## Maths Full length Latest Pattern-2

### PHYSICS

##### Q 1.
If the quadratic equation $$\alpha^2(x + 1)^2 + \beta^2(2x^2 – x + 1) – 5x^2 – 3 = 0$$ is satisfied for all $$x\in R$$, then number of ordered pair $$(\alpha, \beta)$$ which is possible is/are

##### Q 2.
n arithmetic means are inserted between two sets of numbers $$a, a^2$$ & $$b, b^2$$ where$$a, b \in R$$. Suppose $$m^{th}$$mean between these two sets of numbers is same, then a + b equals

##### Q 3.
If the sum of n terms of an A.P. is $$cn^2$$, then the sum of cubes of these n terms is

##### Q 4.
If a, b, c are real and $$x^3–3b^2 x + 2c^3$$ is divisible by (x – a) and (x – b), then $$a + b + c \ is (a \neq b)$$

##### Q 5.
If the equation x – sinx = p has exactly one positive root, then complete set of values of p is

##### Q 6.
If A is a skew symmetric matrix of odd order, then its inverse is

##### Q 7.
The eccentricity of conic $$4(2y – x – 3)^2 + 9(2x + y –1)^2 = 80$$is

##### Q 8.
A dice is rolled 4 times. The probability of getting a larger number than the previous number each time is

##### Q 9.
Area bounded by the curve $$y = tan^{–1}x$$ and $$y = cot^{–1}x$$ between x = 0 and x = 1 is

##### Q 10.
If p is false and q is true, then

##### Q 11.
If the function $$f : [0, 27] \to$$ R is differentiable and $$0 < \alpha < 1 < \beta < 2 < \gamma < 3$$, then $$\int\limits^{27}_0f\ (t)dt$$  is equal to

##### Q 12.
Least positive root of the equation tan $$x = x^2 + 1 – 2x$$ lies in the interval

##### Q 13.

Given that f satisfies $$|f(x) – f(y)| \leq |x – y|$$ for all x and y in [a, b], then

##### Q 14.
If $$x, y, z, t \in R$$ and $$sin^{-1}x+ cos^{-1} y+ cosec^{-1} z\geq t^2,-t\sqrt\pi+\frac{9\pi}4$$ , then principal value of $$sin ^{-1}(sin \ t^2)$$ is

##### Q 15.
In a $$\Delta$$ABC, if $$\angle C =30$$º and $$\angle B=22\frac12$$°, then the ratio in which the line joining A and circumcentre O divides BC is

##### Q 16.
Let $$f_1(x)$$ and $$f_2(x)$$ be even and odd functions respectively, where $$x^2f_1(x)-2f_1(\frac1x)=f_2(x)$$ , then value of $$f_1(3)$$ =

##### Q 17.
If the vectors $$\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$$ and $$\vec{b}=3\hat{i}-4\hat{j}$$ are the adjacent sides of a parallelogram. Then the vector along the bisector of the angle between $$\vec{a}$$ and $$\vec{b}$$ having magnitude $$5 \sqrt3$$ is

##### Q 18.
If the function$$f(x) = x^3 + e^{x^/4}$$ and $$g(x) = f^{–1}(x)$$, then the value of g'(1) =

##### Q 19.
The distance between the line $$\vec{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})$$ and the plane $$\vec{r.}(\hat{i}+5\hat{j}+\hat{k})=5$$ is

##### Q 20.
Equation of common tangent with positive slope to the circle $$(x – 4)^2 + y^2 = 4^2$$ and hyperbola $$4x^2 – 9y^2 = 36$$ is

##### Q 21.
A curve passing through (1, 2) and satisfying the differential equation $$\int\limits_0^xt^2y(t)dt=x^3y(x)(x>0)$$ is $$x^2y=c$$. find the value of c.
##### Q 22.
For all complex numbers $$z_1, z_2$$ satisfying$$|z_1| = 15$$ and $$|z_2 – 4 – 3i| = 5$$ then the minimum value of $$|z1 – z2|$$ is
##### Q 23.
If $$f(x)$$ is a polynomial function and $$f(a) = f '(a) = 0$$ and$$f ''(a)$$is non-zero, then $$\lim\limits_{x \to a} \frac{f(x)}{f'(x)}[\frac{f'(x)}{f(x)}]=$$(where [·] denotes G.I.F.)
##### Q 24.
Let $$f(x)$$ and $$g(x)$$ be differentiable for $$– 1 \leq x \leq3$$ such that$$f ( -1)= 1, g(- 1)= 2$$ and $$g(3)= 5$$ . Let there exists a real number c in [–1, 3] such that f’ (c) = 2g’ (c) then the value of f (3) is
##### Q 25.
If $$\vec x,\vec y$$ , and $$\vec z$$ are unit vectors such that $$|\vec{x}-\vec{y}|^2+|\vec{y}-\vec{z}|^2+|\vec{z}-\vec{x}|^2=9$$, then $$|3\vec x+5\vec y+5\vec z |=$$
##### Q 26.
If for 15 number of observations, $$\sum x =150,\sum x^2=3000$$ . But in the above calculation one observation i.e. 10 was taken wrong and was replaced by its correct value 40. Then the correct variance is
##### Q 27.
Let $$^pC_q$$ be the number of ways in which 5 tickets are selected from 20 tickets from 1 to 20 so that no two consecutive numbered tickets are selected, then $$p – q$$ may be
##### Q 28.
The value of $$2^{10}C_0+\frac{2^2}{2} ^{10}C_1+\frac{2^3}3 ^{10}C_2+\frac{2^4}4^{10}C_3+......+\frac{2^{11}}{11} ^{10}C_{10}=\frac {a{^b}-1}{11}$$,then $$bc_a$$ is,
##### Q 29.
Let $$I=\int\limits_0^2\frac{15+4x-2x{^2}}{e{^6}^{(x-1)}+1}dx,$$then$$\frac6{49}I=$$
##### Q 30.
Total number of ways of selecting four letters from the word ‘ELLIPSE’