IITJEE Mains Mock test -1

IITJEE MAINS MOCK TEST 1

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SECTION – I   Straight Objective Type

 

Question 1. The roots of the equation  $x^4-x^2+1=0$

(1)   are concyclic points in the Argand diagram.

(2)   are vertices of a square.

(3)  are collinear.

(4)   none of the above

 

Question 2.  If  $(1+x+2.x^2)^{50}=\sum_{r=0}^{100} a_r.x^r$   then        $ \sum_{r=0}^{33} a_{3r}= ?$   equal to 

(1) $\cfrac{2^{100}+1}{3}$ $$ $$

(2)  $\cfrac{2^{100}-1}{3}$      $$   $$

(3)  $\cfrac{4^{100}-1}{3}$          $$ $$

(4)   $\cfrac{4^{100}+1}{3}$

 

Question 3.  If   $I=\int_{\cfrac{1}{4}}^{\cfrac{1}{2}} \; \sqrt{log_{\frac{1}{2}}x} dx \;$  then

(1)   $l\lt \cfrac{1}{4}$

(2)  $I\gt \frac{\sqrt{2}}{4}$

(3)  $I\lt \frac{\sqrt{2}}{4}$

(4)  None of these.

 

Question 4.  If a , b, c are distinct numbers in AP and  $\alpha\,,\,\beta$   and  $\gamma$   are distinct angles in AP then  $\begin{vmatrix}a&b&c\\ Cos\alpha & Cos\beta & Cos\gamma \\ Sin\alpha &  Sin\beta & Sin\gamma \end{vmatrix}=0$ is satisfied by 

(1) $\gamma – \beta = \cfrac{\pi}{2}$

(2) $\gamma – \beta = \pi$

(3) $\gamma – \beta = \cfrac{3\pi}{2} $

(4) None of these.

 

Question 5. let  $\lim_{x\to \cfrac{\pi}{4}} (\cfrac{(tan^{-1}x)^3-(Cot^{-1})^3}{(x-\cfrac{\pi}{4})})$

(1)  $\cfrac{6}{{\pi}^2+16}$

(2)  $\cfrac{-6}{{\pi}^2+16}$

(3) $\cfrac{3}{8}$

(4)  None of these.

 

Question 6.  The equation of the tangent to the curve  y(x-2)(x-5)-x+10=0  at the point where it cuts the x-axis is 

(1)  x=40y+10

(2)  x=y+10

(3) x=10y+10

(4) None of these.

 

Question 7.  Find mean deviation from mean of the first (2n+1) natural numbers.

(1)   $\cfrac{n(n+1)}{2(2n+1)}$

(2)   0

(3)   $\cfrac{n(n+1)}{2(2n+1)}$

(4) None of these.

 

Question 8.  In the following figure , AB , DE and  GF are parallel to each other and AD , BG and EF  are parallel to each other .

If CD : CE = CG : CB = 2 :1  then the value area $(\Delta AEG)$ : area $(\Delta  ABD )$ is equal  to

(1)  7 : 2

(2)   3 : 1

(3)   9 : 2

(4)  4 : 1

 

Question 9.   $\int Sin^{-1}(\cfrac{x}{\sqrt{x^2+1}})dx $   is equals 

(1)  $x.Sin^{-1}\Bigl(\cfrac{x}{\sqrt{1+x^2}}\Bigr)+C$

(2) $xtan^{-1}x-\cfrac{1}{2}.log(1+x^2)+C$

(3) $x.tan^{-1}x + C$

(4) None of these.

 

Question 10.  If x + y = d , ax + by = c and bx + ay = c form a triangle then the centroid of the triangle lies on the line.

(1) x-y=d

(2)  x=y

(3) ax+by=0

(4) None of these.

 

Question 11.   The line x+y=10  intersects the circle  $x^2+y^2-14x-16y+88=0$  at the points

(1)  (2,8) ,(4,6)

(2)  (7,3),(2,8)

(3)  (5,5) , (0,10)

(4) None of these.

