca}$, then angle $A$ is

A.$90^\circ$B.$45^\circ$C.$30^\circ$D.$60^\circ$

Mathematics - Inverse Trigonometry

2. The relation between two angles $A = 2\tan^{-1}(2\sqrt{2} - 1)$ & $B = 3\sin^{-1}(1/3) + \sin^{-1}(3/5)$ is

A.$B > A$B.$A > B$C.$A = B$D.None

Mathematics - Complex Numbers

3. The region of Argand plane defined by $|z - 1| + |z + 1| \le 4$ is

A.interior of an ellipseB.exterior of a circleC.interior and boundary of an ellipseD.none

Mathematics - Sequences and Series

4. If $a_1, a_2, a_3, \dots, a_n$ are in H.P., then $a_1 a_2 + a_2 a_3 + \dots + a_{n-1} a_n =$

A.$n - 1$B.$(n - 1)a_1$C.$n a_1 a_n$D.$(n - 1)a_1 a_n$

Mathematics - Quadratic Equations

5. Let $\alpha$ and $\beta$ be the roots of the equation $(x - a)(x - b) = c$, $c \ne 0$. Then the roots of the equation $(x - \alpha)(x - \beta) + c = 0$ are

A.$a, c$B.$b, c$C.$a, b$D.$a + c, b + c$

Mathematics - Sequences and Series

6. If $\log_3 2$, $\log_3(2^x - 5)$ and $\log_3(2^x - 7/2)$ are in A.P., then $x$ is

A.1B.2C.3D.0

Mathematics - Permutations and Combinations

7. In a hockey championship, there were played 153 matches. Every two teams played one match with each other. The number of teams participating in the championship is

A.17B.16C.-17D.18

Mathematics - Binomial Theorem

8. The coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^n$ is

A.${}^n C_4$B.${}^n C_4 + {}^n C_2$C.${}^n C_4 + {}^n C_1 + {}^n C_4 \cdot {}^n C_2$D.${}^n C_4 + {}^n C_2 + {}^n C_1 \cdot {}^n C_2$

Mathematics - Matrices and Determinants

9. $\Delta = \begin{vmatrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0$ if

A.$a, b, c$ are in A.P.B.$a, b, c$ are in G.P.C.$a, b, c$ are in H.P.D.$(x - \alpha)$ is a factor of $ax^2 + 2bx + c$

Mathematics - Probability

10. A single letter is selected at random from the word "probability". The probability that it is a vowel is

A.3/11B.4/11C.2/11D.0

Mathematics - Vectors

11. Let $\mathbf{a} = (2, -1, 1)$, $\mathbf{b} = (1, 2, -1)$ and $\mathbf{c} = (1, 1, -2)$. A vector in a plane of $\mathbf{b}$ and $\mathbf{c}$ whose projection on $\mathbf{a}$ of magnitude $\sqrt{\frac{2}{3}}$ units is

A.(2, 3, -3)B.(2, 3, 5)C.(-2, -1, 5)D.(-2, 3, -5)

Mathematics - 3D Geometry

12. Equation of the plane through the point $(2\hat{i} + 3\hat{j} - \hat{k})$ and perpendicular to the vector $(3\hat{i} - 4\hat{j} + 7\hat{k})$ is

A.$\mathbf{r} \cdot (3\hat{i} + 4\hat{j} + 7\hat{k}) = 13$B.$\mathbf{r} \cdot (3\hat{i} - 4\hat{j} + 7\hat{k}) = 17$C.$\mathbf{r} \cdot (-3\hat{i} + 4\hat{j} - 7\hat{k}) = 13$D.$\mathbf{r} \cdot (3\hat{i} + 4\hat{j} - 7\hat{k}) = 13$

Mathematics - Calculus

13. If $x^y = e^{x - y}$, then $\frac{dy}{dx} =$

A.$\frac{y}{(1 + \log x)}$B.$\frac{(x - y)}{(1 + \log x)^2}$C.$\frac{(x - y)}{(1 + \log x)}$D.$\frac{(\log x)}{(1 + \log x)^2}$

