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An integer is chosen at random. The probability that, sum of the digits of its square is 24 is:

In $\Delta ABC$ if $\sin A\sin B=\frac{ab}{c^2},$ then $\angle C$ is

The coefficient of the term independent of x in the expansion of $(1+x+2x^3){(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$

If $f\left(x\right)=\{\begin{array}{cc}\frac{(1-\cos 2x)(3+\cos x)}{x\tan ax}& x<0;\\ \frac{x(e^x-1)}{1-\cos x}& x>0;\end{array}$ then find the value of $a$ such that $\underset{x\to 0}{lim}f\left(x\right)$ exists

Examine that sin |x| is a continuous function.

The value of $y=(0.36{)}^{{\log }_{0.25}(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\cdots \infty )}$ is

The value of integral $\underset{-1/\sqrt{3}}{\overset{1/\sqrt{3}}{\int }}\frac{x^4}{1-x^4}\cdot {\cos }^{-1}\left(\frac{2x}{1-x^2}\right)dx$ is

Find
the value of a,
b, c,
and d from
the equation:

The area (in sq. units) of the region bounded by the curve $x^2=4y$ and the straight line $x=4y-2$ is:

The number of ways of selecting $7$ cards of same suit from a pack of $52$ is

If $|\overrightarrow{a}|=2,|\overrightarrow{b}|=7$ and $\overrightarrow{a}\times \overrightarrow{b}=3\hat{i}+2\hat{j}+6\hat{k},$ then $\overrightarrow{a}\cdot \overrightarrow{b}=$

Find the radian measure corresponding to the following degree measures:

(i) ${300}^{\circ }$

(ii) ${35}^{\circ }$

(iii) $-{56}^{\circ }$

(iv) ${135}^{\circ }$

(v) $-{300}^{\circ }$

(vi) $7^{\circ }{30}^{\prime }$

(vii) ${125}^{\circ }{30}^{\prime }$

(viii) $-{47}^{\circ }{30}^{\prime }$

If f (x) = 3x² + 15x + 5, then
the approximate value of f (3.02) is

A.
47.66 B. 57.66 C. 67.66 D.
77.66

Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?

If the coefficients of $r^{th}$, $(r+1{)}^{th}$ and $(r+2{)}^{th}$ terms in the binomial expansion of $(1+y{)}^m$are in $A.P.$, then $m$ and $r$ will satisfy the equation

The normal at a point P(x, y) of a curve meets the x-axis at Q and N is the foot of the perpendicular from P on x-axis, If $NQ=\frac{x(1+y^2)}{1+x^2}$, then equation of the curve passing through (3, 1) is

Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(-4, 0, 0) is equal to 10.

If $\frac{1+4p}{p},\frac{1-p}{4},\frac{1-2p}{2}$ are probabilities of three mutually exclusive events, then

$\frac{co{t}^2{15}^{\circ }-1}{co{t}^2{15}^{\circ }+1}$ =

The equation of the conic with focus at (1,-1) directrix x-y+1=0 and eccentricity$\sqrt{2}$ is

Which of the following options are always true in their domain?

The value of the integral $\underset{\frac{-\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}(x^2+\ln \frac{\pi +x}{\pi -x})\cos xdx$ is

All the integral values of $a$ for which the quadratic equation $(x-a)(x-10)+1=0$ has integral roots, are

The general solution of the differential equation $\left(2x\ln y\right)dx+(\frac{x^2}{y}+3y^2)dy=0$ is

(where $c$ is constant of integration)

Find the area of the region bounded by the curves $y^2=9xandy=3x$.

Complex numbers $z$ satisfy the equation $|z-\frac{4}{z}|=2$. Then the difference between the least and the greatest moduli of complex numbers is

Two vectors having equal magnitudes of A make an angle θ with each other. Find the magnitude B of the resultant vector and the angle it makes with any one of the vectors?

For each
of the differential equations given below, indicate its order and
degree (if defined).

(i)

(ii)

(iii)

prove that Sin of a + b + sin of a minus b upon cost of a + b + cos of a minus b is equal to 10

In the following exericise determine whether the given planes are parallel or perpendicular and in case they are neither, find the angle between them.

7x +5y +6z+ 30 =0 and 3x -y- 10z+ 4=0

2x+y+3z-2=0 and x-2y+5=0

2x- 2y+ 4z+ 5= 0 and 3x- 3y+ 6z- 1= 0

2x-y+3z-1=0 and 2x-y+3z+3=0

4x+8y+z-8=0 and y+z-4=0

Let f:
R → R be the Signum Function defined as

and g:
R → R be the Greatest Integer Function given by
g(x) = [x], where [x] is greatest integer
less than or equal to x. Then does fog and gof
coincide in (0, 1]?

The function $f\left(x\right)=\frac{9x}{5}+32$ is the formula to convert $x^{\circ }C$ to Fahrenheit units. Then at what ${}^{\circ }C$ both Celsius and Fahrenheit is same?

If $A=\{5,7,9,11\},B=\{9,10\}$ and $aRb$ means $a<b$ where $a\in A,b\in B,$ then which of the following are true?

If $a{x}_1^2+b{y}_1^2+c{z}_1^2=a{x}_2^2+b{y}_2^2+c{z}_2^2=a{x}_3^2+b{y}_3^2+c{z}_3^2=d,$ and

$a{x}_2{x}_3+b{y}_2{y}_3+c{z}_2{z}_3=a{x}_3{x}_1+b{y}_3{y}_1+c{z}_3{z}_1=a{x}_1{x}_2+b{y}_1{y}_2+c{z}_1{z}_2=f$ then the value of $\left|\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right|$ is:

The
general solution of a differential equation of the type
is

A.

B.

C.

D.

Find the distance of the point (2, 3, -5) from the plane $\overrightarrow{r}.(\hat{i}+2\hat{j}-2\hat{k})=9$.