1
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are three vectors, $$|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=1, |\bar{b} \times \bar{c}|=\sqrt{15}$$ and $$\bar{b}=2 \bar{c}+\lambda \bar{a}$$, then the value of $$\lambda$$ is

A
2
B
2$$\sqrt2$$
C
1
D
4
2
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

The centroid of tetrahedron with vertices at $$\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)$$ and $$\mathrm{D}(-1,-2,4)$$ is

A
$$\left(\frac{3}{4}, \frac{-1}{4}, \frac{11}{4}\right)$$
B
$$\left(\frac{5}{4}, \frac{-3}{4}, \frac{7}{4}\right)$$
C
$$\left(\frac{-3}{4}, \frac{-1}{4}, \frac{11}{4}\right)$$
D
$$\left(\frac{-5}{4}, \frac{-3}{4}, \frac{-7}{4}\right)$$
3
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

Two adjacent sides of a parallelogram $$\mathrm{ABCD}$$ are given by $$\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}$$ and $$\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$$. The side $$\mathrm{AD}$$ is rotated by an acute angle $$\alpha$$ in the plane of parallelogram so that $$\mathrm{AD}$$ becomes $$\mathrm{AD}^{\prime}$$. If $$\mathrm{AD}^{\prime}$$ makes a right angle with side AB, then the cosine of the angle $$\alpha$$ is given by

A
$$\frac{8}{9}$$
B
$$\frac{\sqrt{17}}{9}$$
C
$$\frac{1}{9}$$
D
$$\frac{4 \sqrt{5}}{9}$$
4
MHT CET 2023 12th May Morning Shift
MCQ (Single Correct Answer)
+2
-0

The values of $$a$$ and $$b$$, so that the function

$$f(x)=\left\{\begin{array}{l} x+a \sqrt{2} \sin x, 0 \leq x \leq \frac{\pi}{4} \\ 2 x \cot x+b, \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x, \frac{\pi}{2} < x \leq \pi \end{array}\right.$$

is continuous for $$0 \leq x \leq \pi$$, are respectively given by

A
$$-\frac{\pi}{12}, \frac{\pi}{6}$$
B
$$-\frac{\pi}{6},-\frac{\pi}{12}$$
C
$$\frac{\pi}{6}, \frac{\pi}{12}$$
D
$$\frac{\pi}{6},-\frac{\pi}{12}$$
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