The real number $k$ for which the equation, $2x^3 + 3x + k = 0$ has two distinct real roots in $[0,1]$ belongs to (1) lies between - 1 and 0 . (2) does not exist. (3) lies between 1 and 2 . (4) lies between 2 and 3 .
Q62
Discriminant and conditions for rootsRoot relationships and Vieta's formulasView
If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then $a : b : c$ is: (1) $1 : 3 : 2$ (2) $3 : 1 : 2$ (3) $1 : 2 : 3$ (4) $3 : 2 : 1$
Q63
Complex Numbers ArithmeticTrigonometric/Polar Form and De Moivre's TheoremView
If $z$ is a complex number of unit modulus and argument $\theta$, then $\arg\left(\frac{1+z}{1+\bar{z}}\right)$ can be equal to (given $z \neq -1$) (1) $\theta$ (2) $\pi - \theta$ (3) $-\theta$ (4) $\frac{\pi}{2} - \theta$
Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1} - T_n = 10$, then the value of $n$ is: (1) 10 (2) 8 (3) 5 (4) 7
Q65
Arithmetic Sequences and SeriesProperties of AP Terms under TransformationView
If $x, y, z$ are positive numbers in A.P. and $\tan^{-1}x, \tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then which of the following is correct. (1) $6x = 3y = 2z$ (2) $6x = 4y = 3z$ (3) $x = y = z$ (4) $2x = 3y = 6z$
Q66
Arithmetic Sequences and SeriesTelescoping or Non-Standard Summation Involving an APView
The sum of first 20 terms of the sequence $0.7, 0.77, 0.777, \ldots\ldots$, is: (1) $\frac{7}{81}\left(179 + 10^{-20}\right)$ (2) $\frac{7}{9}\left(99 + 10^{-20}\right)$ (3) $\frac{7}{81}\left(179 - 10^{-20}\right)$ (4) $\frac{7}{9}\left(99 - 10^{-20}\right)$
Q67
Binomial Theorem (positive integer n)Find a Specific Coefficient in a Single Binomial ExpansionView
The term independent of $x$ in the expansion of $\left(\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right)^{10}$ is (1) 210 (2) 310 (3) 4 (4) 120
The expression $\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}$ can be written as: (1) $\tan A + \cot A$ (2) $\sec A + \operatorname{cosec} A$ (3) $\sin A \cos A + 1$ (4) $\sec A \operatorname{cosec} A + 1$
Q69
Straight Lines & Coordinate GeometryReflection and Image in a LineView
A ray of light along $x + \sqrt{3}y = \sqrt{3}$ gets reflected upon reaching $X$-axis, the equation of the reflected ray is (1) $y = \sqrt{3}x - \sqrt{3}$ (2) $\sqrt{3}y = x - 1$ (3) $y = x + \sqrt{3}$ (4) $\sqrt{3}y = x - \sqrt{3}$
Q70
Straight Lines & Coordinate GeometryTriangle Properties and Special PointsView
The $x$-coordinate of the incentre of the triangle that has the coordinates of midpoints of its sides as $(0,1)$, $(1,1)$ and $(1,0)$ is (1) $1 + \sqrt{2}$ (2) $1 - \sqrt{2}$ (3) $2 + \sqrt{2}$ (4) $2 - \sqrt{2}$
The circle passing through $(1, -2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point (1) $(5, -2)$ (2) $(-2, 5)$ (3) $(-5, 2)$ (4) $(2, -5)$
Given: A circle, $2x^2 + 2y^2 = 5$ and a parabola, $y^2 = 4\sqrt{5}x$. Statement-I: An equation of a common tangent to these curves is $y = x + \sqrt{5}$. Statement-II: If the line, $y = mx + \frac{\sqrt{5}}{m}$ $(m \neq 0)$ is their common tangent, then $m$ satisfies $m^4 - 3m^2 + 2 = 0$. (1) Statement-I is true; Statement-II is false. (2) Statement-I is false; Statement-II is true. (3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I. (4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and having centre at $(0, 3)$ is (1) $x^2 + y^2 - 6y - 5 = 0$ (2) $x^2 + y^2 - 6y + 5 = 0$ (3) $x^2 + y^2 - 6y - 7 = 0$ (4) $x^2 + y^2 - 6y + 7 = 0$
All the students of a class performed poorly in Mathematics. The teacher decided to give grace marks of 10 to each of the students. Which of the following statistical measures will not change even after the grace marks were given? (1) mode (2) variance (3) mean (4) median
Q77
