# Sample Paper Class 12 Mathematics Set E

**1. Let L _{1} be a straight line passing through the origin and L_{2 }be the straight line x + y = 1. If the intercepts made by the circle x^{2} + y^{2} – x + 3y = 0 on L_{1} and L_{2} are equal, then which of the following equation can represent L_{1} ?**

(a) x + 7y = 0

(b) x – y = 0

(c) x – 7y = 0

(d) Both (a) and (b)

**Answer**

D

**2. **

**then incorrect about the matrix (PQ) ^{–1} is**

(a) nilpotent

(b) idempotent

(c) involutory

(d) symmetric

**Answer**

B

**3. The equation sin x + x cos x = 0 has at least one root in**

(a) (−π/2,0)

(b) (0, p)

(c) (π, 3π/2)

(d) (0,π/2)

**Answer**

B

**4. The area enclosed by the parabola y ^{2} = 12x and its latus rectum is**

(a) 36

(b) 24

(c) 18

(d) 12

**Answer**

B

**5. Number of permutations 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time are such that the digit. 1 appearing somewhere to the left of 2, 3 appearing to the left of 4 and 5 somewhere to the left of 6, is**

(e.g., 815723946 would be one such permutation)

(a) 9 . 7!

(b) 8!

(c) 5! . 4!

(d) 8! . 4!

**Answer**

A

**6. If the function f : [0, 16] **→** R is differentiable. If 0 < α < 1 and 1 < β < 2, then ∫ ^{16}_{0} f (t) dt is equal to**

(a) 4 [a

^{3}f (a

^{4}) – b

^{3}f (b

^{4})]

(b) 4 [a

^{3}f (a

^{4}) + b

^{3}f (b

^{4})]

(c) 4 [a

^{4}f (a

^{3}) + b

^{4}f (b

^{3})]

(d) 4 [a

^{2}f (a

^{2}) + b

^{2}f (b

^{2})]

**Answer**

B

**7. Three distinct points P (3u ^{2}, 2u^{3}), Q (3v^{2}, 2v^{3}) and R (3w^{2}, 2w^{3}) are collinear then**

(a) uv + vw + wu = 0

(b) uv + vw + wu = 3

(c) uv + vw + wu = 2

(d) uv + vw + wu = 1

**Answer**

A

**8. Let ‘a’ denote the root of equation****cos (cos ^{–1} x) + sin^{–1} sin (1 + x^{2}/2) = 2 sec^{–1} (sec x)**

**then possible values of [ | 10a | ] where [ . ] denotes the greatest integer function will be**

(a) 1

(b) 5

(c) 10

(d) Both (a) and (c)

**Answer**

D

**9. The two of the straight lines represented by the equation ax ^{3} + bx^{2} y + cxy^{2} + dy^{3} = 0 will be at right angle if**

(a) a

^{2}+ c

^{2}= 0

(b) a

^{2}+ ac + bd + d

^{2}= 0

(c) a

^{2}c

^{2}+ bd + d

^{2}= 0

(d) None of these

**Answer**

B

**10. If x ^{2} – 2x cos θ + 1 = 0, then the value of x^{2n} – 2x^{n} cos nθ + 1, n ∈ N is equal to**

(a) cos 2nq

(b) sin 2nq

(c) 0

(d) some real number greater than 0

**Answer**

C

**11. Evaluate ∫8/(x + 2)(x ^{2} + 4) dx**

(a) log | x + 2 | – 1/2 log (x

^{2}+ 4) + tan

^{–1}x/2 + C

(b) log | x + 2 | – 1/2 log (x

^{2}+ 4) + sin

^{–1}x/2 + C

(c) log | x + 2 | – 1/2 log (x

^{2}+ 4) + cos

^{–1}x/2 + C

(d) log | x + 2 | – 1/2 log (x

^{3}+ 4) + tan

^{–1}x/2 + C

**Answer**

A

**12. Vertices of a parallelogram taken in order are A (2, –1, 4), B (1, 0, –1), C (1, 2, 3) and D. Distance of the point P (8, 2, –12) from the plane of the parallelogram is **

(a) 4√6/9

(b) 32√6/9

(c) 16√6/9

(d) None of these

**Answer**

B

**13. Given A̅ = 2î + 3ĵ + 6k̂, B̅ = î + ĵ – 2k̂ and C̅ = î + 2ĵ + k̂. Compute the value of | A̅×[A̅×(A̅×B̅)].C̅ |., **

(a) 343

(b) 512

(c) 221

(d) 243

**Answer**

A

**14. The value of the definite integral**

(a) 2(√2 +1)

(b) 2(√2 –1)

(c) √2 +1

(d) √2 –1

**Answer**

A

**15. Area of triangle formed by common tangents to the circle x ^{2} + y2 – 6x = 0 and x^{2} + y^{2} + 2x = 0 is**

(a) 3√3

(b) 2√3

(c) 9√3

(d) 6√3

**Answer**

A

**16. The locus of the centres of the circles which cut the circles x ^{2} + y^{2} + 4x – 6y + 9 = 0 and x^{2} + y^{2} – 5x + 4y – 2 = 0 orthogonally is**