 

Question 12.  Two tangents are drawn from the point (-2,0) to the parabola   $y^2=8x$. Then, the angle between the tangents is 

(1) $\cfrac{\pi}{4}$

(2) $\cfrac{\pi}{3}$

(3) $\cfrac{\pi}{2}$

(4) None of these.

 

Question 13.  Let d = distance of a point on the ellipse  $\cfrac{x^2}{3}+\cfrac{y^2}{1}=1$ from the origin , then 

(1)  $d\ge \sqrt{3}$

(2)  $d\le 1.$

(3)   $1\le d \le \sqrt{3}$

(4) None of these.

 

Question 14.   The solution of the D.E  $\cfrac{dy}{dx}=\cfrac{3x^2y^4+2xy}{x^2-2x^3y^3}$   is

(1) $\cfrac{y^2}{x}-x^3y^2+c=0$

(2) $\cfrac{x^2}{y}+x^3y^2+c=0$

(3)  $\cfrac{x^2}{y^2}+x^3y^3+c=0$

(4)  $\cfrac{x^2}{3y}-2x^3y^2+c=0$ 

 

Question 15.  For the curve  $y=be^{\cfrac{x}{a}}$

(1)  the sub tangent is of constant length.

(2) the subnormal is of constant length.

(3) the subnormal varies as the square of ordinate.

(4) the sub tangent varies as the radius vector.

 

Question 16.   The domain of  $f(x)=\cfrac{3}{4-x^2}\,+\,log_{10}(x^3-x)$  is

(1)   (1,2)

(2)   $(-1,2)\cup (2,\infty)$

(3)  $(-1,0)\cup (1,2)$

(4)  $(-1,0)\cup(1,2)\cup(2,\infty)$

 

Question 17.   $Cos\Bigl(4tan^{-1}\cfrac{2}{3}\Bigr)\,=$

(1)  $-\cfrac{6}{\sqrt{11}}$

(2)  $\cfrac{6}{\sqrt{11}}$ 

(3)  $\cfrac{3}{\sqrt{11}}$

(4)  None of these.

 

Question 18.   Let  R={(1,3),(4,2),(2,4),(2,3),(3,1)}  be a relation on the set  A = { 1,2,3,4 } . The relation R is 

(1)  Reflexive

(2)  Transitive

(3)  Not Symmetric

(4)  A function

 

Question 19.  If  P ⇒ ( q ∨ r )  is false , then the truth value of p , q , r are respectively 

(1) T , F , F

(2)  F , F , F

(3) F , T ,T

(4) T , T , F

 

Question 20.  If  $A^3=O$  , then      $I\, +\, A +\, A^2$  equals  ( Where I is the unit matrix of order same as that of square matrix A )

(1)  I – A

(2)  $(I\, – \,A)^{-1}$ 

(3)  $(I\,+\,A)^{-1}$

(4)  None of these.

 

SECTION – II   Numerical Value Type ( xxxxxxxxx.xxx form )

 

Question 21.  If a , b and  c  are all  positive , then 

$$  $$ $tan^{-1}\sqrt{\cfrac{a(a+b+c)}{bc}}+tan^{-1}\sqrt{\cfrac{b(a+b+c)}{ac}}+tan^{-1}\sqrt{\cfrac{c(a+b+c)}{ab}}$   equal to  

 

Question 22.   If the sides of a triangle are in the ratio  3 : 4 : 5, then 

$$  $$  $CosA\,+\,CosB\,+\,CosC$  equal to 

 

Question 23.   The sum of the real roots of the equation  (x+1)(x+2)(x+3)(x+4) = 105 , is 

 

Question 24.  A fair dice is tossed repeatedly until a six is obtained. Let X denote the number of tosses required . Then , the conditional probability that  $X\ge 6$  given  $X\gt 3$  equals to 

 

Question 25.  $\cfrac{Sin10°\,+\,Cos10°}{Cos10°\,-\,Sin20°}$  equals to 

 

End of test 1.

iitjee maths mock test 2 is on 24 March 2020 at 12.00 pm.

 

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