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MHT-CET 2025 (19 April Shift II), Paper I, Mathematics question paper

50 questions · 100 marks · 90 minutes · +2 per correct answer, no negative marking

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  1. 1
    The last column in the truth table of the statement pattern [p→(q∧∼p)]∨[(p∨∼q)∧p]\lbrack p\rightarrow (q\land \sim p)\rbrack \vee \lbrack(p\vee \sim q) \land p\rbrack is

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  2. 2
    A straight line through the origin OO meets the line 3y=10−4x3y= 10 - 4x and 8x+6y+5=08x+ 6y+ 5 = 0 at the points A and B respectively. Then O divides the segment AB in the ratio

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  3. 3
    The position of a point in time tt is given by x=a+bt−ct2,y=at+bt2x= a + bt -ct^{2},y= at +bt^{2}. It's resultant acceleration at time tt in seconds is given by

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  4. 4
    Let OA‾=a‾,OB‾=b‾\overline{OA}=\overline{a},\overline{OB}=\overline{b} and if the vector along the angle bisector of ∠AOB\angle AOB is given by xa‾∣a‾∣+y b‾∣ b‾∣x\frac{\overline{a}}{|\overline{a}|}+y\frac{\overline{\text{ }b}}{|\overline{\text{ }b}|} then

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  5. 5
    In triangle ABC , the point P divides BC internally in the ratio 3:43:4 and Q divides CA internally in the ratio 5:35:3. If AP and BQ intersect in a point GG, then GG divides APAP internally in the ratio

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  6. 6
    The derivative of y=(1−x)(2−x)y= (1 -x)(2 -x)(n-x) at x=1x= 1 is

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  7. 7
    If X∼B(n,p)X\sim B(n,p) then P(X=k)P(X=k−1)=\frac{P(X=k)}{P(X=k- 1)}=

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  8. 8
    The differential equation of all straight lines passing through the point (1,−1)(1, - 1) is

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  9. 9
    The first derivative of the function (cos⁡−1(sin⁡1+x2)+xx)\left( \cos^{- 1}\left( \sin\sqrt{\frac{1 +x}{2}} \right)+x^{x} \right) with respect to xx at x=1x= 1 is

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  10. 10
    Let u‾,v‾,w‾\overline{u},\overline{v},\overline{w} be the vectors such that ∣u‾∣=1,∣v‾∣=2,∣w‾∣=3.|\overline{u}| = 1,|\overline{v}| = 2,|\overline{w}| = 3. If the projection v‾\overline{v} along u‾\overline{u} is equal to that of w‾\overline{w} along u‾\overline{u} and the vectors vˉ,wˉ\bar{v},\bar{w} are perpendicular to each other then ∣u‾−v‾+w‾∣|\overline{u}-\overline{v}+\overline{w}| equals

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  11. 11
    The area enclosed between the curves y2=4xy^{2}= 4x and y=∣x∣y= |x| is

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  12. 12
    If tan⁡A=1x(x2+x+1),tan⁡B=xx2+x+1\tan A =\frac{1}{\sqrt{x\left( x^{2}+x+ 1 \right)}},\tan B =\frac{\sqrt{x}}{\sqrt{x^{2}+x+ 1}} and tan⁡C=x−1+x−2+x−3\tan C =\sqrt{x^{- 1}+x^{- 2}+x^{- 3}} then

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  13. 13
    Let XX be a discrete random variable. The probability distribution of X is given below
    XX3010−10-10
    P(X)P(X)15\frac{1}{5}AABB
    and E(X)=4E(X) = 4, then the value of ABAB is equal to

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  14. 14
    The projection of the line segment joining the points (2,1,−3)(2,1, - 3) and (−1,0,2)( - 1,0,2) on the line whose direction ratios are 3,2,63,2,6 is

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  15. 15
    If x25+y25=a25x^{\frac{2}{5}}+y^{\frac{2}{5}}=a^{\frac{2}{5}} then dy dx=\frac{dy}{\text{ }dx}=

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  16. 16
    The perpendicular distance between the lines given by (x−2y+1)2+k(x−2y+1)=0(x- 2y+ 1)^{2}+ k(x- 2y+ 1) = 0 is 5\sqrt{5}, then k=k =

