List of practice Questions

For any real number x, let [ x ] denote the largest integer less than equal to x Let f be a real valued function defined on the interval [-10,10] by $f(x)=\begin{cases} x-[x], & \text { if }(x) \text { is odd } \\ 1+[x]-x & \text { if }(x) \text { is even }\end{cases}$Then the value of$ \frac{\pi^2}{10} \int\limits_{-10}^{10} f(x) \cos \pi x d x$ is :

The slope of the tangent to a curve C : y=y(x) at any point [x, y) on it is $\frac{2 e ^{2 x }-6 e ^{- x }+9}{2+9 e ^{-2 x }}$ If C passes through the points $\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right) $and $\left(\alpha, \frac{1}{2} e ^{2 \alpha}\right)$$ then $e ^\alpha$ is equal to :

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 :

The general solution of the differential equation $\left(x-y^2\right) d x+y\left(5 x+y^2\right) d y=0$ is :

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 :

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

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 :

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 :

For $n \in N$, let $S _{ n }=\left\{ z \in C :| z -3+2 i |=\frac{ n }{4}\right\}$ and $T_n=\left\{z \in C:|z-2+3 i|=\frac{1}{n}\right\}$ Then the number of elements in the set $\left\{ n \in N : S _{ n } \cap T _{ n }=\phi\right\}$ is :

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 :

The number of solutions of $|\cos x |=\sin x$, such that $-4 \pi \leq x \leq 4 \pi$ is :

Which of the following statements is a tautology?

Let $f : R \rightarrow R$ be a continuous function such that $f (3x)- f(x)= x$. If $f(8)=7$, then $f (14)$ is equal to :

The minimum value of the twice differentiable function $f(x)=\int\limits_0^x e^{x-1} f^{\prime}(t) d t-\left(x^2-x+1\right) e^x, x \in R$, is :

Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?

The value of 2 sin(12°) – sin(72°) is

The negation of the Boolean expression ((~ q) ∧ p) ⇒ ((~ p) ∨ q) is logically equivalent to:

If
$\begin{array}{l}\sum_{K=1}^{10}K^2\left(^{10}C_K\right)^2 = 22000L,\end{array}$then L is equal to _____.

Let AB be a chord of length 12 of the circle
$\begin{array}{l}(x-2)^2 + (y+1)^2=\frac{169}{4}.\end{array}$
If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _____ .

Let the eccentricity of the hyperbola
$H : \frac{x²}{a²} - \frac{y²}{b²} = 1$
be √(5/2) and length of its latus rectum be 6√2, If y = 2x + c is a tangent to the hyperbola H. then the value of c² is equal to

If the tangents drawn at the points O(0, 0) and P(1 + √5, 2) on the circle x² + y² – 2x – 4y = 0 intersect at the point Q, then the area of the triangle OPQ is equal to

If two distinct points Q, R lie on the line of intersection of the planes –x + 2y – z = 0 and 3x – 5y + 2z = 0 and
$PQ = PR = \sqrt{18}$
where the point P is (1, –2, 3), then the area of the triangle PQR is equal to

The acute angle between the planes P₁ and P₂, when P₁ and P₂ are the planes passing through the intersection of the planes 5x + 8y + 13z – 29 = 0 and 8x – 7y + z – 20 = 0 and the points (2, 1, 3) and (0, 1, 2), respectively, is

Let l be a line which is normal to the curve $y = 2x^2 + x + 2$ at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .