Mathematics (25) Full Length

PHYSICS

Q 1.

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in

Q 2.

Let S be the sum of the first 9 term of the series :

$$(x+ka)\;+\{(x^2+(k+2)\ a\}\;+\{x^3+(k+4)\ a\}+\{x^4+(k+6)\ a\}........... \;Where\;a\neq0\;and\;x\neq1$$$$if\; S={x^{10}-x+45a(x-1)\over{x-1}},\;then\;k\;is \;equal\;to$$

Q 3.
Let y = y(x) be the solution of the differential equation, $$xy'-y=x^2(xcosx+sinx),x>0.$$. If $$y(\pi)=\pi,$$ then $$y'' \Bigg(\frac\pi2\Bigg)+y\Bigg(\frac\pi2\Bigg)$$ is equal to :

Q 4.

Let f : $$(0,\infty)\rightarrow(0,\infty)$$ be a differentiable function such that f(1) = $$\lim\limits_{t\to x}{t^2f^2(x)-x^2f^2(t)\over{t-x}}=0.$$ if f(x) is equal to :

Q 5.
Box I contains 30 cards numbered I to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

Q 6.

The imaginary part of

$$\Big(3+2\sqrt{-54}\Big)^{\frac12}-\Big(3-2\sqrt{-54}\Big)^{\frac12}$$  can be

Q 7.
A triangle ABC lying in the first quadrant has two vertices as $$A(1, \ 2)$$ and $$B(3,\ 1)$$. If $$\angle BAC=90^{\circ}$$, and ar$$(\triangle ABC)=5\sqrt5\ sq$$ . units, then the abscissa of the vertex $$C$$is :

Q 8.

Contrapositive o the statement :

'If a function f is differentiable at a, then it is also continuous at a', is :

Q 9.
Area (in sq. units) of the region outside $${\mid x\mid\over2}+{\mid y\mid\over3}=1$$ and inside the ellipse $${x^2\over4}+{y^2\over9}=1$$ is :

Q 10.
A plane passing through the point (3,1,1) contains two lines whose direction ratios are 1, – 2, 2 and 2, 3,–1 respectively. If this plane also passes through the point($$\alpha$$,–3,5), then $$\alpha$$ is equal to

Q 11.
Let f(x) = |x – 2| and g(x) = f(f(x)), x $$\in$$ [0, 4]. Then $$\int\limits_0^3$$(g(x) f(x)) dx is equal to :

Q 12.
The circle passing through the intersection of the circles, x2 + y2 – 6x = 0 and x2 + y2 – 4y = 0, having its centre on the line, 2x – 3y + 12 = 0, also passes through the point :

Q 13.
The value of $$\begin{pmatrix}{1+sin{2\pi\over9}+icos{2\pi\over9}}\over{1+sin{2\pi\over9}-icos{2\pi\over9}}\end{pmatrix}^3$$ is :

Q 14.

Let $$a,\;b,\;c,\;\in\;R$$ be all non-zero satisfy $$a^3+b^3+c^3=2$$. If the matrix

$$A=\begin{bmatrix}a&b&c\\b&c&a\\c&a&b\end{bmatrix}$$ stratifies $$A^TA=I$$ , then a value of abc can be :

Q 15.
Two vertical poles AB = 15 m and CD = 10 m are standing apart on a horizontal ground with points A and C on the ground. If P is the point of intersection of BC and AD, then the height of P (in m) above the line AC is :

Q 16.
The function f(x) = $$\begin{cases}\frac\pi4+tan^{-1}x,&|x|\leq1\\\frac12(|x|-1),&|x|>1\end{cases}$$is:

Q 17.
If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $$\Bigg(0,\frac32\Bigg)$$, and $$\Bigg(\frac12,\;2\Bigg)$$ , then

Q 18.
Let n > 2 be an integer. Suppose that there are n Metro stations in a city located around a circular path. Each pair of nearest stations is connected by a straight track only. Further, each pair of nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of n is

Q 19.
The value of $$\displaystyle\sum_{r=0}^{20}\;^{50-r} C_6$$ is equal to :

Q 20.

If the system of equations
$$x+y+z=2\\2x+4y-z=6\\3x+2y+\lambda z=\mu$$

Q 21.
If the equation of a plane P, passing through the intersection of the planes, x + 4y – z + 7 = 0 and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b $$\in$$ R, then the distance of the point (3, 2, –1) from the plane P is ……
Q 22.
Let $$\overline{a},\;\overline{b}$$ and $$\overline{c}$$ be three unit vectors such that $$|a-c|^2+|a-c|^2=8$$. Then $$|a+2\overline{b}|^2+|a+2\overline{c}|^2$$ is equal to
Q 23.
Let {x} and [x] denote the fractional part of x and the greatest integer $$\leq$$ x respectively of a real number x. if $$\int\limits_0^n \left\{x\right\}dx,\;\int\limits_0^n[x]dx$$ and $$10(n^2-n),(n\in N,n>1)$$ are three consecutive terms of a G.P. then n is equal to ……
Q 24.
If $$y=\displaystyle\sum_{k=1}^6k\cos^{-1}\left\{\frac 35 \cos k x-\frac45\sin k x \right\}$$ then $${dy\over d x}$$ at $$x=0$$is
Q 25.
The integral $$\int\limits_0^2||x-1|-x|dx$$ is equal to ;