Mathematics - Calculus

14. The angle of intersection of the curve $y = x^2$ and $6y = 7 - x^3$ at $(1, 1)$ is

A.$\frac{\pi}{4}$B.$\frac{\pi}{3}$C.$\frac{\pi}{2}$D.None

Mathematics - Calculus

15. The largest value of $2x^3 - 3x^2 - 12x + 5$ for $-2 \le x \le 4$ occurs at $x =$

A.-2B.-1C.2D.4

Mathematics - Functions

16. Domain of $f(x) = \sin^{-1}[\log_2(x^2/2)]$ is

A.$[-2, -1] \cup [1, 2]$B.$(-2, -1] \cup [1, 2]$C.$[-2, 2]$D.$[-1, 1]$

Mathematics - Calculus

17. $\lim_{x \to 0} \sqrt{\frac{x - \sin x}{x + \cos^2 x}} =$

A.1B.does not existC.0D.$\infty$

Mathematics - Calculus

18. Let $y = x^2 e^{-x}$, then the interval in which $y$ increases with respect to $x$ is

A.$\mathbb{R}$B.$(-2, 0)$C.$(2, \infty)$D.$(0, 2)$

Mathematics - Differential Equations

19. Solution of the differential equation $\frac{dy}{dx} - y\tan x = -y^2\sec x$ is

A.$y = \frac{\sec x}{x + c}$B.$y = \frac{\sec x}{c - x}$C.$y\sec x = x + c$D.$y\sec x = c - x$

Mathematics - Integral Calculus

20. $\int e^x \left[\frac{x^2 + 1}{(x + 1)^2}\right] dx =$

A.$\frac{e^x}{x + 1} + c$B.$\frac{x - 1}{x + 1}e^x + c$C.$\frac{1 - x}{x + 1}e^x + c$D.$-\frac{e^x}{x + 1} + c$

Mathematics - Parabola

21. Equation of the common tangent to the parabolas $y^2 = 4ax$ and $x^2 = 4ay$ is

A.$x - y + a = 0$B.$x + y - a = 0$C.$x + y + a = 0$D.$x - y - a = 0$

Mathematics - Parabola

22. Focus of the parabola $4x^2 - 12x - 8y + 17 = 0$ is

A.(3/2, 1)B.(3/2, 5/4)C.(3/2, -1)D.(3/2, 5/8)

Mathematics - Coordinate Geometry

23. Area of the triangle formed by $ax^2 + 2hxy + by^2 = 0$ and $lx + my = 1$ is

A.$\frac{\sqrt{h^2 - ab}}{am^2 - 2hlm + bl^2}$B.$\frac{2\sqrt{h^2 - ab}}{am^2 - 2hlm + bl^2}$C.$\frac{\sqrt{h^2 - ab}}{|am^2 - 2hlm + bl^2|}$D.$\frac{\sqrt{h^2 - ab}}{2|am^2 - 2hlm + bl^2|}$

Mathematics - Circles

24. Locus of point of intersection of two mutually perpendicular tangents to the circle $x^2 + y^2 = a^2$ is

A.$x^2 + y^2 = a^2$B.$x^2 + y^2 = 2a^2$C.$x^2 + y^2 = 4a^2$D.$x^2 + y^2 = a$

Mathematics - Circles

25. Condition for two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ being orthogonal is

A.$g_1 g_2 + f_1 f_2 = c_1 + c_2$B.$2g_1 g_2 + 2f_1 f_2 = c_1 + c_2$C.$g_1 g_2 + f_1 f_2 = c_1 - c_2$D.$2g_1 g_2 + 2f_1 f_2 = c_1 - c_2$

Mathematics - Coordinate Geometry

26. In $\triangle ABC$, internal angle bisector of $\angle A$ divides opposite side $BC$ at point $D$. Locus of $D$ is

A.EllipseB.Straight lineC.CircleD.none of these

Mathematics - Ellipse

27. Angle subtended by chord $x + y = 1$ at center of the ellipse $2x^2 + 3y^2 = 6$ is

A.$90^\circ$B.$\tan^{-1}(3\sqrt{2})$C.$\tan^{-1}(2\sqrt{3})$D.None

Mathematics - Conic Sections

28. Eccentricity of conic represented by point $(5\cos\theta, 3\sin\theta)$ is

A.4/5B.5/4C.$\sqrt{34}/5$D.None

Mathematics - Conic Sections

29. Asymptotes of rectangular hyperbola are at an angle of

A.$45^\circ$B.$90^\circ$C.$135^\circ$D.None

Mathematics - Conic Sections

30. Distance between foci of $x^2/16 - y^2/9 = 1$ is

A.10B.5C.12D.8

1. If in a triangle $ABC$, $\frac{2\cos A}{a} = \frac{\cos B}{b} = \frac{\cos C}{c} = \frac{a}{bc} + \frac{b}{ca}$, then angle $A$ is