Sine and Cosine RulesMulti-step composite figure problemView
$ABCD$ is a trapezium such that $AB$ and $CD$ are parallel and $BC \perp CD$. If $\angle ADB = \theta$, $BC = p$ and $CD = q$, then $AB$ is equal to (1) $\frac{p^2 + q^2}{p^2 \cos\theta + q^2 \sin\theta}$ (2) $\frac{\left(p^2 + q^2\right)\sin\theta}{(p\cos\theta + q\sin\theta)^2}$ (3) $\frac{\left(p^2 + q^2\right)\sin\theta}{p\cos\theta + q\sin\theta}$ (4) $\frac{p^2 + q^2\cos\theta}{p\cos\theta + q\sin\theta}$
Q78
Combinations & SelectionSubset Counting with Set-Theoretic ConditionsView
Let $A$ and $B$ be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is: (1) 219 (2) 211 (3) 256 (4) 220
Q79
3x3 MatricesDeterminant of Parametric or Structured MatrixView
If $P = \left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to (1) 5 (2) 0 (3) 4 (4) 11
The number of values of $k$, for which the system of equations: $(k+1)x + 8y = 4k$ $kx + (k+3)y = 3k - 1$ has no solution, is: (1) 2 (2) 3 (3) Infinite (4) 1
Q81
Chain RuleChain Rule with Composition of Explicit FunctionsView
If $y = \sec\left(\tan^{-1}x\right)$, then $\frac{dy}{dx}$ at $x = 1$ is equal to (1) 1 (2) $\sqrt{2}$ (3) $\frac{1}{\sqrt{2}}$ (4) $\frac{1}{2}$
Q82
Tangents, normals and gradientsFind tangent line with a specified slope or from an external pointView
The intercepts on the $x$-axis made by tangents to the curve, $y = \int_0^x |t|\, dt, x \in R$, which are parallel to the line $y = 2x$, are equal to (1) $\pm 3$ (2) $\pm 4$ (3) $\pm 1$ (4) $\pm 2$
Q83
Integration by PartsMultiple-Choice Primitive IdentificationView
Q84
Indefinite & Definite IntegralsIntegral Equation with Symmetry or SubstitutionView
Statement-I: The value of the integral $\int_{\pi/6}^{\pi/3} \frac{dx}{1 + \sqrt{\tan x}}$ is equal to $\frac{\pi}{6}$. Statement-II: $\int_a^b f(x)\, dx = \int_a^b f(a + b - x)\, dx$. (1) Statement-I is true; Statement-II is false. (2) Statement-I is false; Statement-II is true. (3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I. (4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
The area (in square units) bounded by the curves $y = \sqrt{x}$, $2y - x + 3 = 0$, $X$-axis and lying in the first quadrant is (1) 18 sq. units (2) $\frac{27}{4}$ sq. units (3) 9 sq. units (4) 36 sq. units
Q86
Applied differentiationApplied modeling with differentiationView
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 $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is (1) 3500 (2) 4500 (3) 2500 (4) 3000
Q87
Vectors Introduction & 2DMagnitude of Vector ExpressionView
If the vectors $\overrightarrow{AB} = 3\hat{i} + 4\hat{k}$ and $\overrightarrow{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ are the sides of a triangle $ABC$, then the length of the median through $A$ is: (1) $\sqrt{33}$ (2) $\sqrt{45}$ (3) $\sqrt{18}$ (4) $\sqrt{72}$
Q88
Vectors 3D & LinesMCQ: Relationship Between Two LinesView
If the lines $\frac{x-2}{1} = \frac{y-3}{1} = \frac{z-4}{-k}$ and $\frac{x-1}{k} = \frac{y-4}{2} = \frac{z-5}{1}$ are coplanar, then $k$ can have (1) exactly two values. (2) exactly three values. (3) any value. (4) exactly one value.
Q89
Vectors 3D & LinesShortest Distance Between Two LinesView
Distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is (1) $\frac{7}{2}$ (2) $\frac{9}{2}$ (3) $\frac{3}{2}$ (4) $\frac{5}{2}$
Q90
Binomial DistributionCompute Cumulative or Complement Binomial ProbabilityView
A multiple choice examination has 5 questions. Each question has three alternative answers out of which exactly one is correct. The probability that a student will get 4 or more correct answers just by guessing is: (1) $\frac{11}{3^5}$ (2) $\frac{10}{3^5}$ (3) $\frac{17}{3^5}$ (4) $\frac{13}{3^5}$