# Sample Paper Class 12 Mathematics Set M

**1. The equation of the tangent to the curve y = 9 − 2x ^{2} at the point where the ordinate and the abscissa are equal, is**

(a) 2x + y – 3√3 = 0

(b) 2x + y + √3 = 0

(c) 2x + y – √3 = 0

(d) None of these

## Answer

A

**2. **

(a) 0, 1

(b) 1, 2

(c) 1, 3

(d) None of these

## Answer

A

**3. The approximate value of f(5.001), where****f(x) = x ^{3} – 7x^{2} + 15, is**

(a) –34.995

(b) –33.995

(c) –33.335

(d) –35.993

## Answer

A

**4. **

**decreasing function of x for all x ∈ R and b ∈ R, b being independent of x, then**

(a) a ∈ (0, √6)

(b) a ∈ (− √6, √6)

(c) a ∈ (− √6,0)

(d) None of these

## Answer

B

**5. The minimum intercepts made by the axes on the tangent to the ellipse x ^{2}/16 + y^{2}/9 + = 1 is **

(a) 25

(b) 7

(c) 1

(d) None of these

## Answer

B

**6. A foot of the normal from the point (4, 3) to a circle is (2, 1) and a diameter of the circle has equation 2x – y = 2. Then the equation of the circle is**

(a) x^{2 }+ y^{2 }+ 2x – 1 = 0

(b) x^{2 }+ y^{2 }– 2x – 1 = 0

(c) x^{2 }+ y^{2 }– 2y – 1 = 0

(d) None of these

## Answer

B

**7. If the chord of contact of tangents from a point P to the parabola y ^{2 }= 4ax, touches the parabola x^{2 }= 4by, then the locus of P is a/an**

(a) circle

(b) parabola

(c) ellipse

(d) hyperbola

## Answer

D

**8. **

**will represent the ellipse, if r lies in the interval**

(a) (– ∞, 2)

(b) (3, ∞)

(c) (5, ∞)

(d) (1, ∞)

## Answer

C

**9. The equation of the asymptotes of the hyperbola 2x ^{2} + 5xy + 2y^{2} – 11x – 7y – 4 = 0, are**

(a) 2x

^{2}+ 5xy + 2y

^{2 }– 11x – 7y – 5 = 0

(b) 2x

^{2}+ 4xy + 2y

^{2 }– 7x – 11y + 5 = 0

(c) 2x

^{2}+ 5xy + 2y

^{2 }– 11x – 7y + 5 = 0

(d) None of the above

## Answer

C

**10. **

(a) 1

(b) –1

(c) 1/2

(d) None of these

## Answer

A

**11. The probability of getting qualified in IIT/JEE and EAM/CET by a student are respectivley 1/5 and 3/5 The probability that the student gets qualified for at least one of these test, is **

(a) 3/25

(b) 8/25

(c) 17/25

(d) 22/25

## Answer

C

**12. If the mean of a poisson distribution is 1/2 ,then one ratio of P(X = 3) to P(X = 2) is**

(a) 1 : 2

(b) 1 : 4

(c) 1 : 6

(d) 1 : 8

## Answer

C

**13. In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/3 . The probability that he copies is 1/6 and the probability that his answer is correct given that he copied it is 1/8. The probability that he knew the answer to the question given that he correctly answered it, is**

(a) 24/29

(b) 1/4

(c) 3/4

(d) 1/2

## Answer

A

**14. Let S be a non-empty subset of R. Consider the following statement:****P : There is a rational number x ∈ S such that x > 0.****Which of the following statements is the negation of the statement P ?**

(a) There is a rational number x ∈ S such that x ≤ 0.

(b) There is no rational number x ∈ S such that x ≤ 0.

(c) Every rational number x ∈ S satisfies x ≤ 0.

(d) x ∈ S and x ≤ 0 ⇒ x is not rational.

## Answer

C

**15. Consider the following statements****P : Suman is brilliant****Q : Suman is rich****R : Suman is honest****The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as**

(a) ~ Q ↔ ~ P ∧ R

(b) ~ (P ∧ ~ R) ↔ Q

(c) ~ P ∧ (Q ↔ ~ R)

(d) ~ (Q ↔ (P ∧ ~ R))

## Answer

D

**16. If a̅, b̅ and c̅ are unit vectors, then |a̅ − b̅| ^{2} + |b̅ − c̅| ^{2} + |c̅ − a̅| ^{2} does not exceed **

(a) 4

(b) 9

(c) 8

(d) 6

## Answer

B

**17. The diagonals of a parallelogram are given by d̅ _{1} = 2î + 3j − 6k^ and d̅_{2} = 3î + 4j − k^ then area is**

(a) 1/2 √50 sq. units

(b) 1/2 √1005 sq. units

(c) 1/2 √1105 sq. units

(d) None of these

## Answer

D

**18. Equation of the sphere for which the circle x ^{2} + y^{2} + z^{2} + 7y – 2z + 2 = 0, on the plane 2x + 3y + 4z – 8 = 0 be great circle, must be**