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  17. 17
    The value of sin⁡−1(−12)+cos⁡−1(−12)− cot⁡−1(−13)+tan⁡−1(−3) is\sin^{- 1}\left( -\frac{1}{\sqrt{2}} \right)+\cos^{- 1}\left( -\frac{1}{2} \right)- \ \cot^{- 1}\left( -\frac{1}{\sqrt{3}} \right)+\tan^{- 1}( -\sqrt{3})\text{~is}

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  18. 18
    If triangle ABC is a right angled at A and tan⁡B2\tan\frac{B}{2}, tan⁡C2\tan\frac{C}{2} are roots of the equation ax2+bx+c=0ax^{2}+bx+c= 0, a≠0a\neq 0, then

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  19. 19
    If a2+b2+c2=r2a^{2}+b^{2}+c^{2}=r^{2}, then the value of tan⁡−1(abcr)+tan⁡−1(bcar)+tan⁡−1(cabr)=\tan^{- 1}\left( \frac{ab}{cr} \right)+\tan^{- 1}\left( \frac{bc}{ar} \right)+\tan^{- 1}\left( \frac{ca}{br} \right)=

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  20. 20
    The value of ∫131(x−x3)13x4dx\int_{\frac{1}{3}}^{1} \frac{\left( x-x^{3} \right)^{\frac{1}{3}}}{x^{4}}dx is

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  21. 21
    The feasible region for the constraints x−2⩽y,x⩾y−1,x⩾2,y⩽4,x,y⩾0x- 2 \leqslant y,x\geqslant y- 1,x\geqslant 2,y\leqslant 4,x,y\geqslant 0, is____.\_\_\_\_.

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  22. 22
    A plane passes through (2,1,2)(2,1,2) and (1,2,1)(1,2,1) and parallel to the line 2x=3y2x= 3y and z=1z= 1, then the plane also passes through the point

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  23. 23
    In a game, 3 coins are tossed. A person is paid ₹150 if he gets all heads or all tails and he is supposed to pay ₹50 if he gets one head or two heads. The amount he can expect to win / lose on an average per game in ₹ is

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  24. 24
    If sin⁡(sin⁡−115+cos⁡−1x)=1\sin\left( \sin^{- 1}\frac{1}{5}+\cos^{- 1}x \right)= 1, then the value of xx is

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  25. 25
    If two sides of a triangle are 3−2\sqrt{3}- 2 and 3+2\sqrt{3}+ 2 units and their included angle is 60∘60^{\circ}, then the third side of the triangle is

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  26. 26
    The principal increases continuously in a newly opened bank at the rate of 10%10\% per year. An amount of Rs. 2000 is deposited with this bank. How much will it become after 5 years? ( e0.5=1.648e^{0.5}= 1.648 )

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  27. 27
    If A=[12−14]A=\begin{bmatrix} 1 & 2 \\ - 1 & 4 \end{bmatrix} and A−1=αI+βAα,β∈RA^{- 1}=\alpha I+\beta A\alpha,\beta\in R where II is the identity matrix of order 2 , hen 4(α+β)=4(\alpha+\beta) =

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  28. 28
    Which of the following are pairs of equivalent switching circuits? (A) (p∧q)∨(p∧r)(p \wedge q) \vee (p \wedge r) (B) p∨(q∧r)p \vee (q \wedge r) (C) p∧(q∨r)p \wedge (q \vee r) (D) p∧q∧rp \wedge q \wedge r (E) (p∧q)∨r(p \wedge q) \vee r

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  29. 29
    The solution of dy dx=(x+y)2\frac{dy}{\text{ }dx}= (x+y)^{2} is

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  30. 30
    The equation of the plane passing through the line of intersection of the planes x+y+z=1x+y+z= 1 and 3x+4y+5z=23x+ 4y+ 5z= 2 and perpendicular to the XY - plane is

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  31. 31
    A normal is drawn at a point P(x,y)P(x,y) of a curve y=f(x)y= f(x). The normal meets the X axis at Q . l(PQ)=kl(PQ) = k. ( k is a constant) Then equation of the curve through (0,k)(0,k) is

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  32. 32
    If f(x)=(27−2x)13−39−3(243+5x)15,x≠0f(x) =\frac{(27 - 2x)^{\frac{1}{3}}- 3}{9 - 3(243 + 5x)^{\frac{1}{5}}},x\neq 0 is continuous at x=0x= 0, then the value of f(0)f(0) is