2. The relation between two angles $A = 2\tan^{-1}(2\sqrt{2} - 1)$ & $B = 3\sin^{-1}(1/3) + \sin^{-1}(3/5)$ is

3. The region of Argand plane defined by $|z - 1| + |z + 1| \le 4$ is

4. If $a_1, a_2, a_3, \dots, a_n$ are in H.P., then $a_1 a_2 + a_2 a_3 + \dots + a_{n-1} a_n =$

5. Let $\alpha$ and $\beta$ be the roots of the equation $(x - a)(x - b) = c$, $c \ne 0$. Then the roots of the equation $(x - \alpha)(x - \beta) + c = 0$ are

6. If $\log_3 2$, $\log_3(2^x - 5)$ and $\log_3(2^x - 7/2)$ are in A.P., then $x$ is

7. In a hockey championship, there were played 153 matches. Every two teams played one match with each other. The number of teams participating in the championship is

8. The coefficient of $x^4$ in the expansion of $(1 + x + x^2 + x^3)^n$ is

9. $\Delta = \begin{vmatrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0$ if

10. A single letter is selected at random from the word "probability". The probability that it is a vowel is

11. Let $\mathbf{a} = (2, -1, 1)$, $\mathbf{b} = (1, 2, -1)$ and $\mathbf{c} = (1, 1, -2)$. A vector in a plane of $\mathbf{b}$ and $\mathbf{c}$ whose projection on $\mathbf{a}$ of magnitude $\sqrt{\frac{2}{3}}$ units is

12. Equation of the plane through the point $(2\hat{i} + 3\hat{j} - \hat{k})$ and perpendicular to the vector $(3\hat{i} - 4\hat{j} + 7\hat{k})$ is

13. If $x^y = e^{x - y}$, then $\frac{dy}{dx} =$

14. The angle of intersection of the curve $y = x^2$ and $6y = 7 - x^3$ at $(1, 1)$ is

15. The largest value of $2x^3 - 3x^2 - 12x + 5$ for $-2 \le x \le 4$ occurs at $x =$

16. Domain of $f(x) = \sin^{-1}[\log_2(x^2/2)]$ is

17. $\lim_{x \to 0} \sqrt{\frac{x - \sin x}{x + \cos^2 x}} =$

18. Let $y = x^2 e^{-x}$, then the interval in which $y$ increases with respect to $x$ is

19. Solution of the differential equation $\frac{dy}{dx} - y\tan x = -y^2\sec x$ is

20. $\int e^x \left[\frac{x^2 + 1}{(x + 1)^2}\right] dx =$

21. Equation of the common tangent to the parabolas $y^2 = 4ax$ and $x^2 = 4ay$ is

22. Focus of the parabola $4x^2 - 12x - 8y + 17 = 0$ is

23. Area of the triangle formed by $ax^2 + 2hxy + by^2 = 0$ and $lx + my = 1$ is

24. Locus of point of intersection of two mutually perpendicular tangents to the circle $x^2 + y^2 = a^2$ is

25. Condition for two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ being orthogonal is

26. In $\triangle ABC$, internal angle bisector of $\angle A$ divides opposite side $BC$ at point $D$. Locus of $D$ is

27. Angle subtended by chord $x + y = 1$ at center of the ellipse $2x^2 + 3y^2 = 6$ is

28. Eccentricity of conic represented by point $(5\cos\theta, 3\sin\theta)$ is

29. Asymptotes of rectangular hyperbola are at an angle of

30. Distance between foci of $x^2/16 - y^2/9 = 1$ is

Test Result