(a) x

^{2}+ y

^{2}+ z

^{2}+ 4x + 2y – 6z + 10 = 0

(b) x

^{2}+ y

^{2}+ z

^{2}– 4x + 2y – 6z + 10 = 0

(c) x

^{2}+ y

^{2}+ z

^{2}– 6x – 4y – 2z – 10 = 0

(d) x

^{2}+ y

^{2}+ z

^{2}– 2x + 4y – 6z + 10 = 0

## Answer

D

**19. The equations of the perpendicular from the origin to the 2x + 3y + 4z + 5 = 0 and x + 2y + 3z + 4 = 0 must be**

(a) x + 2y – z = 0 = 3x – 2y – z

(b) 2x + y + z = 0 = x – 2y – z

(c) x + 2y – z = 0 = 3x + 2y + z

(d) x – 2y + z = 0 = 3x + 2y + z

## Answer

D

**20. Find the value of l if the following equations are consistent****x + y – 3 = 0, (1 + λ)x + (2 + λ)y – 8 = 0,****x – (1 + λ)y + (2 + λ) = 0.**

(a) 0,−1/2

(b) 1,−5/3

(c) 2,9/5

(d) 4, –3

## Answer

B

**21. Evaluate**

## Answer

C

**22. Evaluate **

**where k (≠ 0) is a function and n ∈ N.**

(a) 1/ke^{−1}

(b) − 1/ke^{−1}

(c) ke^{−1}

(d) – k e

## Answer

A

**23. If f(x) be a continuous function such that f(a – x) + f(x) = 0 for all x ∈ [0, a], then evaluate**

(a) −1/2 a

(b) 1/2 a

(c) 3a

(d) 2a

## Answer

B

**24. If A _{1} is the area of the parabola y^{2} = 4ax lying between vertex and the latus rectum and A_{2} is the area between the latus rectum and the double ordinate x = 2a, then A_{1}/A_{2} = **

(a) 2√2 − 1

(b) 1/7 (2√2 + 1)

(c) 1/7 (2√2 − 1)

(d) none of these

## Answer

B

**25. Solution of the differential equation tan y****sec ^{2}x dx + tan x sec^{2}y dy = 0 is**

(a) tanx/tany = K

(b) tanx tany = K

(c) tanx + tany = K

(d) tanx – tany = K

## Answer

B

**26. Which of the following functions is a solution of the differential equation?**

(a) y = 2x^{2} – 4

(b) y = 2x – 4

(c) y = 2x

(d) y = 2

## Answer

B

**27. For two independent events A and B, P(A∩B) = 3/25, P(A′ ∩B) = 8/25, then P(B) =**

(a) 3/11

(b) 7/25

(c) 11/25

(d) none of these

## Answer

C

**28. If the line ax + by + c = 0 is a normal to the curve xy = 1, then**

(a) a > 0, b > 0

(b) a > 0, b < 0

(c) a < 0, b < 0

(d) none of these

## Answer

B

**29. **

## Answer

A

**30. **

(a) x – a, x – b and x + a + b.

(b) x + a, x + b and x + a + b.

(c) x + a, x + b and x – a – b.

(d) x – a, x – b and x – a – b.

## Answer

A

**31. If A _{ij} is the cofactor of the element aij of the determinant**

**,then write the value of a _{32}·A_{32}.**

(a) 200

(b) 150

(c) 110

(d) 90

## Answer

C

**32. A balloon, which always remains spherical on inflation is being inflated by pumping in 900 cm ^{3}/s of gas. Find the rate at which theradius of the balloon increases when the radius is 15 cm.**

(a) 11 cm/s

(b) 2π cm/s

(c) 1/π cm/s

(d) π

^{2}cm/s

## Answer

C

**33. Find the point on the curve y = x ^{3} – 11x + 5, at which the tangent is y = x – 11.**

(a) (4, –7)

(b) (0, 3)

(c) (–2, –13)

(d) (2, –9)

## Answer

D

**34. **

(a) 0

(b) –2

(c) –1

(d) 2

## Answer

A

**35. Sketch the region lying in the first quadrant and bounded by y = 9x ^{2}, x = 0, y = 1 and y = 4. Find the area of region using integration. **

(a) 5/3 sq. units

(b) 10 sq. units

(c) 14/9 sq. units

(d) 9 sq. units

## Answer

C

**36. Solve the following differential equation**

## Answer

B

**37. Let p be real and |p|≥ 2. If A, B and C are variable angles such that **

**then the minimum value of tan ^{2}A + tan^{2}B + tan^{2}C is**

(a) 8

(b) 12

(c) 18

(d) 6

## Answer

B

**38. Let a̅ and b̅ be two non-collinear unit vectors. ****If α¯ = a̅ − (a̅ ⋅ b̅)b̅ and β¯ = a̅ × b̅, then |β¯| is**

(a) |α¯|

(b) |α¯|+|α¯⋅ a̅|

(c) |α¯|+|α¯⋅ b̅|

(d) |α¯|+ α¯ .(α¯+ b̅)

## Answer

A,C

**39. The odds in favour of a book reviewed by three independent critics are, respectively, 5 : 2, 4 : 3 and 3 : 4. The probability that majority of the critics give favourable remark is **

(a) 210/343

(b) 209/343

(c) 211/343

(d) 205/343

## Answer

C

**40. Bag A contains 5 white and 3 black balls. Bag B is empty. Four balls are taken at random from A and transferred to empty bag B. From B, a ball is drawn at random and is found to be black. Then, the probability that among the transferred balls three are black and one is white is**

(a) 1/8

(b) 7/8

(c) 6/7

(d) 1/7

## Answer

D