List of practice Questions

If n integers taken at random are multiplied together, the probability that the last digit of the product is 1, 3, 7 or 9 is
If \[ \tan\beta = 2\sin\alpha \sin\gamma \cdot \csc(\alpha+\gamma), \] then \(\cot\alpha,\; \cot\beta,\; \cot\gamma\) are in:
If \[ \cos^{-1}\alpha+\cos^{-1}\beta+\cos^{-1}\gamma = 3\pi, \] then the value of \(\alpha\beta+\beta\gamma+\gamma\alpha\) is:
A mixture of CO and CO$_2$ has vapour density 20 at STP. 100 g of this mixture contains ____ mole of CO
if ideal gas expands at constant temperature
The hybridization of atomic orbitals of N in NO$_2^-$, NO$_3^-$ and NH$_4^+$ are respectively
A line cuts the \(x\)-axis at \(A(7,0)\) and the \(y\)-axis at \(B(0,-5)\). A variable line \(PQ\) is drawn perpendicular to \(AB\) cutting the \(x\)-axis and \(y\)-axis at \(P\) and \(Q\) respectively. If \(AQ\) and \(BP\) intersect at \(R\), then the locus of \(R\) is:
A straight line through the origin \(O\) meets the parallel lines \(4x+2y=9\) and \(2x+y+6=0\) at points \(P\) and \(Q\) respectively. The point \(O\) divides the segment \(PQ\) in the ratio:
The number of values of c such that the straight line \(y=4x+c\) touches the curve \(\frac{x^2}{4}+y^2=1\) is
The plane x - 2y + 3z = 17 divides the line joining the points (-2, 4, 7) and (3, -5, 8) in the ratio
The ratio of the distances from the points (1,–1,3) and (3,3,3) to plane \(5x+2y-7z+9=0\) is
If the mean deviation of the numbers 1, 1+d, 1+2d, … , 1+100d from their mean is 255, then the common difference d is
If \[ y=(1-x)(1+x^2)(1+x^4)\cdots(1+x^{2^n}), \] then \(\dfrac{dy}{dx}\) at \(x=0\) is equal to
Consider \(p(x)\) a polynomial of degree 5 having extremum at \(x=-1,1\). Given \[ \lim_{x\to0}\left(\frac{p(x)}{x}-2\right)=4, \] the value of \(p[1]\) (greatest integer function) is}
The integral \[ \int\!\frac{\sin^2x\cos^2x}{(\sin^5x+\cos^3x\sin^2x+\sin^3x\cos^2x+x^5+x\cos^5x)}dx \] is of the form}
If \[ \int \sin(101x)\,\sin^{99}x\,dx = \frac{\sin(100x)\sin^{100}x}{k+5}+c, \] then \(\dfrac{k}{19}=\)
If \(g(x)=\cos x^2\), \(f(x)=\sqrt{x}\) and \(\alpha,\beta\;(\alpha<\beta)\) are the roots of \(18x^2-9\pi x+\pi^2=0\), then the area bounded by the curve \(y=(g\circ f)(x)\) and the lines \(x=\alpha,\;x=\beta\) and \(y=0\) is:
If \(y=f(x)\) passing through \((1,2)\) satisfies the differential equation \[ y(1+xy)\,dx - x\,dy = 0, \] then \(f(x)=\)
If \[ \Delta= \begin{vmatrix} f(x) & f\!\left(\dfrac{1}{x}\right)+f(x)\\ 1 & f\!\left(\dfrac{1}{x}\right) \end{vmatrix} =0 \] where \(f(x)\) is a polynomial and \(f(2)=17\), then \(f(5)=\) ?
The distance between line \( \vec r =2\hat i-2\hat j+3\hat k+\lambda(\hat i-\hat j+4\hat k) \) and plane \( \vec r\cdot(\hat i+\hat j+\hat k)=5 \) is
The symbolic form of logic of the circuit given below is
The number of 4 digit even numbers whose sum of digits is 34
The number of ordered triplets of positive integers satisfying \(20\le x+y+z\le50\) is
If \[ \sum_{r=1}^{n} a_r = \frac{n(n+1)(n+2)}{6}\quad \forall\, n \ge 1, \] then \[ \lim_{n\to\infty}\sum_{r=1}^{n}\frac{1}{a_r} = \]
Value of \[ \sum_{k=1}^{\infty}\sum_{r=0}^{k}\frac{1}{3^{k}}\binom{k}{r} \] is: