Q61
Roots of polynomialsVieta's formulas: compute symmetric functions of rootsView
If $\alpha , \beta , \gamma , \delta$ are the roots of the equation $x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x + 1 = 0$, then $\alpha ^ { 2021 } + \beta ^ { 2021 } + \gamma ^ { 2021 } + \delta ^ { 2021 }$ is equal to (1) 4 (2) 1 (3) - 4 (4) - 1
Q62
Complex Numbers Argand & LociIntersection of Loci and Simultaneous Geometric ConditionsView
For $n \in N$, let $S _ { n } = \left\{ z \in C : \left| z - 3 + 2i \right| = \frac { n } { 4 } \right\}$ and $T _ { n } = \left\{ z \in C : \left| z - 2 + 3i \right| = \frac { 1 } { n } \right\}$. Then the number of elements in the set $\left\{ n \in N : S _ { n } \cap T _ { n } = \phi \right\}$ is (1) 0 (2) 2 (3) 3 (4) 4
Q63
Standard trigonometric equationsSolve trigonometric equation for solutions in an intervalView
The number of solutions of $\cos x = \sin x$, such that $- 4 \pi \leq x \leq 4 \pi$ is (1) 4 (2) 6 (3) 8 (4) 12
Q64
Straight Lines & Coordinate GeometryLine Equation and Parametric RepresentationView
A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line (1) $2x + y = 9$ (2) $3x - 2y = 7$ (3) $x + 2y = 6$ (4) $2x - 3y = 3$
Let the locus of the centre $(\alpha , \beta),\ \beta > 0$, of the circle which touches the circle $x ^ { 2 } + (y - 1) ^ { 2 } = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is (1) $\frac { 32 \sqrt { 2 } } { 3 }$ (2) $\frac { 40 \sqrt { 2 } } { 3 }$ (3) $\frac { 64 } { 3 }$ (4) $\frac { 32 } { 3 }$
Q66
Sequences and series, recurrence and convergenceConvergence proof and limit determinationView
If $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } - n - 1 } + n\alpha + \beta \right) = 0$ then $8\alpha + \beta$ is equal to (1) 4 (2) - 8 (3) - 4 (4) 8
Q68
Sine and Cosine RulesHeights and distances / angle of elevation problemView
A tower $PQ$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower in two parts such that $QR = 15$ m. If from a point $A$ on the ground the angle of elevation of $R$ is $60 ^ { \circ }$ and the part $PR$ of the tower subtends an angle of $15 ^ { \circ }$ at $A$, then the height of the tower is (1) $5(2\sqrt { 3 } + 3)$ m (2) $5(\sqrt { 3 } + 3)$ m (3) $10(\sqrt { 3 } + 1)$ m (4) $10(2\sqrt { 3 } + 1)$ m
Q69
3x3 MatricesLinear System Existence and Uniqueness via DeterminantView
The number of $\theta \in (0,4\pi)$ for which the system of linear equations $3(\sin 3\theta) x - y + z = 2$ $3(\cos 2\theta) x + 4y + 3z = 3$ $6x + 7y + 7z = 9$ has no solution is (1) 6 (2) 7 (3) 8 (4) 9
Q70
Permutations & ArrangementsCounting Functions with ConstraintsView
The total number of functions, $f : \{1,2,3,4\} \rightarrow \{1,2,3,4,5,6\}$ such that $f(1) + f(2) = f(3)$, is equal to (1) 60 (2) 90 (3) 108 (4) 126
Q71
Stationary points and optimisationFind absolute extrema on a closed interval or domainView
If the absolute maximum value of the function $f(x) = (x ^ { 2 } - 2x + 7) e ^ { (4x ^ { 3 } - 12x ^ { 2 } - 180x + 31)}$ in the interval $[-3,0]$ is $f(\alpha)$, then (1) $\alpha = 0$ (2) $\alpha = - 3$ (3) $\alpha \in (-1,0)$ (4) $\alpha \in (-3,-1)$
Q72
Stationary points and optimisationFind critical points and classify extrema of a given functionView
The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is (1) $\frac { 27 } { 4 }$ (2) $\frac { 29 } { 4 }$ (3) $\frac { 37 } { 4 }$ (4) $\frac { 9 } { 2 }$
Q73
Indefinite & Definite IntegralsPiecewise/Periodic Function IntegrationView
For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x) = \begin{cases} x - \lfloor x \rfloor, & \text{if } \lfloor x \rfloor \text{ is odd} \\ 1 + \lfloor x \rfloor - x, & \text{if } \lfloor x \rfloor \text{ is even} \end{cases}$ Then, the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f(x) \cos(\pi x)\, dx$ is (1) 4 (2) 2 (3) 1 (4) 0
Let $ABC$ be a triangle such that $\overrightarrow { BC } = \vec { a }$, $\overrightarrow { CA } = \vec { b }$, $\overrightarrow { AB } = \vec { c }$, $|\vec{a}| = 6\sqrt{2}$, $|\vec{b}| = 2\sqrt{3}$ and $\vec{b} \cdot \vec{c} = 12$. Consider the statements: S1: $|\vec{a} \times \vec{b} + \vec{c} \times \vec{b}| - |\vec{c}| = 6(2\sqrt{2} - 1)$ S2: $\angle ABC = \cos^{-1}\sqrt{\frac{2}{3}}$. Then (1) both $S1$ and $S2$ are true (2) only $S1$ is true (3) only $S2$ is true (4) both $S1$ and $S2$ are false
Q78
Vectors: Lines & PlanesDistance Computation (Point-to-Plane or Line-to-Line)View
Let $P$ be the plane containing the straight line $\frac { x - 3 } { 9 } = \frac { y + 4 } { - 1 } = \frac { z - 7 } { - 5 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 5 }$ and $\frac { x } { 3 } = \frac { y } { 7 } = \frac { z } { 8 }$. If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^ { 2 }$ is equal to (1) $\frac { 147 } { 2 }$ (2) 96 (3) $\frac { 32 } { 3 }$ (4) 54
Q79
Binomial DistributionFind Parameters from Moment ConditionsView
If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is: (1) $\frac { 33 } { 2 ^ { 32 } }$ (2) $\frac { 33 } { 2 ^ { 29 } }$ (3) $\frac { 33 } { 2 ^ { 28 } }$ (4) $\frac { 33 } { 2 ^ { 27 } }$
Q80
Discriminant and conditions for rootsProbability involving discriminant conditionsView
If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x ^ { 2 } + \alpha x + \beta > 0$, for all $x \in R$, is (1) $\frac { 17 } { 36 }$ (2) $\frac { 4 } { 9 }$ (3) $\frac { 1 } { 2 }$ (4) $\frac { 19 } { 36 }$
Q81
Arithmetic Sequences and SeriesArithmetic-Geometric Hybrid ProblemView
Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^ { 2 } - 8ax + 2a = 0$ and $q$ and $s$ are the roots of the equation $x ^ { 2 } + 12bx + 6b = 0$, such that $\frac { 1 } { p }, \frac { 1 } { q }, \frac { 1 } { r }, \frac { 1 } { s }$ are in A.P., then $a ^ { - 1 } - b ^ { - 1 }$ is equal to $\_\_\_\_$.
Q82
Permutations & ArrangementsDictionary Order / Rank of a PermutationView
The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is $\_\_\_\_$.
Q83
Arithmetic Sequences and SeriesSummation of Derived Sequence from APView
Let $a _ { 1 } = b _ { 1 } = 1$, $a _ { n } = a _ { n - 1 } + 2$ and $b _ { n } = a _ { n } + b _ { n - 1 }$ for every natural number $n \geq 2$. Then $\sum _ { n = 1 } ^ { 15 } a _ { n } \cdot b _ { n }$ is equal to $\_\_\_\_$.
Q84
Binomial Theorem (positive integer n)Find the Largest Term or Coefficient in a Binomial ExpansionView
If the maximum value of the term independent of $t$ in the expansion of $\left( t ^ { 2 } x ^ { \frac { 1 } { 5 } } + \frac { 1 - x ^ { \frac { 1 } { 10 } } } { t } \right)^{10}$, $x \geq 0$, is $K$, then $8K$ is equal to $\_\_\_\_$.
Q85
CirclesCircles Tangent to Each Other or to AxesView
The sum of diameters of the circles that touch (i) the parabola $75x ^ { 2 } = 64(5y - 3)$ at the point $\left(\frac { 8 } { 5 }, \frac { 6 } { 5 }\right)$ and (ii) the $y$-axis, is equal to $\_\_\_\_$.
Let the equation of two diameters of a circle $x ^ { 2 } + y ^ { 2 } - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0,\infty)$ of the tangent to the hyperbola $3x ^ { 2 } - y ^ { 2 } = 3$ passing through the centre of the circle is equal to $\_\_\_\_$.
Q87
MatricesMatrix Power Computation and ApplicationView
Let $A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$ and $B = A - I$. If $\omega = \frac { \sqrt { 3 }\, i - 1 } { 2 }$, then the number of elements in the set $\left\{ n \in \{1,2,\ldots,100\} : A ^ { n } + \omega B ^ { n } = A + B \right\}$ is equal to $\_\_\_\_$.
Q88
Curve SketchingContinuity and Discontinuity Analysis of Piecewise FunctionsView
Let $f(x) = \begin{cases} \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 \geq 0 \\ \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 < 0 \end{cases}$, where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is $\_\_\_\_$.
Q89
Sequences and series, recurrence and convergenceConvergence proof and limit determinationView
If $\lim _ { n \rightarrow \infty } \frac { (n+1)^{k-1} } { n ^ { k + 1 } } \left[ (nk+1) + (nk+2) + \ldots + (nk+n) \right] = 33 \cdot \lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { k + 1 } } \cdot \left( 1 ^ { k } + 2 ^ { k } + 3 ^ { k } + \ldots + n ^ { k } \right)$, then the integral value of $k$ is equal to $\_\_\_\_$.
Q90
Vectors: Lines & PlanesDihedral Angle or Angle Between Planes/LinesView
The line of shortest distance between the lines $\frac { x - 2 } { 0 } = \frac { y - 1 } { 1 } = \frac { z } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 5 } { 2 } = \frac { z - 1 } { 1 }$ makes an angle of $\sin ^ { - 1 } \sqrt { \frac { 2 } { 27 } }$ with the plane $P : ax - y - z = 0$, $a > 0$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta - \gamma$ is equal to $\_\_\_\_$.