(a) 9x + 10y – 7 = 0

(b) x – y + 2 = 0

(c) 9x – 10y + 11 = 0

(d) 9x + 10y + 7 = 0

**Answer**

C

**17. The sum to infinity of the series**

(a) 3

(b) 1

(c) 2

(d) 3/2

**Answer**

C

**18. The straight line ĵoining any point P on the parabola y ^{2} = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is**

(a) x

^{2}+ 2y

^{2}– ax = 0

(b) 2x

^{2}+ y

^{2}– 2ax = 0

(c) 2x

^{2}+ 2y

^{2}– ay = 0

(d) 2x

^{2}+ y

^{2}– 2ay = 0

**Answer**

B

**19. A box contains 6 red, 5 blue and 4 white marbles. Four marbles are chosen at random without replacement. The probability that there is atleast one marble of each colour among the four chosen, is**

(a) 48/91

(b) 44/91

(c) 88/91

(d) 24/91

**Answer**

A

**20. Statement-1 : If a, b, c are non real complex and α, β are the roots of the equation ax ^{2} + bx + c = 0 then Im (αβ) ≠ 0. because**

**Statement-2 : A quadratic equation with non real complex coefficient do not have root which are conĵugate of each other.**

(a) Statement-1 is false, Statement-2 is true.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(d) Statement-1 is true, Statement-2 is false.

**Answer**

A

**21. Statement-1 : The line x/b + y/b = 1 touches the curve y = be ^{–x/a} at some point x = x_{0}. because**

**Statement-2 : dy/dx exists at x = x**

_{0}.(a) Statement-1 is false, Statement-2 is true.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(d) Statement-1 is true, Statement-2 is false.

**Answer**

C

**22. Let C be a circle with centre O and HK is the chord of contact of pair of the tangents from points A. OA intersects the circle C at P and Q and B is the midpoint of HK, then ****Statement-1 : AB is the harmonic mean of AP and A because****Statement-2 : AK is the Geometric mean of AB and AO, OA is the arithmetic mean of AP and A**

(a) Statement-1 is false, Statement-2 is true.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(d) Statement-1 is true, Statement-2 is false.

**Answer**

B

**23. Statement-1 : The statement (p ∨ q) ∧ ~ p and ~ p ∧ q are logically equivalent.****Statement-2 : The end columns of the truth table of both statements are identical.**

(a) Statement-1 is false, Statement-2 is true.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(d) Statement-1 is true, Statement-2 is false.

**Answer**

B

**24. Statement-1 : Period of f(x) = sin 4π {x} + tan π [x], where, [x] & {x} denote the G.I.F. & fractional part respectively is 1.****Statement-2: A function f(x) is said to be periodic if there exist a positive number T independent of x such that f(T + x) = f(x). The smallest such positive value of T is called the period or fundamental period.**

(a) Statement-1 is false, Statement-2 is true.

(b) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(c) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(d) Statement-1 is true, Statement-2 is false.

**Answer**

B

**25. The value of the expression**

**where ω is an imaginary cube root of unity, is**

(a) n (n^{2} – 2)/3

(b) n (n^{2} + 2)/3

(c) n (n^{2} – 1)/3

(d) None of these

**Answer**

B

**26. If s, s’ are the length of the perpendicular on a tangent from the foci, a, a’ are those from the vertices is that from the centre and e is the eccentricity of the ellipse, x ^{2}/a^{2} + y^{2}/b^{2} = 1 then ss’ – c^{2}/ aa’ – c^{2 }=**

(a) e

(b) 1/e

(c) 1/e

^{2}

(d) e

^{2}

**Answer**

D

**27. One percent of the population suffers from a certain disease. There is blood test for this disease, and it is 99% accurate, in other words, the probability that it gives the correct answer is 0.99, regardless of whether the person is sick or healthy. A person takes the blood test, and the result says **

that he has the disease. The probability that he actually has the disease, is

(a) 0.99%

(b) 25%

(c) 50%

(d) 75%

**Answer**

C

**28. Set of values of m for which two points P and Q lie on the line y = mx + 8 so that ∠APB = ∠AQB = π/2 where A ≡ (– 4, 0), B ≡(4, 0) is**

(a) (–∞, – √3)∪( √3,∞) –{–2,2}

(b) [– √3, – √3]–{–2, 2}

(c) (–∞, –1)∪(1,∞) –{–2, 2}

(d) {– √3, √3}

**Answer**

A

**29. The trace T _{r}(A) of a 3 × 3 matrix A = (a_{iĵ} ) is defined by the relation Tr(A) = a_{11} + a_{22} + a_{33} (i.e., T_{r}(A) is sum of the main diagonal elements). Which of the following statements cannot hold ?**

(a) T

_{r}(kA) = kT

_{r}(A) (k is a scalar)

(b) T

_{r}(A + B) = T

_{r}(A) + T

_{r}(B)

(c) T

_{r}(I

_{3}) = 3

(d) T

_{r}(A

^{2}) = T

_{r}(A)

^{2}

**Answer**

D

**30.**

(a) 1/2

(b) 1

(c) 4/3

(d) 3/2

**Answer**

A