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  33. 33
    lim⁡x→0∣x∣∣x∣+x2=\underset{x\rightarrow 0}{\lim} \frac{|x|}{|x| +x^{2}}=

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  34. 34
    The value of ∫−33sin⁡7xcos⁡16x dx\int_{- 3}^{3} \sin^{7}x\cos^{16}x\text{ }dx is

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  35. 35
    The coordinates of the foot of the perpendicular drawn from a point P(−1,1,2)P( - 1,1,2) to the plane 2x−3y+z−11=02x- 3y+z- 11 = 0 are

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  36. 36
    The domain of the function f(x)= 7−xPx−1f(x) =\ ^{7 -x}P_{x- 1} is

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  37. 37
    ∫ x+sin⁡x1+cos⁡xdx=\int_{}^{}\ \frac{x+ \sin x}{1 + \cos x}dx=

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  38. 38
    If a‾,b‾,c‾\overline{a},\overline{b},\overline{c} are three vectors such that a‾∣=3\overline{a}\mid = 3, ∣b‾∣=5,∣c‾∣=7|\overline{b}| = 5,|\overline{c}| = 7 then ∣a‾−b‾∣2+∣b‾−c‾∣2+∣c‾−a‾∣2|\overline{a}-\overline{b}|^{2}+ |\overline{b}-\overline{c}|^{2}+ |\overline{c}-\overline{a}|^{2} does not exceed

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  39. 39
    Total number of 3-digit numbers, whose g.c.d with 36 is 2 , is

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  40. 40
    If z−12z+1\frac{z- 1}{2z+ 1} is an imaginary number and if it represents a circle then its radius is

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  41. 41
    If ∫ tan⁡4x dx=atan⁡3x+btanx+cx+k\int_{}^{}\ \tan^{4}x\text{ }dx= a\tan^{3}x+ btanx+ cx+ k (where k is the constant of integration) then the value of a−b+c=a - b + c =

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  42. 42
    The lines x−31=y−21=z−5−k\frac{x- 3}{1}=\frac{y- 2}{1}=\frac{z- 5}{-k} and x−4k=y−31=z−32\frac{x- 4}{k}=\frac{y- 3}{1}=\frac{z - 3}{2} are coplanar, hence k=k =

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  43. 43
    ∫ x dx(x−1)(x−2)=\int_{}^{}\ \frac{x\text{ }dx}{(x- 1)(x- 2)}=

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  44. 44
    Let A and B are independent events with P(B)=25,P(A∪B)=1120P(B) =\frac{2}{5},P(A \cup B) =\frac{11}{20}, then P(A′∣B)P\left( A^{'}\mid B \right) is root of the equation

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  45. 45
    The equation of tangent to the curve y=cos⁡(x+y)y= \cos(x+y) where −2π≤x≤2π- 2\pi\leq x\leq 2\pi and.which is parallel to the line x+2y=0x+ 2y= 0, is

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  46. 46
    The eccentricity of the ellipse 9x2+5y2−30y=09x^{2}+ 5y^{2}- 30y= 0 is

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  47. 47
    If the tangent at (1,7)(1,7) to the curve x2=y−6x^{2}=y- 6 touches the circle x2+y2+16x+12y+C=0x^{2}+y^{2}+ 16x+ 12y+ C = 0, then C=C =

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  48. 48
    The value of 'a' so that the sum of squares of the roots of the equation x2−(a−2)x−a+1=0x^{2}- (a- 2)x-a+ 1 = 0 assumes the least value is

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  49. 49
    If the shortest distance between the lines r‾1=αi^+2j^+2k^+λ(i^−2j^+2k^),λ∈R,α>0{\overline{r}}_{1}=\alpha\widehat{i}+ 2\widehat{j}+ 2\widehat{k}+\lambda(\widehat{i}- 2\widehat{j}+ 2\widehat{k}),\lambda\in\mathbb{R},\alpha> 0 and rˉ2=−4i^−k^+μ(3i^−2j^−2k^),μ∈R{\bar{r}}_{2}= - 4\widehat{i}-\widehat{k}+\mu(3\widehat{i}- 2\widehat{j}- 2\widehat{k}),\mu\in R, is 9 , then the value of α\alpha is

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  50. 50
    If two curves x2−4y2=2x^{2}- 4y^{2}= 2 and 8x2=40−my28x^{2}= 40 - my^{2} are orthogonal to each other then m=m =

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